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    {
      "id": "9231-2021-mj-11-q02",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 2,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_11.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_11.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2021-mj-11-q02-p01-678fddd988.png",
        "9231-topic-13-summation-of-series/assets/9231-2021-mj-11-q02-p02-5d422dcfc7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-11-q02",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "n\n/\n2 1-r-r 2 (a) Use standard results from the List of formulae (MF19) to find in terms of n,\n` j\nsimplifying your answer. [3] r=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2021-mj-11-q02-cb00d328b4.pdf"
    },
    {
      "id": "9231-2021-mj-11-q03",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 3,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_11.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_11.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2021-mj-11-q03-p01-fe3f677c1f.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2021-mj-11-q03-p02-e0cf1138c3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-11-q03",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "4 3 -2x -1 = x 0 3 The equation has roots a, b, c, d.\n3 3 3 3 3 3 3 3 +b +c +d , , , and state the value of . (a) Find a quartic equation whose roots are b a c d a\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2021-mj-11-q03-a82b1dd116.pdf"
    },
    {
      "id": "9231-2021-mj-11-q04",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_11.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_11.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2021-mj-11-q04-p01-5de37a0a69.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2021-mj-11-q04-p02-1f7a195ab9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-11-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "4 The matrix M represents the sequence of two transformations in the x-y plane given by a rotation of 60°\n! anticlockwise about the origin followed by a one-way stretch in the x-direction, scale factor d (d 0).\n(a) Find M in terms of d. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 2 2 units . d (b) The unit square in the x-y plane is transformed by M onto a parallelogram of area\n2\n= d 2. [2] Show that\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2021-mj-11-q04-2a84097b3f.pdf"
    },
    {
      "id": "9231-2021-mj-11-q05",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_11.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_11.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2021-mj-11-q05-p01-f0a3aa45a7.png",
        "9231-topic-15-polar-coordinates/assets/9231-2021-mj-11-q05-p02-ad5f393878.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-11-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "1 1 G G = r-i 5 The curve C has polar equation , where a is a positive constant and r. 0 r acot i\n6 3 b l\n3. 2 It is given that the greatest distance of a point on C from the pole is\n= a 2. [3] (a) Sketch C and show that\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the exact value of the area of the region bounded by C, the initial line and the half-line\n1 = r. [4] i\n6\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2021-mj-11-q05-65fd946554.pdf"
    },
    {
      "id": "9231-2021-mj-11-q06",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 6,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_11.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_11.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2021-mj-11-q06-p01-4ee39a7412.png",
        "9231-topic-16-vectors/assets/9231-2021-mj-11-q06-p02-8d8ffbddc6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-11-q06",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "6 Let t be a positive constant.\nti+j -2i-j. l passes through the point with position vector and is parallel to the vector The The line\n1\nj+tk -2j+k. line l passes through the point with position vector and is parallel to the vector\n2\nl l and is 21. It is given that the shortest distance between the lines\n1 2\n(a) Find the value of t. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2021-mj-11-q06-488d32ee41.pdf"
    },
    {
      "id": "9231-2021-mj-11-q07",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_11.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_11.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-mj-11-q07-p01-3a27dbfae4.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-mj-11-q07-p02-69fb93fd13.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-mj-11-q07-p03-fce03ef241.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-11-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 +x+9 x\n= y 7 The curve C has equation .\nx+1\n(a) Find the equations of the asymptotes of C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the coordinates of the stationary points on C. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2021-mj-11-q07-c583e3300d.pdf"
    },
    {
      "id": "9231-2021-mj-12-q01",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 1,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_12.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_12.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2021-mj-12-q01-p01-5080cc175d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-12-q01",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "4n n +31 -2 2 1 Prove by mathematical induction that is divisible by 15 for all positive integers n. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2021-mj-12-q01-3d252948d4.pdf"
    },
    {
      "id": "9231-2021-mj-12-q02",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 2,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_12.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_12.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2021-mj-12-q02-p01-678fddd988.png",
        "9231-topic-13-summation-of-series/assets/9231-2021-mj-12-q02-p02-5d422dcfc7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-12-q02",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "n\n/\n2 1-r-r 2 (a) Use standard results from the List of formulae (MF19) to find in terms of n,\n` j\nsimplifying your answer. [3] r=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2021-mj-12-q02-cca4352b40.pdf"
    },
    {
      "id": "9231-2021-mj-12-q03",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 3,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_12.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_12.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2021-mj-12-q03-p01-fe3f677c1f.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2021-mj-12-q03-p02-e0cf1138c3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-12-q03",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "4 3 -2x -1 = x 0 3 The equation has roots a, b, c, d.\n3 3 3 3 3 3 3 3 +b +c +d , , , and state the value of . (a) Find a quartic equation whose roots are b a c d a\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2021-mj-12-q03-a865c2cd06.pdf"
    },
    {
      "id": "9231-2021-mj-12-q04",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_12.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_12.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2021-mj-12-q04-p01-5de37a0a69.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2021-mj-12-q04-p02-1f7a195ab9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-12-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "4 The matrix M represents the sequence of two transformations in the x-y plane given by a rotation of 60°\n! anticlockwise about the origin followed by a one-way stretch in the x-direction, scale factor d (d 0).\n(a) Find M in terms of d. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 2 2 units . d (b) The unit square in the x-y plane is transformed by M onto a parallelogram of area\n2\n= d 2. [2] Show that\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2021-mj-12-q04-1d55d7db36.pdf"
    },
    {
      "id": "9231-2021-mj-12-q05",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_12.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_12.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2021-mj-12-q05-p01-f0a3aa45a7.png",
        "9231-topic-15-polar-coordinates/assets/9231-2021-mj-12-q05-p02-ad5f393878.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-12-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "1 1 G G = r-i 5 The curve C has polar equation , where a is a positive constant and r. 0 r acot i\n6 3 b l\n3. 2 It is given that the greatest distance of a point on C from the pole is\n= a 2. [3] (a) Sketch C and show that\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the exact value of the area of the region bounded by C, the initial line and the half-line\n1 = r. [4] i\n6\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2021-mj-12-q05-13b71ef091.pdf"
    },
    {
      "id": "9231-2021-mj-12-q06",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 6,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_12.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_12.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2021-mj-12-q06-p01-4ee39a7412.png",
        "9231-topic-16-vectors/assets/9231-2021-mj-12-q06-p02-8d8ffbddc6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-12-q06",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "6 Let t be a positive constant.\nti+j -2i-j. l passes through the point with position vector and is parallel to the vector The The line\n1\nj+tk -2j+k. line l passes through the point with position vector and is parallel to the vector\n2\nl l and is 21. It is given that the shortest distance between the lines\n1 2\n(a) Find the value of t. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2021-mj-12-q06-3a2c143367.pdf"
    },
    {
      "id": "9231-2021-mj-12-q07",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_12.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_12.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-mj-12-q07-p01-3a27dbfae4.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-mj-12-q07-p02-69fb93fd13.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-mj-12-q07-p03-fce03ef241.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-12-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 +x+9 x\n= y 7 The curve C has equation .\nx+1\n(a) Find the equations of the asymptotes of C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the coordinates of the stationary points on C. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2021-mj-12-q07-97b920dc3c.pdf"
    },
    {
      "id": "9231-2021-mj-13-q01",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_13.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_13.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2021-mj-13-q01-p01-56a70a161e.png",
        "9231-topic-13-summation-of-series/assets/9231-2021-mj-13-q01-p02-259215324c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-13-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "1 (a) Show that\nsin1\ntan(r+1) -tanr = [2] .\ncos(r+1)cosr\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\n= u . Let\ncos(r+1)cosr r\nn\n/\n(b) Use the method of differences to find . [3] u\nr\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2021-mj-13-q01-432bea1cd3.pdf"
    },
    {
      "id": "9231-2021-mj-13-q02",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 2,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_13.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_13.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2021-mj-13-q02-p01-80c3f5c5d5.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2021-mj-13-q02-p02-c1d2d104f7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-13-q02",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "n n n 3 2 -4x +3 = = +b +c 2x 0 2 The cubic equation has roots a, b, c. Let . S a\nn\nS and find the value of S . [3] (a) State the value of\n1 2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) (i) Express S in terms of S and S . [1]\nn+3 n+2 n\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n(ii) Hence, or otherwise, find the value of S . [2]\n4\n....................................................................................................................................................\n................................................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2021-mj-13-q02-ba2b8bedf0.pdf"
    },
    {
      "id": "9231-2021-mj-13-q03",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 3,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_13.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_13.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2021-mj-13-q03-p01-433b8126fb.png",
        "9231-topic-17-proof-by-induction/assets/9231-2021-mj-13-q03-p02-a9e6af40bf.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-13-q03",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "3 (a) Prove by mathematical induction that, for all positive integers n,\nn\n/\n1 4 2 2 2 +r = (n+1) (2n+1). 5r n [6]\n2 ` j\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2021-mj-13-q03-977d7153f8.pdf"
    },
    {
      "id": "9231-2021-mj-13-q04",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_13.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_13.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2021-mj-13-q04-p01-6ffd8f71b4.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2021-mj-13-q04-p02-b9e9c3bd28.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-13-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "4 The matrices A, B and C are given by\n1 0 2 k k\n0 1 1\n= = = -1 C B 0 1 , and , A 5 3\ne-1 2 0o f p f p\n1 0 1 0 1\nwhere k is a real constant.\n(a) Find CAB. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Given that A is singular, find the value of k. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2021-mj-13-q04-98298db767.pdf"
    },
    {
      "id": "9231-2021-mj-13-q05",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_13.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_13.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2021-mj-13-q05-p01-a8f9b4d840.png",
        "9231-topic-15-polar-coordinates/assets/9231-2021-mj-13-q05-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-13-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "1 1\n1 G G = - 5 The curve C has polar equation r , where r. 0 i\nr-i r 2\n(a) Sketch C. [3]\n3-4ln2\n1 = (b) Show that the area of the region bounded by the half-line and C is . [6] i r\n4r 2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n................................................................................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2021-mj-13-q05-3e1312ba52.pdf"
    },
    {
      "id": "9231-2021-mj-13-q06",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 6,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_13.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_13.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2021-mj-13-q06-p01-394783d664.png",
        "9231-topic-16-vectors/assets/9231-2021-mj-13-q06-p02-528b3251d1.png",
        "9231-topic-16-vectors/assets/9231-2021-mj-13-q06-p03-3b4af3aecc.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-13-q06",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "=-i-2j+k+s(2i-3j) = 3i-2k+t(3i-j+3k) l l r r 6 The lines and have equations and\n1 2\nrespectively.\n-2i-2j+4k. The plane l contains and the point P with position vector P\n1 1\n= a+mb+nc. , giving your answer in the form r [2] (a) Find an equation of P\n1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nl l contains and is parallel to . The plane P\n2 2 1\nax+by+cz = d. , giving your answer in the form [4] (b) Find an equation of P\n2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2021-mj-13-q06-1801d07e33.pdf"
    },
    {
      "id": "9231-2021-mj-13-q07",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_13.pdf?download=true",
      "source_pages": [
        15,
        16,
        17,
        18
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_13.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-mj-13-q07-p01-5c8adc7475.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-mj-13-q07-p02-c5d60f6568.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-mj-13-q07-p03-14450fb6c7.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-mj-13-q07-p04-3ed4353cf2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-13-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 -x-3 x\n= y 7 The curve C has equation .\n2 1+x-x\nC. [2] (a) Find the equations of the asymptotes of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the coordinates of any stationary points on C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2021-mj-13-q07-1c8e190de9.pdf"
    },
    {
      "id": "9231-2021-mj-21-q01",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 1,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_21.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2021-mj-21-q01-p01-da8efb7e88.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-21-q01",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "1 (a) Given that a is an integer, show that the system of equations\nax+3y+ = z 14,\n2x+ y+3z = 0,\n-x+2y-5z = 17,\nhas a unique solution and interpret this situation geometrically. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= = = -2 x 1, y 4, z is the solution to the system of equations in (b) Find the value of a for which\npart (a). [1]\n...................................................................................................................................................\n............................................................................................................................................................\n...........................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2021-mj-21-q01-d871fc98be.pdf"
    },
    {
      "id": "9231-2021-mj-21-q02",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 2,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_21.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2021-mj-21-q02-p01-9ddb817b03.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-21-q02",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "2 The variables x and y are related by the differential equation\n2 d y dy\n+3 +2y = 2x+1.\n2 dx dx\n(a) Find the general solution for y in terms of x. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2021-mj-21-q02-5e3df743bc.pdf"
    },
    {
      "id": "9231-2021-mj-21-q03",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 3,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_21.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2021-mj-21-q03-p01-fbfc2192a8.png",
        "9231-topic-24-integration/assets/9231-2021-mj-21-q03-p02-54911c5c1c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-21-q03",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "3\ny\nx n-1 1 2 0 1\nn n n\n3 G G = y x The diagram shows the curve with equation for 1, together with a set of n rectangles 0 x\n1\nof width .\nn\n1\ny 3 1 x dx U , where (a) By considering the sum of the areas of these rectangles, show that\nn\n0\n2\nn+1\n= U . [4]\n2n n e o\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\ny 3 x dx. L (b) Use a similar method to find, in terms of n, a lower bound for [4]\nn\n0\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2021-mj-21-q03-a478c5dc83.pdf"
    },
    {
      "id": "9231-2021-mj-21-q04",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 4,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_21.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2021-mj-21-q04-p01-59022679e9.png",
        "9231-topic-26-differential-equations/assets/9231-2021-mj-21-q04-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-21-q04",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "4 Find the solution of the differential equation\ndy\n1 +y = sini tan i,\n2 di\n1 1 1 = = = y f(i). 0 r, given that y 1 when r. Give your answer in the form [9] where i i\n2\ny 1 = [You may use without proof the result that i.] cosecidi lntan\n2\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.........................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2021-mj-21-q04-a7d4ab752c.pdf"
    },
    {
      "id": "9231-2021-mj-21-q05",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 5,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_21.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_21.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2021-mj-21-q05-p01-3ecb9c7f37.png",
        "9231-topic-25-complex-numbers/assets/9231-2021-mj-21-q05-p02-2e5e83dc76.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-21-q05",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "2 3 n ! z+z +z +...+z 5 (a) State the sum of the series , for z 1. [1]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nn-1 2 ! 1+z+z +...+z = (b) Given that z is an nth root of unity and 1, deduce that [2] 0. z\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 = (cosi+isini), use de Moivre’s theorem to show that z (c) Given instead that\n3\n3\n3cosi-1 / -m\n= . [7] 3 cosmi\n10-6cosi\nm=1\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...........................................................................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2021-mj-21-q05-f4b5011f40.pdf"
    },
    {
      "id": "9231-2021-mj-21-q06",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 6,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_21.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2021-mj-21-q06-p01-1a939d04f3.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2021-mj-21-q06-p02-f52fccc28a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-21-q06",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "6 The matrix A is given by\n22 - 5 8\n3\n= -6 A 0 . 0\nf p\n0 0 1\n-1 2 = A PDP . [7] (a) Find a matrix P and a diagonal matrix D such that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2021-mj-21-q06-10accd5992.pdf"
    },
    {
      "id": "9231-2021-mj-21-q07",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 7,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_21.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_21.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2021-mj-21-q07-p01-7f8ef196a4.png",
        "9231-topic-21-hyperbolic-functions/assets/9231-2021-mj-21-q07-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-21-q07",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "-1 1 = 7 (a) It is given that . y sech x+\n2 b l\ndy 1\n= - y . [3] Express cosh y in terms of x and hence show that sinh\n2 dx\n1 x+\n2 b l\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 –1 in the form sech x+ (b) Find the first three terms in the Maclaurin’s series for\n2 b l\n2 lna+bx+cx ,\nwhere a, b and c are constants to be determined. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2021-mj-21-q07-be7ab976a9.pdf"
    },
    {
      "id": "9231-2021-mj-21-q08",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 8,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_21.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2021-mj-21-q08-p01-130d92c1e8.png",
        "9231-topic-24-integration/assets/9231-2021-mj-21-q08-p02-96d224d0fe.png",
        "9231-topic-24-integration/assets/9231-2021-mj-21-q08-p03-8b14c535d4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-21-q08",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "8 The curve C has parametric equations\n3 1 G G = t- = y sinh2t, x 2cosht, for 1. 0 t\n2 4\ndy dx\n2 = 1-sinh (a) Find t. and show that [3]\ndt dt\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2r radians about the x-axis is denoted by A. The area of the surface generated when C is rotated through\n1 y 3 1 = t- (1+cosh2t)dt. [4] A sinh2t (b) (i) Show that r\n2 4 b l 0\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n......................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2021-mj-21-q08-9e49b3dbad.pdf"
    },
    {
      "id": "9231-2021-mj-22-q01",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 1,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_22.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2021-mj-22-q01-p01-da8efb7e88.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-22-q01",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "1 (a) Given that a is an integer, show that the system of equations\nax+3y+ = z 14,\n2x+ y+3z = 0,\n-x+2y-5z = 17,\nhas a unique solution and interpret this situation geometrically. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= = = -2 x 1, y 4, z is the solution to the system of equations in (b) Find the value of a for which\npart (a). [1]\n...................................................................................................................................................\n............................................................................................................................................................\n...........................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2021-mj-22-q01-26f5020c59.pdf"
    },
    {
      "id": "9231-2021-mj-22-q02",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 2,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_22.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2021-mj-22-q02-p01-9ddb817b03.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-22-q02",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "2 The variables x and y are related by the differential equation\n2 d y dy\n+3 +2y = 2x+1.\n2 dx dx\n(a) Find the general solution for y in terms of x. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2021-mj-22-q02-7c3f77535e.pdf"
    },
    {
      "id": "9231-2021-mj-22-q03",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 3,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_22.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2021-mj-22-q03-p01-fbfc2192a8.png",
        "9231-topic-24-integration/assets/9231-2021-mj-22-q03-p02-54911c5c1c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-22-q03",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "3\ny\nx n-1 1 2 0 1\nn n n\n3 G G = y x The diagram shows the curve with equation for 1, together with a set of n rectangles 0 x\n1\nof width .\nn\n1\ny 3 1 x dx U , where (a) By considering the sum of the areas of these rectangles, show that\nn\n0\n2\nn+1\n= U . [4]\n2n n e o\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\ny 3 x dx. L (b) Use a similar method to find, in terms of n, a lower bound for [4]\nn\n0\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2021-mj-22-q03-083e2b1cfa.pdf"
    },
    {
      "id": "9231-2021-mj-22-q04",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 4,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_22.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2021-mj-22-q04-p01-59022679e9.png",
        "9231-topic-26-differential-equations/assets/9231-2021-mj-22-q04-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-22-q04",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "4 Find the solution of the differential equation\ndy\n1 +y = sini tan i,\n2 di\n1 1 1 = = = y f(i). 0 r, given that y 1 when r. Give your answer in the form [9] where i i\n2\ny 1 = [You may use without proof the result that i.] cosecidi lntan\n2\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.........................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2021-mj-22-q04-527462a22c.pdf"
    },
    {
      "id": "9231-2021-mj-22-q05",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 5,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_22.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_22.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2021-mj-22-q05-p01-3ecb9c7f37.png",
        "9231-topic-25-complex-numbers/assets/9231-2021-mj-22-q05-p02-2e5e83dc76.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-22-q05",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "2 3 n ! z+z +z +...+z 5 (a) State the sum of the series , for z 1. [1]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nn-1 2 ! 1+z+z +...+z = (b) Given that z is an nth root of unity and 1, deduce that [2] 0. z\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 = (cosi+isini), use de Moivre’s theorem to show that z (c) Given instead that\n3\n3\n3cosi-1 / -m\n= . [7] 3 cosmi\n10-6cosi\nm=1\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...........................................................................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2021-mj-22-q05-6369cbe5de.pdf"
    },
    {
      "id": "9231-2021-mj-22-q06",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 6,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_22.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2021-mj-22-q06-p01-1a939d04f3.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2021-mj-22-q06-p02-f52fccc28a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-22-q06",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "6 The matrix A is given by\n22 - 5 8\n3\n= -6 A 0 . 0\nf p\n0 0 1\n-1 2 = A PDP . [7] (a) Find a matrix P and a diagonal matrix D such that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2021-mj-22-q06-4cb6497d6c.pdf"
    },
    {
      "id": "9231-2021-mj-22-q07",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 7,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_22.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_22.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2021-mj-22-q07-p01-7f8ef196a4.png",
        "9231-topic-21-hyperbolic-functions/assets/9231-2021-mj-22-q07-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-22-q07",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "-1 1 = 7 (a) It is given that . y sech x+\n2 b l\ndy 1\n= - y . [3] Express cosh y in terms of x and hence show that sinh\n2 dx\n1 x+\n2 b l\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 –1 in the form sech x+ (b) Find the first three terms in the Maclaurin’s series for\n2 b l\n2 lna+bx+cx ,\nwhere a, b and c are constants to be determined. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2021-mj-22-q07-2bb124da8c.pdf"
    },
    {
      "id": "9231-2021-mj-22-q08",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 8,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_22.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2021-mj-22-q08-p01-130d92c1e8.png",
        "9231-topic-24-integration/assets/9231-2021-mj-22-q08-p02-96d224d0fe.png",
        "9231-topic-24-integration/assets/9231-2021-mj-22-q08-p03-8b14c535d4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-22-q08",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "8 The curve C has parametric equations\n3 1 G G = t- = y sinh2t, x 2cosht, for 1. 0 t\n2 4\ndy dx\n2 = 1-sinh (a) Find t. and show that [3]\ndt dt\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2r radians about the x-axis is denoted by A. The area of the surface generated when C is rotated through\n1 y 3 1 = t- (1+cosh2t)dt. [4] A sinh2t (b) (i) Show that r\n2 4 b l 0\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n......................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2021-mj-22-q08-b15bf43bca.pdf"
    },
    {
      "id": "9231-2021-mj-23-q01",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 1,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_23.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_23.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2021-mj-23-q01-p01-3750c4e233.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-23-q01",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "1 (a) Find a and b such that\n8 5 3 5 3 -iz -z +i = -a)(z -b). z (z [1]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Hence find the roots of\n8 5 3 -iz -z +i = z 0,\nii 2 1 G , where r 0 and 0 2r. [6] re giving your answers in the form i\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2021-mj-23-q01-a7f7f23c4d.pdf"
    },
    {
      "id": "9231-2021-mj-23-q02",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 2,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_23.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_23.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2021-mj-23-q02-p01-46975b453c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-23-q02",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "4 x lncoshx 2 Find the Maclaurin’s series for up to and including the term in . [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...............................................................................................................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2021-mj-23-q02-28185f8500.pdf"
    },
    {
      "id": "9231-2021-mj-23-q03",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 3,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_23.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2021-mj-23-q03-p01-a4c92e4276.png",
        "9231-topic-24-integration/assets/9231-2021-mj-23-q03-p02-46101ccfe0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-23-q03",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "3\ny\nx – 0 1 2 3 N 1 N\nx\nH = N-1 y The diagram shows the curve for 1, together with a set of rectangles of unit x\n2 -1 2x\nwidth.\n(a) By considering the sum of the areas of these rectangles, show that\nN\n/ r\n1 2 1 -1) +1. ln(2N [7]\n4 2 -1 2r\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2021-mj-23-q03-096c972a76.pdf"
    },
    {
      "id": "9231-2021-mj-23-q04",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 4,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_23.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_23.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2021-mj-23-q04-p01-22aebd5b60.png",
        "9231-topic-25-complex-numbers/assets/9231-2021-mj-23-q04-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-23-q04",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "5 5 1 1\n= cosi+isini, z+ z- z 4 By considering the binomial expansions of and , where use de\nz z b l b l\nMoivre’s theorem to show that\nsin5i-asin3i+bsini\n5 = tan , i\ncos5i+acos3i+bcosi\nwhere a and b are integers to be determined. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..............................................................................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2021-mj-23-q04-be117553d2.pdf"
    },
    {
      "id": "9231-2021-mj-23-q05",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_23.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2021-mj-23-q05-p01-5050d2f338.png",
        "9231-topic-26-differential-equations/assets/9231-2021-mj-23-q05-p02-2e5e83dc76.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-23-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "5 The variables x and y are related by the differential equation\n2 d y dy\n-x -2 -3y = 4e .\n2 dx dx\n-x = (a) Find the value of the constant k such that is a particular integral of the differential y kxe\nequation. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\ndy\n1 = = = y when x 0. [6] (b) Find the solution of the differential equation for which\n2 dx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2021-mj-23-q05-f51726aa65.pdf"
    },
    {
      "id": "9231-2021-mj-23-q06",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_23.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2021-mj-23-q06-p01-fcfaf3067a.png",
        "9231-topic-26-differential-equations/assets/9231-2021-mj-23-q06-p02-2e5e83dc76.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-23-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "6 (a) Starting from the definitions of sinh and cosh in terms of exponentials, prove that\n2 = cosh2x-1. 2sinh x [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the solution to the differential equation\ndy\n+ycothx = 4sinhx\ndx\n= = x ln3. for which y 1 when [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2021-mj-23-q06-b63516c1aa.pdf"
    },
    {
      "id": "9231-2021-mj-23-q07",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 7,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_23.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2021-mj-23-q07-p01-6ea7acc5a3.png",
        "9231-topic-24-integration/assets/9231-2021-mj-23-q07-p02-a07dc86ffe.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-23-q07",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "3 -1 n y 2 2 2 = 4+x I 7 The integral , where n is an integer, is defined by dx. I\nn n ` j\n0\nI , expressing your answer in logarithmic form. [3] (a) Find the exact value of\n1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1 n d\n2 2 4+x x , or otherwise, show that (b) By considering\ndxb l ` j\nn\n3 2 = + (n-1)I 4nI . [5]\nn+2 5 2 n b l\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2021-mj-23-q07-4ae7b07c79.pdf"
    },
    {
      "id": "9231-2021-mj-23-q08",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_23.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2021-mj-23-q08-p01-cb1a5c5ac8.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2021-mj-23-q08-p02-ef47da3a2a.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2021-mj-23-q08-p03-1b957851b8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-23-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "8 (a) Find the value of a for which the system of equations\n13x+18y-28z = 0,\n-4x- ay+ = 8z 0,\n2x+ 6y- = 5z 0,\ndoes not have a unique solution. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nThe matrix A is given by\n-28 13 18\n= -4 -1 A 8 .\nf p\n-5 2 6\n2\n(b) Find the eigenvalue of A corresponding to the eigenvector . [1] 0\nf p\n1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1 = A PDP . [8] (c) Find a matrix P and a diagonal matrix D such that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2021-mj-23-q08-4263f5444e.pdf"
    },
    {
      "id": "9231-2021-mj-31-q01",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 1,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_31.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_31.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2021-mj-31-q01-p01-8e6135fe4b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-31-q01",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "1 A particle P of mass 1 kg is moving along a straight line against a resistive force of magnitude\n10 v -1 = = vms N, where is the speed of P at time t s. When t 0, v 25.\n2 (t+1)\nFind an expression for v in terms of t. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.........................................................................................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2021-mj-31-q01-3dcf533d93.pdf"
    },
    {
      "id": "9231-2021-mj-31-q02",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 2,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_31.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2021-mj-31-q02-p01-fe8d4ddef0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-31-q02",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "2 A hollow hemispherical bowl of radius a has a smooth inner surface and is fixed with its axis vertical.\nA particle P of mass m moves in horizontal circles on the inner surface of the bowl, at a height x above\n8\nthe lowest point of the bowl. The speed of P is ga.\n3\nFind x in terms of a. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2021-mj-31-q02-13b217774f.pdf"
    },
    {
      "id": "9231-2021-mj-31-q03",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 3,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_31.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_31.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2021-mj-31-q03-p01-0dada280dd.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-31-q03",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "3 One end of a light elastic string, of natural length a and modulus of elasticity kmg, is attached to a\nfixed point A. The other end of the string is attached to a particle P of mass 4m. The particle P hangs in\nequilibrium a distance x vertically below A.\n4a\n= (a) Show that k . [1]\nx-a\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nAn additional particle, of mass 2m, is now attached to P and the combined particle is released from rest\n1 a, its at the original equilibrium position of P. When the combined particle has descended a distance\n3\n1 speed is ga.\n3\n(b) Find x in terms of a. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2021-mj-31-q03-e72808a4c4.pdf"
    },
    {
      "id": "9231-2021-mj-31-q04",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_31.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_31.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2021-mj-31-q04-p01-e7ff85cf98.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2021-mj-31-q04-p02-94095fba5b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-31-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "4\nkh\nh\nr\nA uniform solid circular cone has vertical height kh and radius r. A uniform solid cylinder has height h\nand radius r. The base of the cone is joined to one of the circular faces of the cylinder so that the axes\nof symmetry of the two solids coincide (see diagram, which shows a cross-section). The cone and the\ncylinder are made of the same material.\n(a) Show that the distance of the centre of mass of the combined solid from the base of the cylinder\n2 +4k+6) h(k\nis . [4]\n4(3+k)\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................................................................................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2021-mj-31-q04-30b840ac54.pdf"
    },
    {
      "id": "9231-2021-mj-31-q05",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_31.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2021-mj-31-q05-p01-029111d037.png",
        "9231-topic-33-circular-motion/assets/9231-2021-mj-31-q05-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-31-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "5 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end\nof the string is attached to a fixed point O. The particle completes vertical circles with centre O. The\npoints A and B are on the path of P, both on the same side of the vertical through O. OA makes an angle\nwith the downward vertical through O and OB makes an angle with the upward vertical through O. i i\nag. The tensions in the string The speed of P when it is at A is u and the speed of P when it is at B is\n= at A and B are T and T respectively. It is given that T 7T .\nA B A B\nand find an expression for u in terms of a and g. [8] Find the value of i\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2021-mj-31-q05-459048f814.pdf"
    },
    {
      "id": "9231-2021-mj-31-q06",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 6,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_31.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_31.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2021-mj-31-q06-p01-24ecef235e.png",
        "9231-topic-36-momentum/assets/9231-2021-mj-31-q06-p02-40fa01cb88.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-31-q06",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "6\nA B\na b\nm m\nu\nu\nTwo uniform smooth spheres A and B of equal radii each have mass m. The two spheres are each\nmoving with speed u on a horizontal surface when they collide. Immediately before the collision, A’s\ndirection of motion makes an angle with the line of centres, and B’s direction of motion makes an a\n1 angle with the line of centres (see diagram). The coefficient of restitution between the spheres is b\n3\n= and 2cosb cosa.\n(a) Show that the direction of motion of A after the collision is perpendicular to the line of centres.\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2021-mj-31-q06-83060e69a8.pdf"
    },
    {
      "id": "9231-2021-mj-31-q07",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_31.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_31.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-mj-31-q07-p01-a53f9bedc7.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-mj-31-q07-p02-b095ac6c30.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-mj-31-q07-p03-3ed4353cf2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-31-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "7 A particle P is projected from a point O on a horizontal plane and moves freely under gravity. The\n-1 4 = initial velocity of P is 100ms at an angle above the horizontal, where . The two times at tani i\n3\nwhich P’s height above the plane is H m differ by 10 s.\n(a) Find the value of H. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2021-mj-31-q07-7cd638bade.pdf"
    },
    {
      "id": "9231-2021-mj-32-q01",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 1,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_32.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_32.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2021-mj-32-q01-p01-8e6135fe4b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-32-q01",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "1 A particle P of mass 1 kg is moving along a straight line against a resistive force of magnitude\n10 v -1 = = vms N, where is the speed of P at time t s. When t 0, v 25.\n2 (t+1)\nFind an expression for v in terms of t. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.........................................................................................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2021-mj-32-q01-0681e5dfe4.pdf"
    },
    {
      "id": "9231-2021-mj-32-q02",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 2,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_32.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2021-mj-32-q02-p01-fe8d4ddef0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-32-q02",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "2 A hollow hemispherical bowl of radius a has a smooth inner surface and is fixed with its axis vertical.\nA particle P of mass m moves in horizontal circles on the inner surface of the bowl, at a height x above\n8\nthe lowest point of the bowl. The speed of P is ga.\n3\nFind x in terms of a. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2021-mj-32-q02-c7bdfd178b.pdf"
    },
    {
      "id": "9231-2021-mj-32-q03",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 3,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_32.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_32.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2021-mj-32-q03-p01-0dada280dd.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-32-q03",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "3 One end of a light elastic string, of natural length a and modulus of elasticity kmg, is attached to a\nfixed point A. The other end of the string is attached to a particle P of mass 4m. The particle P hangs in\nequilibrium a distance x vertically below A.\n4a\n= (a) Show that k . [1]\nx-a\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nAn additional particle, of mass 2m, is now attached to P and the combined particle is released from rest\n1 a, its at the original equilibrium position of P. When the combined particle has descended a distance\n3\n1 speed is ga.\n3\n(b) Find x in terms of a. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2021-mj-32-q03-39ec0bf91d.pdf"
    },
    {
      "id": "9231-2021-mj-32-q04",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_32.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_32.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2021-mj-32-q04-p01-e7ff85cf98.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2021-mj-32-q04-p02-94095fba5b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-32-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "4\nkh\nh\nr\nA uniform solid circular cone has vertical height kh and radius r. A uniform solid cylinder has height h\nand radius r. The base of the cone is joined to one of the circular faces of the cylinder so that the axes\nof symmetry of the two solids coincide (see diagram, which shows a cross-section). The cone and the\ncylinder are made of the same material.\n(a) Show that the distance of the centre of mass of the combined solid from the base of the cylinder\n2 +4k+6) h(k\nis . [4]\n4(3+k)\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................................................................................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2021-mj-32-q04-f6d39e339c.pdf"
    },
    {
      "id": "9231-2021-mj-32-q05",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_32.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2021-mj-32-q05-p01-029111d037.png",
        "9231-topic-33-circular-motion/assets/9231-2021-mj-32-q05-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-32-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "5 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end\nof the string is attached to a fixed point O. The particle completes vertical circles with centre O. The\npoints A and B are on the path of P, both on the same side of the vertical through O. OA makes an angle\nwith the downward vertical through O and OB makes an angle with the upward vertical through O. i i\nag. The tensions in the string The speed of P when it is at A is u and the speed of P when it is at B is\n= at A and B are T and T respectively. It is given that T 7T .\nA B A B\nand find an expression for u in terms of a and g. [8] Find the value of i\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2021-mj-32-q05-b743e5f348.pdf"
    },
    {
      "id": "9231-2021-mj-32-q06",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 6,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_32.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_32.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2021-mj-32-q06-p01-24ecef235e.png",
        "9231-topic-36-momentum/assets/9231-2021-mj-32-q06-p02-40fa01cb88.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-32-q06",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "6\nA B\na b\nm m\nu\nu\nTwo uniform smooth spheres A and B of equal radii each have mass m. The two spheres are each\nmoving with speed u on a horizontal surface when they collide. Immediately before the collision, A’s\ndirection of motion makes an angle with the line of centres, and B’s direction of motion makes an a\n1 angle with the line of centres (see diagram). The coefficient of restitution between the spheres is b\n3\n= and 2cosb cosa.\n(a) Show that the direction of motion of A after the collision is perpendicular to the line of centres.\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2021-mj-32-q06-2bec7c6a5d.pdf"
    },
    {
      "id": "9231-2021-mj-32-q07",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_32.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_32.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-mj-32-q07-p01-a53f9bedc7.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-mj-32-q07-p02-b095ac6c30.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-mj-32-q07-p03-3ed4353cf2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-32-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "7 A particle P is projected from a point O on a horizontal plane and moves freely under gravity. The\n-1 4 = initial velocity of P is 100ms at an angle above the horizontal, where . The two times at tani i\n3\nwhich P’s height above the plane is H m differ by 10 s.\n(a) Find the value of H. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2021-mj-32-q07-a0c94b946d.pdf"
    },
    {
      "id": "9231-2021-mj-33-q01",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 1,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 3,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_33.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_33.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2021-mj-33-q01-p01-919b2abc59.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-33-q01",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "1\nD\nA C 16a\nO 3a 6a\nB\nA uniform lamina ABCD consists of two isosceles triangles ABD and BCD. The diagonals of ABCD\nmeet at the point O. The length of AO is 3a, the length of OC is 6a and the length of BD is 16a (see\ndiagram).\nFind the distance of the centre of mass of the lamina from DB. [3]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2021-mj-33-q01-2672cf5fda.pdf"
    },
    {
      "id": "9231-2021-mj-33-q02",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 2,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_33.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_33.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2021-mj-33-q02-p01-3a5a804954.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-33-q02",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "2 One end of a light elastic string of natural length 0.8 m and modulus of elasticity 36 N is attached to a\n3 = fixed point O on a smooth plane. The plane is inclined at an angle sina to the horizontal, where . a\n5\nA particle P of mass 2 kg is attached to the other end of the string. The string lies along a line of greatest\n-1\n2ms slope of the plane with the particle below the level of O. The particle is projected with speed\ndirectly down the plane from the position where OP is equal to the natural length of the string.\nFind the maximum extension of the string during the subsequent motion. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.........................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2021-mj-33-q02-7b4ab900cb.pdf"
    },
    {
      "id": "9231-2021-mj-33-q03",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 3,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_33.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2021-mj-33-q03-p01-b2fcc64228.png",
        "9231-topic-33-circular-motion/assets/9231-2021-mj-33-q03-p02-0346fd2e39.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-33-q03",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "3\nR\ni\nB\nA\nParticles A and B, of masses 3m and m respectively, are connected by a light inextensible string of\nlength a that passes through a fixed smooth ring R. Particle B hangs in equilibrium vertically below the\n2 ring. Particle A moves in horizontal circles on a smooth horizontal surface with speed ga. The angle\n5\n12 between AR and BR is (see diagram). The normal reaction between A and the surface is mg. i\n5\n(a) Find cosi. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2021-mj-33-q03-de01cf9b6d.pdf"
    },
    {
      "id": "9231-2021-mj-33-q04",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 4,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_33.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2021-mj-33-q04-p01-c842beb3d3.png",
        "9231-topic-33-circular-motion/assets/9231-2021-mj-33-q04-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-33-q04",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "4\n1 u\n2\nB\nO\ni\na\nu A\nA particle of mass m is attached to one end of a light inextensible string of length a. The other end of\nthe string is attached to a fixed point O. The particle is initially held with the string taut at the point A,\nwhere OA makes an angle with the downward vertical through O. The particle is then projected with i\nspeed u perpendicular to OA and begins to move upwards in part of a vertical circle. The string goes\nslack when the particle is at the point B where angle AOB is a right angle. The speed of the particle\n1 when it is at B is (see diagram). u\n2\nFind the tension in the string at A, giving your answer in terms of m and g. [8]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..............................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2021-mj-33-q04-e493eb0b1e.pdf"
    },
    {
      "id": "9231-2021-mj-33-q05",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 5,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_33.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_33.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2021-mj-33-q05-p01-e96ae5ef08.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2021-mj-33-q05-p02-7f46c841d5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-33-q05",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "-1\n20ms 5 A particle P of mass m kg is projected vertically upwards from a point O, with speed , and\n-1\nmoves under gravity. There is a resistive force of magnitude 2mv N, where vms is the speed of P at\ntime t s after projection.\n(a) Find an expression for v in terms of t, while P is moving upwards. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2021-mj-33-q05-12598202fe.pdf"
    },
    {
      "id": "9231-2021-mj-33-q06",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 6,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_33.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_33.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2021-mj-33-q06-p01-32b3a4e7f6.png",
        "9231-topic-36-momentum/assets/9231-2021-mj-33-q06-p02-23a4285393.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-33-q06",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "6\nA B\ni\nu\nm km\nTwo uniform smooth spheres A and B of equal radii have masses m and km respectively. Sphere A is\nmoving with speed u on a smooth horizontal surface when it collides with sphere B which is at rest.\nImmediately before the collision, A’s direction of motion makes an angle with the line of centres (see i\n1 diagram). The coefficient of restitution between the spheres is .\n3\n4ucosi\n. [3] (a) Show that the speed of B after the collision is\n3(1+k)\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2021-mj-33-q06-f388b6b388.pdf"
    },
    {
      "id": "9231-2021-mj-33-q07",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_33.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_33.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-mj-33-q07-p01-b5ebc8490d.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-mj-33-q07-p02-9dd8f075a4.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-mj-33-q07-p03-ecf01b5418.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-33-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "7 A particle P is projected with speed u at an angle above the horizontal from a point O on a horizontal i\nplane and moves freely under gravity. The horizontal and vertical displacements of P from O at a\nsubsequent time t are denoted by x and y respectively.\n(a) Use the equation of the trajectory given in the List of formulae (MF19), together with the condition\n= y 0, to establish an expression for the range R in terms of u, and g. [2] i\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nand g. [2] (b) Deduce an expression for the maximum height H, in terms of u, i\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2021-mj-33-q07-3a42ce433b.pdf"
    },
    {
      "id": "9231-2021-mj-41-q01",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_41.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-mj-41-q01-p01-cf403a6085.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-41-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 A random sample of 7 observations of a variable X are as follows.\n8.26 7.78 7.92 8.04 8.27 7.95 8.34\nn. The population mean of X is\n= 8.22 against the alternative hypothesis (a) Test, at the 10% significance level, the null hypothesis n\n1 8.22. [6] n\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2021-mj-41-q01-896bf27e74.pdf"
    },
    {
      "id": "9231-2021-mj-41-q02",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_41.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_41.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2021-mj-41-q02-p01-cc675c4564.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2021-mj-41-q02-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-41-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "2 A driving school employs four instructors to prepare people for their driving test. The allocation of\npeople to instructors is random. For each of the instructors, the following table gives the number of\npeople who passed and the number who failed their driving test last year.\nInstructor A Instructor B Instructor C Instructor D Total\nPass 72 42 52 68 234\nFail 33 34 41 58 166\nTotal 105 76 93 126 400\nTest at the 10% significance level whether success in the driving test is independent of the instructor.\n[7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n............",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2021-mj-41-q02-f56c087386.pdf"
    },
    {
      "id": "9231-2021-mj-41-q03",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 3,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_41.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_41.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2021-mj-41-q03-p01-22735b61ae.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2021-mj-41-q03-p02-a6ad0ab6e6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-41-q03",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "3 The continuous random variable X has cumulative distribution function F given by\n1 0, x 0\n1 2 G G = F(x) x 0 x 9,\n81\n*\n2 9. 1 x\n(a) Find [3] E X .\n` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n[2] Var X . (b) Find\n` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2021-mj-41-q03-a82e14e289.pdf"
    },
    {
      "id": "9231-2021-mj-41-q04",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 4,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_41.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-mj-41-q04-p01-1d3bde07b8.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-mj-41-q04-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-41-q04",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "4 A scientist is investigating the lengths of the leaves of birch trees in different regions. He takes a random\nsample of 50 leaves from birch trees in region A and a random sample of 60 leaves from birch trees in\nregion B. He records their lengths in cm, x and y, respectively. His results are summarised as follows.\n2 2 = = = = /x /x /y /y 282 1596 328 1808\nThe population mean lengths of leaves from birch trees in regions A and B are cm and cm n n\nA B\nrespectively.\n= Carry out a test at the 5% significance level to test the null hypothesis against the alternative n n\nA B\n! hypothesis [8] . n n\nA B\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.....................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2021-mj-41-q04-028e1b118d.pdf"
    },
    {
      "id": "9231-2021-mj-41-q05",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 5,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_41.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_41.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2021-mj-41-q05-p01-30ff0cc6ed.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2021-mj-41-q05-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-41-q05",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "5 Georgio has designed two new uniforms X and Y for the employees of an airline company. A random\nsample of 11 employees are each asked to assess each of the two uniforms for practicality and\nappearance, and to give a total score out of 100. The scores are given in the table.\nEmployee A B C D E F G H I J K\nUniform X 82 74 42 59 60 73 94 98 62 36 50\nUniform Y 78 75 63 56 67 82 99 90 72 48 61\n(a) Give a reason why a Wilcoxon signed-rank test may be more appropriate than a t-test for\ninvestigating whether there is any evidence of a preference for one of the uniforms. [1]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Carry out a Wilcoxon matched-pairs signed-rank test at the 10% significance level. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2021-mj-41-q05-f49179ef3b.pdf"
    },
    {
      "id": "9231-2021-mj-41-q06",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 6,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_41.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_41.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2021-mj-41-q06-p01-2ce212c4f5.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2021-mj-41-q06-p02-d10246dba0.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2021-mj-41-q06-p03-a627a98eba.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-41-q06",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "6 Tanji has a bag containing 4 red balls and 2 blue balls. He selects 3 balls at random from the bag,\nwithout replacement. The number of red balls selected by Tanji is denoted by X.\n(a) Find the probability generating function G (t) of X. [2]\nX\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 . Tanji also has two coins, each biased so that the probability of obtaining a head when it is thrown is\n4\nHe throws the two coins at the same time. The number of heads obtained is denoted by Y.\n(b) Find the probability generating function G (t) of Y. [2]\nY\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2021-mj-41-q06-b46c49ed63.pdf"
    },
    {
      "id": "9231-2021-mj-42-q01",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_42.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-mj-42-q01-p01-cf403a6085.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-42-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 A random sample of 7 observations of a variable X are as follows.\n8.26 7.78 7.92 8.04 8.27 7.95 8.34\nn. The population mean of X is\n= 8.22 against the alternative hypothesis (a) Test, at the 10% significance level, the null hypothesis n\n1 8.22. [6] n\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2021-mj-42-q01-3a723f9cc7.pdf"
    },
    {
      "id": "9231-2021-mj-42-q02",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_42.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_42.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2021-mj-42-q02-p01-cc675c4564.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2021-mj-42-q02-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-42-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "2 A driving school employs four instructors to prepare people for their driving test. The allocation of\npeople to instructors is random. For each of the instructors, the following table gives the number of\npeople who passed and the number who failed their driving test last year.\nInstructor A Instructor B Instructor C Instructor D Total\nPass 72 42 52 68 234\nFail 33 34 41 58 166\nTotal 105 76 93 126 400\nTest at the 10% significance level whether success in the driving test is independent of the instructor.\n[7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n............",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2021-mj-42-q02-6676b0b72b.pdf"
    },
    {
      "id": "9231-2021-mj-42-q03",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 3,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_42.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_42.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2021-mj-42-q03-p01-22735b61ae.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2021-mj-42-q03-p02-a6ad0ab6e6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-42-q03",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "3 The continuous random variable X has cumulative distribution function F given by\n1 0, x 0\n1 2 G G = F(x) x 0 x 9,\n81\n*\n2 9. 1 x\n(a) Find [3] E X .\n` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n[2] Var X . (b) Find\n` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2021-mj-42-q03-9295423602.pdf"
    },
    {
      "id": "9231-2021-mj-42-q04",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 4,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_42.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-mj-42-q04-p01-1d3bde07b8.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-mj-42-q04-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-42-q04",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "4 A scientist is investigating the lengths of the leaves of birch trees in different regions. He takes a random\nsample of 50 leaves from birch trees in region A and a random sample of 60 leaves from birch trees in\nregion B. He records their lengths in cm, x and y, respectively. His results are summarised as follows.\n2 2 = = = = /x /x /y /y 282 1596 328 1808\nThe population mean lengths of leaves from birch trees in regions A and B are cm and cm n n\nA B\nrespectively.\n= Carry out a test at the 5% significance level to test the null hypothesis against the alternative n n\nA B\n! hypothesis [8] . n n\nA B\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.....................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2021-mj-42-q04-be558c0e53.pdf"
    },
    {
      "id": "9231-2021-mj-42-q05",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 5,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_42.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_42.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2021-mj-42-q05-p01-30ff0cc6ed.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2021-mj-42-q05-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-42-q05",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "5 Georgio has designed two new uniforms X and Y for the employees of an airline company. A random\nsample of 11 employees are each asked to assess each of the two uniforms for practicality and\nappearance, and to give a total score out of 100. The scores are given in the table.\nEmployee A B C D E F G H I J K\nUniform X 82 74 42 59 60 73 94 98 62 36 50\nUniform Y 78 75 63 56 67 82 99 90 72 48 61\n(a) Give a reason why a Wilcoxon signed-rank test may be more appropriate than a t-test for\ninvestigating whether there is any evidence of a preference for one of the uniforms. [1]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Carry out a Wilcoxon matched-pairs signed-rank test at the 10% significance level. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2021-mj-42-q05-dd53d7b8cb.pdf"
    },
    {
      "id": "9231-2021-mj-42-q06",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 6,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_42.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_42.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2021-mj-42-q06-p01-2ce212c4f5.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2021-mj-42-q06-p02-d10246dba0.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2021-mj-42-q06-p03-a627a98eba.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-42-q06",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "6 Tanji has a bag containing 4 red balls and 2 blue balls. He selects 3 balls at random from the bag,\nwithout replacement. The number of red balls selected by Tanji is denoted by X.\n(a) Find the probability generating function G (t) of X. [2]\nX\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 . Tanji also has two coins, each biased so that the probability of obtaining a head when it is thrown is\n4\nHe throws the two coins at the same time. The number of heads obtained is denoted by Y.\n(b) Find the probability generating function G (t) of Y. [2]\nY\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2021-mj-42-q06-8a65d5c8fc.pdf"
    },
    {
      "id": "9231-2021-mj-43-q01",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_43.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-mj-43-q01-p01-20b16964ac.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-43-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 Farmer A grows apples of a certain variety. Each tree produces 14.8 kg of apples, on average, per year.\nFarmer B grows apples of the same variety and claims that his apple trees produce a higher mass of\napples per year than Farmer A’s trees. The masses of apples from Farmer B’s trees may be assumed to\nbe normally distributed.\nA random sample of 10 trees from Farmer B is chosen. The masses, x kg, of apples produced in a year\nare summarised as follows.\n2 = = /x /x 152.0 2313.0\nTest, at the 5% significance level, whether Farmer B’s claim is justified. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2021-mj-43-q01-ccb86ddabb.pdf"
    },
    {
      "id": "9231-2021-mj-43-q02",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 2,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_43.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_43.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2021-mj-43-q02-p01-4ef26b72bd.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-43-q02",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "2 A company is developing a new flavour of chocolate by varying the quantities of the ingredients.\nA random selection of 9 flavours of chocolate are judged by two tasters who each give marks out of 100\nto each flavour of chocolate.\nChocolate A B C D E F G H I\nTaster 1 72 86 75 92 98 79 87 60 62\nTaster 2 84 72 74 95 85 87 82 75 68\nCarry out a Wilcoxon matched-pairs signed-rank test at the 10% significance level to investigate\nwhether, on average, there is a difference between marks awarded by the two tasters. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.......",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2021-mj-43-q02-812ef661a2.pdf"
    },
    {
      "id": "9231-2021-mj-43-q03",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 3,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_43.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-mj-43-q03-p01-62e3764d97.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-mj-43-q03-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-43-q03",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "3 The heights, x m, of a random sample of 50 adult males from country A were recorded. The heights, y m,\nof a random sample of 40 adult males from country B were also recorded. The results are summarised\nas follows.\n2 2 = = = = /x /x /y /y 89.0 159.4 67.2 113.1\nFind a 95% confidence interval for the difference between the mean heights of adult males from country\nA and adult males from country B. [8]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2021-mj-43-q03-5b611e0b34.pdf"
    },
    {
      "id": "9231-2021-mj-43-q04",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 4,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_43.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_43.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2021-mj-43-q04-p01-b9bb135f7d.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2021-mj-43-q04-p02-ea192c3188.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-43-q04",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "4 X is a discrete random variable which takes the values 0, 2, 4, … . The probability generating function\nof X is given by\n1\n= G (t) .\n2 X 3-2t\n(a) Find E(X) Var (X). and [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2021-mj-43-q04-feadf65963.pdf"
    },
    {
      "id": "9231-2021-mj-43-q05",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 5,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_43.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_43.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2021-mj-43-q05-p01-a68b0525cb.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2021-mj-43-q05-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-43-q05",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "5 Chai packs china mugs into cardboard boxes. Chai’s manager suspects that breakages occur at random\ntimes and that the number of breakages may follow a Poisson distribution. He takes a small sample\nof observations and finds that the number of breakages in a one-hour period has a mean of 2.4 and a\nstandard deviation of 1.5.\n(a) Explain how this information tends to support the manager’s suspicion. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nThe manager now takes a larger sample and claims that the numbers of breakages in a one-hour period\nfollow a Poisson distribution. The numbers of breakages in a random sample of 180 one-hour periods\nare summarised in the following table.\nNumber of breakages 0 1 2 3 4 5 6 7 or more\nFrequency 21 33 46 31 23 16 10 0\nThe mean number of breakages calculated from this sample is 2.5.\n(b) Use the data from this larger sample to carry out a goodness of fit test, at the 10% significance\nlevel, to test the claim. [8]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2021-mj-43-q05-5a8f326bde.pdf"
    },
    {
      "id": "9231-2021-mj-43-q06",
      "subject": "9231",
      "year": 2021,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 6,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-may-june/9231_s21_qp_43.pdf?download=true",
      "source_pages": [
        10,
        11,
        12
      ],
      "local_pdf": "_source-pdfs/2021-May-June/qp/9231_s21_qp_43.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2021-mj-43-q06-p01-3ff469d571.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2021-mj-43-q06-p02-3bfa38bfb7.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2021-mj-43-q06-p03-c6bf354cb4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-mj-43-q06",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "6 The continuous random variable X has probability density function f given by\n1 1 G x 1, 0\n8\n1 = G G (8-x) f(x) x 8, 1\n*28\n0 otherwise.\n(a) Find the cumulative distribution function of X. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n5 G = P(X a) . [3] (b) Find the value of the constant a such that\n7\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2021-mj-43-q06-19d1ac360e.pdf"
    },
    {
      "id": "9231-2021-on-11-q01",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 1,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_11.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_11.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2021-on-11-q01-p01-3b8566aca5.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2021-on-11-q01-p02-b1d31dc673.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-11-q01",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "1 It is given that\n2 2 2 3 3 3 a+b+c = +b +c +b +c = = 3, 5, 6. a a\n3 2 +bx +cx+d = x 0 has roots a, b, c. The cubic equation\nFind the values of b, c and d. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2021-on-11-q01-4dc59c55e1.pdf"
    },
    {
      "id": "9231-2021-on-11-q02",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 2,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_11.pdf?download=true",
      "source_pages": [
        3,
        4
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_11.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2021-on-11-q02-p01-c6fcee1221.png",
        "9231-topic-13-summation-of-series/assets/9231-2021-on-11-q02-p02-f8566a460f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-11-q02",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "n\n/\nr(r+1)(r+2) in terms of n, 2 (a) Use standard results from the list of formulae (MF19) to find\nr=1\nfully factorising your answer. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\n(b) Express in partial fractions a",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2021-on-11-q02-003c2232b5.pdf"
    },
    {
      "id": "9231-2021-on-11-q03",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 3,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_11.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_11.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2021-on-11-q03-p01-df55c10974.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-11-q03",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "= a a a 3 The sequence of real numbers , , , … is such that a 1 and\n1 2 3 1\n3\n1\n= + a a .\nn+1 a n\ne o n\nn-1 H H (a) Prove by mathematical induction that lna 3 ln2 for all integers 2. [6] n\nn\n1\n2 2 x+ ln lnx for x 0.] [You may use the fact that\nx b l\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nn-1 2 H -lna lna 3 ln4 for 2. [2] n (b) Show that\nn+1 n\n............................................................................................................................................................\n............................................................................................................................................................\n.........................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2021-on-11-q03-f407a11b36.pdf"
    },
    {
      "id": "9231-2021-on-11-q04",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_11.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_11.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2021-on-11-q04-p01-f1beba77d5.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2021-on-11-q04-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-11-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "-sini cosi 3 0\n= 4 The matrix M is given by . M\nesini cosi oe0 1o\n(a) The matrix M represents a sequence of two geometrical transformations.\nState the type of each transformation, and make clear the order in which they are applied. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nG G i, for r, for which the transformation represented by M has exactly 0 (b) Find the values of i\none invariant line through the origin, giving your answers in terms of r. [9]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2021-on-11-q04-787eb4dd43.pdf"
    },
    {
      "id": "9231-2021-on-11-q05",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 5,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_11.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_11.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2021-on-11-q05-p01-42cd7334d4.png",
        "9231-topic-16-vectors/assets/9231-2021-on-11-q05-p02-5631ea56f7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-11-q05",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "= -2i+3j+3k+m(i+k) +n(2i+3j). 5 The plane has equation r P\nax+by+cz = d. P, giving your answer in the form [4] (a) Find a Cartesian equation of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2i-3j+5k and is parallel to the vector k. The line l passes through the point P with position vector\nP. [3] (b) Find the position vector of the point where l meets\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2021-on-11-q05-3339c09d00.pdf"
    },
    {
      "id": "9231-2021-on-11-q06",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_11.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_11.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2021-on-11-q06-p01-34483d7561.png",
        "9231-topic-15-polar-coordinates/assets/9231-2021-on-11-q06-p02-7b25f26cb6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-11-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "1 G G = 2cosi(1+sini), r 6 The curve C has polar equation for r. 0 i\n2\n(a) Find the polar coordinates of the point on C that is furthest from the pole. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2021-on-11-q06-f83bd81072.pdf"
    },
    {
      "id": "9231-2021-on-11-q07",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_11.pdf?download=true",
      "source_pages": [
        12,
        13,
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_11.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-on-11-q07-p01-175fd3c470.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-on-11-q07-p02-7ece10a5eb.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-on-11-q07-p03-14450fb6c7.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-on-11-q07-p04-3ed4353cf2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-11-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "4x+5\n= 7 The curve C has equation y .\n2 4-4x\n(a) Find the equations of the asymptotes of C. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the coordinates of any stationary points on C. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2021-on-11-q07-eec72f6f0a.pdf"
    },
    {
      "id": "9231-2021-on-12-q01",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 1,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_12.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_12.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2021-on-12-q01-p01-9b8a359b10.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-12-q01",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "1 (a) Give full details of the geometrical transformation in the x-y plane represented by the matrix\n6 0\n. [1]\ne0 6o\n............................................................................................................................................................\n............................................................................................................................................................\n3 4\n= Let A .\ne2 2o\n(b) The triangle DEF in the x-y plane is transformed by A onto triangle PQR.\n2 Given that the area of triangle DEF is 13 cm , find the area of triangle PQR. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n6 0\n= AB . [2] (c) Find the matrix B such that\ne0 6o\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(d) Show that the origin is the only invariant point of the transformation in the x-y plane represented\nby A. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2021-on-12-q01-674e4ac0df.pdf"
    },
    {
      "id": "9231-2021-on-12-q02",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_12.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_12.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2021-on-12-q02-p01-242fc91648.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-12-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "ax = 2 It is given that , where a is a constant. y xe\nProve by mathematical induction that, for all positive integers n,\nn d y\nn-1 n ax = x+na a e . [6]\nn ` j dx\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2021-on-12-q02-b5c0cf8f66.pdf"
    },
    {
      "id": "9231-2021-on-12-q03",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 3,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_12.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_12.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2021-on-12-q03-p01-31cd1067ad.png",
        "9231-topic-13-summation-of-series/assets/9231-2021-on-12-q03-p02-8e13466075.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-12-q03",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "n r(r+2)\n/\n= 3 Let . S ln\n2 n (r+1)\nr=1\nn+2\n= (a) Using the method of differences, or otherwise, show that S ln . [4]\n2(n+1) n\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2021-on-12-q03-72e15c8526.pdf"
    },
    {
      "id": "9231-2021-on-12-q04",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 4,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_12.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_12.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2021-on-12-q04-p01-7297f5ed76.png",
        "9231-topic-13-summation-of-series/assets/9231-2021-on-12-q04-p02-66ad171e66.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-12-q04",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "3 2 +2x +3x+3 = x 0 4 The cubic equation has roots a, b, c.\n2 2 2 +b +c . [2] (a) Find the value of a\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n3 3 3 +b +c = 1. [2] (b) Show that a\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2021-on-12-q04-590089ea25.pdf"
    },
    {
      "id": "9231-2021-on-12-q05",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_12.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_12.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2021-on-12-q05-p01-d4573d86f9.png",
        "9231-topic-15-polar-coordinates/assets/9231-2021-on-12-q05-p02-99fac449ca.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-12-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "1 G = 3+2sini, -r r 5 The curve C has polar equation for r. i\n(a) The diagram shows part of C. Sketch the rest of C on the diagram. [1]\nO = 0 i\n= rsini 2. The straight line l has polar equation\n(b) Add l to the diagram in part (a) and find the polar coordinates of the points of intersection of C\nand l. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(c) The region R is enclosed by C and l, and contains the pole.\nFind the area of R, giving your answer in exact form. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2021-on-12-q05-0a9296923f.pdf"
    },
    {
      "id": "9231-2021-on-12-q06",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_12.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_12.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-on-12-q06-p01-96b8c2c670.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-on-12-q06-p02-dc9c4fae33.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-12-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 x\n= y 6 The curve C has equation .\nx-3\n(a) Find the equations of the asymptotes of C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 1 0 y 12. [4] (b) Show that there is no point on C for which\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2021-on-12-q06-2d55fd2884.pdf"
    },
    {
      "id": "9231-2021-on-12-q07",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 7,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 17,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_12.pdf?download=true",
      "source_pages": [
        12,
        13,
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_12.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2021-on-12-q07-p01-5c870415e6.png",
        "9231-topic-16-vectors/assets/9231-2021-on-12-q07-p02-255b741e68.png",
        "9231-topic-16-vectors/assets/9231-2021-on-12-q07-p03-14450fb6c7.png",
        "9231-topic-16-vectors/assets/9231-2021-on-12-q07-p04-3ed4353cf2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-12-q07",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "7 The points A, B, C have position vectors\n2i+2j, -j+k 2i+j-7k and\nrespectively, relative to the origin O.\n= r.n p. [3] (a) Find an equation of the plane OAB, giving your answer in the form\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nx-3y-2z = has equation 1. The plane P\nfrom the origin. [1] (b) Find the perpendicular distance of P\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2021-on-12-q07-e820846a06.pdf"
    },
    {
      "id": "9231-2021-on-13-q01",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 1,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_13.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_13.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2021-on-13-q01-p01-3b8566aca5.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2021-on-13-q01-p02-b1d31dc673.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-13-q01",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "1 It is given that\n2 2 2 3 3 3 a+b+c = +b +c +b +c = = 3, 5, 6. a a\n3 2 +bx +cx+d = x 0 has roots a, b, c. The cubic equation\nFind the values of b, c and d. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2021-on-13-q01-00f0412a55.pdf"
    },
    {
      "id": "9231-2021-on-13-q02",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 2,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_13.pdf?download=true",
      "source_pages": [
        3,
        4
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_13.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2021-on-13-q02-p01-c6fcee1221.png",
        "9231-topic-13-summation-of-series/assets/9231-2021-on-13-q02-p02-f8566a460f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-13-q02",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "n\n/\nr(r+1)(r+2) in terms of n, 2 (a) Use standard results from the list of formulae (MF19) to find\nr=1\nfully factorising your answer. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\n(b) Express in partial fractions a",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2021-on-13-q02-afa6e98f29.pdf"
    },
    {
      "id": "9231-2021-on-13-q03",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 3,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_13.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_13.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2021-on-13-q03-p01-df55c10974.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-13-q03",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "= a a a 3 The sequence of real numbers , , , … is such that a 1 and\n1 2 3 1\n3\n1\n= + a a .\nn+1 a n\ne o n\nn-1 H H (a) Prove by mathematical induction that lna 3 ln2 for all integers 2. [6] n\nn\n1\n2 2 x+ ln lnx for x 0.] [You may use the fact that\nx b l\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nn-1 2 H -lna lna 3 ln4 for 2. [2] n (b) Show that\nn+1 n\n............................................................................................................................................................\n............................................................................................................................................................\n.........................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2021-on-13-q03-687207d506.pdf"
    },
    {
      "id": "9231-2021-on-13-q04",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_13.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_13.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2021-on-13-q04-p01-f1beba77d5.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2021-on-13-q04-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-13-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "-sini cosi 3 0\n= 4 The matrix M is given by . M\nesini cosi oe0 1o\n(a) The matrix M represents a sequence of two geometrical transformations.\nState the type of each transformation, and make clear the order in which they are applied. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nG G i, for r, for which the transformation represented by M has exactly 0 (b) Find the values of i\none invariant line through the origin, giving your answers in terms of r. [9]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2021-on-13-q04-0d622ac0c2.pdf"
    },
    {
      "id": "9231-2021-on-13-q05",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 5,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_13.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_13.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2021-on-13-q05-p01-42cd7334d4.png",
        "9231-topic-16-vectors/assets/9231-2021-on-13-q05-p02-5631ea56f7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-13-q05",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "= -2i+3j+3k+m(i+k) +n(2i+3j). 5 The plane has equation r P\nax+by+cz = d. P, giving your answer in the form [4] (a) Find a Cartesian equation of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2i-3j+5k and is parallel to the vector k. The line l passes through the point P with position vector\nP. [3] (b) Find the position vector of the point where l meets\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2021-on-13-q05-17edff5596.pdf"
    },
    {
      "id": "9231-2021-on-13-q06",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_13.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_13.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2021-on-13-q06-p01-34483d7561.png",
        "9231-topic-15-polar-coordinates/assets/9231-2021-on-13-q06-p02-7b25f26cb6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-13-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "1 G G = 2cosi(1+sini), r 6 The curve C has polar equation for r. 0 i\n2\n(a) Find the polar coordinates of the point on C that is furthest from the pole. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2021-on-13-q06-97d40fd9f7.pdf"
    },
    {
      "id": "9231-2021-on-13-q07",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_13.pdf?download=true",
      "source_pages": [
        12,
        13,
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_13.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-on-13-q07-p01-175fd3c470.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-on-13-q07-p02-7ece10a5eb.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-on-13-q07-p03-14450fb6c7.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2021-on-13-q07-p04-3ed4353cf2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-13-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "4x+5\n= 7 The curve C has equation y .\n2 4-4x\n(a) Find the equations of the asymptotes of C. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the coordinates of any stationary points on C. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2021-on-13-q07-aca70ac0d8.pdf"
    },
    {
      "id": "9231-2021-on-21-q01",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 1,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_21.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_21.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2021-on-21-q01-p01-f266982d0f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-21-q01",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "2 x x 1 Find the Maclaurin’s series for from first principles up to and including the term in . [5] e tanx\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n........................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2021-on-21-q01-f0e158a7f0.pdf"
    },
    {
      "id": "9231-2021-on-21-q02",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 2,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_21.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2021-on-21-q02-p01-371bd12ecc.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-21-q02",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "2 The matrix A is given by\n-1 2 12\n= A 0 1 0 .\nf p\n0 0 3\nUse the characteristic equation of A to show that\n4 2 = +qI, A pA\nwhere p and q are integers to be determined. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.......................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2021-on-21-q02-378641e34c.pdf"
    },
    {
      "id": "9231-2021-on-21-q03",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 3,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_21.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_21.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2021-on-21-q03-p01-fe12173a83.png",
        "9231-topic-23-differentiation/assets/9231-2021-on-21-q03-p02-ff703ab390.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-21-q03",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "3 The curve C has equation\n3 3 -4x = xy y 3.\ndy\n(-1,1) = 11. (a) Show that, at the point on C, [3]\ndx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2021-on-21-q03-379ccf17b5.pdf"
    },
    {
      "id": "9231-2021-on-21-q04",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 4,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_21.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2021-on-21-q04-p01-0316911d86.png",
        "9231-topic-24-integration/assets/9231-2021-on-21-q04-p02-87b5c9e91e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-21-q04",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "4\ny\nx 0 1 2 3 4 N – 1 N\nlnx\nH = (N-2) y The diagram shows the curve with equation for 2, together with a set of rectangles x\n2 x\nof unit width.\n(a) By considering the sum of the areas of these rectangles, show that\nN\n2+3ln2 1+lnN /lnr\n1 - . [7]\nN 4 2 r\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nN\n/lnr\n(b) Use a similar method to find, in terms of N, a lower bound for . [3]\n2 r\nr=1\n...............................................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2021-on-21-q04-76d3d89fe6.pdf"
    },
    {
      "id": "9231-2021-on-21-q05",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_21.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2021-on-21-q05-p01-31a2e29ee9.png",
        "9231-topic-26-differential-equations/assets/9231-2021-on-21-q05-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-21-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "5 Find the particular solution of the differential equation\n2 d y dy\n-2 +y = 4cosx,\n2 dx dx\ndy\n= = =-4 3. given that, when x 0, y and [11]\ndx\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.....................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2021-on-21-q05-2a43e3c27e.pdf"
    },
    {
      "id": "9231-2021-on-21-q06",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 6,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_21.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_21.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2021-on-21-q06-p01-3b21dd19e5.png",
        "9231-topic-25-complex-numbers/assets/9231-2021-on-21-q06-p02-e9ca87a55b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-21-q06",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "6 (a) Use de Moivre’s theorem to show that\n5 cosec i\n= cosec5i . [6]\n4 2 i-20cosec i+16 5cosec\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2021-on-21-q06-21ca56fc51.pdf"
    },
    {
      "id": "9231-2021-on-21-q07",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_21.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2021-on-21-q07-p01-cc0b9130a4.png",
        "9231-topic-26-differential-equations/assets/9231-2021-on-21-q07-p02-c487cbe949.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-21-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "7 (a) Show that an appropriate integrating factor for\ndy\n2 2 2 -1 +y = -x -1 x x x\ndx\n2 x+ -1. is [4] x\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2021-on-21-q07-509b8326a2.pdf"
    },
    {
      "id": "9231-2021-on-21-q08",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 8,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_21.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2021-on-21-q08-p01-cdd848ac12.png",
        "9231-topic-24-integration/assets/9231-2021-on-21-q08-p02-c32ca0ea68.png",
        "9231-topic-24-integration/assets/9231-2021-on-21-q08-p03-8b14c535d4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-21-q08",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "8 (a) Starting from the definition of cosh in terms of exponentials, prove that\n2 = cosh2A+1. 2cosh A [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nThe curve C has parametric equations\n3 -1 1 G G = cosh2t-4t, = 2cosh2t+3t, y x for t .\n2 2 2\nThe area of the surface generated when C is rotated through 2r radians about the y-axis is denoted\nby A.\n1\n2 y = (2cosh2t+3t)cosh2t (b) (i) Show that dt. [4] A 10r\n-1\n2\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n...................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2021-on-21-q08-bf8ab00b50.pdf"
    },
    {
      "id": "9231-2021-on-22-q01",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 1,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_22.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_22.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2021-on-22-q01-p01-3c78411617.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-22-q01",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "2 2 = +cosh(x y sinh(x ) ). 1 It is given that\n(a) Use standard results from the list of formulae (MF19) to find the Maclaurin’s series for y in terms\n4 of x up to and including the term in x . [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n4 d y\n= when x 0. [1] (b) Deduce the value of\n4 dx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\n2 y\ndx, giving your answer as a rational y (c) Use your answer to part (a) to find an approximation to\n0 fraction in its lowest terms. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................................................................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2021-on-22-q01-a17a0aaf72.pdf"
    },
    {
      "id": "9231-2021-on-22-q02",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 2,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_22.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2021-on-22-q02-p01-b22aca5836.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-22-q02",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "2 Find the solution of the differential equation\n3 4x dy y\n+ = 6x\n4 dx +5 x\n= = = y f(x). for which y 1 when x 1. Give your answer in the form [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2021-on-22-q02-5ef1f2830e.pdf"
    },
    {
      "id": "9231-2021-on-22-q03",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 3,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_22.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2021-on-22-q03-p01-4bcf837dff.png",
        "9231-topic-24-integration/assets/9231-2021-on-22-q03-p02-30ae4f23be.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-22-q03",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "3\ny\n1\nx n-1 0 1 1 2\nn n n\n2 G G = 1-x y The diagram shows the curve with equation for 1, together with a set of n rectangles 0 x\n1\nof width .\nn\n(a) By considering the sum of the areas of the rectangles, show that\n2 1 +3n-1 4n y 2 1 (1-x )dx . [4]\n2 6n 0\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2021-on-22-q03-c278beece4.pdf"
    },
    {
      "id": "9231-2021-on-22-q04",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 4,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_22.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_22.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2021-on-22-q04-p01-60f199679c.png",
        "9231-topic-25-complex-numbers/assets/9231-2021-on-22-q04-p02-476e05463c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-22-q04",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "5 -1 = 4 (a) Write down all the roots of the equation 0. [2] x\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n4 2 = i-8cos i+1. cos4i 8cos [4] (b) Use de Moivre’s theorem to show that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2021-on-22-q04-30f8dd9404.pdf"
    },
    {
      "id": "9231-2021-on-22-q05",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 5,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_22.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2021-on-22-q05-p01-8e3a1c07ae.png",
        "9231-topic-24-integration/assets/9231-2021-on-22-q05-p02-41a4ccb252.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-22-q05",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "5 The curve C has parametric equations\n-1 -1 3 3 3 G G = 3t+2t +at = 4t- +bt x y t , , for 2, 1 t\n2\nwhere a and b are constants.\n2 =-1 = and . a b (a) It is given that\n3 2\n2 2 dy dx -2 25 2 2 + = +t Show that and find the exact length of C. [6] (t )\n4 edto edto\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2021-on-22-q05-a16b438824.pdf"
    },
    {
      "id": "9231-2021-on-22-q06",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 6,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_22.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2021-on-22-q06-p01-6597412ba1.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2021-on-22-q06-p02-e15509d1fc.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-22-q06",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "6 The matrix P is given by\n1 6 6\n= P 0 2 6 .\nf p\n-3 0 0\n-1\n(a) Use the characteristic equation of P to find P . [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2021-on-22-q06-132ce8758a.pdf"
    },
    {
      "id": "9231-2021-on-22-q07",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_22.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2021-on-22-q07-p01-98c2600922.png",
        "9231-topic-26-differential-equations/assets/9231-2021-on-22-q07-p02-baa7a603f1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-22-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "2 = y x w 7 It is given that and\n2 dw d w\n2 2 2 +4x(x+1) + +8x+2)w = +4x+2. x (5x 5x\n2 dx dx\n(a) Show that\n2 d dy y\n2 +4 +5y = +4x+2. 5x [4]\n2 dx dx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2021-on-22-q07-56dcfade6f.pdf"
    },
    {
      "id": "9231-2021-on-22-q08",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 8,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_22.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_22.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2021-on-22-q08-p01-10d823c266.png",
        "9231-topic-21-hyperbolic-functions/assets/9231-2021-on-22-q08-p02-2e5fe26a5c.png",
        "9231-topic-21-hyperbolic-functions/assets/9231-2021-on-22-q08-p03-8b14c535d4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-22-q08",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "8 (a) Starting from the definitions of tanh and sech in terms of exponentials, prove that\n2 2 1-tanh = x sech x. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\ny 2 2 = u tanhx, or otherwise, find dx. [2] sech xtanh x (b) Using the substitution\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nln3\ny n 2 H = 0, I sech xtanh x dx. n It is given that, for\nn\n0\nH 2, n (c) Show that, for\nn-2 3\n3 4 (n+1)I = + (n-2)I . [5]\nn-2 5 5 n b l b l\nd\n=-tanhxsech (sech x) [You may use the result that x.]\ndx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................................................................................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2021-on-22-q08-2a17b6d221.pdf"
    },
    {
      "id": "9231-2021-on-23-q01",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 1,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_23.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_23.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2021-on-23-q01-p01-f266982d0f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-23-q01",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "2 x x 1 Find the Maclaurin’s series for from first principles up to and including the term in . [5] e tanx\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n........................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2021-on-23-q01-16af3094c4.pdf"
    },
    {
      "id": "9231-2021-on-23-q02",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 2,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_23.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2021-on-23-q02-p01-371bd12ecc.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-23-q02",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "2 The matrix A is given by\n-1 2 12\n= A 0 1 0 .\nf p\n0 0 3\nUse the characteristic equation of A to show that\n4 2 = +qI, A pA\nwhere p and q are integers to be determined. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.......................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2021-on-23-q02-cfc17047f5.pdf"
    },
    {
      "id": "9231-2021-on-23-q03",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 3,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_23.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_23.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2021-on-23-q03-p01-fe12173a83.png",
        "9231-topic-23-differentiation/assets/9231-2021-on-23-q03-p02-ff703ab390.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-23-q03",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "3 The curve C has equation\n3 3 -4x = xy y 3.\ndy\n(-1,1) = 11. (a) Show that, at the point on C, [3]\ndx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2021-on-23-q03-b733c4dcb9.pdf"
    },
    {
      "id": "9231-2021-on-23-q04",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 4,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_23.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2021-on-23-q04-p01-0316911d86.png",
        "9231-topic-24-integration/assets/9231-2021-on-23-q04-p02-87b5c9e91e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-23-q04",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "4\ny\nx 0 1 2 3 4 N – 1 N\nlnx\nH = (N-2) y The diagram shows the curve with equation for 2, together with a set of rectangles x\n2 x\nof unit width.\n(a) By considering the sum of the areas of these rectangles, show that\nN\n2+3ln2 1+lnN /lnr\n1 - . [7]\nN 4 2 r\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nN\n/lnr\n(b) Use a similar method to find, in terms of N, a lower bound for . [3]\n2 r\nr=1\n...............................................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2021-on-23-q04-407c961a39.pdf"
    },
    {
      "id": "9231-2021-on-23-q05",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_23.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2021-on-23-q05-p01-31a2e29ee9.png",
        "9231-topic-26-differential-equations/assets/9231-2021-on-23-q05-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-23-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "5 Find the particular solution of the differential equation\n2 d y dy\n-2 +y = 4cosx,\n2 dx dx\ndy\n= = =-4 3. given that, when x 0, y and [11]\ndx\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.....................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2021-on-23-q05-e237c4d7bc.pdf"
    },
    {
      "id": "9231-2021-on-23-q06",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 6,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_23.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_23.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2021-on-23-q06-p01-3b21dd19e5.png",
        "9231-topic-25-complex-numbers/assets/9231-2021-on-23-q06-p02-e9ca87a55b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-23-q06",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "6 (a) Use de Moivre’s theorem to show that\n5 cosec i\n= cosec5i . [6]\n4 2 i-20cosec i+16 5cosec\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2021-on-23-q06-7869fd6bce.pdf"
    },
    {
      "id": "9231-2021-on-23-q07",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_23.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2021-on-23-q07-p01-cc0b9130a4.png",
        "9231-topic-26-differential-equations/assets/9231-2021-on-23-q07-p02-c487cbe949.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-23-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "7 (a) Show that an appropriate integrating factor for\ndy\n2 2 2 -1 +y = -x -1 x x x\ndx\n2 x+ -1. is [4] x\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2021-on-23-q07-1b0fbe8308.pdf"
    },
    {
      "id": "9231-2021-on-23-q08",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 8,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_23.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2021-on-23-q08-p01-cdd848ac12.png",
        "9231-topic-24-integration/assets/9231-2021-on-23-q08-p02-c32ca0ea68.png",
        "9231-topic-24-integration/assets/9231-2021-on-23-q08-p03-8b14c535d4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-23-q08",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "8 (a) Starting from the definition of cosh in terms of exponentials, prove that\n2 = cosh2A+1. 2cosh A [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nThe curve C has parametric equations\n3 -1 1 G G = cosh2t-4t, = 2cosh2t+3t, y x for t .\n2 2 2\nThe area of the surface generated when C is rotated through 2r radians about the y-axis is denoted\nby A.\n1\n2 y = (2cosh2t+3t)cosh2t (b) (i) Show that dt. [4] A 10r\n-1\n2\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n....................................................................................................................................................\n...................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2021-on-23-q08-d2c1ed9e18.pdf"
    },
    {
      "id": "9231-2021-on-31-q01",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 1,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_31.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2021-on-31-q01-p01-b1a3f96626.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-31-q01",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "1 One end of a light elastic string, of natural length a and modulus of elasticity 3mg, is attached to a fixed\npoint O on a smooth horizontal plane. A particle P of mass m is attached to the other end of the string\n4 and moves in a horizontal circle with centre O. The speed of P is ga.\n3\nFind the extension of the string. [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2021-on-31-q01-ca4f8193d4.pdf"
    },
    {
      "id": "9231-2021-on-31-q02",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 2,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_31.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_31.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2021-on-31-q02-p01-ff26a74854.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-31-q02",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "-2\nams 2 A particle P of mass m kg moves along a horizontal straight line with acceleration given by\n2 v(1-2t )\n= a ,\nt\n-1\nvms is the velocity of P at time t s. where\n(a) Find an expression for v in terms of t and an arbitrary constant. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= = (b) Given that a 5 when t 1, find an expression, in terms of m and t, for the horizontal force acting\non P at time t. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2021-on-31-q02-dfcdabd824.pdf"
    },
    {
      "id": "9231-2021-on-31-q03",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 3,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_31.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_31.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2021-on-31-q03-p01-bbe731c7eb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-31-q03",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "3 A light elastic string has natural length a and modulus of elasticity 12mg. One end of the string is\nattached to a fixed point O. The other end of the string is attached to a particle of mass m. The particle\nhangs in equilibrium vertically below O. The particle is pulled vertically down and released from rest\n1 2 with the extension of the string equal to e, where a. In the subsequent motion the particle has e\n3\n1 when it has ascended a distance a. 2ga speed\n3\nFind e in terms of a. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2021-on-31-q03-ada1f8445c.pdf"
    },
    {
      "id": "9231-2021-on-31-q04",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_31.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_31.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2021-on-31-q04-p01-5a7c7e24d8.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2021-on-31-q04-p02-d1115fdbca.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-31-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "4\n3a D C\nh\nF\n3a\nh\nA E B\nA uniform lamina AECF is formed by removing two identical triangles BCE and CDF from a square\n= = lamina ABCD. The square has side 3a and EB DF h (see diagram).\n(a) Find the distance of the centre of mass of the lamina AECF from AD and from AB, giving your\nanswers in terms of a and h. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2021-on-31-q04-6750520dea.pdf"
    },
    {
      "id": "9231-2021-on-31-q05",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 5,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_31.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_31.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-on-31-q05-p01-7314dfabd6.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-on-31-q05-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-31-q05",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "5 A particle P is projected from a point O on a horizontal plane and moves freely under gravity. Its initial\n-1 -1 4 ums and its angle of projection is above the horizontal. At time 8 s after projection, sin ( ) speed is\n5\nP is at the point A. At time 32 s after projection, P is at the point B. The direction of motion of P at B is\nperpendicular to its direction of motion at A.\nFind the value of u. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n........................................................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2021-on-31-q05-24f405de9f.pdf"
    },
    {
      "id": "9231-2021-on-31-q06",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 6,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_31.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2021-on-31-q06-p01-2521c675e5.png",
        "9231-topic-33-circular-motion/assets/9231-2021-on-31-q06-p02-31e1c29bb3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-31-q06",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "6 A particle P, of mass m, is attached to one end of a light inextensible string of length a. The other end\nof the string is attached to a fixed point O. The particle P moves in complete vertical circles about O\nwith the string taut. The points A and B are on the path of P with AB a diameter of the circle. OA makes\nan angle with the downward vertical through O and OB makes an angle with the upward vertical i i\n5ag. through O. The speed of P when it is at A is\nThe ratio of the tension in the string when P is at A to the tension in the string when P is at B is 9 : 5.\ncosi. [6] (a) Find the value of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2021-on-31-q06-777b212b12.pdf"
    },
    {
      "id": "9231-2021-on-31-q07",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 7,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_31.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_31.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2021-on-31-q07-p01-a96ccf7f05.png",
        "9231-topic-36-momentum/assets/9231-2021-on-31-q07-p02-48fb63cd21.png",
        "9231-topic-36-momentum/assets/9231-2021-on-31-q07-p03-dbfefa0085.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-31-q07",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "7\nA B\nc\nb\na\nu\nP\nC\nThe smooth vertical walls AB and CB are at right angles to each other. A particle P is moving with\nspeed u on a smooth horizontal floor and strikes the wall CB at an angle a. It rebounds at an angle to b\nthe wall CB. The particle then strikes the wall AB and rebounds at an angle to that wall (see diagram). c\nThe coefficient of restitution between each wall and P is e.\n= (a) Show that tanb etana. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nin terms of and explain what this result means about the final direction of motion of P. (b) Express c a\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2021-on-31-q07-ddf4483903.pdf"
    },
    {
      "id": "9231-2021-on-32-q01",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 1,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_32.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_32.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-on-32-q01-p01-ef85580def.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-32-q01",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "1 A particle is projected with speed u at an angle above the horizontal from a point O on a horizontal a\nplane. The particle moves freely under gravity.\n(a) Write down the horizontal and vertical components of the velocity of the particle at time T after\nprojection. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nAt time T after projection, the direction of motion of the particle is perpendicular to the direction of\nprojection.\n(b) Express T in terms of u, g and a. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nu\n2 T . [1] (c) Deduce that\ng\n............................................................................................................................................................\n............................................................................................................................................................\n............................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2021-on-32-q01-1c4f4bf125.pdf"
    },
    {
      "id": "9231-2021-on-32-q02",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 2,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_32.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_32.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2021-on-32-q02-p01-502e1feeab.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-32-q02",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "2 A light spring AB has natural length a and modulus of elasticity 5mg. The end A of the spring is attached\nto a fixed point on a smooth horizontal surface. A particle P of mass m is attached to the end B of the\nspring. The spring and particle P are at rest on the surface.\nAnother particle Q of mass km is moving with speed along the horizontal surface towards P in 4ga\nthe direction BA. The particles P and Q collide directly and coalesce. In the subsequent motion the\n1 greatest amount by which the spring is compressed is a.\n5\nFind the value of k. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n......................................................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2021-on-32-q02-e16308cfd5.pdf"
    },
    {
      "id": "9231-2021-on-32-q03",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 3,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_32.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2021-on-32-q03-p01-ba3cc2c574.png",
        "9231-topic-33-circular-motion/assets/9231-2021-on-32-q03-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-32-q03",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "3\nR\ni\nA m 3m B\nParticles A and B, of masses m and 3m respectively, are connected by a light inextensible string of\nlength a that passes through a fixed smooth ring R. Particle B hangs in equilibrium vertically below the\nring. Particle A moves in horizontal circles with speed v. Particles A and B are at the same horizontal\nlevel. The angle between AR and BR is (see diagram). i\n1 = . [2] cosi (a) Show that\n3\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find an expression for v in terms of a and g. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2021-on-32-q03-45c5aa9a51.pdf"
    },
    {
      "id": "9231-2021-on-32-q04",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_32.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_32.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2021-on-32-q04-p01-51b638f3c2.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2021-on-32-q04-p02-675e454aaa.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-32-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "4\n1 B A a\n2\nka\na\n1 An object is formed by removing a solid cylinder, of height ka and radius a, from a uniform solid\n2\nhemisphere of radius a. The axes of symmetry of the hemisphere and the cylinder coincide and one\ncircular face of the cylinder coincides with the plane face of the hemisphere. AB is a diameter of the\ncircular face of the hemisphere (see diagram).\n2 3a(2-k )\n(a) Show that the distance of the centre of mass of the object from AB is . [4]\n2(8-3k)\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2021-on-32-q04-42b90ea68e.pdf"
    },
    {
      "id": "9231-2021-on-32-q05",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 5,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_32.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_32.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2021-on-32-q05-p01-f4b7bf6fb2.png",
        "9231-topic-36-momentum/assets/9231-2021-on-32-q05-p02-a37ede79f6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-32-q05",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "5\nA B\nu\n60°\n3\nm m\n2\nu\n3\nm Two uniform smooth spheres A and B of equal radii have masses m and respectively. The two\n2\nspheres are each moving with speed u on a horizontal surface when they collide. Immediately before\nthe collision A’s direction of motion is along the line of centres, and B’s direction of motion makes an\nangle of 60° with the line of centres (see diagram). The coefficient of restitution between the spheres\n2 is .\n3\n(a) Find the angle through which the direction of motion of B is deflected by the collision. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2021-on-32-q05-8ab72fc47e.pdf"
    },
    {
      "id": "9231-2021-on-32-q06",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 6,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_32.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_32.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2021-on-32-q06-p01-64ac2e95f4.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2021-on-32-q06-p02-fb686d72d3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-32-q06",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "6 A particle P of mass 2 kg moves along a horizontal straight line. The point O is a fixed point on this\n-1\nline. At time t s the velocity of P is vms and the displacement of P from O is x m.\n128\n= = = -15. 8x- t 0, x 8 and v N acts on P in the direction OP. When A force of magnitude\n3 x e o\n2\n2 = - -4). (a) Show that v (x [5]\nx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2021-on-32-q06-9c63786b53.pdf"
    },
    {
      "id": "9231-2021-on-32-q07",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 7,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_32.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2021-on-32-q07-p01-ddde35786d.png",
        "9231-topic-33-circular-motion/assets/9231-2021-on-32-q07-p02-79115867d7.png",
        "9231-topic-33-circular-motion/assets/9231-2021-on-32-q07-p03-dbfefa0085.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-32-q07",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "7 One end of a light inextensible string of length a is attached to a fixed point O. The other end of the\nstring is attached to a particle P of mass m. The particle P is held vertically below O with the string\ntaut and then projected horizontally. When the string makes an angle of 60° with the upward vertical,\nP becomes detached from the string. In its subsequent motion, P passes through the point A which is a\ndistance a vertically above O.\n(a) The speed of P when it becomes detached from the string is V. Use the equation of the trajectory\nof a projectile to find V in terms of a and g. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2021-on-32-q07-90ad346523.pdf"
    },
    {
      "id": "9231-2021-on-33-q01",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 1,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_33.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2021-on-33-q01-p01-b1a3f96626.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-33-q01",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "1 One end of a light elastic string, of natural length a and modulus of elasticity 3mg, is attached to a fixed\npoint O on a smooth horizontal plane. A particle P of mass m is attached to the other end of the string\n4 and moves in a horizontal circle with centre O. The speed of P is ga.\n3\nFind the extension of the string. [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2021-on-33-q01-7d5c698491.pdf"
    },
    {
      "id": "9231-2021-on-33-q02",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 2,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_33.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_33.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2021-on-33-q02-p01-ff26a74854.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-33-q02",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "-2\nams 2 A particle P of mass m kg moves along a horizontal straight line with acceleration given by\n2 v(1-2t )\n= a ,\nt\n-1\nvms is the velocity of P at time t s. where\n(a) Find an expression for v in terms of t and an arbitrary constant. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= = (b) Given that a 5 when t 1, find an expression, in terms of m and t, for the horizontal force acting\non P at time t. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2021-on-33-q02-b34a727948.pdf"
    },
    {
      "id": "9231-2021-on-33-q03",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 3,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_33.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_33.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2021-on-33-q03-p01-bbe731c7eb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-33-q03",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "3 A light elastic string has natural length a and modulus of elasticity 12mg. One end of the string is\nattached to a fixed point O. The other end of the string is attached to a particle of mass m. The particle\nhangs in equilibrium vertically below O. The particle is pulled vertically down and released from rest\n1 2 with the extension of the string equal to e, where a. In the subsequent motion the particle has e\n3\n1 when it has ascended a distance a. 2ga speed\n3\nFind e in terms of a. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2021-on-33-q03-56476f1d45.pdf"
    },
    {
      "id": "9231-2021-on-33-q04",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_33.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_33.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2021-on-33-q04-p01-5a7c7e24d8.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2021-on-33-q04-p02-d1115fdbca.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-33-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "4\n3a D C\nh\nF\n3a\nh\nA E B\nA uniform lamina AECF is formed by removing two identical triangles BCE and CDF from a square\n= = lamina ABCD. The square has side 3a and EB DF h (see diagram).\n(a) Find the distance of the centre of mass of the lamina AECF from AD and from AB, giving your\nanswers in terms of a and h. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2021-on-33-q04-e774e0f80d.pdf"
    },
    {
      "id": "9231-2021-on-33-q05",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 5,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_33.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_33.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-on-33-q05-p01-7314dfabd6.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2021-on-33-q05-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-33-q05",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "5 A particle P is projected from a point O on a horizontal plane and moves freely under gravity. Its initial\n-1 -1 4 ums and its angle of projection is above the horizontal. At time 8 s after projection, sin ( ) speed is\n5\nP is at the point A. At time 32 s after projection, P is at the point B. The direction of motion of P at B is\nperpendicular to its direction of motion at A.\nFind the value of u. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n........................................................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2021-on-33-q05-c33b59c25a.pdf"
    },
    {
      "id": "9231-2021-on-33-q06",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 6,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_33.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2021-on-33-q06-p01-2521c675e5.png",
        "9231-topic-33-circular-motion/assets/9231-2021-on-33-q06-p02-31e1c29bb3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-33-q06",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "6 A particle P, of mass m, is attached to one end of a light inextensible string of length a. The other end\nof the string is attached to a fixed point O. The particle P moves in complete vertical circles about O\nwith the string taut. The points A and B are on the path of P with AB a diameter of the circle. OA makes\nan angle with the downward vertical through O and OB makes an angle with the upward vertical i i\n5ag. through O. The speed of P when it is at A is\nThe ratio of the tension in the string when P is at A to the tension in the string when P is at B is 9 : 5.\ncosi. [6] (a) Find the value of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2021-on-33-q06-a02fbfd8a5.pdf"
    },
    {
      "id": "9231-2021-on-33-q07",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 7,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_33.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_33.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2021-on-33-q07-p01-a96ccf7f05.png",
        "9231-topic-36-momentum/assets/9231-2021-on-33-q07-p02-48fb63cd21.png",
        "9231-topic-36-momentum/assets/9231-2021-on-33-q07-p03-dbfefa0085.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-33-q07",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "7\nA B\nc\nb\na\nu\nP\nC\nThe smooth vertical walls AB and CB are at right angles to each other. A particle P is moving with\nspeed u on a smooth horizontal floor and strikes the wall CB at an angle a. It rebounds at an angle to b\nthe wall CB. The particle then strikes the wall AB and rebounds at an angle to that wall (see diagram). c\nThe coefficient of restitution between each wall and P is e.\n= (a) Show that tanb etana. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nin terms of and explain what this result means about the final direction of motion of P. (b) Express c a\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2021-on-33-q07-f454576c86.pdf"
    },
    {
      "id": "9231-2021-on-41-q01",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_41.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-on-41-q01-p01-2b02587960.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-41-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 The times taken for students at a college to run 200 m have a normal distribution with mean s. The n\ntimes, x s, are recorded for a random sample of 10 students from the college. The results are summarised\nas follows, where is the sample mean. x\n/(x-x)2 = = x 25.6 78.5\n(a) Find a 90% confidence interval for n. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= k is carried out on this sample, using a 10% significance level. The A test of the null hypothesis n\n1 test does not support the alternative hypothesis k. n\n(b) Find the greatest possible value of k. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2021-on-41-q01-7fd91d5dd1.pdf"
    },
    {
      "id": "9231-2021-on-41-q02",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 2,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_41.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_41.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2021-on-41-q02-p01-3cd20780cf.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2021-on-41-q02-p02-45e5d522b0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-41-q02",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "2 The continuous random variable X has cumulative distribution function F given by\nZ\n1-1, 0 x ]\n] 1 2 G G ] (1+x) -1 0, x\n2 = F(x) [\n1 2 1 G 1- (1-x) 0 1, x\n] 2\n]] 2 1 x 1.\n\\\n(a) Find the probability density function of X. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1 1 G G . [2] P X (b) Find\n2 2 b l\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2021-on-41-q02-5027e24902.pdf"
    },
    {
      "id": "9231-2021-on-41-q03",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 3,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_41.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_41.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2021-on-41-q03-p01-99313580ff.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2021-on-41-q03-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-41-q03",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "3 A supermarket sells pears in packs of 8. Some of the pears in a pack may not be ripe, and the supermarket\nmanager claims that the number of unripe pears in a pack can be modelled by the distribution B(8, 0.15).\nA random sample of 150 packs was selected and the number of unripe pears in each pack was recorded.\nThe following table shows the observed frequencies together with some of the expected frequencies\nusing the manager’s binomial distribution.\nNumber of unripe pears per pack 0 1 2 3 4 5 H6\nObserved frequency 35 48 43 15 6 3 0\nExpected frequency 40.874 35.641 12.579 2.775 0.392 p q\n(a) Find the values of p and q. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Carry out a goodness of fit test, at the 5% significance level, to test whether the manager’s claim is\njustified. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2021-on-41-q03-c9f7da6dbb.pdf"
    },
    {
      "id": "9231-2021-on-41-q04",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 4,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_41.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-on-41-q04-p01-062ea52ca4.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-on-41-q04-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-41-q04",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "4 Manet has developed a new training course to help athletes improve their time taken to run 800 m.\nManet claims that his course will decrease an athlete’s time by more than 2 s on average. For a random\nsample of 10 athletes the times taken, in seconds, before and after the course are given in the following\ntable.\nAthlete A B C D E F G H I J\nBefore 150 146 131 135 126 142 130 129 137 134\nAfter 145 138 129 135 122 135 132 128 127 137\nUse a t-test, at the 5% significance level, to test whether Manet’s claim is justified, stating any\nassumption that you make. [8]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...........................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2021-on-41-q04-4d9dc7de78.pdf"
    },
    {
      "id": "9231-2021-on-41-q05",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 5,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_41.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_41.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2021-on-41-q05-p01-eacd189d53.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2021-on-41-q05-p02-3dca2fbe51.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-41-q05",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "5 Nine balls labelled 1, 2, 3, 4, 5, 6, 7, 8, 9 are placed in a bag. Kai selects three balls at random from\nthe bag, without replacement. The random variable X is the number of balls selected by Kai that are\nlabelled with a multiple of 3.\n(a) Find the probability generating function G (t) of X. [3]\nX\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nThe balls are replaced in the bag.\nJacob now selects two balls at random from the bag, without replacement. The random variable Y is the\nnumber of balls selected by Jacob that are labelled with an even number.\nG (t) of Y. [2] (b) Find the probability generating function\nY\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2021-on-41-q05-e84ffe6b11.pdf"
    },
    {
      "id": "9231-2021-on-41-q06",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 6,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_41.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_41.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2021-on-41-q06-p01-b60b0c147c.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2021-on-41-q06-p02-0bc12c5b27.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2021-on-41-q06-p03-a5443110d8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-41-q06",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "6 The blood cholesterol levels, measured in suitable units, of a random sample of 11 women and a random\nsample of 12 men are shown below.\nWomen 51 55 242 167 152 256 75 137 98 238 235\nMen 311 262 170 302 175 320 220 260 72 351 86 333\nCarry out a Wilcoxon rank-sum test, at the 5% significance level, to test whether, on average, there is a\ndifference in cholesterol levels between women and men. [9]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.............................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2021-on-41-q06-ff4e06ee33.pdf"
    },
    {
      "id": "9231-2021-on-42-q01",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_42.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-on-42-q01-p01-ffc9c160eb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-42-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 The number, x, of pine trees was counted in each of 40 randomly chosen regions of equal size in\ncountry A. The number, y, of pine trees was counted in each of 60 randomly chosen regions of the same\nequal size in country B. The results are summarised as follows.\n2 2 = = = = /x /x /y /y 752 14320 1548 40200\nFind a 95% confidence interval for the difference between the mean number of pine trees in regions of\nthis size in countries A and B. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..............................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2021-on-42-q01-891d34a7a7.pdf"
    },
    {
      "id": "9231-2021-on-42-q02",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_42.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_42.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2021-on-42-q02-p01-44ada38c6d.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2021-on-42-q02-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-42-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "2 It is claimed that the heights of a particular age group of boys follow a normal distribution with mean\n125 cm and standard deviation 12 cm. Observations for a randomly chosen group of 60 boys in this age\ngroup are summarised in the following table. The table also gives the expected frequencies, correct to\n2 decimal places, based on the normal distribution with mean 125 cm and standard deviation 12 cm.\n1 1 1 1 1 H G G G G Height, x 140 x 100 100 x 110 110 x 120 120 x 130 130 x 140\nx cm\nObserved 0 3 15 23 11 8\nfrequency\nExpected 1.12 5.22 13.97 19.38 13.97 6.34\nfrequency\n1 G 130 x 140 is obtained. [2] (a) Show how the expected frequency for\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Carry out a goodness of fit test, at the 5% significance level, to determine whether the claim is\nsupported by the data. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2021-on-42-q02-0b2eb3bbef.pdf"
    },
    {
      "id": "9231-2021-on-42-q03",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 3,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_42.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_42.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2021-on-42-q03-p01-e7fdb94160.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2021-on-42-q03-p02-0226a783c8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-42-q03",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "3 The continuous random variable X has probability density function f given by\n1 a+ x 1 G 0 1, x\n5\n= 1 f(x) G G 2a- 1 2, x x\n5 *\n0 otherwise,\nwhere a is a constant.\n(a) Find the value of a. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 E(X ). [2] (b) Find\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2021-on-42-q03-490ac2ef33.pdf"
    },
    {
      "id": "9231-2021-on-42-q04",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 4,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_42.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_42.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2021-on-42-q04-p01-95b630726e.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2021-on-42-q04-p02-4c865b7f5e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-42-q04",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "4 Applicants for a particular college take a written test when they attend for interview. There are two\ndifferent written tests, A and B, and each applicant takes one or the other. The interviewer wants\nto determine whether the medians of the distribution of marks obtained in the two tests are equal.\nThe marks obtained by a random sample of 8 applicants who took test A and a random sample of\n8 applicants who took test B are as follows.\n46 32 29 12 33 18 25 40 Test A\n36 28 49 37 48 35 41 31 Test B\n(a) Carry out a Wilcoxon rank-sum test at the 5% significance level to determine whether there is a\ndifference in the population median marks obtained in the two tests. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2021-on-42-q04-03f5ed706b.pdf"
    },
    {
      "id": "9231-2021-on-42-q05",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 5,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_42.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_42.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2021-on-42-q05-p01-e1b3150240.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2021-on-42-q05-p02-74ed0dc5be.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-42-q05",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "2 = = = P(X r) kr 5 The random variable X is such that for r 1,2,3,4, where k is a constant.\n(a) Find the value of k. [1]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nG (t ) of X. [2] (b) Find the probability generating function\nX\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 1 1 2 = + t+ . G (t ) t The random variable Y has probability generating function\n4 2 4 Y\nThe random variable Z is the sum of X and Y.\nG (t ) of Z as a (c) Assuming that X and Y are independent, find the probability generating function\nZ\npolynomial in t. [3]\n............................................................................................................................................................\n.........................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2021-on-42-q05-8fd1f67430.pdf"
    },
    {
      "id": "9231-2021-on-42-q06",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 6,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_42.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-on-42-q06-p01-da44ea6b8e.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-on-42-q06-p02-0bc12c5b27.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-on-42-q06-p03-a627a98eba.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-42-q06",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "6 A scientist is investigating the masses of a particular type of fish found in lakes A and B. He chooses a\nrandom sample of 10 fish of this type from lake A and records their masses, x kg, as follows.\n2.1 1.8 0.9 3.0 2.4 2.6 1.8 2.2 1.9 2.5\nThe scientist also chooses a random sample of 12 fish of this type from lake B, but he only has a\nsummary of their masses, y kg, as follows.\n2 = = /y /y 24.48 53.75\nTest at the 10% significance level whether the mean mass of fish of this type in lake A is greater than\nthe mean mass of fish of this type in lake B. You should state any assumptions that you need to make\nfor the test to be valid. [10]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...............................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2021-on-42-q06-7362227664.pdf"
    },
    {
      "id": "9231-2021-on-43-q01",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_43.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-on-43-q01-p01-2b02587960.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-43-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 The times taken for students at a college to run 200 m have a normal distribution with mean s. The n\ntimes, x s, are recorded for a random sample of 10 students from the college. The results are summarised\nas follows, where is the sample mean. x\n/(x-x)2 = = x 25.6 78.5\n(a) Find a 90% confidence interval for n. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= k is carried out on this sample, using a 10% significance level. The A test of the null hypothesis n\n1 test does not support the alternative hypothesis k. n\n(b) Find the greatest possible value of k. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2021-on-43-q01-ba1ad3dba1.pdf"
    },
    {
      "id": "9231-2021-on-43-q02",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 2,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_43.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_43.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2021-on-43-q02-p01-3cd20780cf.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2021-on-43-q02-p02-45e5d522b0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-43-q02",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "2 The continuous random variable X has cumulative distribution function F given by\nZ\n1-1, 0 x ]\n] 1 2 G G ] (1+x) -1 0, x\n2 = F(x) [\n1 2 1 G 1- (1-x) 0 1, x\n] 2\n]] 2 1 x 1.\n\\\n(a) Find the probability density function of X. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1 1 G G . [2] P X (b) Find\n2 2 b l\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2021-on-43-q02-48436dc4af.pdf"
    },
    {
      "id": "9231-2021-on-43-q03",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 3,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_43.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_43.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2021-on-43-q03-p01-99313580ff.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2021-on-43-q03-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-43-q03",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "3 A supermarket sells pears in packs of 8. Some of the pears in a pack may not be ripe, and the supermarket\nmanager claims that the number of unripe pears in a pack can be modelled by the distribution B(8, 0.15).\nA random sample of 150 packs was selected and the number of unripe pears in each pack was recorded.\nThe following table shows the observed frequencies together with some of the expected frequencies\nusing the manager’s binomial distribution.\nNumber of unripe pears per pack 0 1 2 3 4 5 H6\nObserved frequency 35 48 43 15 6 3 0\nExpected frequency 40.874 35.641 12.579 2.775 0.392 p q\n(a) Find the values of p and q. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Carry out a goodness of fit test, at the 5% significance level, to test whether the manager’s claim is\njustified. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2021-on-43-q03-b8d5949a4e.pdf"
    },
    {
      "id": "9231-2021-on-43-q04",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 4,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_43.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-on-43-q04-p01-062ea52ca4.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2021-on-43-q04-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-43-q04",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "4 Manet has developed a new training course to help athletes improve their time taken to run 800 m.\nManet claims that his course will decrease an athlete’s time by more than 2 s on average. For a random\nsample of 10 athletes the times taken, in seconds, before and after the course are given in the following\ntable.\nAthlete A B C D E F G H I J\nBefore 150 146 131 135 126 142 130 129 137 134\nAfter 145 138 129 135 122 135 132 128 127 137\nUse a t-test, at the 5% significance level, to test whether Manet’s claim is justified, stating any\nassumption that you make. [8]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...........................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2021-on-43-q04-9746fe88e9.pdf"
    },
    {
      "id": "9231-2021-on-43-q05",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 5,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_43.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_43.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2021-on-43-q05-p01-eacd189d53.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2021-on-43-q05-p02-3dca2fbe51.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-43-q05",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "5 Nine balls labelled 1, 2, 3, 4, 5, 6, 7, 8, 9 are placed in a bag. Kai selects three balls at random from\nthe bag, without replacement. The random variable X is the number of balls selected by Kai that are\nlabelled with a multiple of 3.\n(a) Find the probability generating function G (t) of X. [3]\nX\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nThe balls are replaced in the bag.\nJacob now selects two balls at random from the bag, without replacement. The random variable Y is the\nnumber of balls selected by Jacob that are labelled with an even number.\nG (t) of Y. [2] (b) Find the probability generating function\nY\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2021-on-43-q05-94f3f09826.pdf"
    },
    {
      "id": "9231-2021-on-43-q06",
      "subject": "9231",
      "year": 2021,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 6,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2021-oct-nov/9231_w21_qp_43.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2021-Oct-Nov/qp/9231_w21_qp_43.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2021-on-43-q06-p01-b60b0c147c.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2021-on-43-q06-p02-0bc12c5b27.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2021-on-43-q06-p03-a5443110d8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2021-on-43-q06",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "6 The blood cholesterol levels, measured in suitable units, of a random sample of 11 women and a random\nsample of 12 men are shown below.\nWomen 51 55 242 167 152 256 75 137 98 238 235\nMen 311 262 170 302 175 320 220 260 72 351 86 333\nCarry out a Wilcoxon rank-sum test, at the 5% significance level, to test whether, on average, there is a\ndifference in cholesterol levels between women and men. [9]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.............................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2021-on-43-q06-c8789ff489.pdf"
    },
    {
      "id": "9231-2022-mj-11-q01",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_11.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_11.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2022-mj-11-q01-p01-c3cc4aa5d2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-11-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "1 Let a be a positive constant.\nn\n/ 1\nin terms of n and a. [4] (a) Use the method of differences to find\n(ar+1)(ar+a+1)\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n3\n/ 1 1\n= . [3] (b) Find the value of a for which\n(ar+1)(ar+a+1) 6\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2022-mj-11-q01-49a71df0e1.pdf"
    },
    {
      "id": "9231-2022-mj-11-q02",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 2,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_11.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_11.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2022-mj-11-q02-p01-39b7ae04f7.png",
        "9231-topic-16-vectors/assets/9231-2022-mj-11-q02-p02-90740f2c5c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-11-q02",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "2 The points A, B, C have position vectors\n4i-4j+k, -4i+3j-4k, 4i-j-2k,\nrespectively, relative to the origin O.\nax+by+cz = d. [5] (a) Find the equation of the plane ABC, giving your answer in the form\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2022-mj-11-q02-0a4c0926aa.pdf"
    },
    {
      "id": "9231-2022-mj-11-q03",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 3,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_11.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_11.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2022-mj-11-q03-p01-f1859c57a1.png",
        "9231-topic-17-proof-by-induction/assets/9231-2022-mj-11-q03-p02-ac32b6fe5d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-11-q03",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "2 H u u u 3 The sequence of positive numbers , , , … is such that u 4 and, for 1, n\n1 2 3 1\n2 +u +12 u\nn n = u .\nn+1 2u\nn\n2 -4, (a) By considering u or otherwise, prove by mathematical induction that u 4 for all\nn+1 n\npositive integers n. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2022-mj-11-q03-10d9f0a26e.pdf"
    },
    {
      "id": "9231-2022-mj-11-q04",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 4,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_11.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_11.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2022-mj-11-q04-p01-83a0863bc3.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2022-mj-11-q04-p02-9897b3bdd1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-11-q04",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "3 2 +5x -6 = 2x 0 4 The cubic equation has roots a, b, c.\n1 1 1\n, , . [3] (a) Find a cubic equation whose roots are\n3 3 3 a b c\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 1 1\n+ + . [3] (b) Find the value of\n6 6 6 a b c\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2022-mj-11-q04-d3684285d1.pdf"
    },
    {
      "id": "9231-2022-mj-11-q05",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 5,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_11.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_11.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-mj-11-q05-p01-cb6e1b2e64.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-mj-11-q05-p02-b27f730d95.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-11-q05",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 -x-1 2x\n= y 5 The curve C has equation .\n2 +x+1 x\n(a) Show that C has no vertical asymptotes and state the equation of the horizontal asymptote of C.\n[3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the coordinates of the stationary points on C. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2022-mj-11-q05-7df92be8a5.pdf"
    },
    {
      "id": "9231-2022-mj-11-q06",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_11.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_11.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2022-mj-11-q06-p01-964513205d.png",
        "9231-topic-15-polar-coordinates/assets/9231-2022-mj-11-q06-p02-240ad47e66.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-11-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "-1 1 2 G G = 6 The curve C has polar equation , where 2. 0 r tan i i\n2 b l\n(a) Sketch C and state, in exact form, the greatest distance of a point on C from the pole. [3]\n............................................................................................................................................................\n= (b) Find the exact value of the area of the region bounded by C and the half‑line 2. [5] i\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2022-mj-11-q06-faf5851d54.pdf"
    },
    {
      "id": "9231-2022-mj-11-q07",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 7,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 17,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_11.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_11.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-mj-11-q07-p01-ff262773a2.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-mj-11-q07-p02-33c4a98c2e.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-mj-11-q07-p03-f511404415.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-11-q07",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "1 2 3\n= A 4 k 6 7 The matrix A is given by .\nf p\n7 8 9\n(a) Find the set of values of k for which A is non‑singular. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1\nA . [4] (b) Given that A is non‑singular, find, in terms of k, the entries in the top row of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2022-mj-11-q07-8075a0d5de.pdf"
    },
    {
      "id": "9231-2022-mj-12-q01",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_12.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_12.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2022-mj-12-q01-p01-c3cc4aa5d2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-12-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "1 Let a be a positive constant.\nn\n/ 1\nin terms of n and a. [4] (a) Use the method of differences to find\n(ar+1)(ar+a+1)\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n3\n/ 1 1\n= . [3] (b) Find the value of a for which\n(ar+1)(ar+a+1) 6\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2022-mj-12-q01-d7d6d56808.pdf"
    },
    {
      "id": "9231-2022-mj-12-q02",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 2,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_12.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_12.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2022-mj-12-q02-p01-39b7ae04f7.png",
        "9231-topic-16-vectors/assets/9231-2022-mj-12-q02-p02-90740f2c5c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-12-q02",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "2 The points A, B, C have position vectors\n4i-4j+k, -4i+3j-4k, 4i-j-2k,\nrespectively, relative to the origin O.\nax+by+cz = d. [5] (a) Find the equation of the plane ABC, giving your answer in the form\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2022-mj-12-q02-100b7c40d7.pdf"
    },
    {
      "id": "9231-2022-mj-12-q03",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 3,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_12.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_12.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2022-mj-12-q03-p01-f1859c57a1.png",
        "9231-topic-17-proof-by-induction/assets/9231-2022-mj-12-q03-p02-ac32b6fe5d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-12-q03",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "2 H u u u 3 The sequence of positive numbers , , , … is such that u 4 and, for 1, n\n1 2 3 1\n2 +u +12 u\nn n = u .\nn+1 2u\nn\n2 -4, (a) By considering u or otherwise, prove by mathematical induction that u 4 for all\nn+1 n\npositive integers n. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2022-mj-12-q03-a30b46619c.pdf"
    },
    {
      "id": "9231-2022-mj-12-q04",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 4,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_12.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_12.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2022-mj-12-q04-p01-83a0863bc3.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2022-mj-12-q04-p02-9897b3bdd1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-12-q04",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "3 2 +5x -6 = 2x 0 4 The cubic equation has roots a, b, c.\n1 1 1\n, , . [3] (a) Find a cubic equation whose roots are\n3 3 3 a b c\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 1 1\n+ + . [3] (b) Find the value of\n6 6 6 a b c\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2022-mj-12-q04-12b44eb650.pdf"
    },
    {
      "id": "9231-2022-mj-12-q05",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 5,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_12.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_12.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-mj-12-q05-p01-cb6e1b2e64.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-mj-12-q05-p02-b27f730d95.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-12-q05",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 -x-1 2x\n= y 5 The curve C has equation .\n2 +x+1 x\n(a) Show that C has no vertical asymptotes and state the equation of the horizontal asymptote of C.\n[3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the coordinates of the stationary points on C. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2022-mj-12-q05-2b8c340cbb.pdf"
    },
    {
      "id": "9231-2022-mj-12-q06",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_12.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_12.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2022-mj-12-q06-p01-964513205d.png",
        "9231-topic-15-polar-coordinates/assets/9231-2022-mj-12-q06-p02-240ad47e66.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-12-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "-1 1 2 G G = 6 The curve C has polar equation , where 2. 0 r tan i i\n2 b l\n(a) Sketch C and state, in exact form, the greatest distance of a point on C from the pole. [3]\n............................................................................................................................................................\n= (b) Find the exact value of the area of the region bounded by C and the half‑line 2. [5] i\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2022-mj-12-q06-7fb0efbda2.pdf"
    },
    {
      "id": "9231-2022-mj-12-q07",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 7,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 17,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_12.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_12.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-mj-12-q07-p01-ff262773a2.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-mj-12-q07-p02-33c4a98c2e.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-mj-12-q07-p03-f511404415.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-12-q07",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "1 2 3\n= A 4 k 6 7 The matrix A is given by .\nf p\n7 8 9\n(a) Find the set of values of k for which A is non‑singular. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1\nA . [4] (b) Given that A is non‑singular, find, in terms of k, the entries in the top row of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2022-mj-12-q07-456afe70df.pdf"
    },
    {
      "id": "9231-2022-mj-13-q01",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 1,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_13.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_13.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-mj-13-q01-p01-809eb401a6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-13-q01",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "x+1\n= 1 (a) Sketch the curve with equation y . [2]\nx-1\n+1 +1 x x\n1-2. = (b) Sketch the curve with equation and find the set of values of x for which y\n-1 -1 x x\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2022-mj-13-q01-9ff29406f0.pdf"
    },
    {
      "id": "9231-2022-mj-13-q02",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 2,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_13.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_13.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-mj-13-q02-p01-da0ea96ff6.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-mj-13-q02-p02-e8e1742b86.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-13-q02",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "3 2 +5x +10x-2 = x 0 2 The cubic equation has roots a, b, c.\n2 2 2 +b +c . [3] (a) Find the value of a\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2022-mj-13-q02-42875d7534.pdf"
    },
    {
      "id": "9231-2022-mj-13-q03",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 3,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_13.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_13.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-mj-13-q03-p01-acde63ea8e.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-mj-13-q03-p02-100397068a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-13-q03",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 +x-1 ax\n= y 3 A curve C has equation , where a is a positive constant.\nx-1\n(a) Find the equations of the asymptotes of C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 1 1+4a. 1 y [4] (b) Show that there is no point on C for which\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................................................................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2022-mj-13-q03-54e500cb34.pdf"
    },
    {
      "id": "9231-2022-mj-13-q04",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 4,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_13.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_13.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2022-mj-13-q04-p01-8cbef6e5e0.png",
        "9231-topic-13-summation-of-series/assets/9231-2022-mj-13-q04-p02-04d750e431.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-13-q04",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "rx 2x x = -2e +1). u e (e 4 Let\nr\nn\n/ u in terms of n and x. [3] (a) Using the method of differences, or otherwise, find\nr\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Deduce the set of non-zero values of x for which the infinite series\n+u +u +f u\n1 2 3\nis convergent and give the sum to infinity when this exists. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2022-mj-13-q04-35580bc71d.pdf"
    },
    {
      "id": "9231-2022-mj-13-q05",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 5,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_13.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_13.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-mj-13-q05-p01-43ac3a4f59.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-mj-13-q05-p02-a1e4fc44ba.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-13-q05",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "1 a\n= A 5 Let , where a is a positive constant.\ne0 1o\n(a) State the type of the geometrical transformation in the x–y plane represented by A. [1]\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Prove by mathematical induction that, for all positive integers n,\n1 na\nn = A . [5]\ne0 1 o\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2022-mj-13-q05-6dd6642664.pdf"
    },
    {
      "id": "9231-2022-mj-13-q06",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_13.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_13.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2022-mj-13-q06-p01-defb22ec2b.png",
        "9231-topic-15-polar-coordinates/assets/9231-2022-mj-13-q06-p02-3d0d8736aa.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-13-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "2 2 +xy+y = x a, 6 The curve C has Cartesian equation where a is a positive constant.\n2a\n2 = r . [3] (a) Show that the polar equation of C is\n2+sin2i\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 G G r. [2] 0 (b) Sketch the part of C for i\n4\n1 = The region R is enclosed by this part of C, the initial line and the half-line r. i\n4\n2tani\nsin2i may be expressed as . Use this result to show that the area of R is (c) It is given that\n2 1+tan i\n1 2 r 1+tan i 4 y 1 a di\n2 2 1+tani+tan i 0\n= and use the substitution t tani to find the exact value of this area. [8]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............................................................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2022-mj-13-q06-46f891f225.pdf"
    },
    {
      "id": "9231-2022-mj-13-q07",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 7,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 18,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_13.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_13.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2022-mj-13-q07-p01-2b916b30a1.png",
        "9231-topic-16-vectors/assets/9231-2022-mj-13-q07-p02-bc9395ed9f.png",
        "9231-topic-16-vectors/assets/9231-2022-mj-13-q07-p03-158f6dedae.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-13-q07",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "7 The position vectors of the points A, B, C, D are\n7i+4j-k, 11i+3j, 2i+6j+3k, 2i+7j+mk\nrespectively.\n(a) Given that the shortest distance between the line AB and the line CD is 3, show that\n2 -5m+4 = 0. [7] m\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2022-mj-13-q07-1274aec3ad.pdf"
    },
    {
      "id": "9231-2022-mj-21-q01",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 1,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_21.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2022-mj-21-q01-p01-3f69c2a046.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-21-q01",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "3 i G G = i a. r e4 1 The curve C has polar equation for 0\nin terms of s. [5] Given that the length of C is s, find a\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.............................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2022-mj-21-q01-2c7235be0d.pdf"
    },
    {
      "id": "9231-2022-mj-21-q02",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 2,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_21.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_21.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2022-mj-21-q02-p01-81489dec7d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-21-q02",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "2 (a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that\n2 = x+1. cosh2x 2sinh [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= cosh2x ksinh x has two distinct real roots. [5] (b) Find the set of values of k for which\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................................................................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2022-mj-21-q02-2ebe9b0d7e.pdf"
    },
    {
      "id": "9231-2022-mj-21-q03",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 3,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_21.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2022-mj-21-q03-p01-8379d8b358.png",
        "9231-topic-26-differential-equations/assets/9231-2022-mj-21-q03-p02-6af936bef3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-21-q03",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "3 The variables t and x are related by the differential equation\n2 x dx d\n2 + +x = +1. t\n2 dt dt\n(a) Find the general solution for x in terms of t. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2022-mj-21-q03-663c31f247.pdf"
    },
    {
      "id": "9231-2022-mj-21-q04",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 4,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_21.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2022-mj-21-q04-p01-c777a78e60.png",
        "9231-topic-24-integration/assets/9231-2022-mj-21-q04-p02-c330e08296.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-21-q04",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "x G G = 4 The diagram shows the curve with equation for 1, together with a set of N rectangles 0 x y 2\n1\neach of width .\nN\ny\n2\n1\n0\nx\nN-1 0 1 2 1\nN N N\n1\ny x 1 2 dx U (a) By considering the sum of the areas of these rectangles, show that , where\nN\n0 1\n2N\n= . [4] U\nN 1\n-1 N 2N\n` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\ny x L 2 dx. (b) Use a similar method to find, in terms of N, a lower bound for [4]\nN\n0\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2022-mj-21-q04-9546c3cb0f.pdf"
    },
    {
      "id": "9231-2022-mj-21-q05",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 5,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_21.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_21.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2022-mj-21-q05-p01-910b8f5f96.png",
        "9231-topic-23-differentiation/assets/9231-2022-mj-21-q05-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-21-q05",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "= = 5 The variables x and y are such that y 0 when x 0 and\n3 (x+1)y+ (x+y+1) = 1.\ndy 3\n= - = when x 0. [3] (a) Show that\n4 dx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 x . [7] (b) Find the Maclaurin’s series for y up to and including the term in\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................................................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2022-mj-21-q05-0f14079903.pdf"
    },
    {
      "id": "9231-2022-mj-21-q06",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_21.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2022-mj-21-q06-p01-fdde1d3819.png",
        "9231-topic-26-differential-equations/assets/9231-2022-mj-21-q06-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-21-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "= y vx 6 Use the substitution to find the solution of the differential equation\ndy\n2 2 = y+ +y x 9x\ndx\n= = = y f(x), f(x) for which y 0 when x 1. Give your answer in the form where is a polynomial in x.\n[10]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n........................................................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2022-mj-21-q06-6a29e98285.pdf"
    },
    {
      "id": "9231-2022-mj-21-q07",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 7,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_21.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_21.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2022-mj-21-q07-p01-014f94102a.png",
        "9231-topic-25-complex-numbers/assets/9231-2022-mj-21-q07-p02-64d4e47620.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-21-q07",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "7 (a) Use de Moivre’s theorem to show that\n7 cosec i\n= cosec7i .\n6 4 2 i-56cosec i+112cosec i-64 7cosec\n[6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2022-mj-21-q07-376e3127fa.pdf"
    },
    {
      "id": "9231-2022-mj-21-q08",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_21.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-mj-21-q08-p01-c512cf355f.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-mj-21-q08-p02-13645b42cb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-21-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "8 (a) Find the value of a for which the system of equations\n3x+ = ay 0,\n5x- = y 0,\nx+3y+2z = 0,\ndoes not have a unique solution. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nThe matrix A is given by\n3 0 0\n= -1 A 5 0 .\nf p\n1 3 2\n-1 2 = A PDP . [7] (b) Find a matrix P and a diagonal matrix D such that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2022-mj-21-q08-d0d7330305.pdf"
    },
    {
      "id": "9231-2022-mj-22-q01",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 1,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_22.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2022-mj-22-q01-p01-3f69c2a046.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-22-q01",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "3 i G G = i a. r e4 1 The curve C has polar equation for 0\nin terms of s. [5] Given that the length of C is s, find a\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.............................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2022-mj-22-q01-0c223d31a7.pdf"
    },
    {
      "id": "9231-2022-mj-22-q02",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 2,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_22.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_22.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2022-mj-22-q02-p01-81489dec7d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-22-q02",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "2 (a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that\n2 = x+1. cosh2x 2sinh [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= cosh2x ksinh x has two distinct real roots. [5] (b) Find the set of values of k for which\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................................................................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2022-mj-22-q02-0327e46d54.pdf"
    },
    {
      "id": "9231-2022-mj-22-q03",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 3,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_22.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2022-mj-22-q03-p01-8379d8b358.png",
        "9231-topic-26-differential-equations/assets/9231-2022-mj-22-q03-p02-6af936bef3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-22-q03",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "3 The variables t and x are related by the differential equation\n2 x dx d\n2 + +x = +1. t\n2 dt dt\n(a) Find the general solution for x in terms of t. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2022-mj-22-q03-e3409e8931.pdf"
    },
    {
      "id": "9231-2022-mj-22-q04",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 4,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_22.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2022-mj-22-q04-p01-c777a78e60.png",
        "9231-topic-24-integration/assets/9231-2022-mj-22-q04-p02-c330e08296.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-22-q04",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "x G G = 4 The diagram shows the curve with equation for 1, together with a set of N rectangles 0 x y 2\n1\neach of width .\nN\ny\n2\n1\n0\nx\nN-1 0 1 2 1\nN N N\n1\ny x 1 2 dx U (a) By considering the sum of the areas of these rectangles, show that , where\nN\n0 1\n2N\n= . [4] U\nN 1\n-1 N 2N\n` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\ny x L 2 dx. (b) Use a similar method to find, in terms of N, a lower bound for [4]\nN\n0\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2022-mj-22-q04-2f1d3a1751.pdf"
    },
    {
      "id": "9231-2022-mj-22-q05",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 5,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_22.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_22.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2022-mj-22-q05-p01-910b8f5f96.png",
        "9231-topic-23-differentiation/assets/9231-2022-mj-22-q05-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-22-q05",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "= = 5 The variables x and y are such that y 0 when x 0 and\n3 (x+1)y+ (x+y+1) = 1.\ndy 3\n= - = when x 0. [3] (a) Show that\n4 dx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 x . [7] (b) Find the Maclaurin’s series for y up to and including the term in\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................................................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2022-mj-22-q05-d55d54e063.pdf"
    },
    {
      "id": "9231-2022-mj-22-q06",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_22.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2022-mj-22-q06-p01-fdde1d3819.png",
        "9231-topic-26-differential-equations/assets/9231-2022-mj-22-q06-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-22-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "= y vx 6 Use the substitution to find the solution of the differential equation\ndy\n2 2 = y+ +y x 9x\ndx\n= = = y f(x), f(x) for which y 0 when x 1. Give your answer in the form where is a polynomial in x.\n[10]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n........................................................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2022-mj-22-q06-d882abff89.pdf"
    },
    {
      "id": "9231-2022-mj-22-q07",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 7,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_22.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_22.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2022-mj-22-q07-p01-014f94102a.png",
        "9231-topic-25-complex-numbers/assets/9231-2022-mj-22-q07-p02-64d4e47620.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-22-q07",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "7 (a) Use de Moivre’s theorem to show that\n7 cosec i\n= cosec7i .\n6 4 2 i-56cosec i+112cosec i-64 7cosec\n[6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2022-mj-22-q07-9fb9dd05c5.pdf"
    },
    {
      "id": "9231-2022-mj-22-q08",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_22.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-mj-22-q08-p01-c512cf355f.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-mj-22-q08-p02-13645b42cb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-22-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "8 (a) Find the value of a for which the system of equations\n3x+ = ay 0,\n5x- = y 0,\nx+3y+2z = 0,\ndoes not have a unique solution. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nThe matrix A is given by\n3 0 0\n= -1 A 5 0 .\nf p\n1 3 2\n-1 2 = A PDP . [7] (b) Find a matrix P and a diagonal matrix D such that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2022-mj-22-q08-63a8cc987b.pdf"
    },
    {
      "id": "9231-2022-mj-23-q01",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 1,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_23.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_23.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2022-mj-23-q01-p01-2f759a1d89.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-23-q01",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "ii 3 2 = -7i, z 7 3 1 Find the roots of the equation giving your answers in the form , where r 0 and re\n1 G -r r. [5] i\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...........................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2022-mj-23-q01-5048d603f7.pdf"
    },
    {
      "id": "9231-2022-mj-23-q02",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 2,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_23.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2022-mj-23-q02-p01-cc7c34aedf.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-23-q02",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "2 -lncosx. x 2 (a) Find the coefficient of in the Maclaurin’s series for [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= -lncosx = (b) Find the length of the arc of the curve with equation y from the point where x 0 to\n1 = the point where r. [4] x\n4\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2022-mj-23-q02-dc3f76b09c.pdf"
    },
    {
      "id": "9231-2022-mj-23-q03",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 3,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_23.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-mj-23-q03-p01-5c50e68d6b.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-mj-23-q03-p02-d433af3d9f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-23-q03",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "3 The matrix A is given by\n-9 6 5\n= -8 A 5 5 .\nf p\n-1 1 2\n(a) Find the eigenvalues of A. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2022-mj-23-q03-86ac4a6b2a.pdf"
    },
    {
      "id": "9231-2022-mj-23-q04",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 4,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_23.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_23.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2022-mj-23-q04-p01-ec74e10121.png",
        "9231-topic-23-differentiation/assets/9231-2022-mj-23-q04-p02-f391575340.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-23-q04",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "4 It is given that\n-1 -1 = -t+tan = t+sinh x t y t. and\n2 2 +1+ +1 dy t t\n= - (a) Show that . [4]\n2 dx t\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2022-mj-23-q04-d826f74458.pdf"
    },
    {
      "id": "9231-2022-mj-23-q05",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_23.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2022-mj-23-q05-p01-48db2877a4.png",
        "9231-topic-26-differential-equations/assets/9231-2022-mj-23-q05-p02-9acf9e8874.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-23-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "5 Find the solution of the differential equation\ndy\nx(x+7) +7y = x\ndx\n= = = y 7 y f(x). for which when x 1. Give your answer in the form [9]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n......................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2022-mj-23-q05-eadf9b6db8.pdf"
    },
    {
      "id": "9231-2022-mj-23-q06",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 6,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_23.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2022-mj-23-q06-p01-881526ab14.png",
        "9231-topic-24-integration/assets/9231-2022-mj-23-q06-p02-468c48aa62.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-23-q06",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "6\ny\nx\nn-1 0 1 1 2\nn n n\nG G = ln(1+x) y The diagram shows the curve with equation for 1, together with a set of n 0 x\n1\nrectangles each of width .\nn\n1\ny 1 ln(1+x)dx U , where (a) By considering the sum of the areas of these rectangles, show that\nn\n0\n(2n)! 1\n= -lnn. U ln [4]\nn n! n\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\ny ln(1+x)dx. L (b) Use a similar method to find, in terms of n, a lower bound for [4]\nn\n0\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2022-mj-23-q06-e64938ccfb.pdf"
    },
    {
      "id": "9231-2022-mj-23-q07",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_23.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2022-mj-23-q07-p01-80d1f28560.png",
        "9231-topic-26-differential-equations/assets/9231-2022-mj-23-q07-p02-0db0be78fb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-23-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "7 The variables x and y are related by the differential equation\n2 d y\n-y = 4 3.\n2 dx\ndy\n= = =-3 It is given that, when 2. x 0, y and\ndx\n(a) Find y in terms of x. [8]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2022-mj-23-q07-c5dba106e3.pdf"
    },
    {
      "id": "9231-2022-mj-23-q08",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 8,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_23.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2022-mj-23-q08-p01-6ea80f5cb3.png",
        "9231-topic-24-integration/assets/9231-2022-mj-23-q08-p02-d2ec6d1872.png",
        "9231-topic-24-integration/assets/9231-2022-mj-23-q08-p03-0a259ef903.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-23-q08",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "y n ! -1. 8 (a) Find idi, where n [2] sinicos\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 r\n2 y m n = idi. I sin icos Let\nm,n\n0\nH H and 0, m 2 n (b) Show that, for\nm-1\n= I I . [5]\nm+n m-2,n m,n\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................",
      "pdf_path": "9231-topic-24-integration/print/9231-2022-mj-23-q08-d17a020364.pdf"
    },
    {
      "id": "9231-2022-mj-31-q01",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 1,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_31.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_31.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2022-mj-31-q01-p01-c291a38c76.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-31-q01",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "1\nA\ni\n7.5 N\nA particle of weight 10 N is attached to one end of a light elastic string. The other end of the string is\nattached to a fixed point A on a horizontal ceiling. A horizontal force of 7.5 N acts on the particle. In the\nequilibrium position, the string makes an angle with the ceiling (see diagram). The string has natural i\nlength 0.8 m and modulus of elasticity 50 N.\n(a) Find the tension in the string. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the vertical distance between the particle and the ceiling. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2022-mj-31-q01-9b537f6af4.pdf"
    },
    {
      "id": "9231-2022-mj-31-q02",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 2,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_31.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2022-mj-31-q02-p01-79ab318978.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-31-q02",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "2 One end of a light inextensible string of length a is attached to a fixed point O. A particle of mass m is\nattached to the other end of the string. The particle is held at the point A with the string taut. The angle\n4 = between OA and the downward vertical is equal to a, where . The particle is projected from A, cosa\n5\nperpendicular to the string in an upwards direction, with a speed 3ga. It then moves along a circular\npath in a vertical plane. The string first goes slack when it makes an angle with the upward vertical i\nthrough O.\nFind the value of cosi. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2022-mj-31-q02-00df73c9c7.pdf"
    },
    {
      "id": "9231-2022-mj-31-q03",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 3,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_31.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_31.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2022-mj-31-q03-p01-354cdc691c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-31-q03",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "3 A particle P is moving in a horizontal straight line. Initially P is at the point O on the line and is moving\n4000 -1 -2\nwith velocity 25ms . At time t s after passing through O, the acceleration of P is ms in\n3 (5t+4)\nthe direction PO. The displacement of P from O at time t is x m.\nFind an expression for x in terms of t. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..............................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2022-mj-31-q03-2713c90684.pdf"
    },
    {
      "id": "9231-2022-mj-31-q04",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_31.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_31.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2022-mj-31-q04-p01-5d13a6ee35.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2022-mj-31-q04-p02-6defba9987.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-31-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "4\n2a\nh\na\nA B\nAn object is composed of a hemispherical shell of radius 2a attached to a closed hollow circular\ncylinder of height h and base radius a. The hemispherical shell and the hollow cylinder are made of the\nsame uniform material. The axes of symmetry of the shell and the cylinder coincide. AB is a diameter\nof the lower end of the cylinder (see diagram).\n(a) Find, in terms of a and h, an expression for the distance of the centre of mass of the object from AB.\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 = The object is placed on a rough plane which is inclined to the horizontal at an angle i, where . tani\n3\nThe object is in equilibrium with AB in contact with the plane and lying along a line of greatest slope of\nthe plane.\n(b) Find the set of possible values of h, in terms of a. [4]\n....................................................................................................................................................................\n.................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2022-mj-31-q04-f414841e02.pdf"
    },
    {
      "id": "9231-2022-mj-31-q05",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_31.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2022-mj-31-q05-p01-5a8de8bd23.png",
        "9231-topic-33-circular-motion/assets/9231-2022-mj-31-q05-p02-b4d61e0185.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-31-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "5\nC D\ni\na\ni B\n3a\nA\nA light inextensible string AB passes through two small holes C and D in a smooth horizontal table\n= = where AC 3a and DB a. A particle of mass m is attached at the end A and moves in a horizontal\n3\ncircle with angular velocity m ~. A particle of mass is attached to the end B and moves in a horizontal\n4\ncircle with angular velocity k~. AC makes an angle with the downward vertical and DB makes an i\nangle with the horizontal (see diagram). i\nFind the value of k. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.....................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2022-mj-31-q05-2b939e19a5.pdf"
    },
    {
      "id": "9231-2022-mj-31-q06",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 6,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_31.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_31.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2022-mj-31-q06-p01-aa06087d83.png",
        "9231-topic-36-momentum/assets/9231-2022-mj-31-q06-p02-af355189fd.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-31-q06",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "6 Two uniform smooth spheres A and B of equal radii have masses m and km respectively. The two\nspheres are on a horizontal surface. Sphere A is travelling with speed u towards sphere B which is at\nrest. The spheres collide. Immediately before the collision, the direction of motion of A makes an angle\n1 with the line of centres. The coefficient of restitution between the spheres is . a\n2\n3ucosa\nand find also an expression for the speed of (a) Show that the speed of B after the collision is\n2(1+k)\nA along the line of centres after the collision, in terms of k, u and a. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2022-mj-31-q06-fbdd4ba5b4.pdf"
    },
    {
      "id": "9231-2022-mj-31-q07",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_31.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_31.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2022-mj-31-q07-p01-9a146efd7e.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2022-mj-31-q07-p02-fab369731a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-31-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "7 Particles P and Q are projected in the same vertical plane from a point O at the top of a cliff. The\nheight of the cliff exceeds 50 m. Both particles move freely under gravity. Particle P is projected with\n-1 35 4 = speed ms at an angle above the horizontal, where . Particle Q is projected with speed tana a\n3 2\n-1 1 = ums at an angle above the horizontal, where . Particle Q is projected one second after the tanb b\n2\nprojection of particle P. The particles collide T s after the projection of particle Q.\n(a) Write down expressions, in terms of T, for the horizontal displacements of P and Q from O when\n= 5(T+1). they collide and hence show that [4] 4uT 21\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2022-mj-31-q07-bc7a9fdd18.pdf"
    },
    {
      "id": "9231-2022-mj-32-q01",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 1,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_32.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_32.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2022-mj-32-q01-p01-c291a38c76.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-32-q01",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "1\nA\ni\n7.5 N\nA particle of weight 10 N is attached to one end of a light elastic string. The other end of the string is\nattached to a fixed point A on a horizontal ceiling. A horizontal force of 7.5 N acts on the particle. In the\nequilibrium position, the string makes an angle with the ceiling (see diagram). The string has natural i\nlength 0.8 m and modulus of elasticity 50 N.\n(a) Find the tension in the string. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the vertical distance between the particle and the ceiling. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2022-mj-32-q01-3c4637c036.pdf"
    },
    {
      "id": "9231-2022-mj-32-q02",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 2,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_32.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2022-mj-32-q02-p01-79ab318978.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-32-q02",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "2 One end of a light inextensible string of length a is attached to a fixed point O. A particle of mass m is\nattached to the other end of the string. The particle is held at the point A with the string taut. The angle\n4 = between OA and the downward vertical is equal to a, where . The particle is projected from A, cosa\n5\nperpendicular to the string in an upwards direction, with a speed 3ga. It then moves along a circular\npath in a vertical plane. The string first goes slack when it makes an angle with the upward vertical i\nthrough O.\nFind the value of cosi. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2022-mj-32-q02-ffd4f6951b.pdf"
    },
    {
      "id": "9231-2022-mj-32-q03",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 3,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_32.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_32.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2022-mj-32-q03-p01-354cdc691c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-32-q03",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "3 A particle P is moving in a horizontal straight line. Initially P is at the point O on the line and is moving\n4000 -1 -2\nwith velocity 25ms . At time t s after passing through O, the acceleration of P is ms in\n3 (5t+4)\nthe direction PO. The displacement of P from O at time t is x m.\nFind an expression for x in terms of t. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..............................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2022-mj-32-q03-d175896125.pdf"
    },
    {
      "id": "9231-2022-mj-32-q04",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_32.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_32.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2022-mj-32-q04-p01-5d13a6ee35.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2022-mj-32-q04-p02-6defba9987.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-32-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "4\n2a\nh\na\nA B\nAn object is composed of a hemispherical shell of radius 2a attached to a closed hollow circular\ncylinder of height h and base radius a. The hemispherical shell and the hollow cylinder are made of the\nsame uniform material. The axes of symmetry of the shell and the cylinder coincide. AB is a diameter\nof the lower end of the cylinder (see diagram).\n(a) Find, in terms of a and h, an expression for the distance of the centre of mass of the object from AB.\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 = The object is placed on a rough plane which is inclined to the horizontal at an angle i, where . tani\n3\nThe object is in equilibrium with AB in contact with the plane and lying along a line of greatest slope of\nthe plane.\n(b) Find the set of possible values of h, in terms of a. [4]\n....................................................................................................................................................................\n.................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2022-mj-32-q04-281f93c39c.pdf"
    },
    {
      "id": "9231-2022-mj-32-q05",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_32.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2022-mj-32-q05-p01-5a8de8bd23.png",
        "9231-topic-33-circular-motion/assets/9231-2022-mj-32-q05-p02-b4d61e0185.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-32-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "5\nC D\ni\na\ni B\n3a\nA\nA light inextensible string AB passes through two small holes C and D in a smooth horizontal table\n= = where AC 3a and DB a. A particle of mass m is attached at the end A and moves in a horizontal\n3\ncircle with angular velocity m ~. A particle of mass is attached to the end B and moves in a horizontal\n4\ncircle with angular velocity k~. AC makes an angle with the downward vertical and DB makes an i\nangle with the horizontal (see diagram). i\nFind the value of k. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.....................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2022-mj-32-q05-be1bf19cce.pdf"
    },
    {
      "id": "9231-2022-mj-32-q06",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 6,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_32.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_32.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2022-mj-32-q06-p01-aa06087d83.png",
        "9231-topic-36-momentum/assets/9231-2022-mj-32-q06-p02-af355189fd.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-32-q06",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "6 Two uniform smooth spheres A and B of equal radii have masses m and km respectively. The two\nspheres are on a horizontal surface. Sphere A is travelling with speed u towards sphere B which is at\nrest. The spheres collide. Immediately before the collision, the direction of motion of A makes an angle\n1 with the line of centres. The coefficient of restitution between the spheres is . a\n2\n3ucosa\nand find also an expression for the speed of (a) Show that the speed of B after the collision is\n2(1+k)\nA along the line of centres after the collision, in terms of k, u and a. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2022-mj-32-q06-974307358c.pdf"
    },
    {
      "id": "9231-2022-mj-32-q07",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_32.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_32.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2022-mj-32-q07-p01-9a146efd7e.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2022-mj-32-q07-p02-fab369731a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-32-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "7 Particles P and Q are projected in the same vertical plane from a point O at the top of a cliff. The\nheight of the cliff exceeds 50 m. Both particles move freely under gravity. Particle P is projected with\n-1 35 4 = speed ms at an angle above the horizontal, where . Particle Q is projected with speed tana a\n3 2\n-1 1 = ums at an angle above the horizontal, where . Particle Q is projected one second after the tanb b\n2\nprojection of particle P. The particles collide T s after the projection of particle Q.\n(a) Write down expressions, in terms of T, for the horizontal displacements of P and Q from O when\n= 5(T+1). they collide and hence show that [4] 4uT 21\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2022-mj-32-q07-f193d04f93.pdf"
    },
    {
      "id": "9231-2022-mj-33-q01",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 1,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_33.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_33.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2022-mj-33-q01-p01-16bf9fee5d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-33-q01",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "1 A uniform lamina OABC is a trapezium whose vertices can be represented by coordinates in the x-y\nplane. The coordinates of the vertices are O (0, 0), A (15, 0), B (9, 4) and C (3, 4).\nFind the x-coordinate of the centre of mass of the lamina. [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...............................................................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2022-mj-33-q01-35ec8ae069.pdf"
    },
    {
      "id": "9231-2022-mj-33-q02",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 2,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_33.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_33.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2022-mj-33-q02-p01-dd36ef545a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-33-q02",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "2 A particle P of mass m is attached to one end of a light elastic string of natural length a and modulus of\n4 elasticity mg. The other end of the string is attached to a fixed point O on a rough horizontal surface.\n3\nThe particle is at rest on the surface with the string at its natural length. The coefficient of friction\n1 between P and the surface is . The particle is projected along the surface in the direction OP with a\n3\n1 speed of ga.\n2\nFind the greatest extension of the string during the subsequent motion. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2022-mj-33-q02-3740cf773e.pdf"
    },
    {
      "id": "9231-2022-mj-33-q03",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 3,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_33.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_33.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2022-mj-33-q03-p01-0957c9d4c3.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2022-mj-33-q03-p02-93399d4418.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-33-q03",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "-1\n25ms 3 A particle P is projected with speed at an angle above the horizontal from a point O on a i\n-1\nhorizontal plane and moves freely under gravity. After 2 s the speed of P is 15ms .\nsini. [5] (a) Find the value of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............................................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2022-mj-33-q03-b3a0ab0afa.pdf"
    },
    {
      "id": "9231-2022-mj-33-q04",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 4,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_33.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2022-mj-33-q04-p01-3ed5eadb2e.png",
        "9231-topic-33-circular-motion/assets/9231-2022-mj-33-q04-p02-73386a397e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-33-q04",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "4 One end of a light inextensible string of length a is attached to a fixed point O. A particle of mass m\nis attached to the other end of the string and is held with the string taut at the point A. At A the string\nmakes an angle with the upward vertical through O. The particle is projected perpendicular to the i\nstring in a downward direction from A with a speed u. It moves along a circular path in the vertical\nplane.\nWhen the string makes an angle with the downward vertical through O, the speed of the particle is 2u a\nand the magnitude of the tension in the string is 10 times its magnitude at A.\n2 = It is given that ga. u\n3\n(a) Find, in terms of m and g, the magnitude of the tension in the string at A. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2022-mj-33-q04-9b34dd79aa.pdf"
    },
    {
      "id": "9231-2022-mj-33-q05",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 5,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_33.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_33.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2022-mj-33-q05-p01-276916f2ef.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2022-mj-33-q05-p02-fb686d72d3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-33-q05",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "-1\nvms 5 A particle P of mass 4 kg is moving in a horizontal straight line. At time t s the velocity of P is\nand the displacement of P from a fixed point O on the line is x m. The only force acting on P is a\n-x -x = = = +12)e resistive force of magnitude N. When t 0, x 0 and v 4. (4e\nx 1+3e\n= . [4] v (a) Show by integration that\nx e\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2022-mj-33-q05-c749de76b1.pdf"
    },
    {
      "id": "9231-2022-mj-33-q06",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 6,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_33.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_33.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2022-mj-33-q06-p01-dc50361a2f.png",
        "9231-topic-36-momentum/assets/9231-2022-mj-33-q06-p02-7a5c2a1656.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-33-q06",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "6\nC\nb\nu\ni 60°\nA\nB\n= AB and BC are two fixed smooth vertical barriers on a smooth horizontal surface, with angle ABC 60° .\nA particle of mass m is moving with speed u on the surface. The particle strikes AB at an angle with i\nAB. It then strikes BC and rebounds at an angle with BC (see diagram). The coefficient of restitution b\n= tani between the particle and each barrier is e and 2.\nThe kinetic energy of the particle after the first collision is 40% of its kinetic energy before the first\ncollision.\n(a) Find the value of e. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2022-mj-33-q06-6d4f3a5f8d.pdf"
    },
    {
      "id": "9231-2022-mj-33-q07",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 7,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_33.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_33.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2022-mj-33-q07-p01-a9ab442b8e.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2022-mj-33-q07-p02-5b6ccbb651.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-33-q07",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "7\nB\nD\ni\nA\na\nE\na\nC\nA uniform cylinder with a rough surface and of radius a is fixed with its axis horizontal. Two identical\nuniform rods AB and BC, each of weight W and length 2a, are rigidly joined at B with AB perpendicular\nto BC. The rods rest on the cylinder in a vertical plane perpendicular to the axis of the cylinder with\nAB at an angle to the horizontal. D and E are the midpoints of AB and BC respectively and also the i\npoints of contact of the rods with the cylinder (see diagram). The rods are about to slip in a clockwise\ndirection. The coefficient of friction between each rod and the cylinder is n.\nThe normal reaction between AB and the cylinder is R and the normal reaction between BC and the\ncylinder is N.\n(a) Find the ratio R : N in terms of n. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2022-mj-33-q07-8c9446348a.pdf"
    },
    {
      "id": "9231-2022-mj-41-q01",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_41.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-mj-41-q01-p01-e2f56108c0.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-mj-41-q01-p02-75fa747ed4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-41-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 A manager is investigating the times taken by employees to complete a particular task as a result of the\nintroduction of new technology. He claims that the mean time taken to complete the task is reduced by\nmore than 0.4 minutes. He chooses a random sample of 10 employees. The times taken, in minutes,\nbefore and after the introduction of the new technology are recorded in the table.\nEmployee A B C D E F G H I J\nTime before\n10.2 9.8 12.4 11.6 10.8 11.2 14.6 10.6 12.3 11.0\nnew technology\nTime after\n9.6 8.5 12.4 10.9 10.2 10.6 12.8 10.8 12.5 10.6\nnew technology\n(a) Test at the 10% significance level whether the manager’s claim is justified. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2022-mj-41-q01-8e963585cc.pdf"
    },
    {
      "id": "9231-2022-mj-41-q02",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 2,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_41.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_41.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2022-mj-41-q02-p01-886dfc1278.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2022-mj-41-q02-p02-744201acea.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-41-q02",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "2 The probability generating function, G (t), of the random variable Y is given by\nY\n2 3 4 = 0.04+0.2t+0.37t +0.3t +0.09t G (t) .\nY\n(a) Find Var(Y). [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2022-mj-41-q02-8d2eb975a8.pdf"
    },
    {
      "id": "9231-2022-mj-41-q03",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 3,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_41.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_41.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2022-mj-41-q03-p01-66982c8a86.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2022-mj-41-q03-p02-466aa9e24a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-41-q03",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "3 The continuous random variable X has probability density function f given by\n1 G kx(4-x) 2, 0 x\nG G k(6-x) = f(x) 2 x 6,\n*\notherwise, 0\nwhere k is a constant.\n3 = k . [1] (a) Show that\n40\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= E(X) 2.5, find Var(X). [3] (b) Given that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2022-mj-41-q03-348b79b281.pdf"
    },
    {
      "id": "9231-2022-mj-41-q04",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 4,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_41.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_41.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2022-mj-41-q04-p01-48554439d7.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2022-mj-41-q04-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-41-q04",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "4 A scientist is investigating the numbers of a particular type of butterfly in a certain region. He claims\nthat the numbers of these butterflies found per square metre can be modelled by a Poisson distribution\nwith mean 2.5. He takes a random sample of 120 areas, each of one square metre, and counts the\nnumber of these butterflies in each of these areas. The following table shows the observed frequencies\ntogether with some of the expected frequencies using the scientist’s Poisson distribution.\nH Number per square metre 0 1 2 3 4 5 6 7\nObserved frequency 12 20 36 32 13 6 1 0\nExpected frequency 9.85 24.63 30.78 25.65 p 8.02 3.34 q\n(a) Find the values of p and q, correct to 2 decimal places. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Carry out a goodness of fit test, at the 10% significance level, to test the scientist’s claim. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2022-mj-41-q04-48e05ca344.pdf"
    },
    {
      "id": "9231-2022-mj-41-q05",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 5,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_41.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-mj-41-q05-p01-35ef0b6c89.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-mj-41-q05-p02-44b14b80d5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-41-q05",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "5 Raman is researching the heights of male giraffes in a particular region. Raman assumes that the heights\nof male giraffes in this region are normally distributed. He takes a random sample of 8 male giraffes\nfrom the region and measures the height, in metres, of each giraffe. These heights are as follows.\n5.2 5.8 4.9 6.1 5.5 5.9 5.4 5.6\n(a) Find a 90% confidence interval for the population mean height of male giraffes in this region. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2022-mj-41-q05-6821ae09be.pdf"
    },
    {
      "id": "9231-2022-mj-41-q06",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 6,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_41.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_41.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2022-mj-41-q06-p01-755ca73d55.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2022-mj-41-q06-p02-0bc12c5b27.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-41-q06",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "6 A teacher at a large college gave a mathematical puzzle to all the students. The median time taken by a\nrandom sample of 24 students to complete the puzzle was 18.0 minutes. The students were then given\npractice in solving puzzles. Two weeks later, the students were given another mathematical puzzle of\nthe same type as the first. The times, in minutes, taken by the random sample of 24 students to complete\nthis puzzle are as follows.\n18.2 17.5 16.4 15.1 20.5 26.5 19.2 23.2\n17.9 18.8 25.8 19.9 17.7 16.2 17.3 16.6\n17.1 20.1 20.3 12.6 16.0 21.4 22.7 18.4\nThe teacher claims that the practice has not made any difference to the average time taken to complete\na puzzle of this type.\nCarry out a Wilcoxon signed-rank test, at the 10% significance level, to test whether there is sufficient\nevidence to reject the teacher’s claim. [10]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2022-mj-41-q06-e686749b3e.pdf"
    },
    {
      "id": "9231-2022-mj-42-q01",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_42.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-mj-42-q01-p01-e2f56108c0.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-mj-42-q01-p02-75fa747ed4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-42-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 A manager is investigating the times taken by employees to complete a particular task as a result of the\nintroduction of new technology. He claims that the mean time taken to complete the task is reduced by\nmore than 0.4 minutes. He chooses a random sample of 10 employees. The times taken, in minutes,\nbefore and after the introduction of the new technology are recorded in the table.\nEmployee A B C D E F G H I J\nTime before\n10.2 9.8 12.4 11.6 10.8 11.2 14.6 10.6 12.3 11.0\nnew technology\nTime after\n9.6 8.5 12.4 10.9 10.2 10.6 12.8 10.8 12.5 10.6\nnew technology\n(a) Test at the 10% significance level whether the manager’s claim is justified. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2022-mj-42-q01-2f48ee81fc.pdf"
    },
    {
      "id": "9231-2022-mj-42-q02",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 2,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_42.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_42.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2022-mj-42-q02-p01-886dfc1278.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2022-mj-42-q02-p02-744201acea.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-42-q02",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "2 The probability generating function, G (t), of the random variable Y is given by\nY\n2 3 4 = 0.04+0.2t+0.37t +0.3t +0.09t G (t) .\nY\n(a) Find Var(Y). [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2022-mj-42-q02-188e9ad94d.pdf"
    },
    {
      "id": "9231-2022-mj-42-q03",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 3,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_42.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_42.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2022-mj-42-q03-p01-66982c8a86.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2022-mj-42-q03-p02-466aa9e24a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-42-q03",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "3 The continuous random variable X has probability density function f given by\n1 G kx(4-x) 2, 0 x\nG G k(6-x) = f(x) 2 x 6,\n*\notherwise, 0\nwhere k is a constant.\n3 = k . [1] (a) Show that\n40\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= E(X) 2.5, find Var(X). [3] (b) Given that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2022-mj-42-q03-08369c67f5.pdf"
    },
    {
      "id": "9231-2022-mj-42-q04",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 4,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_42.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_42.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2022-mj-42-q04-p01-48554439d7.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2022-mj-42-q04-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-42-q04",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "4 A scientist is investigating the numbers of a particular type of butterfly in a certain region. He claims\nthat the numbers of these butterflies found per square metre can be modelled by a Poisson distribution\nwith mean 2.5. He takes a random sample of 120 areas, each of one square metre, and counts the\nnumber of these butterflies in each of these areas. The following table shows the observed frequencies\ntogether with some of the expected frequencies using the scientist’s Poisson distribution.\nH Number per square metre 0 1 2 3 4 5 6 7\nObserved frequency 12 20 36 32 13 6 1 0\nExpected frequency 9.85 24.63 30.78 25.65 p 8.02 3.34 q\n(a) Find the values of p and q, correct to 2 decimal places. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Carry out a goodness of fit test, at the 10% significance level, to test the scientist’s claim. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2022-mj-42-q04-f88bfd139c.pdf"
    },
    {
      "id": "9231-2022-mj-42-q05",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 5,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_42.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-mj-42-q05-p01-35ef0b6c89.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-mj-42-q05-p02-44b14b80d5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-42-q05",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "5 Raman is researching the heights of male giraffes in a particular region. Raman assumes that the heights\nof male giraffes in this region are normally distributed. He takes a random sample of 8 male giraffes\nfrom the region and measures the height, in metres, of each giraffe. These heights are as follows.\n5.2 5.8 4.9 6.1 5.5 5.9 5.4 5.6\n(a) Find a 90% confidence interval for the population mean height of male giraffes in this region. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2022-mj-42-q05-32618ea0ea.pdf"
    },
    {
      "id": "9231-2022-mj-42-q06",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 6,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_42.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_42.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2022-mj-42-q06-p01-755ca73d55.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2022-mj-42-q06-p02-0bc12c5b27.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-42-q06",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "6 A teacher at a large college gave a mathematical puzzle to all the students. The median time taken by a\nrandom sample of 24 students to complete the puzzle was 18.0 minutes. The students were then given\npractice in solving puzzles. Two weeks later, the students were given another mathematical puzzle of\nthe same type as the first. The times, in minutes, taken by the random sample of 24 students to complete\nthis puzzle are as follows.\n18.2 17.5 16.4 15.1 20.5 26.5 19.2 23.2\n17.9 18.8 25.8 19.9 17.7 16.2 17.3 16.6\n17.1 20.1 20.3 12.6 16.0 21.4 22.7 18.4\nThe teacher claims that the practice has not made any difference to the average time taken to complete\na puzzle of this type.\nCarry out a Wilcoxon signed-rank test, at the 10% significance level, to test whether there is sufficient\nevidence to reject the teacher’s claim. [10]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2022-mj-42-q06-e28e5cf598.pdf"
    },
    {
      "id": "9231-2022-mj-43-q01",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_43.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-mj-43-q01-p01-a32b304e01.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-43-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 The times taken by members of a large quiz club to complete a challenge have a normal distribution\nwith mean minutes. The times, x minutes, are recorded for a random sample of 8 members of the n\nclub. The results are summarised as follows, where is the sample mean. x\n2 = /(x-x) = x 33.8 94.5\nFind a 95% confidence interval for n. [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.......................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2022-mj-43-q01-e2a6c0b6a6.pdf"
    },
    {
      "id": "9231-2022-mj-43-q02",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_43.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_43.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2022-mj-43-q02-p01-e0ed9e8b85.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-43-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "2 A scientist is investigating the size of shells at various beach locations. She selects four beach locations\nand takes a random sample of shells from each of these beaches. She classifies each shell as large or\nsmall. Her results are summarised in the following table.\nBeach location\nTotal A B C D\nLarge 68 69 96 81 314\nSize of shell\nSmall 28 55 64 39 186\nTotal 96 124 160 120 500\nTest, at the 10% significance level, whether the size of shell is independent of the beach location. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.....................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2022-mj-43-q02-da328e9b88.pdf"
    },
    {
      "id": "9231-2022-mj-43-q03",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 3,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_43.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_43.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2022-mj-43-q03-p01-8e9d4d5039.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2022-mj-43-q03-p02-f5ec863655.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-43-q03",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "3 George throws two coins, A and B, at the same time. Coin A is biased so that the probability of\n1 obtaining a head is a. Coin B is biased so that the probability of obtaining a head is b, where b a .\nThe probability generating function of X, the number of heads obtained by George, is G (t). The\nX\n5 1 2 coefficients of t and t G (t) in are and respectively.\n12 12 X\n(a) Find the value of a. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nThe random variable Y is the sum of two independent observations of X.\n(b) Find the probability generating function of Y, giving your answer as a polynomial in t. [3]\n................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2022-mj-43-q03-9dc5416d35.pdf"
    },
    {
      "id": "9231-2022-mj-43-q04",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 4,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_43.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_43.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2022-mj-43-q04-p01-360cb04732.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2022-mj-43-q04-p02-9342dc3d69.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-43-q04",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "4 The continuous random variable X has probability density function f given by\n1 3\nG G 1+ 1 x 3,\n2 8 = f(x) x e o\n*\notherwise. 0\n(a) Find . [3] E X\n` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 = Y X . The random variable Y is given by\n(b) Find the probability density function of Y. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2022-mj-43-q04-a6cf10c20a.pdf"
    },
    {
      "id": "9231-2022-mj-43-q05",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 5,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_43.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_43.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2022-mj-43-q05-p01-815bdeafc9.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2022-mj-43-q05-p02-573cf8d44c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-43-q05",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "5 A manager claims that the lengths of the rubber tubes that his company produces have a median of\n5.50 cm. The lengths, in cm, of a random sample of 11 tubes produced by this company are as follows.\n5.56 5.45 5.47 5.58 5.54 5.52 5.60 5.35 5.59 5.51 5.62\nIt is required to test at the 10% significance level the null hypothesis that the population median length\nis 5.50 cm against the alternative hypothesis that the population median length is not equal to 5.50 cm.\nShow that both a sign test and a Wilcoxon signed-rank test give the same conclusion and state this\nconclusion. [9]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n............................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2022-mj-43-q05-8f230c4ea8.pdf"
    },
    {
      "id": "9231-2022-mj-43-q06",
      "subject": "9231",
      "year": 2022,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 6,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-may-june/9231_s22_qp_43.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-May-June/qp/9231_s22_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-mj-43-q06-p01-6507ef63c1.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-mj-43-q06-p02-0c92ec06ea.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-mj-43-q06",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "6 A company has two machines, A and B, which independently fill small bottles with a liquid. The\nvolumes of liquid per bottle, in suitable units, filled by machines A and B are denoted by x and y\nrespectively. A scientist at the company takes a random sample of 40 bottles filled by machine A and a\nrandom sample of 50 bottles filled by machine B. The results are summarised as follows.\n2 2 = = = = /x /y /x /y 31400 37600 1120 1370\nThe population means of the volumes of liquid in the bottles filled by machines A and B are denoted by\nand . n n\nA B\nand . [8] (a) Test at the 2% significance level whether there is any difference between n n\nA B\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2022-mj-43-q06-72034dcfe8.pdf"
    },
    {
      "id": "9231-2022-on-11-q01",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 1,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_11.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_11.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2022-on-11-q01-p01-07f7761a36.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2022-on-11-q01-p02-2543aed1c7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-11-q01",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "3 2 ! = +bx +d = x 0 d 0. 1 The cubic equation has roots a, b, c, where and a b\n3 +27d = 4b 0. [5] (a) Show that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 2 +c = 2a 3b, find the values of b and d. [3] (b) Given that\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2022-on-11-q01-c021e7d202.pdf"
    },
    {
      "id": "9231-2022-on-11-q02",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_11.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_11.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2022-on-11-q02-p01-9b58a1b215.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-11-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "2n n +97 -50 7 is divisible by 48. [6] 2 Prove by mathematical induction that, for all positive integers n,\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.......................................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2022-on-11-q02-b48a54b3af.pdf"
    },
    {
      "id": "9231-2022-on-11-q03",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 3,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_11.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_11.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2022-on-11-q03-p01-54d8de4524.png",
        "9231-topic-13-summation-of-series/assets/9231-2022-on-11-q03-p02-585ad9f080.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-11-q03",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "3 3\n2r+1 - 2r-1 3 (a) By considering , use the method of differences to prove that\n` j ` j\nn\n/\n1 2 = n(n+1)(2n+1). [5] r\n6\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2022-on-11-q03-dc24237e7b.pdf"
    },
    {
      "id": "9231-2022-on-11-q04",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 4,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_11.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_11.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2022-on-11-q04-p01-7d0bb141b1.png",
        "9231-topic-16-vectors/assets/9231-2022-on-11-q04-p02-6df8b58921.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-11-q04",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "= 4i+4j+2k+n(3i+2j-k). = 3i-2j+k+m(-i+2j+k) P r 4 The plane contains the lines r and\nax+by+cz = P, d. giving your answer in the form [4] (a) Find a Cartesian equation of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....",
      "pdf_path": "9231-topic-16-vectors/print/9231-2022-on-11-q04-c41c532158.pdf"
    },
    {
      "id": "9231-2022-on-11-q05",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 5,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_11.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_11.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-on-11-q05-p01-2aaedefba2.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-on-11-q05-p02-392d57dd7c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-11-q05",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "-1 1 2 2 k 1\n2 2 = M 5 The matrix M is given by , where k is a constant.\n1 1 2pe0 1o 2 f\n2 2\n(a) The matrix M represents a sequence of two geometrical transformations.\nState the type of each transformation, and make clear the order in which they are applied. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) The triangle ABC in the x–y plane is transformed by M onto triangle DEF.\nFind, in terms of k, the single matrix which transforms triangle DEF onto triangle ABC. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2022-on-11-q05-a5ef03fa7f.pdf"
    },
    {
      "id": "9231-2022-on-11-q06",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_11.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_11.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2022-on-11-q06-p01-aec3d5714a.png",
        "9231-topic-15-polar-coordinates/assets/9231-2022-on-11-q06-p02-f82b6f91db.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-11-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "6 (a) Show that the curve with Cartesian equation\n2\n2 2 2 2 +y = -y x 36 x\n` j ` j\n2 = r 36cos2i. has polar equation [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1 1 2 G G = r 36cos2i, for r. The curve C has polar equation i r\n4 4\n(b) Sketch C and state the maximum distance of a point on C from the pole. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n(c) Find the area of the region enclosed by C. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(d) Find the maximum distance of a point on C from the initial line, giving the answer in exact form.\n[6]\n............................................................................................................................................................\n.............................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2022-on-11-q06-0c4a5fa9d3.pdf"
    },
    {
      "id": "9231-2022-on-11-q07",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_11.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_11.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-on-11-q07-p01-33963c27e7.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-on-11-q07-p02-ce1a831c57.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-on-11-q07-p03-14450fb6c7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-11-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 5x\n= y 7 The curve C has equation .\n5x-2\n(a) Find the equations of the asymptotes of C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the coordinates of the stationary points on C. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2022-on-11-q07-244c711aee.pdf"
    },
    {
      "id": "9231-2022-on-12-q01",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_12.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_12.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2022-on-12-q01-p01-a5467618b4.png",
        "9231-topic-13-summation-of-series/assets/9231-2022-on-12-q01-p02-1eda26552a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-12-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "n\n/\nr(r+2) 1 (a) Use the list of formulae (MF19) to find in terms of n, simplifying your answer. [2]\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nn\n/ 1 1\nin partial fractions and hence find in terms of n. [4] (b) Express\nr(r+2) r(r+2)\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2022-on-12-q01-11309dbf42.pdf"
    },
    {
      "id": "9231-2022-on-12-q02",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 2,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_12.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_12.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2022-on-12-q02-p01-ee31588ff9.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2022-on-12-q02-p02-54fb4054fc.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-12-q02",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "4 2 +3x +2x+6 = a, b, c, x 0 2 The equation has roots d.\n1 1 1 1 1 1 1 1\n+ + + , , , and state the value of . (a) Find a quartic equation whose roots are\n2 2 2 2 2 2 2 2 a b c d a b c d\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2022-on-12-q02-b5e6d66726.pdf"
    },
    {
      "id": "9231-2022-on-12-q03",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 3,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_12.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_12.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-on-12-q03-p01-4be755073f.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-on-12-q03-p02-fb552d7940.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-12-q03",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "1 0 1 0\n! = M k 0 3 The matrix M is given by , where k is a constant and or 1.\ne0 koek 1o\n(a) The matrix M represents a sequence of two geometrical transformations.\nState the type of each transformation, and make clear the order in which they are applied. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1\nM as the product of two matrices, neither of which is I. [2] (b) Write\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 k\n= y x. (c) Show that the invariant points of the transformation represented by M lie on the line\n1-k\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2022-on-12-q03-6cfb7f8acc.pdf"
    },
    {
      "id": "9231-2022-on-12-q04",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 4,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_12.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_12.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2022-on-12-q04-p01-8c3b592bdb.png",
        "9231-topic-17-proof-by-induction/assets/9231-2022-on-12-q04-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-12-q04",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "m(x) = f f(x). 4 The function f is such that\nProve by mathematical induction that, for every positive integer n,\n2n-1\nd\nxfl(x) = + (2n-1)f(x). xf(x) [7]\n2n-1` j dx\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...............................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2022-on-12-q04-19bbe48184.pdf"
    },
    {
      "id": "9231-2022-on-12-q05",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_12.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_12.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2022-on-12-q05-p01-e20799fd50.png",
        "9231-topic-15-polar-coordinates/assets/9231-2022-on-12-q05-p02-7355868ef7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-12-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "1 2 G G = r asec 5 The curve C has polar equation i, where a is a positive constant and r. 0 i\n4\n(a) Sketch C, stating the polar coordinates of the point of intersection of C with the initial line and\n1 = also with the half-line . [3] i r\n4\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the maximum distance of a point of C from the initial line. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 = r. [4] (c) Find the area of the region enclosed by C, the initial line and the half-line i\n4\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2022-on-12-q05-d256d776ec.pdf"
    },
    {
      "id": "9231-2022-on-12-q06",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 6,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_12.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_12.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2022-on-12-q06-p01-9fc62fb027.png",
        "9231-topic-16-vectors/assets/9231-2022-on-12-q06-p02-75b74f283a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-12-q06",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "= 2j+6k+n(i+2j-2k) = 2i+k+m(i-j+2k) l l r 6 The lines and have equations r and\n1 2\nrespectively.\nThe point P on l l l l and the point Q on are such that PQ is perpendicular to both and .\n1 2 1 2\n(a) Find the length PQ. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2022-on-12-q06-e1721a8fcb.pdf"
    },
    {
      "id": "9231-2022-on-12-q07",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_12.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_12.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-on-12-q07-p01-7333ba50ab.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-on-12-q07-p02-612def2d4c.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-on-12-q07-p03-14450fb6c7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-12-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 -x x\n= y 7 The curve C has equation .\nx+1\n(a) Find the equations of the asymptotes of C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the exact coordinates of the stationary points on C. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2022-on-12-q07-2e270354ce.pdf"
    },
    {
      "id": "9231-2022-on-13-q01",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 1,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_13.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_13.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2022-on-13-q01-p01-07f7761a36.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2022-on-13-q01-p02-2543aed1c7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-13-q01",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "3 2 ! = +bx +d = x 0 d 0. 1 The cubic equation has roots a, b, c, where and a b\n3 +27d = 4b 0. [5] (a) Show that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 2 +c = 2a 3b, find the values of b and d. [3] (b) Given that\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2022-on-13-q01-13542deffd.pdf"
    },
    {
      "id": "9231-2022-on-13-q02",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_13.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_13.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2022-on-13-q02-p01-9b58a1b215.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-13-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "2n n +97 -50 7 is divisible by 48. [6] 2 Prove by mathematical induction that, for all positive integers n,\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.......................................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2022-on-13-q02-909c6dcbc5.pdf"
    },
    {
      "id": "9231-2022-on-13-q03",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 3,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_13.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_13.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2022-on-13-q03-p01-54d8de4524.png",
        "9231-topic-13-summation-of-series/assets/9231-2022-on-13-q03-p02-585ad9f080.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-13-q03",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "3 3\n2r+1 - 2r-1 3 (a) By considering , use the method of differences to prove that\n` j ` j\nn\n/\n1 2 = n(n+1)(2n+1). [5] r\n6\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2022-on-13-q03-3c366055fe.pdf"
    },
    {
      "id": "9231-2022-on-13-q04",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 4,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_13.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_13.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2022-on-13-q04-p01-7d0bb141b1.png",
        "9231-topic-16-vectors/assets/9231-2022-on-13-q04-p02-6df8b58921.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-13-q04",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "= 4i+4j+2k+n(3i+2j-k). = 3i-2j+k+m(-i+2j+k) P r 4 The plane contains the lines r and\nax+by+cz = P, d. giving your answer in the form [4] (a) Find a Cartesian equation of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....",
      "pdf_path": "9231-topic-16-vectors/print/9231-2022-on-13-q04-e33a105a1d.pdf"
    },
    {
      "id": "9231-2022-on-13-q05",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 5,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_13.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_13.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-on-13-q05-p01-2aaedefba2.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2022-on-13-q05-p02-392d57dd7c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-13-q05",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "-1 1 2 2 k 1\n2 2 = M 5 The matrix M is given by , where k is a constant.\n1 1 2pe0 1o 2 f\n2 2\n(a) The matrix M represents a sequence of two geometrical transformations.\nState the type of each transformation, and make clear the order in which they are applied. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) The triangle ABC in the x–y plane is transformed by M onto triangle DEF.\nFind, in terms of k, the single matrix which transforms triangle DEF onto triangle ABC. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2022-on-13-q05-f0c40c613f.pdf"
    },
    {
      "id": "9231-2022-on-13-q06",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_13.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_13.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2022-on-13-q06-p01-aec3d5714a.png",
        "9231-topic-15-polar-coordinates/assets/9231-2022-on-13-q06-p02-f82b6f91db.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-13-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "6 (a) Show that the curve with Cartesian equation\n2\n2 2 2 2 +y = -y x 36 x\n` j ` j\n2 = r 36cos2i. has polar equation [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1 1 2 G G = r 36cos2i, for r. The curve C has polar equation i r\n4 4\n(b) Sketch C and state the maximum distance of a point on C from the pole. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n(c) Find the area of the region enclosed by C. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(d) Find the maximum distance of a point on C from the initial line, giving the answer in exact form.\n[6]\n............................................................................................................................................................\n.............................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2022-on-13-q06-d13f861b35.pdf"
    },
    {
      "id": "9231-2022-on-13-q07",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_13.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_13.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-on-13-q07-p01-33963c27e7.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-on-13-q07-p02-ce1a831c57.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2022-on-13-q07-p03-14450fb6c7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-13-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 5x\n= y 7 The curve C has equation .\n5x-2\n(a) Find the equations of the asymptotes of C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the coordinates of the stationary points on C. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2022-on-13-q07-45a1a39106.pdf"
    },
    {
      "id": "9231-2022-on-21-q01",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 1,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_21.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_21.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2022-on-21-q01-p01-32faf6c97d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-21-q01",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "2 x ln(1+e x 1 Find the Maclaurin’s series for up to and including the term in . [5] )\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n............................................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2022-on-21-q01-7d5e3751ea.pdf"
    },
    {
      "id": "9231-2022-on-21-q02",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 2,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_21.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-on-21-q02-p01-0414429715.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-21-q02",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "2 (a) Show that the system of equations\nx-y+2z = 4,\nx-y-3z = a,\nx-y+7z = 13,\nwhere a is a constant, does not have a unique solution. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= -5, a show that the system of equations in part (a) is consistent. Interpret this (b) Given that\nsituation geometrically. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n! -5, show that the system of equations in part (a) is inconsistent. Interpret a (c) Given instead that\nthis situation geometrically. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2022-on-21-q02-bcbb2e587a.pdf"
    },
    {
      "id": "9231-2022-on-21-q03",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 3,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_21.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2022-on-21-q03-p01-16646ad8f7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-21-q03",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "3 The curve C has parametric equations\n1 1 t 3 t G G = (t-2), = - , for 2. y 4e2 0 t x e t\n3\nFind, in terms of e, the length of C. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n............................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2022-on-21-q03-2fb61531db.pdf"
    },
    {
      "id": "9231-2022-on-21-q04",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 4,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_21.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_21.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2022-on-21-q04-p01-889b4576a2.png",
        "9231-topic-21-hyperbolic-functions/assets/9231-2022-on-21-q04-p02-d9565e0505.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-21-q04",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "cosh sinh 4 (a) Starting from the definitions of and in terms of exponentials, prove that\n2 2 x-sinh = cosh x 1. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nd -1 = (sinhx) sechx. tan [3] (b) Show that\ndx` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2022-on-21-q04-0f1810da1f.pdf"
    },
    {
      "id": "9231-2022-on-21-q05",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_21.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2022-on-21-q05-p01-d88feb252c.png",
        "9231-topic-26-differential-equations/assets/9231-2022-on-21-q05-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-21-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "5 Find the particular solution of the differential equation\n2 d y dy\n2 +2 +y = +3x+3, 2 4x\n2 dx dx\ndy\n= = = y 0. given that, when x 0, [10]\ndx\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2022-on-21-q05-7613bd2537.pdf"
    },
    {
      "id": "9231-2022-on-21-q06",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 6,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_21.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-on-21-q06-p01-32597a2319.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-on-21-q06-p02-f7e5871c11.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-21-q06",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "6 The matrix A is given by\n-3 -7 2\n= A 0 5 7 .\nf p\n-2 0 0\n-1 5 = (a) Find a matrix P and a diagonal matrix D such that A PDP . [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2022-on-21-q06-445a5d4a9c.pdf"
    },
    {
      "id": "9231-2022-on-21-q07",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 7,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_21.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_21.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2022-on-21-q07-p01-d3e6cee348.png",
        "9231-topic-25-complex-numbers/assets/9231-2022-on-21-q07-p02-92e4b8af0e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-21-q07",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "n-1 2 3 ! 1+w+w +w +f+w 7 (a) State the sum of the series , for w 1. [1]\n............................................................................................................................................................\n............................................................................................................................................................\n1 k k (1+itani) = i(coski+isinki), where is not an integer multiple of r. [2] sec (b) Show that i\n2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nn-1\n/\nk (1+itani) , show that (c) By considering\nk=0\nn-1\n/\nk n = coti(1-sec icosni), sec isinki\nk=0\n1 provided is not an integer multiple of r. [5] i\n2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2022-on-21-q07-81b386ca75.pdf"
    },
    {
      "id": "9231-2022-on-21-q08",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 8,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_21.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2022-on-21-q08-p01-c05dfa3af2.png",
        "9231-topic-26-differential-equations/assets/9231-2022-on-21-q08-p02-14450fb6c7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-21-q08",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "2 = 1- (i-1) u 8 (a) Use the substitution to find\ni-1\ny\ndi. [3]\n2 1- (i-1)\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the solution of the differential equation\ndy\n-1 2 -y = (i-1), sin i i\ndi\n1 1 = = = y f(i). 0 2, given that y 1 when 1. Give your answer in the form [11] where i i\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2022-on-21-q08-b020b400ec.pdf"
    },
    {
      "id": "9231-2022-on-22-q01",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 1,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_22.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-on-22-q01-p01-916e3dd38d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-22-q01",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "1 (a) Find the set of values of k for which the system of equations\n+2y+3z = x 1,\nkx+4y+6z = 0,\n7x+8y+9z = 3,\nhas a unique solution. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Interpret the situation geometrically in the case where the system of equations does not have a\nunique solution. [2]\n............................................................................................................................................................\n.........................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2022-on-22-q01-5573a4bb13.pdf"
    },
    {
      "id": "9231-2022-on-22-q02",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 2,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_22.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_22.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2022-on-22-q02-p01-71eef5530f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-22-q02",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "2 A curve has equation\n2 (x+1)y+y = 2.\ndy 2\n= - -2). (0, at the point [3] (a) Show that\n3 dx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 d y\n-2). (0, at the point [4] (b) Find the value of\n2 dx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2022-on-22-q02-d56776736b.pdf"
    },
    {
      "id": "9231-2022-on-22-q03",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 3,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_22.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2022-on-22-q03-p01-e1a957ea08.png",
        "9231-topic-24-integration/assets/9231-2022-on-22-q03-p02-5ed70ea2cf.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-22-q03",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "-x 1 x G G = + 3 (a) A curve has equation , for 1. Find, in terms of and e, the area of the 0 x y e e r\n4\nsurface generated when the curve is rotated through 2r radians about the x-axis. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2022-on-22-q03-c10ed41b02.pdf"
    },
    {
      "id": "9231-2022-on-22-q04",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 4,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_22.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2022-on-22-q04-p01-b1d3c9e777.png",
        "9231-topic-26-differential-equations/assets/9231-2022-on-22-q04-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-22-q04",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "4 Find the solution of the differential equation\ndx\n2 2 -1) +4x = -1 (4t 4t\ndt\n= = = x f(t). for which x 3 when t 1. Give your answer in the form [9]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.............................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2022-on-22-q04-358246b781.pdf"
    },
    {
      "id": "9231-2022-on-22-q05",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 5,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_22.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_22.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2022-on-22-q05-p01-e00a379af8.png",
        "9231-topic-25-complex-numbers/assets/9231-2022-on-22-q05-p02-933e13f411.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-22-q05",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "5 (a) Write down the fourth roots of unity. [1]\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Use de Moivre’s theorem to show that\n4 2 = i-8cos i+1. cos4i 8cos [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2022-on-22-q05-bac1b3dbf2.pdf"
    },
    {
      "id": "9231-2022-on-22-q06",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 6,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_22.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2022-on-22-q06-p01-d3561b89e7.png",
        "9231-topic-24-integration/assets/9231-2022-on-22-q06-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-22-q06",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "6\ny\nx 2 3 n – 1 n 0 1\n1\n2 (n-1) = The diagram shows the curve y for x 0, together with a set of rectangles of unit\n2 +2x x width.\nBy considering the sum of the areas of these rectangles, show that\nn\n1 / 1 2 1 n+1+ +2n + -ln 2+ . [10] ln n 3 3\n3 ` j 2 ` j +2r r\nr=1\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...............................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2022-on-22-q06-9024fb1859.pdf"
    },
    {
      "id": "9231-2022-on-22-q07",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 7,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_22.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-on-22-q07-p01-b3b729ba7d.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-on-22-q07-p02-189e61d79d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-22-q07",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "7 (a) It is given that is an eigenvalue of the non-singular square matrix A, with corresponding m\neigenvector e.\n-1 -1\nShow that A is an eigenvalue of for which e is a corresponding eigenvector. [2] m\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nThe matrix A is given by\n2 0 3\n= -4 A 15 3 .\nf p\n3 0 2\n-1 (b) Given that is an eigenvalue of A, find a corresponding eigenvector. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n0 1\nand are eigenvectors of A. Find the corresponding eigenvalues. [2] 1 2 (c) It is also given that\nf p f p\n0 1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2022-on-22-q07-503412afe8.pdf"
    },
    {
      "id": "9231-2022-on-22-q08",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 8,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_22.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2022-on-22-q08-p01-44be7ae7f9.png",
        "9231-topic-26-differential-equations/assets/9231-2022-on-22-q08-p02-14450fb6c7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-22-q08",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "2 = y coshu, 8 It is given that where u 0, and\n2 2 d u du du -x 2 u-1 + +coshu -2coshu = cosh 4e .\n2 dxo edxo edx\n(a) Show that\n2 d y dy\n-x + -2y = 4e . [4]\n2 dx dx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\ndu\n= = = 3. x 0, u ln3 and [10] (b) Find u in terms of x, given that, when\ndx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2022-on-22-q08-e56f72f799.pdf"
    },
    {
      "id": "9231-2022-on-23-q01",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 1,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_23.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_23.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2022-on-23-q01-p01-32faf6c97d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-23-q01",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "2 x ln(1+e x 1 Find the Maclaurin’s series for up to and including the term in . [5] )\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n............................................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2022-on-23-q01-0662def89a.pdf"
    },
    {
      "id": "9231-2022-on-23-q02",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 2,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_23.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-on-23-q02-p01-0414429715.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-23-q02",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "2 (a) Show that the system of equations\nx-y+2z = 4,\nx-y-3z = a,\nx-y+7z = 13,\nwhere a is a constant, does not have a unique solution. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= -5, a show that the system of equations in part (a) is consistent. Interpret this (b) Given that\nsituation geometrically. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n! -5, show that the system of equations in part (a) is inconsistent. Interpret a (c) Given instead that\nthis situation geometrically. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2022-on-23-q02-30ced6e7d4.pdf"
    },
    {
      "id": "9231-2022-on-23-q03",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 3,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_23.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2022-on-23-q03-p01-16646ad8f7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-23-q03",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "3 The curve C has parametric equations\n1 1 t 3 t G G = (t-2), = - , for 2. y 4e2 0 t x e t\n3\nFind, in terms of e, the length of C. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n............................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2022-on-23-q03-86272d877b.pdf"
    },
    {
      "id": "9231-2022-on-23-q04",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 4,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_23.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_23.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2022-on-23-q04-p01-889b4576a2.png",
        "9231-topic-21-hyperbolic-functions/assets/9231-2022-on-23-q04-p02-d9565e0505.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-23-q04",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "cosh sinh 4 (a) Starting from the definitions of and in terms of exponentials, prove that\n2 2 x-sinh = cosh x 1. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nd -1 = (sinhx) sechx. tan [3] (b) Show that\ndx` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2022-on-23-q04-ff04aada36.pdf"
    },
    {
      "id": "9231-2022-on-23-q05",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_23.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2022-on-23-q05-p01-d88feb252c.png",
        "9231-topic-26-differential-equations/assets/9231-2022-on-23-q05-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-23-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "5 Find the particular solution of the differential equation\n2 d y dy\n2 +2 +y = +3x+3, 2 4x\n2 dx dx\ndy\n= = = y 0. given that, when x 0, [10]\ndx\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2022-on-23-q05-5f92de1e8e.pdf"
    },
    {
      "id": "9231-2022-on-23-q06",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 6,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_23.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-on-23-q06-p01-32597a2319.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2022-on-23-q06-p02-f7e5871c11.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-23-q06",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "6 The matrix A is given by\n-3 -7 2\n= A 0 5 7 .\nf p\n-2 0 0\n-1 5 = (a) Find a matrix P and a diagonal matrix D such that A PDP . [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2022-on-23-q06-8f51440f74.pdf"
    },
    {
      "id": "9231-2022-on-23-q07",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 7,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_23.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_23.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2022-on-23-q07-p01-d3e6cee348.png",
        "9231-topic-25-complex-numbers/assets/9231-2022-on-23-q07-p02-92e4b8af0e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-23-q07",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "n-1 2 3 ! 1+w+w +w +f+w 7 (a) State the sum of the series , for w 1. [1]\n............................................................................................................................................................\n............................................................................................................................................................\n1 k k (1+itani) = i(coski+isinki), where is not an integer multiple of r. [2] sec (b) Show that i\n2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nn-1\n/\nk (1+itani) , show that (c) By considering\nk=0\nn-1\n/\nk n = coti(1-sec icosni), sec isinki\nk=0\n1 provided is not an integer multiple of r. [5] i\n2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2022-on-23-q07-6fcfb05828.pdf"
    },
    {
      "id": "9231-2022-on-23-q08",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 8,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_23.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2022-on-23-q08-p01-c05dfa3af2.png",
        "9231-topic-26-differential-equations/assets/9231-2022-on-23-q08-p02-14450fb6c7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-23-q08",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "2 = 1- (i-1) u 8 (a) Use the substitution to find\ni-1\ny\ndi. [3]\n2 1- (i-1)\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the solution of the differential equation\ndy\n-1 2 -y = (i-1), sin i i\ndi\n1 1 = = = y f(i). 0 2, given that y 1 when 1. Give your answer in the form [11] where i i\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2022-on-23-q08-599b8a020f.pdf"
    },
    {
      "id": "9231-2022-on-31-q01",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 1,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 3,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_31.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2022-on-31-q01-p01-1de63fe820.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-31-q01",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "1 A particle of mass 2 kg is attached to one end of a light inextensible string of length 0.6 m. The other\nend of the string is attached to a fixed point on a smooth horizontal surface. The particle is moving in a\ncircular path on the surface. The tension in the string is 20 N.\nFind how many revolutions the particle makes per minute. [3]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2022-on-31-q01-2c64425de0.pdf"
    },
    {
      "id": "9231-2022-on-31-q02",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 2,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_31.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_31.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2022-on-31-q02-p01-d50aad5527.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-31-q02",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "2 A light elastic string has natural length a and modulus of elasticity 4 mg. One end of the string is fixed\nto a point O on a smooth horizontal surface. A particle P of mass m is attached to the other end of the\nstring. The particle P is projected along the surface in the direction OP. When the length of the string is\n3 5 1 a, a, the speed of P is v. When the length of the string is the speed of P is v.\n2 2 4\n(a) Find an expression for v in terms of a and g. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n3\na. [2] (b) Find, in terms of g, the acceleration of P when the stretched length of the string is\n2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2022-on-31-q02-89394c4655.pdf"
    },
    {
      "id": "9231-2022-on-31-q03",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 3,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_31.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_31.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2022-on-31-q03-p01-bf1b4b2373.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2022-on-31-q03-p02-25af68c5b5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-31-q03",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "3\nB\nC\ni A\nA smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform\nrod AB, of length 4a and weight W, rests against the surface of the cylinder. The end A of the rod is in\ncontact with the horizontal surface. The vertical plane containing the rod AB is perpendicular to the\n= axis of the cylinder. The point of contact between the rod and the cylinder is C, where AC 3a. The\n3 = angle between the rod and the horizontal surface is tani where (see diagram). The coefficient of i\n4\n6\nfriction between the rod and the horizontal surface is .\n7\nA particle of weight kW is attached to the rod at B. The rod is about to slip. The normal reaction between\nthe rod and the cylinder is N.\n8 = W(1+2k). (a) Show that N [2]\n15\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2022-on-31-q03-2e0246a77c.pdf"
    },
    {
      "id": "9231-2022-on-31-q04",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 4,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_31.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_31.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2022-on-31-q04-p01-54c46ac0ef.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2022-on-31-q04-p02-e3ab0fc32c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-31-q04",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "-1\nvms 4 A particle of mass 0.5 kg moves along a horizontal straight line. Its velocity is at time t s. The\n2 forces acting on the particle are a driving force of magnitude 50 N and a resistance of magnitude 2v N .\n-1\nThe initial velocity of the particle is 3ms .\n(a) Find an expression for v in terms of t. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2022-on-31-q04-8df4ce930e.pdf"
    },
    {
      "id": "9231-2022-on-31-q05",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_31.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2022-on-31-q05-p01-a2baa78501.png",
        "9231-topic-33-circular-motion/assets/9231-2022-on-31-q05-p02-d097de3aac.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-31-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "5 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end\nof the string is attached to a fixed point O. The string is held taut with OP horizontal. The particle P\n1 is projected vertically downwards with speed and starts to move in a vertical circle. P passes ag\n3\nthrough the lowest point of the circle and reaches the point Q where OQ makes an angle with the i\n11 downward vertical. At Q the speed of P is and the tension in the string is mg. kag\n6\ncosi. [4] (a) Find the value of k and the value of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2022-on-31-q05-cd6c054787.pdf"
    },
    {
      "id": "9231-2022-on-31-q06",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 6,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_31.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_31.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2022-on-31-q06-p01-23f5345c37.png",
        "9231-topic-36-momentum/assets/9231-2022-on-31-q06-p02-117e4f6784.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-31-q06",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "6\nA B\na u\n5\nu\n8\nm km\nTwo uniform smooth spheres A and B of equal radii have masses m and km respectively. The two\n5\nspheres are moving on a horizontal surface with speeds u and u respectively. Immediately before the\n8\nspheres collide, A is travelling along the line of centres, and B’s direction of motion makes an angle a\n2 with the line of centres (see diagram). The coefficient of restitution between the spheres is and\n3\n3 = tana .\n4\nAfter the collision, the direction of motion of B is perpendicular to the line of centres.\n(a) Find the value of k. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2022-on-31-q06-d119c3f130.pdf"
    },
    {
      "id": "9231-2022-on-31-q07",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_31.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_31.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2022-on-31-q07-p01-2d250e240c.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2022-on-31-q07-p02-1ad3b23ce7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-31-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "-1\nVms 7 A particle P is projected with speed at an angle 75° above the horizontal from a point O on a\nhorizontal plane. It then moves freely under gravity.\n2V\nsin75°. (a) Show that the total time of flight, in seconds, is [2]\ng\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nA smooth vertical barrier is now inserted with its lower end on the plane at a distance 15 m from O. The\nparticle is projected as before but now strikes the barrier, rebounds and returns to O. The coefficient of\n3\n. restitution between the barrier and the particle is\n5\n(b) Explain why the total time of flight is unchanged. [1]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2022-on-31-q07-e902bb001e.pdf"
    },
    {
      "id": "9231-2022-on-32-q01",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 1,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_32.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2022-on-32-q01-p01-22a65d077f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-32-q01",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "1 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end\nof the string is attached to a fixed point O. The string is held taut with OP making an angle with the a\n2 = downward vertical, where . The particle P is projected perpendicular to OP in an upwards cosa\n3\ndirection with speed 3ag. It then starts to move along a circular path in a vertical plane.\nFind the cosine of the angle between the string and the upward vertical when the string first becomes\nslack. [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.........",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2022-on-32-q01-69a5f02a4f.pdf"
    },
    {
      "id": "9231-2022-on-32-q02",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 2,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_32.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_32.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2022-on-32-q02-p01-4e0cf3bb78.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-32-q02",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "2\nA\n9a\nB D C\nx\n6a\n= A uniform lamina is in the form of a triangle ABC in which angle B is a right angle, AB 9a and\n= = BD x BC 6a. The point D is on BC such that (see diagram). The region ABD is removed from the\nlamina. The resulting shape ADC is placed with the edge DC on a horizontal surface and the plane ADC\nis vertical.\nFind the set of values of x, in terms of a, for which the shape is in equilibrium. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n................................................................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2022-on-32-q02-008df6e642.pdf"
    },
    {
      "id": "9231-2022-on-32-q03",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 3,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_32.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_32.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2022-on-32-q03-p01-a25630337b.png",
        "9231-topic-34-hooke-s-law/assets/9231-2022-on-32-q03-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-32-q03",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "16\nMg, 3 One end of a light elastic string, of natural length a and modulus of elasticity is attached to a\n3\nfixed point O. A particle P of mass 4M is attached to the other end of the string and hangs vertically in\nequilibrium. Another particle of mass 2M is attached to P and the combined particle is then released\n1 from rest. The speed of the combined particle when it has descended a distance is v. a\n4\nFind an expression for v in terms of g and a. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2022-on-32-q03-8366a2fddf.pdf"
    },
    {
      "id": "9231-2022-on-32-q04",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 4,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_32.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_32.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2022-on-32-q04-p01-543380b31d.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2022-on-32-q04-p02-cdb3a570f7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-32-q04",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "-1\nvms 4 A particle P of mass 5 kg moves along a horizontal straight line. At time t s, the velocity of P is\nand its displacement from a fixed point O on the line is x m. The forces acting on P are a force of\n500\n1 2 = = magnitude N in the direction OP and a resistive force of magnitude N. When t 0, x 0 and v\nv 2\n= v 5.\n(a) Find an expression for v in terms of x. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................................................................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2022-on-32-q04-0c78819dd6.pdf"
    },
    {
      "id": "9231-2022-on-32-q05",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 5,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_32.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_32.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2022-on-32-q05-p01-011bde79ed.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2022-on-32-q05-p02-c2259c9077.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-32-q05",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "-1\nums 5 A particle P is projected with speed at an angle of above the horizontal from a point O on a i\nhorizontal plane and moves freely under gravity. The horizontal and vertical displacements of P from O\nat a subsequent time t s are denoted by x m and y m respectively.\n(a) Show that the equation of the trajectory is given by\n2 gx\n2 = xtani- (1+tan y i). [4]\n2 2u\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2022-on-32-q05-34b4405ac6.pdf"
    },
    {
      "id": "9231-2022-on-32-q06",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 6,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_32.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2022-on-32-q06-p01-133d36d877.png",
        "9231-topic-33-circular-motion/assets/9231-2022-on-32-q06-p02-29fc66530c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-32-q06",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "6\nR\nB h\nA\nA light inextensible string is threaded through a fixed smooth ring R which is at a height h above a\nsmooth horizontal surface. One end of the string is attached to a particle A of mass m. The other end\n6\nof the string is attached to a particle B of mass m. The particle A moves in a horizontal circle on the\n7\nsurface. The particle B hangs in equilibrium below the ring and above the surface (see diagram).\nWhen A has constant angular speed ~, the angle between AR and BR is and the normal reaction i\nbetween A and the surface is N.\n3\nWhen A has constant angular speed ~, the angle between AR and BR is and the normal reaction a\n2\n1 between A and the surface is . N\n2\n4 = (a) Show that cosa. [5] cosi\n9\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2022-on-32-q06-f780827f3e.pdf"
    },
    {
      "id": "9231-2022-on-32-q07",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 7,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_32.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_32.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2022-on-32-q07-p01-1430a87568.png",
        "9231-topic-36-momentum/assets/9231-2022-on-32-q07-p02-b5d0df0f9a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-32-q07",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "7\nu\n2u\nA B\nβ α\n1\nm m\n2\n1 Two uniform smooth spheres A and B of equal radii have masses m and respectively. The two m\n2\nspheres are moving on a horizontal surface when they collide. Immediately before the collision,\nsphere A is travelling with speed u and its direction of motion makes an angle with the line of centres. a\nSphere B is travelling with speed 2u and its direction of motion makes an angle with the line of b\n5 a+b = centres (see diagram). The coefficient of restitution between the spheres is 90°. and\n8\n(a) Find the component of the velocity of B parallel to the line of centres after the collision, giving\nyour answer in terms of u and a. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n................................................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2022-on-32-q07-b9cd2b707b.pdf"
    },
    {
      "id": "9231-2022-on-33-q01",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 1,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 3,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_33.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2022-on-33-q01-p01-1de63fe820.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-33-q01",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "1 A particle of mass 2 kg is attached to one end of a light inextensible string of length 0.6 m. The other\nend of the string is attached to a fixed point on a smooth horizontal surface. The particle is moving in a\ncircular path on the surface. The tension in the string is 20 N.\nFind how many revolutions the particle makes per minute. [3]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2022-on-33-q01-a87c01a976.pdf"
    },
    {
      "id": "9231-2022-on-33-q02",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 2,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_33.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_33.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2022-on-33-q02-p01-d50aad5527.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-33-q02",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "2 A light elastic string has natural length a and modulus of elasticity 4 mg. One end of the string is fixed\nto a point O on a smooth horizontal surface. A particle P of mass m is attached to the other end of the\nstring. The particle P is projected along the surface in the direction OP. When the length of the string is\n3 5 1 a, a, the speed of P is v. When the length of the string is the speed of P is v.\n2 2 4\n(a) Find an expression for v in terms of a and g. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n3\na. [2] (b) Find, in terms of g, the acceleration of P when the stretched length of the string is\n2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2022-on-33-q02-d1fd2be15c.pdf"
    },
    {
      "id": "9231-2022-on-33-q03",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 3,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_33.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_33.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2022-on-33-q03-p01-bf1b4b2373.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2022-on-33-q03-p02-25af68c5b5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-33-q03",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "3\nB\nC\ni A\nA smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform\nrod AB, of length 4a and weight W, rests against the surface of the cylinder. The end A of the rod is in\ncontact with the horizontal surface. The vertical plane containing the rod AB is perpendicular to the\n= axis of the cylinder. The point of contact between the rod and the cylinder is C, where AC 3a. The\n3 = angle between the rod and the horizontal surface is tani where (see diagram). The coefficient of i\n4\n6\nfriction between the rod and the horizontal surface is .\n7\nA particle of weight kW is attached to the rod at B. The rod is about to slip. The normal reaction between\nthe rod and the cylinder is N.\n8 = W(1+2k). (a) Show that N [2]\n15\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2022-on-33-q03-a6777273c9.pdf"
    },
    {
      "id": "9231-2022-on-33-q04",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 4,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_33.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_33.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2022-on-33-q04-p01-54c46ac0ef.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2022-on-33-q04-p02-e3ab0fc32c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-33-q04",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "-1\nvms 4 A particle of mass 0.5 kg moves along a horizontal straight line. Its velocity is at time t s. The\n2 forces acting on the particle are a driving force of magnitude 50 N and a resistance of magnitude 2v N .\n-1\nThe initial velocity of the particle is 3ms .\n(a) Find an expression for v in terms of t. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2022-on-33-q04-5e7a7f5f09.pdf"
    },
    {
      "id": "9231-2022-on-33-q05",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_33.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2022-on-33-q05-p01-a2baa78501.png",
        "9231-topic-33-circular-motion/assets/9231-2022-on-33-q05-p02-d097de3aac.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-33-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "5 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end\nof the string is attached to a fixed point O. The string is held taut with OP horizontal. The particle P\n1 is projected vertically downwards with speed and starts to move in a vertical circle. P passes ag\n3\nthrough the lowest point of the circle and reaches the point Q where OQ makes an angle with the i\n11 downward vertical. At Q the speed of P is and the tension in the string is mg. kag\n6\ncosi. [4] (a) Find the value of k and the value of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2022-on-33-q05-df8493cfea.pdf"
    },
    {
      "id": "9231-2022-on-33-q06",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 6,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_33.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_33.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2022-on-33-q06-p01-23f5345c37.png",
        "9231-topic-36-momentum/assets/9231-2022-on-33-q06-p02-117e4f6784.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-33-q06",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "6\nA B\na u\n5\nu\n8\nm km\nTwo uniform smooth spheres A and B of equal radii have masses m and km respectively. The two\n5\nspheres are moving on a horizontal surface with speeds u and u respectively. Immediately before the\n8\nspheres collide, A is travelling along the line of centres, and B’s direction of motion makes an angle a\n2 with the line of centres (see diagram). The coefficient of restitution between the spheres is and\n3\n3 = tana .\n4\nAfter the collision, the direction of motion of B is perpendicular to the line of centres.\n(a) Find the value of k. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2022-on-33-q06-784ca10eeb.pdf"
    },
    {
      "id": "9231-2022-on-33-q07",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_33.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_33.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2022-on-33-q07-p01-2d250e240c.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2022-on-33-q07-p02-1ad3b23ce7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-33-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "-1\nVms 7 A particle P is projected with speed at an angle 75° above the horizontal from a point O on a\nhorizontal plane. It then moves freely under gravity.\n2V\nsin75°. (a) Show that the total time of flight, in seconds, is [2]\ng\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nA smooth vertical barrier is now inserted with its lower end on the plane at a distance 15 m from O. The\nparticle is projected as before but now strikes the barrier, rebounds and returns to O. The coefficient of\n3\n. restitution between the barrier and the particle is\n5\n(b) Explain why the total time of flight is unchanged. [1]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2022-on-33-q07-c5ebd79d21.pdf"
    },
    {
      "id": "9231-2022-on-41-q01",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_41.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-on-41-q01-p01-a24b86240d.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-on-41-q01-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-41-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 Jasmine is researching the heights of pine trees in forests in two regions A and B. She chooses a random\nsample of 50 pine trees in region A and records their heights, x m. She also chooses a random sample of\n60 pine trees in region B and records their heights, y m. Her results are summarised as follows.\n2 2 = = = = /x /y /x /y 53200 57900 1625 1854\nFind a 95% confidence interval for the difference between the population mean heights of pine trees in\nregions A and B. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...............................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2022-on-41-q01-19ad37e7c7.pdf"
    },
    {
      "id": "9231-2022-on-41-q02",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_41.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_41.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2022-on-41-q02-p01-dcc03d12b1.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2022-on-41-q02-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-41-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "2 An organisation runs courses to train students to become engineers. These students are taught in groups\nof 8. The director of the organisation claims that on average 60% of the students in a group achieve a\npass. A random sample of 150 groups of 8 students is chosen. The following table shows the observed\nfrequencies together with some of the expected frequencies using the appropriate binomial distribution.\nNumber of\npasses per 0 1 2 3 4 5 6 7 8\ngroup\nObserved\n0 0 8 24 45 36 26 10 1\nfrequency\nExpected\np 1.180 6.193 18.579 34.836 q r 13.437 2.519\nfrequency\n(a) Find the values of p, q and r giving your answers correct to 3 decimal places. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Carry out a goodness of fit test, at the 10% significance level, to test whether there is evidence to\nreject the director’s claim. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2022-on-41-q02-f77284fb76.pdf"
    },
    {
      "id": "9231-2022-on-41-q03",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 3,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_41.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_41.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2022-on-41-q03-p01-7d625a2a8a.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2022-on-41-q03-p02-d59d611497.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-41-q03",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "3 A large college is holding a piano competition. Each student has played a particular piece of music and\ntwo judges have each awarded a mark out of 80. The marks awarded to a random sample of 14 students\nare shown in the following table.\nStudent A B C D E F G H I J K L M N\nJudge 1 79 54 63 74 69 52 50 57 55 42 63 55 56 48\nJudge 2 75 62 60 73 76 41 31 51 45 55 49 50 65 36\n(a) One of the students claims that on average Judge 1 is awarding higher marks than Judge 2. Carry\nout a Wilcoxon matched-pairs signed-rank test at the 5% significance level to test whether the data\nsupports the student’s claim. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2022-on-41-q03-b9e3573db4.pdf"
    },
    {
      "id": "9231-2022-on-41-q04",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 4,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_41.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_41.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2022-on-41-q04-p01-913751abd1.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2022-on-41-q04-p02-0b73e7c3a5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-41-q04",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "2 4 Jason has three biased coins. For each coin the probability of obtaining a head when it is thrown is .\n3\nJason throws all three coins. The number of heads obtained is denoted by X.\n(a) Find the probability generating function G (t) of X. [3]\nX\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nJason also has two unbiased coins. He throws all five coins. The number of heads obtained from the\n1 1 1 2 = + t+ . The random variable Z is the G (t) t two unbiased coins is denoted by Y. It is given that\n4 2 4 Y\ntotal number of heads obtained when Jason throws all five coins.\n(b) Find the probability generating function of Z, expressing your answer as a polynomial. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2022-on-41-q04-4f58c9e81e.pdf"
    },
    {
      "id": "9231-2022-on-41-q05",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 5,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_41.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_41.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2022-on-41-q05-p01-56cb2f2b81.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2022-on-41-q05-p02-371a09f526.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-41-q05",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "5 The continuous random variable X has cumulative distribution function F given by\n1 x 0 0,\n1 2 G G = - (12-x) F(x) 1 x 12, 0\n144\n*\n2 1 x 12.\n(a) Find the upper quartile of X. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 Var(X ). [5] (b) Find\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2022-on-41-q05-9741c9c114.pdf"
    },
    {
      "id": "9231-2022-on-41-q06",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 6,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_41.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-on-41-q06-p01-4d1e0865a6.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-on-41-q06-p02-0bc12c5b27.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-41-q06",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "6 A company manufactures copper pipes. The pipes are produced by two different machines, A and B.\nAn inspector claims that the mean diameter of the pipes produced by machine A is greater than the\nmean diameter of the pipes produced by machine B. He takes a random sample of 12 pipes produced by\nmachine A and measures their diameters, x cm. His results are summarised as follows.\n2 = = /x /x 3.26 6.24\nHe also takes a random sample of 10 pipes produced by machine B and measures their diameters in cm.\nHis results are as follows.\n0.48 0.53 0.47 0.54 0.54 0.55 0.46 0.55 0.50 0.48\nThe diameters of the pipes produced by each machine are assumed to be normally distributed with\nequal population variances.\nTest at the 2.5% significance level whether the data supports the inspector’s claim. [9]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..............................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2022-on-41-q06-766a6a615c.pdf"
    },
    {
      "id": "9231-2022-on-42-q01",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_42.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-on-42-q01-p01-b8e13752c2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-42-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 A basketball club has a large number of players. The heights, x m, of a random sample of 10 of these\nplayers are measured. A 90% confidence interval for the population mean height, m, of players in n\nthis club is calculated. It is assumed that heights are normally distributed. The confidence interval is\nG G 2.02. 1.78 n\n2 /x /x and for this sample. [6] Find the values of\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.....................................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2022-on-42-q01-a47ac107de.pdf"
    },
    {
      "id": "9231-2022-on-42-q02",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_42.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_42.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2022-on-42-q02-p01-eda2b2b2a9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-42-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "2 In the colleges in three regions of a particular country, students are given individual targets to achieve.\nTheir performance is measured against their individual target and graded as ‘above target’, ‘on target’\nor ‘below target’. For a random sample of students from each of the three regions, the observed\nfrequencies are summarised in the following table.\nRegion\nA B C Total\nAbove target 62 41 44 147\nPerformance On target 102 94 95 291\nBelow target 56 45 61 162\nTotal 220 180 200 600\nTest, at the 10% significance level, whether performance is independent of region. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.................................................................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2022-on-42-q02-99c4dda00c.pdf"
    },
    {
      "id": "9231-2022-on-42-q03",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 3,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_42.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-on-42-q03-p01-4cf56b6fcd.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-on-42-q03-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-42-q03",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "3 A scientist is investigating the masses of birds of a certain species in country X and country Y. She\ntakes a random sample of 50 birds of this species from country X and a random sample of 80 birds\nof this species from country Y. She records their masses in kg, x and y, respectively. Her results are\nsummarised as follows.\n2 2 = = = = /x /y /x /y 115.2 172.6 75.5 116.8\nkg kg The population mean masses of these birds in countries X and Y are and respectively. n n\nx y\n= against the alternative hypothesis Test, at the 5% significance level, the null hypothesis n n\nx y\n2 . State your conclusion in the context of the question. [8] n n\nx y\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..............................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2022-on-42-q03-8e38f278a4.pdf"
    },
    {
      "id": "9231-2022-on-42-q04",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 4,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_42.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_42.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2022-on-42-q04-p01-15c5eb1c99.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2022-on-42-q04-p02-91d95d8928.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-42-q04",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "4 The continuous random variable X has probability density function f given by\n1 G k x 1, 0\nG G = f(x) kx x 2, 1\n*\notherwise, 0\nwhere k is a constant.\n2 = . [1] k (a) Show that\n5\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the interquartile range of X. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2022-on-42-q04-53ddd8821e.pdf"
    },
    {
      "id": "9231-2022-on-42-q05",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 5,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_42.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_42.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2022-on-42-q05-p01-ea64b3cd57.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2022-on-42-q05-p02-bbc9febd42.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-42-q05",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "5 A 6‑sided dice, A, with faces numbered 1, 2, 3, 4, 5, 6 is biased so that the probability of throwing a 6 is\n1 . The random variable X is the number of 6s obtained when dice A is thrown twice.\n4\n(a) Find the probability generating function of X. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nA second dice, B, with faces numbered 1, 2, 3, 4, 5, 6 is unbiased. The random variable Y is the number\nof 6s obtained when dice B is thrown twice.\nThe random variable Z is the total number of 6s obtained when both dice are thrown twice.\n(b) Find the probability generating function of Z, expressing your answer as a polynomial. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2022-on-42-q05-bf1036ec27.pdf"
    },
    {
      "id": "9231-2022-on-42-q06",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 6,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_42.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_42.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2022-on-42-q06-p01-7ebdaf6271.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2022-on-42-q06-p02-ac343671bf.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-42-q06",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "6 The manager of a technology company A claims that his employees earn more per year than the\nemployees at technology company B. The amounts earned per year, in hundreds of dollars, by a random\nsample of 12 employees from company A and an independent random sample of 12 employees from\ncompany B are shown below.\nCompany A 461 482 374 512 415 452 502 427 398 545 612 359\nCompany B 454 506 491 384 361 443 401 472 414 342 355 437\n(a) Carry out a Wilcoxon rank‑sum test at the 5% significance level to test whether the manager’s\nclaim is supported by the data. [9]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2022-on-42-q06-c711d4177d.pdf"
    },
    {
      "id": "9231-2022-on-43-q01",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_43.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-on-43-q01-p01-a24b86240d.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-on-43-q01-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-43-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 Jasmine is researching the heights of pine trees in forests in two regions A and B. She chooses a random\nsample of 50 pine trees in region A and records their heights, x m. She also chooses a random sample of\n60 pine trees in region B and records their heights, y m. Her results are summarised as follows.\n2 2 = = = = /x /y /x /y 53200 57900 1625 1854\nFind a 95% confidence interval for the difference between the population mean heights of pine trees in\nregions A and B. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...............................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2022-on-43-q01-4c61a1415e.pdf"
    },
    {
      "id": "9231-2022-on-43-q02",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_43.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_43.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2022-on-43-q02-p01-dcc03d12b1.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2022-on-43-q02-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-43-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "2 An organisation runs courses to train students to become engineers. These students are taught in groups\nof 8. The director of the organisation claims that on average 60% of the students in a group achieve a\npass. A random sample of 150 groups of 8 students is chosen. The following table shows the observed\nfrequencies together with some of the expected frequencies using the appropriate binomial distribution.\nNumber of\npasses per 0 1 2 3 4 5 6 7 8\ngroup\nObserved\n0 0 8 24 45 36 26 10 1\nfrequency\nExpected\np 1.180 6.193 18.579 34.836 q r 13.437 2.519\nfrequency\n(a) Find the values of p, q and r giving your answers correct to 3 decimal places. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Carry out a goodness of fit test, at the 10% significance level, to test whether there is evidence to\nreject the director’s claim. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2022-on-43-q02-cc6e195938.pdf"
    },
    {
      "id": "9231-2022-on-43-q03",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 3,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_43.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_43.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2022-on-43-q03-p01-7d625a2a8a.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2022-on-43-q03-p02-d59d611497.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-43-q03",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "3 A large college is holding a piano competition. Each student has played a particular piece of music and\ntwo judges have each awarded a mark out of 80. The marks awarded to a random sample of 14 students\nare shown in the following table.\nStudent A B C D E F G H I J K L M N\nJudge 1 79 54 63 74 69 52 50 57 55 42 63 55 56 48\nJudge 2 75 62 60 73 76 41 31 51 45 55 49 50 65 36\n(a) One of the students claims that on average Judge 1 is awarding higher marks than Judge 2. Carry\nout a Wilcoxon matched-pairs signed-rank test at the 5% significance level to test whether the data\nsupports the student’s claim. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2022-on-43-q03-0e07fbb67a.pdf"
    },
    {
      "id": "9231-2022-on-43-q04",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 4,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_43.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_43.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2022-on-43-q04-p01-913751abd1.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2022-on-43-q04-p02-0b73e7c3a5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-43-q04",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "2 4 Jason has three biased coins. For each coin the probability of obtaining a head when it is thrown is .\n3\nJason throws all three coins. The number of heads obtained is denoted by X.\n(a) Find the probability generating function G (t) of X. [3]\nX\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nJason also has two unbiased coins. He throws all five coins. The number of heads obtained from the\n1 1 1 2 = + t+ . The random variable Z is the G (t) t two unbiased coins is denoted by Y. It is given that\n4 2 4 Y\ntotal number of heads obtained when Jason throws all five coins.\n(b) Find the probability generating function of Z, expressing your answer as a polynomial. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2022-on-43-q04-9609fd6bc6.pdf"
    },
    {
      "id": "9231-2022-on-43-q05",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 5,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_43.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_43.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2022-on-43-q05-p01-56cb2f2b81.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2022-on-43-q05-p02-371a09f526.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-43-q05",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "5 The continuous random variable X has cumulative distribution function F given by\n1 x 0 0,\n1 2 G G = - (12-x) F(x) 1 x 12, 0\n144\n*\n2 1 x 12.\n(a) Find the upper quartile of X. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 Var(X ). [5] (b) Find\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2022-on-43-q05-71f68e7865.pdf"
    },
    {
      "id": "9231-2022-on-43-q06",
      "subject": "9231",
      "year": 2022,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 6,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2022-oct-nov/9231_w22_qp_43.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2022-Oct-Nov/qp/9231_w22_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-on-43-q06-p01-4d1e0865a6.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2022-on-43-q06-p02-0bc12c5b27.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2022-on-43-q06",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "6 A company manufactures copper pipes. The pipes are produced by two different machines, A and B.\nAn inspector claims that the mean diameter of the pipes produced by machine A is greater than the\nmean diameter of the pipes produced by machine B. He takes a random sample of 12 pipes produced by\nmachine A and measures their diameters, x cm. His results are summarised as follows.\n2 = = /x /x 3.26 6.24\nHe also takes a random sample of 10 pipes produced by machine B and measures their diameters in cm.\nHis results are as follows.\n0.48 0.53 0.47 0.54 0.54 0.55 0.46 0.55 0.50 0.48\nThe diameters of the pipes produced by each machine are assumed to be normally distributed with\nequal population variances.\nTest at the 2.5% significance level whether the data supports the inspector’s claim. [9]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..............................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2022-on-43-q06-f7fcfd8fa8.pdf"
    },
    {
      "id": "9231-2023-mj-11-q01",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 1,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_11.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_11.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2023-mj-11-q01-p01-3352f28bc1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-11-q01",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "3 0\n= A 1 Let .\ne1 1o\n(a) Prove by mathematical induction that, for all positive integers n,\nn 2#3 0\nn = 2A . [5]\nn -1 e3 2o\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2023-mj-11-q01-9d62e725b9.pdf"
    },
    {
      "id": "9231-2023-mj-11-q02",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 2,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_11.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_11.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2023-mj-11-q02-p01-6c37e0df71.png",
        "9231-topic-13-summation-of-series/assets/9231-2023-mj-11-q02-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-11-q02",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "3 2 +4x +6x+1 = x 0 2 The cubic equation has roots a, b, c.\n2 2 2 +b +c . [2] (a) Find the value of a\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Use standard results from the list of formulae (MF19) to show that\nn\n/\n2 2 2 2 (a+r) + (b+r) + (c+r) = +an+b), n(n\n` j\nr=1\nwhere a and b are constants to be determined. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2023-mj-11-q02-b3bdc09bda.pdf"
    },
    {
      "id": "9231-2023-mj-11-q03",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 3,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_11.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_11.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2023-mj-11-q03-p01-5cd060f08a.png",
        "9231-topic-13-summation-of-series/assets/9231-2023-mj-11-q03-p02-1d2d71c902.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-11-q03",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "n\n/ 1\n3 (a) Use the method of differences to find in terms of n and k , where k is a\n(kr+1)(kr-k+1)\nr=1\npositive constant. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2023-mj-11-q03-55b60740a4.pdf"
    },
    {
      "id": "9231-2023-mj-11-q04",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_11.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_11.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2023-mj-11-q04-p01-dc0758b7dc.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2023-mj-11-q04-p02-90e6e7879c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-11-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "2 a b\n! = M b 0. 4 The matrix M is given by , where a, b, c are real constants and\n2 e o c a\n(a) Show that M does not represent a rotation about the origin. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the equations of the invariant lines, through the origin, of the transformation in the x – y plane\nrepresented by M. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2023-mj-11-q04-aa2b482467.pdf"
    },
    {
      "id": "9231-2023-mj-11-q05",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_11.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_11.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2023-mj-11-q05-p01-362699ed0b.png",
        "9231-topic-15-polar-coordinates/assets/9231-2023-mj-11-q05-p02-1ecd781a54.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-11-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "1\n2 G G = 5 The curve C has polar equation , for r. r 0 i\n2 +1 i\n(a) Sketch C and state the polar coordinates of the point of C furthest from the pole. [3]\n............................................................................................................................................................\n= r. [4] (b) Find the area of the region enclosed by C, the initial line, and the half-line i\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(c) Show that, at the point of C furthest from the initial line,\n1\ni+ coti-1 = 0\nio e\nand verify that this equation has a root between 1.1 and 1.2. [5]\n............................................................................................................................................................\n......................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2023-mj-11-q05-5df7e6e64c.pdf"
    },
    {
      "id": "9231-2023-mj-11-q06",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_11.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_11.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-mj-11-q06-p01-372d22ce40.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-mj-11-q06-p02-098bf705ba.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-mj-11-q06-p03-efe5cdf827.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-11-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 +2x-15 x\n= y 6 The curve C has equation .\nx-2\n(a) Find the equations of the asymptotes of C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Show that C has no stationary points. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2023-mj-11-q06-5ba4dbc55d.pdf"
    },
    {
      "id": "9231-2023-mj-11-q07",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 7,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_11.pdf?download=true",
      "source_pages": [
        16,
        17
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_11.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2023-mj-11-q07-p01-7800dfcf37.png",
        "9231-topic-16-vectors/assets/9231-2023-mj-11-q07-p02-ec0ca3bee5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-11-q07",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "= -4j-3k+m(i-j+k) +n(i+j-k). 7 The plane has equation r P\n1\npx+qy+rz = (a) Obtain an equation of d. in the form [4] P\n1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nr.(-5i+3j+5k) = 4. has equation (b) The plane P\n2\nand . [4] Find a vector equation of the line of intersection of P P\n1 2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2023-mj-11-q07-059f5947fe.pdf"
    },
    {
      "id": "9231-2023-mj-12-q01",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 1,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_12.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_12.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2023-mj-12-q01-p01-3352f28bc1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-12-q01",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "3 0\n= A 1 Let .\ne1 1o\n(a) Prove by mathematical induction that, for all positive integers n,\nn 2#3 0\nn = 2A . [5]\nn -1 e3 2o\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2023-mj-12-q01-5b875caa98.pdf"
    },
    {
      "id": "9231-2023-mj-12-q02",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 2,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_12.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_12.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2023-mj-12-q02-p01-6c37e0df71.png",
        "9231-topic-13-summation-of-series/assets/9231-2023-mj-12-q02-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-12-q02",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "3 2 +4x +6x+1 = x 0 2 The cubic equation has roots a, b, c.\n2 2 2 +b +c . [2] (a) Find the value of a\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Use standard results from the list of formulae (MF19) to show that\nn\n/\n2 2 2 2 (a+r) + (b+r) + (c+r) = +an+b), n(n\n` j\nr=1\nwhere a and b are constants to be determined. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2023-mj-12-q02-6ca0999b5b.pdf"
    },
    {
      "id": "9231-2023-mj-12-q03",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 3,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_12.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_12.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2023-mj-12-q03-p01-5cd060f08a.png",
        "9231-topic-13-summation-of-series/assets/9231-2023-mj-12-q03-p02-1d2d71c902.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-12-q03",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "n\n/ 1\n3 (a) Use the method of differences to find in terms of n and k , where k is a\n(kr+1)(kr-k+1)\nr=1\npositive constant. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2023-mj-12-q03-b3bf265596.pdf"
    },
    {
      "id": "9231-2023-mj-12-q04",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_12.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_12.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2023-mj-12-q04-p01-dc0758b7dc.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2023-mj-12-q04-p02-90e6e7879c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-12-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "2 a b\n! = M b 0. 4 The matrix M is given by , where a, b, c are real constants and\n2 e o c a\n(a) Show that M does not represent a rotation about the origin. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the equations of the invariant lines, through the origin, of the transformation in the x – y plane\nrepresented by M. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2023-mj-12-q04-451eb8cb8d.pdf"
    },
    {
      "id": "9231-2023-mj-12-q05",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_12.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_12.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2023-mj-12-q05-p01-362699ed0b.png",
        "9231-topic-15-polar-coordinates/assets/9231-2023-mj-12-q05-p02-1ecd781a54.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-12-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "1\n2 G G = 5 The curve C has polar equation , for r. r 0 i\n2 +1 i\n(a) Sketch C and state the polar coordinates of the point of C furthest from the pole. [3]\n............................................................................................................................................................\n= r. [4] (b) Find the area of the region enclosed by C, the initial line, and the half-line i\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(c) Show that, at the point of C furthest from the initial line,\n1\ni+ coti-1 = 0\nio e\nand verify that this equation has a root between 1.1 and 1.2. [5]\n............................................................................................................................................................\n......................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2023-mj-12-q05-c7bc502da4.pdf"
    },
    {
      "id": "9231-2023-mj-12-q06",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_12.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_12.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-mj-12-q06-p01-372d22ce40.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-mj-12-q06-p02-098bf705ba.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-mj-12-q06-p03-efe5cdf827.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-12-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 +2x-15 x\n= y 6 The curve C has equation .\nx-2\n(a) Find the equations of the asymptotes of C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Show that C has no stationary points. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2023-mj-12-q06-0c55969595.pdf"
    },
    {
      "id": "9231-2023-mj-12-q07",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 7,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_12.pdf?download=true",
      "source_pages": [
        16,
        17
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_12.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2023-mj-12-q07-p01-7800dfcf37.png",
        "9231-topic-16-vectors/assets/9231-2023-mj-12-q07-p02-ec0ca3bee5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-12-q07",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "= -4j-3k+m(i-j+k) +n(i+j-k). 7 The plane has equation r P\n1\npx+qy+rz = (a) Obtain an equation of d. in the form [4] P\n1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nr.(-5i+3j+5k) = 4. has equation (b) The plane P\n2\nand . [4] Find a vector equation of the line of intersection of P P\n1 2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2023-mj-12-q07-de798f6ea7.pdf"
    },
    {
      "id": "9231-2023-mj-13-q01",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 1,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_13.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_13.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2023-mj-13-q01-p01-9933dfc453.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-13-q01",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "3n n +32 -33 5 1 Prove by mathematical induction that, for all positive integers n, is divisible by 31. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.......................................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2023-mj-13-q01-00e0d6e66c.pdf"
    },
    {
      "id": "9231-2023-mj-13-q02",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 2,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_13.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_13.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2023-mj-13-q02-p01-88237dcafa.png",
        "9231-topic-13-summation-of-series/assets/9231-2023-mj-13-q02-p02-0c16b4bceb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-13-q02",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "2 (a) Use standard results from the list of formulae (MF19) to show that\nn\n/\n2 3 2 +6r-5) = +bn +cn, (6r an\nr=1\nwhere a, b and c are integers to be determined. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nn 2 +6r-5 / 6r\nin terms of n. [4] (b) Use the method of differences to find\n2 +r r\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2023-mj-13-q02-c6d53f681b.pdf"
    },
    {
      "id": "9231-2023-mj-13-q03",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 3,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_13.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_13.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2023-mj-13-q03-p01-368788c7e0.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2023-mj-13-q03-p02-4f733ed14e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-13-q03",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "4 2 -x +2x+5 = a, b, c, d. x 0 3 The equation has roots\n2 2 2 2 2 2 2 2 +b +c +d , , , (a) Find a quartic equation whose roots are and state the value of . a b c d a\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2023-mj-13-q03-f3d9a58370.pdf"
    },
    {
      "id": "9231-2023-mj-13-q04",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_13.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_13.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2023-mj-13-q04-p01-bb4151352f.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2023-mj-13-q04-p02-5ad657d66b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-13-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "-sin2i cos2i 1 k\n1 1 = r 4 The matrix M is given by , where 0 and k is a non-zero constant. M i\nesin2i cos2ioe0 1o\nThe matrix M represents a sequence of two geometrical transformations, one of which is a shear.\n(a) Describe fully the other transformation and state the order in which the transformations are\napplied. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1\nM as the product of two matrices, neither of which is I. [2] (b) Write\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nM-I tani for which is singular. [5] (c) Find, in terms of k, the value of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2023-mj-13-q04-ff9fb813ad.pdf"
    },
    {
      "id": "9231-2023-mj-13-q05",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_13.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_13.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2023-mj-13-q05-p01-9506f7ebe1.png",
        "9231-topic-15-polar-coordinates/assets/9231-2023-mj-13-q05-p02-c984fb5ef9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-13-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "5 (a) Show that the curve with Cartesian equation\n2 2 -y = x a,\n2 = r asec2i. [3] where a is a positive constant, has polar equation\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 2 1 G = r. r asec2i, where a is a positive constant, for 0 The curve C has polar equation i\n4\n(b) Sketch C and state the minimum distance of C from the pole. [3]\n............................................................................................................................................................\n(c) Find, in terms of a, the exact value of the area of the region enclosed by C, the initial line, and the\n1 = i r. half-line [You may use any result from the list of formulae (MF19) without proof.] [4]\n12\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................................................................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2023-mj-13-q05-bd393b126e.pdf"
    },
    {
      "id": "9231-2023-mj-13-q06",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 6,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_13.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_13.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2023-mj-13-q06-p01-bb71b73a55.png",
        "9231-topic-16-vectors/assets/9231-2023-mj-13-q06-p02-6c819d8255.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-13-q06",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "6 The points A, B, C have position vectors\ni+j, -i+2j+4k, -2i+j+3k,\nrespectively, relative to the origin O.\nax+by+cz = d. [5] (a) Find the equation of the plane ABC, giving your answer in the form\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the perpendicular distance from O to the plane ABC. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2023-mj-13-q06-57fca6463a.pdf"
    },
    {
      "id": "9231-2023-mj-13-q07",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_13.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_13.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-mj-13-q07-p01-bfcdbe7efa.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-mj-13-q07-p02-f0d1ae544b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-13-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 +2x+1 x\n= y 7 The curve C has equation .\nx-3\n(a) Find the equations of the asymptotes of C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the coordinates of the turning points on C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2023-mj-13-q07-ff0c0c555b.pdf"
    },
    {
      "id": "9231-2023-mj-21-q01",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 1,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_21.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-mj-21-q01-p01-7f86ecc830.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-21-q01",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "1 (a) Show that the system of equations\nx+2y+3z = 1,\n4x+5y+6z = 1,\n7x+8y+9z = 1,\ndoes not have a unique solution. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Show that the system of equations in part (a) is consistent. Interpret this situation geometrically.\n[3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2023-mj-21-q01-76746920d6.pdf"
    },
    {
      "id": "9231-2023-mj-21-q02",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 2,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_21.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2023-mj-21-q02-p01-1a75bd7edf.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-21-q02",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "= x+y 2 Use the substitution to find the solution of the differential equation z\n1+3x+3y dy\n=\n3x+3y-1 dx\naln(x+y) +b(x-y) +c = = = 0, for which y 0 when x 1. Give your answer in the form where a, b\nand c are constants to be determined. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n........................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2023-mj-21-q02-8b3374372e.pdf"
    },
    {
      "id": "9231-2023-mj-21-q03",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 3,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_21.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_21.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2023-mj-21-q03-p01-5dc679e395.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-21-q03",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "-1 4 (z+z = cosi+isini, ) 3 (a) By considering the binomial expansion of , where use de Moivre’s z\n1 4 = (cos4i+4cos2i+3). i cos theorem to show that [5]\n8\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\n3 2 y 2 = (1-x sini to find the exact value of dx. [3] x )2 (b) Use the substitution\n0\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2023-mj-21-q03-b9509df684.pdf"
    },
    {
      "id": "9231-2023-mj-21-q04",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 4,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_21.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2023-mj-21-q04-p01-03bbe4f7e7.png",
        "9231-topic-24-integration/assets/9231-2023-mj-21-q04-p02-497469905f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-21-q04",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "1\ny 5 n = (1+x I ) dx. I 4 The integral is defined by\nn n\n0\nd\n5 n x(1+x (a) By considering ) , or otherwise, show that\ndx` j\nn (5n+1)I = +5nI . [5] 2\nn-1 n\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................",
      "pdf_path": "9231-topic-24-integration/print/9231-2023-mj-21-q04-b5f64aa586.pdf"
    },
    {
      "id": "9231-2023-mj-21-q05",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 5,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_21.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-mj-21-q05-p01-ccbb92ec7d.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-mj-21-q05-p02-e5bf6fe381.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-21-q05",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "5 The matrix A is given by\n-11 18 5\n= -4 A 8 6 .\nf p\n-20 32 10\n3 2 -4m -20m+48 = m 0 and hence find the (a) Show that the characteristic equation of A is\neigenvalues of A. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2023-mj-21-q05-e21222696b.pdf"
    },
    {
      "id": "9231-2023-mj-21-q06",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_21.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2023-mj-21-q06-p01-ad566981cc.png",
        "9231-topic-26-differential-equations/assets/9231-2023-mj-21-q06-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-21-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "6 Find the particular solution of the differential equation\n2 d dx x\n-12 +36x = 37sint,\n2 dt dt\ndx\n= = = 0, 0. given that, when t x [11]\ndt\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.......................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2023-mj-21-q06-e502e77aed.pdf"
    },
    {
      "id": "9231-2023-mj-21-q07",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 7,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_21.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2023-mj-21-q07-p01-80ec8ece17.png",
        "9231-topic-24-integration/assets/9231-2023-mj-21-q07-p02-b927e87ed2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-21-q07",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "x y 2 = -1 dx. u x 7 (a) Use the substitution to find [3]\n2 -1 x\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\ny\nx N-1 O 1 2 3 N\n-1 = (N-1) y cosh x The diagram shows the curve with equation together with a set of rectangles of\nunit width.\n(b) By considering the sum of the areas of these rectangles, show that\nN\n/\n2 2 2 2 r+ -1 N+ -1 - -1. ln r Nln N N [5]\n` j ` j\nr=2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......",
      "pdf_path": "9231-topic-24-integration/print/9231-2023-mj-21-q07-ecfcb8d7b6.pdf"
    },
    {
      "id": "9231-2023-mj-21-q08",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 8,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_21.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_21.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2023-mj-21-q08-p01-ea787c2b1c.png",
        "9231-topic-23-differentiation/assets/9231-2023-mj-21-q08-p02-21c2245655.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-21-q08",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "8 (a) Starting from the definitions of sech and tanh in terms of exponentials, prove that\n2 2 1-sech = t tanh t. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nThe curve C has parametric equations\n1 2 4 2 = t+lnsecht, = 1+tanh t 0. x tanh y t, for\n2\ndy\n2 =-4sech t. (b) Show that [5]\ndx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2023-mj-21-q08-ba92b4fe3c.pdf"
    },
    {
      "id": "9231-2023-mj-22-q01",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 1,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_22.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-mj-22-q01-p01-7f86ecc830.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-22-q01",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "1 (a) Show that the system of equations\nx+2y+3z = 1,\n4x+5y+6z = 1,\n7x+8y+9z = 1,\ndoes not have a unique solution. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Show that the system of equations in part (a) is consistent. Interpret this situation geometrically.\n[3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2023-mj-22-q01-f5e677e8a7.pdf"
    },
    {
      "id": "9231-2023-mj-22-q02",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 2,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_22.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2023-mj-22-q02-p01-1a75bd7edf.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-22-q02",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "= x+y 2 Use the substitution to find the solution of the differential equation z\n1+3x+3y dy\n=\n3x+3y-1 dx\naln(x+y) +b(x-y) +c = = = 0, for which y 0 when x 1. Give your answer in the form where a, b\nand c are constants to be determined. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n........................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2023-mj-22-q02-2ae5add855.pdf"
    },
    {
      "id": "9231-2023-mj-22-q03",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 3,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_22.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_22.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2023-mj-22-q03-p01-5dc679e395.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-22-q03",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "-1 4 (z+z = cosi+isini, ) 3 (a) By considering the binomial expansion of , where use de Moivre’s z\n1 4 = (cos4i+4cos2i+3). i cos theorem to show that [5]\n8\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\n3 2 y 2 = (1-x sini to find the exact value of dx. [3] x )2 (b) Use the substitution\n0\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2023-mj-22-q03-f06cdf7204.pdf"
    },
    {
      "id": "9231-2023-mj-22-q04",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 4,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_22.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2023-mj-22-q04-p01-03bbe4f7e7.png",
        "9231-topic-24-integration/assets/9231-2023-mj-22-q04-p02-497469905f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-22-q04",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "1\ny 5 n = (1+x I ) dx. I 4 The integral is defined by\nn n\n0\nd\n5 n x(1+x (a) By considering ) , or otherwise, show that\ndx` j\nn (5n+1)I = +5nI . [5] 2\nn-1 n\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................",
      "pdf_path": "9231-topic-24-integration/print/9231-2023-mj-22-q04-fe024b2cb5.pdf"
    },
    {
      "id": "9231-2023-mj-22-q05",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 5,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_22.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-mj-22-q05-p01-ccbb92ec7d.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-mj-22-q05-p02-e5bf6fe381.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-22-q05",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "5 The matrix A is given by\n-11 18 5\n= -4 A 8 6 .\nf p\n-20 32 10\n3 2 -4m -20m+48 = m 0 and hence find the (a) Show that the characteristic equation of A is\neigenvalues of A. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2023-mj-22-q05-a06199b1ec.pdf"
    },
    {
      "id": "9231-2023-mj-22-q06",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_22.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2023-mj-22-q06-p01-ad566981cc.png",
        "9231-topic-26-differential-equations/assets/9231-2023-mj-22-q06-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-22-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "6 Find the particular solution of the differential equation\n2 d dx x\n-12 +36x = 37sint,\n2 dt dt\ndx\n= = = 0, 0. given that, when t x [11]\ndt\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.......................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2023-mj-22-q06-50f6bc98a3.pdf"
    },
    {
      "id": "9231-2023-mj-22-q07",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 7,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_22.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2023-mj-22-q07-p01-80ec8ece17.png",
        "9231-topic-24-integration/assets/9231-2023-mj-22-q07-p02-b927e87ed2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-22-q07",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "x y 2 = -1 dx. u x 7 (a) Use the substitution to find [3]\n2 -1 x\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\ny\nx N-1 O 1 2 3 N\n-1 = (N-1) y cosh x The diagram shows the curve with equation together with a set of rectangles of\nunit width.\n(b) By considering the sum of the areas of these rectangles, show that\nN\n/\n2 2 2 2 r+ -1 N+ -1 - -1. ln r Nln N N [5]\n` j ` j\nr=2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......",
      "pdf_path": "9231-topic-24-integration/print/9231-2023-mj-22-q07-1c082c589c.pdf"
    },
    {
      "id": "9231-2023-mj-22-q08",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 8,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_22.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_22.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2023-mj-22-q08-p01-ea787c2b1c.png",
        "9231-topic-23-differentiation/assets/9231-2023-mj-22-q08-p02-21c2245655.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-22-q08",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "8 (a) Starting from the definitions of sech and tanh in terms of exponentials, prove that\n2 2 1-sech = t tanh t. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nThe curve C has parametric equations\n1 2 4 2 = t+lnsecht, = 1+tanh t 0. x tanh y t, for\n2\ndy\n2 =-4sech t. (b) Show that [5]\ndx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2023-mj-22-q08-b89de6deef.pdf"
    },
    {
      "id": "9231-2023-mj-23-q01",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 1,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_23.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_23.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2023-mj-23-q01-p01-d324bd3a0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-23-q01",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "-1 3 sin x x 1 (a) Find the Maclaurin series for up to and including the term in . [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2023-mj-23-q01-6ee1fc7ad9.pdf"
    },
    {
      "id": "9231-2023-mj-23-q02",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 2,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_23.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2023-mj-23-q02-p01-21204c42bd.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-23-q02",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "2 The variables x and y are related by the differential equation\n2 dx d x\n2 +5 +x = +10t+13. 6 t\n2 dt dt\n(a) Find the general solution for x in terms of t. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2023-mj-23-q02-8b272f6e52.pdf"
    },
    {
      "id": "9231-2023-mj-23-q03",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 3,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_23.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_23.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2023-mj-23-q03-p01-eb71640053.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-23-q03",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "4 4 1 1\nz+ z- = cosi+isini, z 3 By considering the binomial expansions of and , where use de\nz z b l b l\nMoivre’s theorem to show that\ncos4i+acos2i+b\n4 = cot , i\ncos4i-acos2i+b\nwhere a and b are integers to be determined. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n......................................................................................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2023-mj-23-q03-50451dfe75.pdf"
    },
    {
      "id": "9231-2023-mj-23-q04",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 4,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_23.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_23.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2023-mj-23-q04-p01-fbae68eeb5.png",
        "9231-topic-23-differentiation/assets/9231-2023-mj-23-q04-p02-0738aedce1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-23-q04",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "4 The curve C has equation\n3 6 + (x+y) = 4y 109.\ndy 1\n= (-4, 3) on C, . [3] (a) Show that, at the point\n17 dx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2023-mj-23-q04-ac13538988.pdf"
    },
    {
      "id": "9231-2023-mj-23-q05",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_23.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2023-mj-23-q05-p01-d2a7265ca5.png",
        "9231-topic-26-differential-equations/assets/9231-2023-mj-23-q05-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-23-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "5 (a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that\n2 = cosh2x+1. 2cosh x [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the solution of the differential equation\ndy\n+2ytanhx = 1\ndx\n= = = y f(x). for which y 1 when x 0. Give your answer in the form [8]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2023-mj-23-q05-ef6c28f062.pdf"
    },
    {
      "id": "9231-2023-mj-23-q06",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 6,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_23.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2023-mj-23-q06-p01-921d2953a8.png",
        "9231-topic-24-integration/assets/9231-2023-mj-23-q06-p02-08d72fae09.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-23-q06",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "6\ny\n1\nx O n-1 1 2 1\nn n n\n2 G G = (1-x) y The diagram shows the curve with equation for 1, together with a set of n 0 x\n1\nrectangles of width .\nn\n1\ny 2 1 (1-x) dx U , where (a) By considering the sum of the areas of these rectangles, show that\nn\n0\n2 +3n+1 2n\n= U . [5]\n2 n 6n\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\ny 2 (1-x) dx. L (b) Use a similar method to find, in terms of n, a lower bound for [4]\nn\n0\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2023-mj-23-q06-3c1f8d9ec0.pdf"
    },
    {
      "id": "9231-2023-mj-23-q07",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 7,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_23.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2023-mj-23-q07-p01-e8bc46e0c2.png",
        "9231-topic-24-integration/assets/9231-2023-mj-23-q07-p02-b2df8975c6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-23-q07",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "4\n1 3 y 2 n = (1+x I I )2 dx. 7 The integral , where n is an integer, is defined by\nn n\n0\nI lna, giving your answer in the form where a is an integer to be (a) Find the exact value of\n-1\ndetermined. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nd 1 2 n x(1+x )2 , or otherwise, show that (b) By considering\ndxb l\nn\n5 4 (n+1)I = + nI . [5]\nn-2 3 3 n b l\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................",
      "pdf_path": "9231-topic-24-integration/print/9231-2023-mj-23-q07-9393db864a.pdf"
    },
    {
      "id": "9231-2023-mj-23-q08",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_23.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-mj-23-q08-p01-7d02647a99.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-mj-23-q08-p02-c68660f640.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-23-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "8 The matrix A is given by\n-6a 2a+2 a\n= 1-a 0 A 0\nf p\n2-a -1 0\n! ! where a is a constant with a 0 and a 1.\nx 1\n= (a) Show that the equation has a unique solution and interpret this situation geometrically. A y 2\nf p f p\nz 3\n[3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1-a -1. and [2] (b) Show that the eigenvalues of A are a,\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2023-mj-23-q08-2e9531dee8.pdf"
    },
    {
      "id": "9231-2023-mj-31-q01",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 1,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_31.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_31.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2023-mj-31-q01-p01-1b259cb133.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-31-q01",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "1 One end of a light elastic string, of natural length a and modulus of elasticity 3mg, is attached to a\nfixed point O. The other end of the string is attached to a particle P of mass m. The string hangs with P\nvertically below O. The particle P is pulled vertically downwards so that the extension of the string is\n2a. The particle P is then released from rest.\n3\n(a) Find the speed of P when it is at a distance a below O. [3]\n4\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n(b) Find the initial acceleration of P when it is released from rest. [2]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...............................................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2023-mj-31-q01-22a1839a73.pdf"
    },
    {
      "id": "9231-2023-mj-31-q02",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 2,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_31.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_31.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2023-mj-31-q02-p01-2ca8ff4731.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-31-q02",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "2\nP\nu\na\ni\nA particle P of mass m is moving with speed u on a fixed smooth horizontal surface. It collides at an\nangle with a fixed smooth vertical barrier. After the collision, P moves at an angle with the barrier, i a\n1 = where (see diagram). The coefficient of restitution between P and the barrier is e. The particle tani\n2\nP loses 20% of its kinetic energy as a result of the collision.\nFind the value of e. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2023-mj-31-q02-82be7116e6.pdf"
    },
    {
      "id": "9231-2023-mj-31-q03",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 3,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_31.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2023-mj-31-q03-p01-6b8d3547c7.png",
        "9231-topic-33-circular-motion/assets/9231-2023-mj-31-q03-p02-af809a6b4f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-31-q03",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "3 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end\nof the string is attached to a fixed point O. The particle P is held at the point A, where OA makes an\nangle with the downward vertical through O, and with the string taut. The particle P is projected i\nperpendicular to OA in an upwards direction with speed u. It then starts to move along a circular path\nin a vertical plane. The string goes slack when P is at B, where angle AOB is 90° and the speed of\n4 P is ag.\n5\nsini. [2] (a) Find the value of\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n(b) Find, in terms of m and g, the tension in the string when P is at A. [5]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2023-mj-31-q03-44e335e6dc.pdf"
    },
    {
      "id": "9231-2023-mj-31-q04",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_31.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_31.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2023-mj-31-q04-p01-eaee0d2478.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2023-mj-31-q04-p02-a435de16be.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-31-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "4\ny\na\nd\n2a\nx O C\nAn object is formed from a solid hemisphere, of radius 2a, and a solid cylinder, of radius a and height\nd. The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane\nface of the hemisphere. The line OC forms a diameter of the base of the cylinder, where C is the centre\nof the plane face of the hemisphere and O is common to both circumferences (see diagram). Relative to\naxes through O, parallel and perpendicular to OC as shown, the centre of mass of the object is (x,y).\n2 +3ad 32a\n= x (a) Show that and find an expression, in terms of a and d, for [5] y.\n16a+3d\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n1 = The object is placed on a rough plane which is inclined to the horizontal at an angle where . sini i\n6\nThe object is in equilibrium with CO horizontal, where CO lies in a vertical plane through a line of\ngreatest slope.\n(b) Find d in terms of a. [3]\n...................................................................................................................................................\n.................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2023-mj-31-q04-6cb8a54ed4.pdf"
    },
    {
      "id": "9231-2023-mj-31-q05",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_31.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2023-mj-31-q05-p01-9cd0cfec03.png",
        "9231-topic-33-circular-motion/assets/9231-2023-mj-31-q05-p02-dd8e02240e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-31-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "5 A light elastic string of natural length a and modulus of elasticity mmg has one end attached to a fixed\npoint O on a smooth horizontal surface. When a particle of mass m is attached to the free end of the\nstring, it moves with speed v in a horizontal circle with centre O and radius x. When, instead, a particle\n1 of mass 2m is attached to the free end of the string, this particle moves with speed in a horizontal v\n2\n3\ncircle with centre O and radius x.\n4\n(a) Find x in terms of a. [5]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n.............................................................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2023-mj-31-q05-1189a12326.pdf"
    },
    {
      "id": "9231-2023-mj-31-q06",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 6,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_31.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_31.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2023-mj-31-q06-p01-04355bc494.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2023-mj-31-q06-p02-2d6c785828.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-31-q06",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "6 A particle P moving in a straight line has displacement x m from a fixed point O on the line and velocity\n-1 -2 = = = v+9. vms ms at time t s. The acceleration of P, in , is given by When t 0, x 2 and v 72. 6v\n(a) Find an expression for v in terms of x. [4]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n............................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2023-mj-31-q06-79097324d6.pdf"
    },
    {
      "id": "9231-2023-mj-31-q07",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_31.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_31.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2023-mj-31-q07-p01-07f4192f1e.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2023-mj-31-q07-p02-2451f30b8b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-31-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "-1\n40ms 7 At time t s, a particle P is projected with speed at an angle above the horizontal from a i\npoint O on a horizontal plane and moves freely under gravity. The greatest height achieved by P during\nits flight is H m and the corresponding time is T s.\n(a) Obtain expressions for H and T in terms of i. [2]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n1 = = During the time between t T and t 3, P descends a distance . H\n4\ni. [4] (b) Find the value of\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n........................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2023-mj-31-q07-70d933841f.pdf"
    },
    {
      "id": "9231-2023-mj-32-q01",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 1,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_32.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_32.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2023-mj-32-q01-p01-1b259cb133.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-32-q01",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "1 One end of a light elastic string, of natural length a and modulus of elasticity 3mg, is attached to a\nfixed point O. The other end of the string is attached to a particle P of mass m. The string hangs with P\nvertically below O. The particle P is pulled vertically downwards so that the extension of the string is\n2a. The particle P is then released from rest.\n3\n(a) Find the speed of P when it is at a distance a below O. [3]\n4\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n(b) Find the initial acceleration of P when it is released from rest. [2]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...............................................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2023-mj-32-q01-0045246971.pdf"
    },
    {
      "id": "9231-2023-mj-32-q02",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 2,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_32.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_32.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2023-mj-32-q02-p01-2ca8ff4731.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-32-q02",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "2\nP\nu\na\ni\nA particle P of mass m is moving with speed u on a fixed smooth horizontal surface. It collides at an\nangle with a fixed smooth vertical barrier. After the collision, P moves at an angle with the barrier, i a\n1 = where (see diagram). The coefficient of restitution between P and the barrier is e. The particle tani\n2\nP loses 20% of its kinetic energy as a result of the collision.\nFind the value of e. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2023-mj-32-q02-f1964bf428.pdf"
    },
    {
      "id": "9231-2023-mj-32-q03",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 3,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_32.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2023-mj-32-q03-p01-6b8d3547c7.png",
        "9231-topic-33-circular-motion/assets/9231-2023-mj-32-q03-p02-af809a6b4f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-32-q03",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "3 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end\nof the string is attached to a fixed point O. The particle P is held at the point A, where OA makes an\nangle with the downward vertical through O, and with the string taut. The particle P is projected i\nperpendicular to OA in an upwards direction with speed u. It then starts to move along a circular path\nin a vertical plane. The string goes slack when P is at B, where angle AOB is 90° and the speed of\n4 P is ag.\n5\nsini. [2] (a) Find the value of\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n(b) Find, in terms of m and g, the tension in the string when P is at A. [5]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2023-mj-32-q03-c365100d21.pdf"
    },
    {
      "id": "9231-2023-mj-32-q04",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_32.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_32.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2023-mj-32-q04-p01-eaee0d2478.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2023-mj-32-q04-p02-a435de16be.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-32-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "4\ny\na\nd\n2a\nx O C\nAn object is formed from a solid hemisphere, of radius 2a, and a solid cylinder, of radius a and height\nd. The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane\nface of the hemisphere. The line OC forms a diameter of the base of the cylinder, where C is the centre\nof the plane face of the hemisphere and O is common to both circumferences (see diagram). Relative to\naxes through O, parallel and perpendicular to OC as shown, the centre of mass of the object is (x,y).\n2 +3ad 32a\n= x (a) Show that and find an expression, in terms of a and d, for [5] y.\n16a+3d\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n1 = The object is placed on a rough plane which is inclined to the horizontal at an angle where . sini i\n6\nThe object is in equilibrium with CO horizontal, where CO lies in a vertical plane through a line of\ngreatest slope.\n(b) Find d in terms of a. [3]\n...................................................................................................................................................\n.................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2023-mj-32-q04-9bc1f70bca.pdf"
    },
    {
      "id": "9231-2023-mj-32-q05",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_32.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2023-mj-32-q05-p01-9cd0cfec03.png",
        "9231-topic-33-circular-motion/assets/9231-2023-mj-32-q05-p02-dd8e02240e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-32-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "5 A light elastic string of natural length a and modulus of elasticity mmg has one end attached to a fixed\npoint O on a smooth horizontal surface. When a particle of mass m is attached to the free end of the\nstring, it moves with speed v in a horizontal circle with centre O and radius x. When, instead, a particle\n1 of mass 2m is attached to the free end of the string, this particle moves with speed in a horizontal v\n2\n3\ncircle with centre O and radius x.\n4\n(a) Find x in terms of a. [5]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n.............................................................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2023-mj-32-q05-b0aa3094cc.pdf"
    },
    {
      "id": "9231-2023-mj-32-q06",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 6,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_32.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_32.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2023-mj-32-q06-p01-04355bc494.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2023-mj-32-q06-p02-2d6c785828.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-32-q06",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "6 A particle P moving in a straight line has displacement x m from a fixed point O on the line and velocity\n-1 -2 = = = v+9. vms ms at time t s. The acceleration of P, in , is given by When t 0, x 2 and v 72. 6v\n(a) Find an expression for v in terms of x. [4]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n............................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2023-mj-32-q06-b9b135595e.pdf"
    },
    {
      "id": "9231-2023-mj-32-q07",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_32.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_32.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2023-mj-32-q07-p01-07f4192f1e.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2023-mj-32-q07-p02-2451f30b8b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-32-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "-1\n40ms 7 At time t s, a particle P is projected with speed at an angle above the horizontal from a i\npoint O on a horizontal plane and moves freely under gravity. The greatest height achieved by P during\nits flight is H m and the corresponding time is T s.\n(a) Obtain expressions for H and T in terms of i. [2]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n1 = = During the time between t T and t 3, P descends a distance . H\n4\ni. [4] (b) Find the value of\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n........................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2023-mj-32-q07-0cc88be75f.pdf"
    },
    {
      "id": "9231-2023-mj-33-q01",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 1,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_33.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2023-mj-33-q01-p01-62827dd2f3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-33-q01",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "1 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end\nof the string is attached to a fixed point O. The particle P is held at the point A, where OA makes an\nangle with the downward vertical through O, and with the string taut. The particle P is projected a\nperpendicular to OA in an upwards direction with speed 3ag. It then starts to move along a circular\npath in a vertical plane. The string goes slack when P is at B, where OB makes an angle with the i\nupward vertical.\n4 = Given that , find the value of cosi. [4] cosa\n5\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.................................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2023-mj-33-q01-fee4cda1c5.pdf"
    },
    {
      "id": "9231-2023-mj-33-q02",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 2,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_33.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_33.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2023-mj-33-q02-p01-3b0837487b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-33-q02",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "2 One end of a light elastic string, of natural length a and modulus of elasticity mmg, is attached to a\nfixed point O. The string lies on a smooth horizontal surface. A particle P of mass m is attached to the\nother end of the string. The particle P is projected in the direction OP. When the length of the string is\n5 4 1 a, a, the speed of P is 2ag. When the length of the string is the speed of P is 2ag.\n3 2 3\nm. [4] Find the value of\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n......................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2023-mj-33-q02-82a449440d.pdf"
    },
    {
      "id": "9231-2023-mj-33-q03",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 3,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_33.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_33.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2023-mj-33-q03-p01-fe756f1bcd.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2023-mj-33-q03-p02-827606a94c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-33-q03",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "3\nA\n3a\nD\n5a\nx\nC B\nE\n6a\n= = = A uniform lamina is in the form of a triangle ABC, with AC 8a, BC 6a and angle ACB 90°. The\n= = point D on AC is such that AD 3a. The point E on CB is such that CE x (see diagram). The triangle\nCDE is removed from the lamina.\n(a) Find, in terms of a and x, the distance of the centre of mass of the remaining object ADEB from\nAC. [4]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\nThe object ADEB is on the point of toppling about the point E when the object is in the vertical plane\nwith its edge EB on a smooth horizontal surface.\n(b) Find x in terms of a. [3]\n.......................................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2023-mj-33-q03-bcb8a8815a.pdf"
    },
    {
      "id": "9231-2023-mj-33-q04",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 4,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_33.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_33.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2023-mj-33-q04-p01-250598a4c1.png",
        "9231-topic-36-momentum/assets/9231-2023-mj-33-q04-p02-93f4d24b2d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-33-q04",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "4\n2u\nu\nA B\n30°\nm m\nTwo identical smooth uniform spheres A and B each have mass m. The two spheres are moving on a\nsmooth horizontal surface when they collide with speeds u and 2u respectively. Immediately before the\ncollision, A’s direction of motion makes an angle of 30° with the line of centres, and B’s direction of\nmotion is perpendicular to the line of centres (see diagram). After the collision, A and B are moving in\nthe same direction. The coefficient of restitution between the spheres is e.\n(a) Find the value of e. [5]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n.....................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2023-mj-33-q04-8b3f8271b8.pdf"
    },
    {
      "id": "9231-2023-mj-33-q05",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_33.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2023-mj-33-q05-p01-2ae0356677.png",
        "9231-topic-33-circular-motion/assets/9231-2023-mj-33-q05-p02-6e228e395c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-33-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "5 One end of a light elastic string, of natural length 12a and modulus of elasticity kmg, is attached to\na fixed point O. The other end of the string is attached to a particle of mass m. The particle moves\n3\nwith constant speed 3ag in a horizontal circle with centre at a distance 12a below O. The string is\n2\ninclined at an angle to the downward vertical through O. i\n(a) Find, in terms of a, the extension of the string. [5]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n.........................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2023-mj-33-q05-a24abcc04b.pdf"
    },
    {
      "id": "9231-2023-mj-33-q06",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 6,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_33.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_33.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2023-mj-33-q06-p01-dea6173d96.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2023-mj-33-q06-p02-972dd2f47f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-33-q06",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "6 A particle of mass m kg falls vertically under gravity, from rest. At time t s, P has fallen x m and has\n-1\nvelocity vms . The only forces acting on P are its weight and a resistance of magnitude kmgv N,\nwhere k is a constant.\n(a) Find an expression for v in terms of t, g and k. [5]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n..................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2023-mj-33-q06-62a2ddd7e2.pdf"
    },
    {
      "id": "9231-2023-mj-33-q07",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_33.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_33.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2023-mj-33-q07-p01-e643df37ed.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2023-mj-33-q07-p02-d59a7834cc.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-33-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "7 The points O and P are on a horizontal plane, a distance 8 m apart. A ball is thrown from O with speed\n-1 4 = ums at an angle above the horizontal, where . At the same instant, a model aircraft is tani i\n3\n-1\nlaunched with speed 5ms parallel to the horizontal plane from a point 4 m vertically above P. The\nmodel aircraft moves in the same vertical plane as the ball and in the same horizontal direction as the\n-1\nball. The model aircraft moves horizontally with a constant speed of 5ms . After T s, the ball and the\nmodel aircraft collide.\n(a) Find the value of T. [6]\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n...................................................................................................................................................\n............................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2023-mj-33-q07-b92ccf11f2.pdf"
    },
    {
      "id": "9231-2023-mj-41-q01",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_41.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-mj-41-q01-p01-d102e8b5b2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-41-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 The lengths of the leaves of a particular type of tree are normally distributed with mean cm. The μ\nlengths, x cm, of a random sample of 12 leaves of this type are recorded. The results are summarised as\nfollows.\n2 = = /x /x 695.8 91.2\nFind a 95% confidence interval for μ. [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2023-mj-41-q01-9eef348ba3.pdf"
    },
    {
      "id": "9231-2023-mj-41-q02",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 2,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_41.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-mj-41-q02-p01-aa85bddd1f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-41-q02",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "2 The children at two large schools, P and Q, are all given the same puzzle to solve. A random sample\nof size 10 is taken from the children at school P. Their individual times to complete the puzzle give a\n2 sample mean of 9.12 minutes and an unbiased variance estimate of 2.16 minutes . A random sample of\nsize 12 is taken from the children at school Q. Their individual times, x minutes, to complete the puzzle\nare summarised by\n2 /(x-x) = = /x 21.5, 99.6\nis the sample mean. Times to complete the puzzle are assumed to be normally distributed with x where\nthe same population variance.\nTest at the 5% significance level whether the population mean time taken to complete the puzzle by\nchildren at school P is greater than the population mean time taken to complete the puzzle by children\nat school Q. [8]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...............................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2023-mj-41-q02-dc99eca4f7.pdf"
    },
    {
      "id": "9231-2023-mj-41-q03",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 3,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_41.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_41.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2023-mj-41-q03-p01-40e2a45dd6.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2023-mj-41-q03-p02-9f391877f5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-41-q03",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "3 A random sample of 50 values of the continuous random variable X was taken. These values are\nsummarised in the following table.\n1 1 1 1 1 G G G G G G G Interval 3.5 x 4 1 x 1.5 1.5 x 2 2 x 2.5 2.5 x 3 3 x 3.5\nObserved frequency 3 3 8 11 13 12\nIt is required to test the goodness of fit of the distribution with probability density function f given by\n1 4\n2 G G +x 1 x 4,\n2 = f(x) 24ex o\n*\n0 otherwise.\nThe expected frequencies, correct to 4 decimal places, are given in the following table.\n1 1 1 1 1 G G G G G G G Interval 3.5 x 4 1 x 1.5 1.5 x 2 2 x 2.5 2.5 x 3 3 x 3.5\nExpected frequency 4.4271 a 6.1285 8.4549 b 14.9678\n= (a) Show that a 4.6007 and find the value of b. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................................................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2023-mj-41-q03-26728d6c1b.pdf"
    },
    {
      "id": "9231-2023-mj-41-q04",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 4,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_41.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_41.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2023-mj-41-q04-p01-52c69285d7.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2023-mj-41-q04-p02-300e9f138c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-41-q04",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "4 A random sample of 13 technology companies is chosen and the numbers of employees in 2018 and in\n2022 are recorded.\nCompany A B C D E F G H I J K L M\nNumber in 2018 104 19 126 234 970 514 35 149 429 12 86 304 1104\nNumber in 2022 106 24 127 228 1012 525 32 156 449 24 78 294 1154\nA researcher claims that there has been an increase in the median number of employees at technology\ncompanies between 2018 and 2022.\n(a) Carry out a Wilcoxon matched-pairs signed-rank test, at the 5% significance level, to test whether\nthe data supports this claim. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2023-mj-41-q04-6df9ec7105.pdf"
    },
    {
      "id": "9231-2023-mj-41-q05",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 5,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_41.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_41.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2023-mj-41-q05-p01-80e3d62148.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2023-mj-41-q05-p02-f5800f28f4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-41-q05",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "5 Harry has three coins.\n1 • One coin is biased so that, when it is thrown, the probability of obtaining a head is .\n3\n1 • The second coin is biased so that, when it is thrown, the probability of obtaining a head is .\n4\n1 • The third coin is biased so that, when it is thrown, the probability of obtaining a head is .\n5\nThe random variable X is the number of heads that Harry obtains when he throws all three coins\ntogether.\n(a) Find the probability generating function of X. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nIsaac has two fair coins. The random variable Y is the number of heads that Isaac obtains when he\nthrows both of his coins together. The random variable Z is the total number of heads obtained when\nHarry throws his three coins and Isaac throws his two coins.\n(b) Find the probability generating function of Z, expressing your answer as a polynomial in t. [4]\n............................................................................................................................................................\n........................................................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2023-mj-41-q05-7cb9c20956.pdf"
    },
    {
      "id": "9231-2023-mj-41-q06",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 6,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_41.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_41.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2023-mj-41-q06-p01-fd70c1d5db.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2023-mj-41-q06-p02-002975c7a6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-41-q06",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "6 The continuous random variable X has probability density function f given by\n1 1 3 - x x G G +4e e2 2ln3, 0 x 2\n28` = j f(x)\n* 0 otherwise.\n(a) Find the cumulative distribution function of X. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 (X) = . Y e2 The random variable Y is defined by\n(b) Find the probability density function of Y. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2023-mj-41-q06-f87d1463a0.pdf"
    },
    {
      "id": "9231-2023-mj-42-q01",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_42.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-mj-42-q01-p01-d102e8b5b2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-42-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 The lengths of the leaves of a particular type of tree are normally distributed with mean cm. The μ\nlengths, x cm, of a random sample of 12 leaves of this type are recorded. The results are summarised as\nfollows.\n2 = = /x /x 695.8 91.2\nFind a 95% confidence interval for μ. [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2023-mj-42-q01-3ac63204d1.pdf"
    },
    {
      "id": "9231-2023-mj-42-q02",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 2,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_42.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-mj-42-q02-p01-aa85bddd1f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-42-q02",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "2 The children at two large schools, P and Q, are all given the same puzzle to solve. A random sample\nof size 10 is taken from the children at school P. Their individual times to complete the puzzle give a\n2 sample mean of 9.12 minutes and an unbiased variance estimate of 2.16 minutes . A random sample of\nsize 12 is taken from the children at school Q. Their individual times, x minutes, to complete the puzzle\nare summarised by\n2 /(x-x) = = /x 21.5, 99.6\nis the sample mean. Times to complete the puzzle are assumed to be normally distributed with x where\nthe same population variance.\nTest at the 5% significance level whether the population mean time taken to complete the puzzle by\nchildren at school P is greater than the population mean time taken to complete the puzzle by children\nat school Q. [8]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...............................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2023-mj-42-q02-1ad980ce9b.pdf"
    },
    {
      "id": "9231-2023-mj-42-q03",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 3,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_42.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_42.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2023-mj-42-q03-p01-40e2a45dd6.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2023-mj-42-q03-p02-9f391877f5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-42-q03",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "3 A random sample of 50 values of the continuous random variable X was taken. These values are\nsummarised in the following table.\n1 1 1 1 1 G G G G G G G Interval 3.5 x 4 1 x 1.5 1.5 x 2 2 x 2.5 2.5 x 3 3 x 3.5\nObserved frequency 3 3 8 11 13 12\nIt is required to test the goodness of fit of the distribution with probability density function f given by\n1 4\n2 G G +x 1 x 4,\n2 = f(x) 24ex o\n*\n0 otherwise.\nThe expected frequencies, correct to 4 decimal places, are given in the following table.\n1 1 1 1 1 G G G G G G G Interval 3.5 x 4 1 x 1.5 1.5 x 2 2 x 2.5 2.5 x 3 3 x 3.5\nExpected frequency 4.4271 a 6.1285 8.4549 b 14.9678\n= (a) Show that a 4.6007 and find the value of b. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................................................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2023-mj-42-q03-5c42f97bf9.pdf"
    },
    {
      "id": "9231-2023-mj-42-q04",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 4,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_42.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_42.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2023-mj-42-q04-p01-52c69285d7.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2023-mj-42-q04-p02-300e9f138c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-42-q04",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "4 A random sample of 13 technology companies is chosen and the numbers of employees in 2018 and in\n2022 are recorded.\nCompany A B C D E F G H I J K L M\nNumber in 2018 104 19 126 234 970 514 35 149 429 12 86 304 1104\nNumber in 2022 106 24 127 228 1012 525 32 156 449 24 78 294 1154\nA researcher claims that there has been an increase in the median number of employees at technology\ncompanies between 2018 and 2022.\n(a) Carry out a Wilcoxon matched-pairs signed-rank test, at the 5% significance level, to test whether\nthe data supports this claim. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2023-mj-42-q04-6804809e06.pdf"
    },
    {
      "id": "9231-2023-mj-42-q05",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 5,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_42.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_42.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2023-mj-42-q05-p01-80e3d62148.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2023-mj-42-q05-p02-f5800f28f4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-42-q05",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "5 Harry has three coins.\n1 • One coin is biased so that, when it is thrown, the probability of obtaining a head is .\n3\n1 • The second coin is biased so that, when it is thrown, the probability of obtaining a head is .\n4\n1 • The third coin is biased so that, when it is thrown, the probability of obtaining a head is .\n5\nThe random variable X is the number of heads that Harry obtains when he throws all three coins\ntogether.\n(a) Find the probability generating function of X. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nIsaac has two fair coins. The random variable Y is the number of heads that Isaac obtains when he\nthrows both of his coins together. The random variable Z is the total number of heads obtained when\nHarry throws his three coins and Isaac throws his two coins.\n(b) Find the probability generating function of Z, expressing your answer as a polynomial in t. [4]\n............................................................................................................................................................\n........................................................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2023-mj-42-q05-8b5f9b089c.pdf"
    },
    {
      "id": "9231-2023-mj-42-q06",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 6,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_42.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_42.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2023-mj-42-q06-p01-fd70c1d5db.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2023-mj-42-q06-p02-002975c7a6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-42-q06",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "6 The continuous random variable X has probability density function f given by\n1 1 3 - x x G G +4e e2 2ln3, 0 x 2\n28` = j f(x)\n* 0 otherwise.\n(a) Find the cumulative distribution function of X. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1 (X) = . Y e2 The random variable Y is defined by\n(b) Find the probability density function of Y. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2023-mj-42-q06-ab75ebce5b.pdf"
    },
    {
      "id": "9231-2023-mj-43-q01",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 1,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_43.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_43.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2023-mj-43-q01-p01-6524effd68.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2023-mj-43-q01-p02-043a7810be.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-43-q01",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "1 The continuous random variable X has probability density function f given by\n-1 -2 1\nG G -x 1 27, x x 3 3\n6` = j f(x)\n* 0 otherwise.\n(a) Find the cumulative distribution function of X. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\n= 3. Y X The random variable Y is defined by\n(b) Find the probability density function of Y. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2023-mj-43-q01-8de1e3a2b1.pdf"
    },
    {
      "id": "9231-2023-mj-43-q02",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 2,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_43.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-mj-43-q02-p01-29f549fb75.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-43-q02",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "2 Shane is studying the lengths of the tails of male red kangaroos. He takes a random sample of 14 male\nred kangaroos and measures the length of the tail, x m, for each kangaroo. He then calculates a 90%\nconfidence interval for the population mean tail length, m, of male red kangaroos. He assumes that n\nG G the tail lengths are normally distributed and finds that 1.14. 1.11 n\n2 /x /x and for this sample. [6] Find the values of\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..............................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2023-mj-43-q02-218d6696d7.pdf"
    },
    {
      "id": "9231-2023-mj-43-q03",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 3,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_43.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_43.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2023-mj-43-q03-p01-fe2e54589c.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2023-mj-43-q03-p02-7ff96747dc.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-43-q03",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "3 A large number of students took two test papers in mathematics. The teacher believes that the marks\nobtained in Paper 1 will be higher than the marks obtained in Paper 2. She chooses a random sample of\n9 students and compares their marks. The marks are shown in the table.\nStudent A B C D E F G H I\nPaper 1 46 73 55 64 86 42 66 68 60\nPaper 2 41 66 61 63 90 40 58 42 70\n(a) Carry out a Wilcoxon matched-pairs signed-rank test, at the 5% significance level, to test whether\nthe data supports the teacher’s belief. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2023-mj-43-q03-67e7970475.pdf"
    },
    {
      "id": "9231-2023-mj-43-q04",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 4,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_43.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-mj-43-q04-p01-fbd7a8ad38.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-mj-43-q04-p02-0bb99619ea.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-43-q04",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "4 An inspector is checking the lengths of metal rods produced by two machines, X and Y. These rods\nshould be of the same length, but the inspector suspects that those made by machine X are shorter, on\naverage, than those made by machine Y. The inspector chooses a random sample of 80 rods made by\nmachine X and a random sample of 60 rods made by machine Y. The lengths of these rods are x cm and\ny cm respectively. Her results are summarised as follows.\n2 2 = = = = /x /x /y /y 164.0 338.1 124.8 261.1\n(a) Test at the 10% significance level whether the data supports the inspector’s suspicion. [8]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2023-mj-43-q04-4565019d18.pdf"
    },
    {
      "id": "9231-2023-mj-43-q05",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 5,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_43.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_43.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2023-mj-43-q05-p01-16685765d5.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2023-mj-43-q05-p02-6758e7be95.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-43-q05",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "G (t) 5 The random variable X has probability generating function given by\nX\n2 = k(1+3t+4t G (t) ),\nX\nwhere k is a constant.\n11 = [3] E(X) . (a) Show that\n8\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nG (t) given by The random variable Y has probability generating function\nY\n1 2 = (1+2t). G (t) t\n3 Y\n= X+Y Z The random variables X and Y are independent and .\n(b) Find the probability generating function of Z, expressing your answer as a polynomial in t. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2023-mj-43-q05-3570cda076.pdf"
    },
    {
      "id": "9231-2023-mj-43-q06",
      "subject": "9231",
      "year": 2023,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 6,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-may-june/9231_s23_qp_43.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-May-June/qp/9231_s23_qp_43.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2023-mj-43-q06-p01-9c83eed338.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2023-mj-43-q06-p02-100d4cb921.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-mj-43-q06",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "6 A scientist is investigating whether the ability to remember depends on age. A random sample of 150\nstudents in different age groups is chosen. Each student is shown a set of 20 objects for thirty seconds\nand then asked to list as many as they can remember. The students are graded A or B according to how\nmany objects they remembered correctly: grade A for 16 or more correct and grade B for fewer than 16\ncorrect. The results are shown in the table.\nAge of students\n11-12 13-14 15-16 years years years\n25 16 19 Grade A\n28 45 17 Grade B\n2 | -test at the 2.5% significance level to test whether grade is independent of age of (a) Carry out a\nstudent. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2023-mj-43-q06-1bdb413fb7.pdf"
    },
    {
      "id": "9231-2023-on-11-q01",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_11.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_11.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2023-on-11-q01-p01-d724edd96e.png",
        "9231-topic-13-summation-of-series/assets/9231-2023-on-11-q01-p02-eb633c6a7c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-11-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "2 2 (r+1) -r 1 (a) By considering , use the method of differences to prove that\nn\n1 / n(n+1). [4] = r\n2\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2023-on-11-q01-2a9d414fa0.pdf"
    },
    {
      "id": "9231-2023-on-11-q02",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_11.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_11.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2023-on-11-q02-p01-2160c318f8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-11-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "2 Prove by mathematical induction that, for all positive integers n,\nn+1 n 1- (n+1)x +nx\nn-1 2 1+2x+3x +f+nx = . [6]\n2 (1-x)\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n......................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2023-on-11-q02-c7d36509f8.pdf"
    },
    {
      "id": "9231-2023-on-11-q03",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 3,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_11.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_11.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2023-on-11-q03-p01-2de3d13558.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-11-q03",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "4 3 2 +bx +cx +dx-2 = x 0 3 The quartic equation has roots a, b, c, d. It is given that\n-1 -1 -1 -1 2 2 2 2 a+b+c+d = +b +c +d = +b +c +d = 3, 5, 6. a a\n(a) Find the values of b, c and d. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n3 3 3 3 4 4 4 4 +b +c +d = -27, +b +c +d find the value of . [2] (b) Given also that a a\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2023-on-11-q03-febf602b51.pdf"
    },
    {
      "id": "9231-2023-on-11-q04",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 4,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_11.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_11.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2023-on-11-q04-p01-9cfbb14daf.png",
        "9231-topic-16-vectors/assets/9231-2023-on-11-q04-p02-0f942ab018.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-11-q04",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "l l 4 The lines and have equations\n1 2\n= 2i-2j+3k+n(2i-3j+k) = -2i-3j-5k+m(-4i+3j+5k) r r and\nrespectively.\nl l and . [5] (a) Find the shortest distance between\n1 2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........",
      "pdf_path": "9231-topic-16-vectors/print/9231-2023-on-11-q04-5fefe29512.pdf"
    },
    {
      "id": "9231-2023-on-11-q05",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 5,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_11.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_11.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2023-on-11-q05-p01-8f3dfc7bef.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2023-on-11-q05-p02-deadd76f06.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-11-q05",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "5 Let k be a constant. The matrices A, B and C are given by\n-2 1 k 3 0\n-2 -1 1\n= = = -1 C A 2 1 3 , B and . 3\n1 1 3o e f p f p\n3 2 5 0 0\nIt is given that A is singular.\n-7 3\n= (a) Show that CAB . [5]\ne-9 3o\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n- (b) Find the equations of the invariant lines, through the origin, of the transformation in the x y plane\nrepresented by CAB. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2023-on-11-q05-95bff42aa2.pdf"
    },
    {
      "id": "9231-2023-on-11-q06",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_11.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_11.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2023-on-11-q06-p01-d1e00350b7.png",
        "9231-topic-15-polar-coordinates/assets/9231-2023-on-11-q06-p02-3099a69ab9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-11-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "6 (a) Show that the curve with Cartesian equation\n2\n1 1 2 x- +y =\n2 4 b l\n= has polar equation r cosi. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nC and C have polar equations The curves\n1 2\n= = r sin2i r cosi and\n1 G G respectively, where r. The curves C and C intersect at the pole and at another point P. 0 i\n2 1 2\n(b) Find the polar coordinates of P. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nC and C , clearly identifying each curve, and mark the point P. [3] (c) In a single diagram sketch\n1 2\n(d) The region R is enclosed by C and C and includes the line OP.\n1 2\nFind, in exact form, the area of R. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2023-on-11-q06-5fa529fe8d.pdf"
    },
    {
      "id": "9231-2023-on-11-q07",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_11.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_11.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-on-11-q07-p01-1cb023337e.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-on-11-q07-p02-098c0a2888.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-on-11-q07-p03-8f3cdb9b49.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-11-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 +2 x\n= = f(x) y f(x), 7 The curve C has equation where .\n2 -x-2 x\n(a) Find the equations of the asymptotes of C. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the coordinates of any stationary points on C, giving your answers correct to 1 decimal place.\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2023-on-11-q07-8007c789f3.pdf"
    },
    {
      "id": "9231-2023-on-12-q01",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_12.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_12.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2023-on-12-q01-p01-e9bfb4ccfb.png",
        "9231-topic-13-summation-of-series/assets/9231-2023-on-12-q01-p02-6dd04e77b2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-12-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "n\n/\n2 +3r+1) 1 (a) Use standard results from the list of formulae (MF19) to find in terms of n, (3r\nr=1\nsimplifying your answer. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2023-on-12-q01-c452f05c97.pdf"
    },
    {
      "id": "9231-2023-on-12-q02",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_12.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_12.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2023-on-12-q02-p01-9231c8cb6f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-12-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "2 Prove by mathematical induction that, for all positive integers n,\nn d\nx x 2 2 = +2nx+n(n-1) [6] x e x e .\nn` j ` j dx\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2023-on-12-q02-104d56c284.pdf"
    },
    {
      "id": "9231-2023-on-12-q03",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 3,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_12.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_12.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2023-on-12-q03-p01-e0abf2aa7f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-12-q03",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "k 0 1 0\n! ! = M k 0 k 1. 3 The matrix M is given by , where k is a constant and and\ne0 1oe1 1o\n(a) The matrix M represents a sequence of two geometrical transformations. State the type of each\ntransformation, and make clear the order in which they are applied. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nThe unit square in the x – y plane is transformed by M onto parallelogram OPQR.\n(b) Find, in terms of k, the area of parallelogram OPQR and the matrix which transforms OPQR onto\nthe unit square. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\nis invariant under the transformation in (c) Show that the line through the origin with gradient\nk-1\nthe x – y plane represented by M. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2023-on-12-q03-3e5f310fe8.pdf"
    },
    {
      "id": "9231-2023-on-12-q04",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 4,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_12.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_12.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2023-on-12-q04-p01-42ce80c6ce.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2023-on-12-q04-p02-16a0214119.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-12-q04",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "3 2 +18x +6x-1 = 27x 0 4 The cubic equation has roots a, b, c.\n3b+1, 3a+1, 3c+1 is (a) Show that a cubic equation with roots\n3 2 -y +y-2 = y 0. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2023-on-12-q04-b9b33a227e.pdf"
    },
    {
      "id": "9231-2023-on-12-q05",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 5,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_12.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_12.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2023-on-12-q05-p01-138796f144.png",
        "9231-topic-16-vectors/assets/9231-2023-on-12-q05-p02-e564781ca2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-12-q05",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "= i-j-2k+m(i-2j-3k) +n(3i-k). 5 The plane has equation r P\n1\nax+by+cz = d. in the form [4] (a) Find an equation for P\n1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n= -3i+k+t(i+j+k). r , has equation The line l, which does not lie in P\n1\n. [2] (b) Show that l is parallel to P\n1\n............................................................................................................................................................\n.................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2023-on-12-q05-ff18c4f15d.pdf"
    },
    {
      "id": "9231-2023-on-12-q06",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_12.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_12.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2023-on-12-q06-p01-c1734bc23c.png",
        "9231-topic-15-polar-coordinates/assets/9231-2023-on-12-q06-p02-958702633a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-12-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "1 -i - r 1 G G = -e 6 The curve C has polar equation , where r. r e 0 i 2\n2\n(a) Sketch C and state, in exact form, the greatest distance of a point on C from the pole. [3]\n............................................................................................................................................................\n(b) Find the exact value of the area of the region bounded by C and the initial line. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2023-on-12-q06-a74e7f2465.pdf"
    },
    {
      "id": "9231-2023-on-12-q07",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_12.pdf?download=true",
      "source_pages": [
        13,
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_12.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-on-12-q07-p01-03bc0f9eb9.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-on-12-q07-p02-3c9924e230.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-on-12-q07-p03-e68fce0ab3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-12-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 x\n= = f(x) y f(x), 7 The curve C has equation where .\nx+1\n(a) Find the equations of the asymptotes of C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the coordinates of any stationary points on C. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2023-on-12-q07-85a9e5c79a.pdf"
    },
    {
      "id": "9231-2023-on-13-q01",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_13.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_13.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2023-on-13-q01-p01-d724edd96e.png",
        "9231-topic-13-summation-of-series/assets/9231-2023-on-13-q01-p02-eb633c6a7c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-13-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "2 2 (r+1) -r 1 (a) By considering , use the method of differences to prove that\nn\n1 / n(n+1). [4] = r\n2\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2023-on-13-q01-d2196a4aaf.pdf"
    },
    {
      "id": "9231-2023-on-13-q02",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_13.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_13.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2023-on-13-q02-p01-2160c318f8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-13-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "2 Prove by mathematical induction that, for all positive integers n,\nn+1 n 1- (n+1)x +nx\nn-1 2 1+2x+3x +f+nx = . [6]\n2 (1-x)\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n......................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2023-on-13-q02-d4dad86575.pdf"
    },
    {
      "id": "9231-2023-on-13-q03",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 3,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_13.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_13.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2023-on-13-q03-p01-2de3d13558.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-13-q03",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "4 3 2 +bx +cx +dx-2 = x 0 3 The quartic equation has roots a, b, c, d. It is given that\n-1 -1 -1 -1 2 2 2 2 a+b+c+d = +b +c +d = +b +c +d = 3, 5, 6. a a\n(a) Find the values of b, c and d. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n3 3 3 3 4 4 4 4 +b +c +d = -27, +b +c +d find the value of . [2] (b) Given also that a a\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2023-on-13-q03-139300606a.pdf"
    },
    {
      "id": "9231-2023-on-13-q04",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 4,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_13.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_13.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2023-on-13-q04-p01-9cfbb14daf.png",
        "9231-topic-16-vectors/assets/9231-2023-on-13-q04-p02-0f942ab018.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-13-q04",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "l l 4 The lines and have equations\n1 2\n= 2i-2j+3k+n(2i-3j+k) = -2i-3j-5k+m(-4i+3j+5k) r r and\nrespectively.\nl l and . [5] (a) Find the shortest distance between\n1 2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........",
      "pdf_path": "9231-topic-16-vectors/print/9231-2023-on-13-q04-e457aa1112.pdf"
    },
    {
      "id": "9231-2023-on-13-q05",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 5,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_13.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_13.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2023-on-13-q05-p01-8f3dfc7bef.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2023-on-13-q05-p02-deadd76f06.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-13-q05",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "5 Let k be a constant. The matrices A, B and C are given by\n-2 1 k 3 0\n-2 -1 1\n= = = -1 C A 2 1 3 , B and . 3\n1 1 3o e f p f p\n3 2 5 0 0\nIt is given that A is singular.\n-7 3\n= (a) Show that CAB . [5]\ne-9 3o\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n- (b) Find the equations of the invariant lines, through the origin, of the transformation in the x y plane\nrepresented by CAB. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2023-on-13-q05-74685543d2.pdf"
    },
    {
      "id": "9231-2023-on-13-q06",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_13.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_13.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2023-on-13-q06-p01-d1e00350b7.png",
        "9231-topic-15-polar-coordinates/assets/9231-2023-on-13-q06-p02-3099a69ab9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-13-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "6 (a) Show that the curve with Cartesian equation\n2\n1 1 2 x- +y =\n2 4 b l\n= has polar equation r cosi. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nC and C have polar equations The curves\n1 2\n= = r sin2i r cosi and\n1 G G respectively, where r. The curves C and C intersect at the pole and at another point P. 0 i\n2 1 2\n(b) Find the polar coordinates of P. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nC and C , clearly identifying each curve, and mark the point P. [3] (c) In a single diagram sketch\n1 2\n(d) The region R is enclosed by C and C and includes the line OP.\n1 2\nFind, in exact form, the area of R. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2023-on-13-q06-b81de9c2f7.pdf"
    },
    {
      "id": "9231-2023-on-13-q07",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_13.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_13.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-on-13-q07-p01-1cb023337e.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-on-13-q07-p02-098c0a2888.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2023-on-13-q07-p03-8f3cdb9b49.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-13-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 +2 x\n= = f(x) y f(x), 7 The curve C has equation where .\n2 -x-2 x\n(a) Find the equations of the asymptotes of C. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the coordinates of any stationary points on C, giving your answers correct to 1 decimal place.\n[4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2023-on-13-q07-925b64ba24.pdf"
    },
    {
      "id": "9231-2023-on-21-q01",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 1,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_21.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-on-21-q01-p01-8a3580147c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-21-q01",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "1 Show that the system of equations\n14x-4y+6z = 5,\nx+ y+ = kz 3,\n-21x+6y-9z = 14,\nwhere k is a constant, does not have a unique solution and interpret this situation geometrically. [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2023-on-21-q01-a11a3b186e.pdf"
    },
    {
      "id": "9231-2023-on-21-q02",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 2,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_21.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_21.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2023-on-21-q02-p01-e3e97ae92f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-21-q02",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "3 (z+5i) = 4+4 rcosi+i(rsini-5), 3i, 2 Find the roots of the equation giving your answers in the form\n2 1 1 where r 0 and 0 2r. [5] i\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.............................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2023-on-21-q02-97b9433930.pdf"
    },
    {
      "id": "9231-2023-on-21-q03",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 3,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_21.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_21.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2023-on-21-q03-p01-a3a7cc66c3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-21-q03",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "-1 1 1 2 x lna+bx+cx 3 Find the first three terms in the Maclaurin’s series for in the form , giving tanh e\n2 2 b l\nthe exact values of the constants a, b and c. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.............................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2023-on-21-q03-d8f569b871.pdf"
    },
    {
      "id": "9231-2023-on-21-q04",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 4,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_21.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2023-on-21-q04-p01-41effd08fc.png",
        "9231-topic-26-differential-equations/assets/9231-2023-on-21-q04-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-21-q04",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "4 Find the particular solution of the differential equation\n2 d y dy\n2 +2 +3y = 27x ,\n2 dx dx\ndy\n= -8. = = given that, when x 0, y 2 and [10]\ndx\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2023-on-21-q04-6045e19cca.pdf"
    },
    {
      "id": "9231-2023-on-21-q05",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 5,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_21.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2023-on-21-q05-p01-3afba81d45.png",
        "9231-topic-24-integration/assets/9231-2023-on-21-q05-p02-b0557b0830.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-21-q05",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "5 The curve C has parametric equations\n3 1 2 1 G = -2t = 2t+5, x t 2, y for 0 t 3. 2\n3\n(a) Find the exact length of C. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2023-on-21-q05-c02da31a5f.pdf"
    },
    {
      "id": "9231-2023-on-21-q06",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_21.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2023-on-21-q06-p01-eeba00dda6.png",
        "9231-topic-26-differential-equations/assets/9231-2023-on-21-q06-p02-c29e09cf1a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-21-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "6 (a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that\n= sinh2x 2sinhxcoshx. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\ny 2 = u sinhx, find dx. [4] sinh 2xcoshx (b) Using the substitution\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2023-on-21-q06-fe24edcc12.pdf"
    },
    {
      "id": "9231-2023-on-21-q07",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 7,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_21.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-on-21-q07-p01-c8e6174f9d.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-on-21-q07-p02-755e0ce754.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-21-q07",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "7 The matrix A is given by\n-6 2 13\n= -2 A 0 5 .\nf p\n0 0 8\n-1 -1 = A PDP . [7] (a) Find a matrix P and a diagonal matrix D such that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2023-on-21-q07-8d7e98e473.pdf"
    },
    {
      "id": "9231-2023-on-21-q08",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 8,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_21.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_21.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2023-on-21-q08-p01-c5318af3f5.png",
        "9231-topic-25-complex-numbers/assets/9231-2023-on-21-q08-p02-5603dd4ceb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-21-q08",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "n-1 2 ! 1+z+z +f+z 8 (a) State the sum of the series , for z 1. [1]\n............................................................................................................................................................\n! = cosi+isini, z where cosi 1, show that (b) By letting\n1 sinnisini\n1+cosi+cos2i+f+cos(n-1)i = 1-cosni+ . [7]\n1-cosi 2e o\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2023-on-21-q08-72b6cba134.pdf"
    },
    {
      "id": "9231-2023-on-22-q01",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 1,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_22.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_22.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2023-on-22-q01-p01-a441385ab4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-22-q01",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "2 2 ln(x+2) +ln(x +5) x 1 Find the Maclaurin’s series for up to and including the term in . [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...................................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2023-on-22-q01-089c158279.pdf"
    },
    {
      "id": "9231-2023-on-22-q02",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 2,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_22.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_22.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2023-on-22-q02-p01-2923ac6b71.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-22-q02",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "2 It is given that\n1\nt = 1+ = x and . y te\nt\ndy\nt 3 2 = -e +t (a) Show that (t ). [3]\ndx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 d y\nin terms of t. [4] (b) Find\n2 dx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2023-on-22-q02-aaf249d75a.pdf"
    },
    {
      "id": "9231-2023-on-22-q03",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 3,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_22.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_22.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2023-on-22-q03-p01-b2d3996712.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-22-q03",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "3 (a) Use de Moivre’s theorem to show that\n5 3 = i-20cos i+5cosi. cos5i 16cos [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Hence obtain the roots of the equation\n5 3 -40x +10x- = 32x 2 0\ncos(qr), where q is a rational number. [4] in the form\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............................................................................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2023-on-22-q03-07dcdf8ecd.pdf"
    },
    {
      "id": "9231-2023-on-22-q04",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 4,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_22.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2023-on-22-q04-p01-e6470e1725.png",
        "9231-topic-26-differential-equations/assets/9231-2023-on-22-q04-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-22-q04",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "4 Find the solution of the differential equation\ndy\n+3y = sinx\ndx\n= = = y f(x). for which y 1 when x 0. Give your answer in the form [9]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2023-on-22-q04-88bd1d9336.pdf"
    },
    {
      "id": "9231-2023-on-22-q05",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 5,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_22.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_22.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2023-on-22-q05-p01-8dbca9f7c2.png",
        "9231-topic-21-hyperbolic-functions/assets/9231-2023-on-22-q05-p02-f50379238b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-22-q05",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "5\ny\nM\nO x\n2 = y xsech x The diagram shows part of the curve and its maximum point M.\n(a) Show that, at M,\n2xtanhx-1 = 0\nand verify that this equation has a root between 0.7 and 0.8. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) By considering a suitable set of rectangles, use the diagram to show that\nn\n/\n2 1 ntanhn+lnsechn-tanh1-lnsech1. [6] rsech r\nr=2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2023-on-22-q05-17839c6fe3.pdf"
    },
    {
      "id": "9231-2023-on-22-q06",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 6,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_22.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-on-22-q06-p01-e7be4af56c.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-on-22-q06-p02-6868956835.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-22-q06",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "6 The matrix P is given by\n-1 1 1\n= P 0 2 1 .\nf p\n-1 0 0\n(a) State the eigenvalues of P. [1]\n............................................................................................................................................................\n............................................................................................................................................................\n-1\nP . [4] (b) Use the characteristic equation of P to find\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2023-on-22-q06-36be1354be.pdf"
    },
    {
      "id": "9231-2023-on-22-q07",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 7,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_22.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2023-on-22-q07-p01-2d3b84b72a.png",
        "9231-topic-24-integration/assets/9231-2023-on-22-q07-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-22-q07",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "7 (a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that\n2 = cosh2A-1. 2sinh A [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 2 G G = y x , for . The area of the surface generated when the curve is 0 x (b) A curve has equation\n3\nrotated through 2r radians about the x-axis is denoted by S.\n820 1 1 = -ln3 = S to show that . [9] x sinhu Use the substitution r\n2 32 81 b l\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2023-on-22-q07-3fa2f4df69.pdf"
    },
    {
      "id": "9231-2023-on-22-q08",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 8,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_22.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2023-on-22-q08-p01-0bd450709c.png",
        "9231-topic-26-differential-equations/assets/9231-2023-on-22-q08-p02-31eb2980d3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-22-q08",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "4 = v y 8 It is given that and\n2 2 d y dy dy\n-2x 3 2 3 4 +3y +y +y = . y e\n2 edxo dx dx\n(a) Show that\n2 v d dv -2x + +4v = 4e . [4]\n2 dx dx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2023-on-22-q08-dfcbfd82cf.pdf"
    },
    {
      "id": "9231-2023-on-23-q01",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 1,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_23.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-on-23-q01-p01-8a3580147c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-23-q01",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "1 Show that the system of equations\n14x-4y+6z = 5,\nx+ y+ = kz 3,\n-21x+6y-9z = 14,\nwhere k is a constant, does not have a unique solution and interpret this situation geometrically. [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2023-on-23-q01-20bb631c92.pdf"
    },
    {
      "id": "9231-2023-on-23-q02",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 2,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_23.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_23.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2023-on-23-q02-p01-e3e97ae92f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-23-q02",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "3 (z+5i) = 4+4 rcosi+i(rsini-5), 3i, 2 Find the roots of the equation giving your answers in the form\n2 1 1 where r 0 and 0 2r. [5] i\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.............................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2023-on-23-q02-d8a0b74b24.pdf"
    },
    {
      "id": "9231-2023-on-23-q03",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 3,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_23.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_23.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2023-on-23-q03-p01-a3a7cc66c3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-23-q03",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "-1 1 1 2 x lna+bx+cx 3 Find the first three terms in the Maclaurin’s series for in the form , giving tanh e\n2 2 b l\nthe exact values of the constants a, b and c. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.............................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2023-on-23-q03-bc6e3f8ba1.pdf"
    },
    {
      "id": "9231-2023-on-23-q04",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 4,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_23.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2023-on-23-q04-p01-41effd08fc.png",
        "9231-topic-26-differential-equations/assets/9231-2023-on-23-q04-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-23-q04",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "4 Find the particular solution of the differential equation\n2 d y dy\n2 +2 +3y = 27x ,\n2 dx dx\ndy\n= -8. = = given that, when x 0, y 2 and [10]\ndx\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2023-on-23-q04-04bddcefc5.pdf"
    },
    {
      "id": "9231-2023-on-23-q05",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 5,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_23.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2023-on-23-q05-p01-3afba81d45.png",
        "9231-topic-24-integration/assets/9231-2023-on-23-q05-p02-b0557b0830.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-23-q05",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "5 The curve C has parametric equations\n3 1 2 1 G = -2t = 2t+5, x t 2, y for 0 t 3. 2\n3\n(a) Find the exact length of C. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2023-on-23-q05-c22826fcf4.pdf"
    },
    {
      "id": "9231-2023-on-23-q06",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_23.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2023-on-23-q06-p01-eeba00dda6.png",
        "9231-topic-26-differential-equations/assets/9231-2023-on-23-q06-p02-c29e09cf1a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-23-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "6 (a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that\n= sinh2x 2sinhxcoshx. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\ny 2 = u sinhx, find dx. [4] sinh 2xcoshx (b) Using the substitution\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2023-on-23-q06-23041bf230.pdf"
    },
    {
      "id": "9231-2023-on-23-q07",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 7,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_23.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-on-23-q07-p01-c8e6174f9d.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2023-on-23-q07-p02-755e0ce754.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-23-q07",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "7 The matrix A is given by\n-6 2 13\n= -2 A 0 5 .\nf p\n0 0 8\n-1 -1 = A PDP . [7] (a) Find a matrix P and a diagonal matrix D such that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2023-on-23-q07-ec16ce18d4.pdf"
    },
    {
      "id": "9231-2023-on-23-q08",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 8,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_23.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_23.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2023-on-23-q08-p01-c5318af3f5.png",
        "9231-topic-25-complex-numbers/assets/9231-2023-on-23-q08-p02-5603dd4ceb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-23-q08",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "n-1 2 ! 1+z+z +f+z 8 (a) State the sum of the series , for z 1. [1]\n............................................................................................................................................................\n! = cosi+isini, z where cosi 1, show that (b) By letting\n1 sinnisini\n1+cosi+cos2i+f+cos(n-1)i = 1-cosni+ . [7]\n1-cosi 2e o\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2023-on-23-q08-e80473d38c.pdf"
    },
    {
      "id": "9231-2023-on-31-q01",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 1,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_31.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_31.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2023-on-31-q01-p01-c1a9c9ecbf.png",
        "9231-topic-36-momentum/assets/9231-2023-on-31-q01-p02-282ade3079.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-31-q01",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "1\nu\nu\nA B\n60° i\nm 2m\nTwo uniform smooth spheres A and B of equal radii have masses m and 2m respectively. The two spheres\nare moving with equal speeds u on a smooth horizontal surface when they collide. Immediately before\nthe collision, A’s direction of motion makes an angle of 60° with the line of centres, and B’s direction of\nmotion makes an angle with the line of centres (see diagram). The coefficient of restitution between i\nthe spheres is e.\nAfter the collision, the component of the velocity of A along the line of centres is v and B moves\nperpendicular to the line of centres. Sphere A now has twice as much kinetic energy as sphere B.\n1 = u(4cosi-1). (a) Show that [1] v\n2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\ncosi. [4] (b) Find the value of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2023-on-31-q01-b82d6807f7.pdf"
    },
    {
      "id": "9231-2023-on-31-q02",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 2,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_31.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_31.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2023-on-31-q02-p01-b42a952980.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2023-on-31-q02-p02-c4f83c6b45.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-31-q02",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "-1\n5ms 2 A ball of mass 2 kg is projected vertically downwards with speed through a liquid. At time t s\n-1\nafter projection, the velocity of the ball is vms and its displacement from its starting point is x m.\n2 The forces acting on the ball are its weight and a resistive force of magnitude 0.2v N.\n(a) Find an expression for v in terms of t. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2023-on-31-q02-a6d805af86.pdf"
    },
    {
      "id": "9231-2023-on-31-q03",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 3,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_31.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_31.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2023-on-31-q03-p01-ff71ca3003.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2023-on-31-q03-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-31-q03",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "3\nP\ni A\n2a E\nB\nD\nC\nA uniform square lamina of side 2a and weight W is suspended from a light inextensible string attached\nto the midpoint E of the side AB. The other end of the string is attached to a fixed point P on a rough\nvertical wall. The vertex B of the lamina is in contact with the wall. The string EP is perpendicular\nto the side AB and makes an angle with the wall (see diagram). The string and the lamina are in i\na vertical plane perpendicular to the wall. The coefficient of friction between the wall and the\n1 lamina is .\n2\ntani. [8] Given that the vertex B is about to slip up the wall, find the value of\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.....................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2023-on-31-q03-225bd15f71.pdf"
    },
    {
      "id": "9231-2023-on-31-q04",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 4,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_31.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_31.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2023-on-31-q04-p01-cc4d8ec6de.png",
        "9231-topic-34-hooke-s-law/assets/9231-2023-on-31-q04-p02-3e12fd7687.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-31-q04",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "4\nA\nL\n12a\nP\nL\nB\nA light elastic string has natural length 8a and modulus of elasticity 5 mg. A particle P of mass m is\nattached to the midpoint of the string. The ends of the string are attached to points A and B which\nare a distance 12a apart on a smooth horizontal table. The particle P is held on the table so that\n= = AP BP L (see diagram). The particle P is released from rest. When P is at the midpoint of AB it has\nspeed 80ag.\n(a) Find L in terms of a. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the initial acc",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2023-on-31-q04-9e44a8eaed.pdf"
    },
    {
      "id": "9231-2023-on-31-q05",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 5,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_31.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_31.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2023-on-31-q05-p01-5f77be2927.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2023-on-31-q05-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-31-q05",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "-1\nums 5 A particle P is projected with speed at an angle above the horizontal from a point O on a i\nhorizontal plane and moves freely under gravity. During its flight P passes through the point which is\n3\na horizontal distance 3a from O and a vertical distance a above the horizontal plane. It is given that\n8\n1 = . tani\n3\n2 = u 8ag. [2] (a) Show that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1\nVms at an angle above the horizontal from O at the instant A particle Q is projected with speed a\nwhen P is at its highest point. Particles P and Q both land at the same point on the horizontal plane at\nthe same time.\n(b) Find V in terms of a and g. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2023-on-31-q05-9c2a0f5a97.pdf"
    },
    {
      "id": "9231-2023-on-31-q06",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 6,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_31.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2023-on-31-q06-p01-002806d151.png",
        "9231-topic-33-circular-motion/assets/9231-2023-on-31-q06-p02-012d18d0e6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-31-q06",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "6 A particle P of mass m is attached to one end of a light inextensible rod of length 3a. An identical\nparticle Q is attached to the other end of the rod. The rod is smoothly pivoted at a point O on the rod,\n= where OQ x. The system, of rod and particles, rotates about O in a vertical plane.\nAt an instant when the rod is vertical, with P above Q, the particle P is moving horizontally with\nag, 2 speed u. When the rod has turned through an angle of 60° from the vertical, the speed of P is\nand the tensions in the two parts of the rod, OP and OQ, have equal magnitudes.\n(a) Show that the speed of Q when the rod has turned through an angle of 60° from the vertical is\n2x\nag. [2]\n3a-x\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find x in terms of a. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2023-on-31-q06-e83385be53.pdf"
    },
    {
      "id": "9231-2023-on-32-q01",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 1,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_32.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2023-on-32-q01-p01-1b98b06b48.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-32-q01",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "1 One end of a light inextensible string of length a is attached to a fixed point O. The other end of the\nstring is attached to a particle of mass m. The string is taut and makes an angle with the downward i\n2 = vertical through O, where . The particle moves in a horizontal circle with speed v. cosi\n3\nFind v in terms of a and g. [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.........................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2023-on-32-q01-15adb38030.pdf"
    },
    {
      "id": "9231-2023-on-32-q02",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 2,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_32.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_32.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2023-on-32-q02-p01-e9387fbcc5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-32-q02",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "-1\nvms 2 A particle P of mass 0.5 kg moves in a straight line. At time t s the velocity of P is and its\ndisplacement from a fixed point O on the line is x m. The only forces acting on P are a force of\n150\nmagnitude in the direction of increasing displacement and a resistive force of magnitude N\n2 (x+1)\n450\n= = = N. When t 0, x 0 and v 20.\n3 (x+1)\nAx+B\n= Find v in terms of x, giving your answer in the form v , where A and B are constants to be\n(x+1)\ndetermined. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n........................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2023-on-32-q02-f370052d09.pdf"
    },
    {
      "id": "9231-2023-on-32-q03",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 3,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_32.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_32.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2023-on-32-q03-p01-6b575226fc.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2023-on-32-q03-p02-138f981cb9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-32-q03",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "3\nB\nD E\nA C\n2a\n= = A uniform lamina is in the form of an isosceles triangle ABC in which AC 2a and angle ABC 90°.\n= The point D on AB is such that the ratio DB : AB 1: k. The point E on CB is such that DE is parallel to\nAC. The triangle DBE is removed from the lamina (see diagram).\n(a) Find, in terms of k, the distance of the centre of mass of the remaining lamina ADEC from the\nmidpoint of AC. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.......................................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2023-on-32-q03-f26670a85a.pdf"
    },
    {
      "id": "9231-2023-on-32-q04",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 4,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_32.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_32.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2023-on-32-q04-p01-d02db5245a.png",
        "9231-topic-36-momentum/assets/9231-2023-on-32-q04-p02-30eb37cabf.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-32-q04",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "4\nu\nB A\nd i\nm m\nd d\nTwo smooth vertical walls meet at right angles. The smooth sphere A, with mass m, is at rest on a\nsmooth horizontal surface and is at a distance d from each wall. An identical smooth sphere B is moving\non the horizontal surface with speed u at an angle with the line of centres when the spheres collide i\n(see diagram). After the collision, the spheres take the same time to reach a wall. The coefficient of\n1 restitution between the spheres is .\n2\ntani. [4] (a) Find the value of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n................................................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2023-on-32-q04-0d5972cd47.pdf"
    },
    {
      "id": "9231-2023-on-32-q05",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_32.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2023-on-32-q05-p01-8ab596f45c.png",
        "9231-topic-33-circular-motion/assets/9231-2023-on-32-q05-p02-539e9eac7e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-32-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "5\nv\nB\nB\ni a\nO\nα\nA\nv\nA\nA bead of mass m moves on a smooth circular wire, with centre O and radius a, in a vertical plane. The\nbead has speed v when it is at the point A where OA makes an angle with the downward vertical a\nA\n3 = through O, and cosa v . Subsequently the bead has speed at the point B, where OB makes an\n5 B\nangle with the upward vertical through O. Angle AOB is a right angle (see diagram). The reaction of i\n1 the wire on the bead at B is in the direction OB and has magnitude equal to of the magnitude of the\n6\nreaction when the bead is at A.\n(a) Find, in terms of m and g, the magnitude of the reaction at B. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2023-on-32-q05-0a3371b252.pdf"
    },
    {
      "id": "9231-2023-on-32-q06",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 6,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_32.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_32.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2023-on-32-q06-p01-e5a20e6278.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2023-on-32-q06-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-32-q06",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "6 A particle P is projected with speed u at an angle above the horizontal from a point O on a horizontal a\nplane and moves freely under gravity. The horizontal and vertical displacements of P from O at a\nsubsequent time t are denoted by x and y respectively.\n(a) Derive the equation of the trajectory of P in the form\n2 gx\n2 = xtana- y sec a. [3]\n2 2u\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nDuring its flight, P must clear an obstacle of height h m that is at a horizontal distance of 32 m from\n-1 -1 = = u 40 2ms u 40ms , P just clears the obstacle. When , P only the point of projection. When\nachieves 80% of the height required to clear the obstacle.\n(b) Find the two possible values of h. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n................................................................................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2023-on-32-q06-28ecf82b48.pdf"
    },
    {
      "id": "9231-2023-on-32-q07",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 7,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_32.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_32.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2023-on-32-q07-p01-e607228b3c.png",
        "9231-topic-34-hooke-s-law/assets/9231-2023-on-32-q07-p02-a6c87b83c5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-32-q07",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "7\n5a\nB A\n3a 4a\nP\nF\nA particle P of mass m is attached to one end of a light rod of length 3a. The other end of the rod is\nable to pivot smoothly about the fixed point A. The particle is also attached to one end of a light spring\nof natural length a and modulus of elasticity kmg. The other end of the spring is attached to a fixed\npoint B. The points A and B are in a horizontal line, a distance 5a apart, and these two points and the\nrod are in a vertical plane.\nInitially, P is held in equilibrium by a vertical force F with the stretched length of the spring equal to 4a\n6\n(see diagram). The particle is released from rest in this position and has a speed of 2ag when the rod\n5\nbecomes horizontal.\n(a) Find the value of k. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2023-on-32-q07-881a002f7f.pdf"
    },
    {
      "id": "9231-2023-on-33-q01",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 1,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_33.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_33.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2023-on-33-q01-p01-c1a9c9ecbf.png",
        "9231-topic-36-momentum/assets/9231-2023-on-33-q01-p02-282ade3079.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-33-q01",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "1\nu\nu\nA B\n60° i\nm 2m\nTwo uniform smooth spheres A and B of equal radii have masses m and 2m respectively. The two spheres\nare moving with equal speeds u on a smooth horizontal surface when they collide. Immediately before\nthe collision, A’s direction of motion makes an angle of 60° with the line of centres, and B’s direction of\nmotion makes an angle with the line of centres (see diagram). The coefficient of restitution between i\nthe spheres is e.\nAfter the collision, the component of the velocity of A along the line of centres is v and B moves\nperpendicular to the line of centres. Sphere A now has twice as much kinetic energy as sphere B.\n1 = u(4cosi-1). (a) Show that [1] v\n2\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\ncosi. [4] (b) Find the value of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2023-on-33-q01-bc6d45a590.pdf"
    },
    {
      "id": "9231-2023-on-33-q02",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 2,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_33.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_33.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2023-on-33-q02-p01-b42a952980.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2023-on-33-q02-p02-c4f83c6b45.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-33-q02",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "-1\n5ms 2 A ball of mass 2 kg is projected vertically downwards with speed through a liquid. At time t s\n-1\nafter projection, the velocity of the ball is vms and its displacement from its starting point is x m.\n2 The forces acting on the ball are its weight and a resistive force of magnitude 0.2v N.\n(a) Find an expression for v in terms of t. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2023-on-33-q02-7f615cc6a6.pdf"
    },
    {
      "id": "9231-2023-on-33-q03",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 3,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_33.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_33.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2023-on-33-q03-p01-ff71ca3003.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2023-on-33-q03-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-33-q03",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "3\nP\ni A\n2a E\nB\nD\nC\nA uniform square lamina of side 2a and weight W is suspended from a light inextensible string attached\nto the midpoint E of the side AB. The other end of the string is attached to a fixed point P on a rough\nvertical wall. The vertex B of the lamina is in contact with the wall. The string EP is perpendicular\nto the side AB and makes an angle with the wall (see diagram). The string and the lamina are in i\na vertical plane perpendicular to the wall. The coefficient of friction between the wall and the\n1 lamina is .\n2\ntani. [8] Given that the vertex B is about to slip up the wall, find the value of\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n.....................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2023-on-33-q03-4ebd3ba9d4.pdf"
    },
    {
      "id": "9231-2023-on-33-q04",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 4,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_33.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_33.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2023-on-33-q04-p01-cc4d8ec6de.png",
        "9231-topic-34-hooke-s-law/assets/9231-2023-on-33-q04-p02-3e12fd7687.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-33-q04",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "4\nA\nL\n12a\nP\nL\nB\nA light elastic string has natural length 8a and modulus of elasticity 5 mg. A particle P of mass m is\nattached to the midpoint of the string. The ends of the string are attached to points A and B which\nare a distance 12a apart on a smooth horizontal table. The particle P is held on the table so that\n= = AP BP L (see diagram). The particle P is released from rest. When P is at the midpoint of AB it has\nspeed 80ag.\n(a) Find L in terms of a. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the initial acc",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2023-on-33-q04-75d2d1e2d4.pdf"
    },
    {
      "id": "9231-2023-on-33-q05",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 5,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_33.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_33.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2023-on-33-q05-p01-5f77be2927.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2023-on-33-q05-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-33-q05",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "-1\nums 5 A particle P is projected with speed at an angle above the horizontal from a point O on a i\nhorizontal plane and moves freely under gravity. During its flight P passes through the point which is\n3\na horizontal distance 3a from O and a vertical distance a above the horizontal plane. It is given that\n8\n1 = . tani\n3\n2 = u 8ag. [2] (a) Show that\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1\nVms at an angle above the horizontal from O at the instant A particle Q is projected with speed a\nwhen P is at its highest point. Particles P and Q both land at the same point on the horizontal plane at\nthe same time.\n(b) Find V in terms of a and g. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2023-on-33-q05-7317e96be1.pdf"
    },
    {
      "id": "9231-2023-on-33-q06",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 6,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_33.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2023-on-33-q06-p01-002806d151.png",
        "9231-topic-33-circular-motion/assets/9231-2023-on-33-q06-p02-012d18d0e6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-33-q06",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "6 A particle P of mass m is attached to one end of a light inextensible rod of length 3a. An identical\nparticle Q is attached to the other end of the rod. The rod is smoothly pivoted at a point O on the rod,\n= where OQ x. The system, of rod and particles, rotates about O in a vertical plane.\nAt an instant when the rod is vertical, with P above Q, the particle P is moving horizontally with\nag, 2 speed u. When the rod has turned through an angle of 60° from the vertical, the speed of P is\nand the tensions in the two parts of the rod, OP and OQ, have equal magnitudes.\n(a) Show that the speed of Q when the rod has turned through an angle of 60° from the vertical is\n2x\nag. [2]\n3a-x\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find x in terms of a. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n......................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2023-on-33-q06-2e8d7e5f70.pdf"
    },
    {
      "id": "9231-2023-on-41-q01",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_41.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-on-41-q01-p01-b5a18851c0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-41-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 Maya is an athlete who competes in 1500-metre races. Last summer her practice run times had mean\n4.22 minutes. Over the winter she has done some intense training to try to improve her times. A random\nsample of 10 of her practice run times, x minutes, this summer are summarised as follows.\n2 = = /x /x 176.83 42.05\nMaya’s new practice run times are normally distributed. She believes that on average her times have\nimproved as a result of her training.\nTest, at the 5% significance level, whether Maya’s belief is supported by the data. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...................................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2023-on-41-q01-4611062464.pdf"
    },
    {
      "id": "9231-2023-on-41-q02",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_41.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_41.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2023-on-41-q02-p01-e730e86467.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-41-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "2 A town council has published its plans for redeveloping the town centre and residents are being asked\nwhether they approve or disapprove. A random sample of 250 responses has been selected from\nresidents in the four main streets in the town: North, East, South and West Streets. The results are\nshown in the table.\nNorth Street East Street South Street West Street\nApprove 33 54 42 26\nDisapprove 19 39 28 9\nTest, at the 5% significance level, whether the opinions of the residents are independent of the streets\non which they live. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n........................................................................................................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2023-on-41-q02-b2fc420db7.pdf"
    },
    {
      "id": "9231-2023-on-41-q03",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 3,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_41.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-on-41-q03-p01-672e659d9f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-41-q03",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "3 Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month\nperiod, a large group of people exercised for 20 minutes each day. For a randomly chosen sample of 10\nof these people, the LDL blood cholesterol levels were measured at the beginning and the end of the\nthree-month period. The results, measured in suitable units, are as follows.\nPerson A B C D E F G H I J\nBeginning 72 84 120 90 102 135 64 75 80 88\nCholesterol\nlevel\nEnd 64 76 105 92 105 115 67 75 75 84\n(a) Test, at the 2.5% significance level, whether there is evidence that the population mean LDL blood\ncholesterol level has reduced by more than 2 units after the three-month period. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2023-on-41-q03-e54700a56b.pdf"
    },
    {
      "id": "9231-2023-on-41-q04",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 4,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_41.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_41.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2023-on-41-q04-p01-f407e911ce.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2023-on-41-q04-p02-376b36f026.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-41-q04",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "4\nf(x)\nx\n0 2 6\nAs shown in the diagram, the continuous random variable X has probability density function f given by\nG G mx 0 x 2,\nk = f(x) G G +c 6, 2 x\n2 x *\n0 otherwise,\nwhere m, k and c are constants.\n1 1 G = = , show that and find the values of k and c. [4] P(X 2) m (a) Given that\n3 6\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the exact numerical value of th",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2023-on-41-q04-fedab4cab6.pdf"
    },
    {
      "id": "9231-2023-on-41-q05",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 5,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_41.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_41.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2023-on-41-q05-p01-6dd3acf92d.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2023-on-41-q05-p02-4482d47d5f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-41-q05",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "5 The random variable X has the geometric distribution Geo(p).\npt\n= 1-p. , where q [3] (a) Show that the probability generating function of X is\n1-qt\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nq\n= Var(X) . [5] (b) Use the probability generating function of X to show that\n2 p\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2023-on-41-q05-9f3c861670.pdf"
    },
    {
      "id": "9231-2023-on-41-q06",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 6,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_41.pdf?download=true",
      "source_pages": [
        10,
        11,
        12
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_41.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2023-on-41-q06-p01-53316cd20c.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2023-on-41-q06-p02-fa3a9fa97e.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2023-on-41-q06-p03-3dafdf6388.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-41-q06",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "6 A school is conducting an experiment to see whether the distance that children can throw a ball increases\nin hot weather. On a cold day, all the children at the school were asked to throw a ball as far as possible.\nThe distances thrown were measured and recorded. The median distance thrown by a random sample\nof 25 of the children was 22.0 m. The children were asked to throw the ball again on a hot day. The\ndistances thrown by the same 25 children were measured and recorded and these distances, in m, are\nshown below.\n21.2 23.5 22.9 18.6 19.4\n22.1 26.5 20.2 25.7 20.6\n22.3 17.4 22.2 27.0 23.9\n28.2 22.6 27.2 23.0 23.7\n19.8 22.7 23.3 21.5 24.3\nThe teacher claims that on average the distances thrown will be further when it is hot.\nCarry out a Wilcoxon signed-rank test, at the 5% significance level, to test whether the data supports\nthe teacher’s claim. [10]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2023-on-41-q06-dc1c4ba1d9.pdf"
    },
    {
      "id": "9231-2023-on-42-q01",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_42.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-on-42-q01-p01-c616d65f3d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-42-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 A factory produces small bottles of natural spring water. Two different machines, X and Y, are used\nto fill empty bottles with the water. A quality control engineer checks the volumes of water in the\nbottles filled by each of the machines. He chooses a random sample of 60 bottles filled by machine X\nand a random sample of 75 bottles filled by machine Y. The volumes of water, x and y respectively, in\nmillilitres, are summarised as follows.\n2 2 /x = /(x-x) /(y-y) = = / = 6345 243.8 384.9 y 7614\nand are the sample means of the volume of water in the bottles filled by machines X and Y x y\nrespectively.\nFind a 95% confidence interval for the difference between the mean volume of water in bottles filled by\nmachine X and the mean volume of water in bottles filled by machine Y. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2023-on-42-q01-4b09a079d5.pdf"
    },
    {
      "id": "9231-2023-on-42-q02",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_42.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_42.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2023-on-42-q02-p01-8303cf3bf9.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2023-on-42-q02-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-42-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "2 The number of breakdowns on a particular section of road is recorded each day over a period of 90 days.\nIt is suggested that the number of breakdowns follows a Poisson distribution with mean 3.5. The data is\nsummarised in the table, together with some of the expected frequencies resulting from the suggested\nPoisson distribution.\nNumber of 8 or\n0 1 2 3 4 5 6 7\nbreakdowns per day more\nObserved frequency 0 5 13 17 21 16 9 5 4\nExpected frequency 2.718 9.512 16.646 16.993 11.895 3.469 2.407\n(a) Complete the table. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Carry out a goodness of fit test, at the 10% significance level, to determine whether or not Po(3.5)\nis a good fit to the data. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2023-on-42-q02-a7279cdd72.pdf"
    },
    {
      "id": "9231-2023-on-42-q03",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 3,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_42.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_42.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2023-on-42-q03-p01-94e2f63e71.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2023-on-42-q03-p02-181824281e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-42-q03",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "3 Toby has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 3 marbles\nfrom the bag, without replacement. The random variable X is the number of red marbles that Toby\nobtains.\n(a) Find the probability generating function of X. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nLing also has a bag which contains 6 red marbles and 3 green marbles. He randomly chooses 2 marbles\nfrom his bag, without replacement. The random variable Y is the number of red marbles that Ling\n1 2 (1+6t+5t ). obtains. It is given that the probability generating function of Y is\n12\nThe random variable Z is the total number of red marbles that Toby and Ling obtain.\n(b) Find the probability generating function of Z, expressing your answer as a polynomial in t. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2023-on-42-q03-d3bb87443e.pdf"
    },
    {
      "id": "9231-2023-on-42-q04",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 4,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_42.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_42.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2023-on-42-q04-p01-b29da6e599.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2023-on-42-q04-p02-6ad358122a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-42-q04",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "4\nf(x)\nx\n0 4 6\nThe diagram shows the continuous random variable X with probability density function f given by\nZ\n1 3 G G (4ax-bx ] ) 0 x 4,\n128\n]\n]\nG G = f(x) c 4 x 6, [\n]\n]]0 otherwise,\n\\\nwhere a, b and c are constants.\nThe upper quartile of X is equal to 4.\n1 = and find the values of a and b. [4] c (a) Show that\n8\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the exact value of the median of X. [3]\n.........................................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2023-on-42-q04-2c9defe0c2.pdf"
    },
    {
      "id": "9231-2023-on-42-q05",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 5,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_42.pdf?download=true",
      "source_pages": [
        10,
        11,
        12
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-on-42-q05-p01-6641ab17b6.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-on-42-q05-p02-3a1beb96fa.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-on-42-q05-p03-216f9af9b8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-42-q05",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "5 A company is deciding which of two machines, X and Y, can make a certain type of electrical component\nmore quickly. The times taken, in minutes, to make one component of this type are recorded for a\nrandom sample of 8 components made by machine X and a random sample of 9 components made by\nmachine Y. These times are as follows.\nMachine X 4.0 4.6 4.7 4.8 5.0 5.2 5.6 5.8\nMachine Y 4.5 4.9 5.1 5.3 5.4 5.7 5.9 6.3 6.4\nThe manager claims that on average the time taken by machine X to make one component is less than\nthat taken by machine Y.\n(a) Carry out a Wilcoxon rank‑sum test at the 5% significance level to test whether the manager’s\nclaim is supported by the data. [6]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.............................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2023-on-42-q05-8ed6961a69.pdf"
    },
    {
      "id": "9231-2023-on-43-q01",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_43.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-on-43-q01-p01-b5a18851c0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-43-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 Maya is an athlete who competes in 1500-metre races. Last summer her practice run times had mean\n4.22 minutes. Over the winter she has done some intense training to try to improve her times. A random\nsample of 10 of her practice run times, x minutes, this summer are summarised as follows.\n2 = = /x /x 176.83 42.05\nMaya’s new practice run times are normally distributed. She believes that on average her times have\nimproved as a result of her training.\nTest, at the 5% significance level, whether Maya’s belief is supported by the data. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...................................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2023-on-43-q01-d98dfdd1f2.pdf"
    },
    {
      "id": "9231-2023-on-43-q02",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_43.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_43.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2023-on-43-q02-p01-e730e86467.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-43-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "2 A town council has published its plans for redeveloping the town centre and residents are being asked\nwhether they approve or disapprove. A random sample of 250 responses has been selected from\nresidents in the four main streets in the town: North, East, South and West Streets. The results are\nshown in the table.\nNorth Street East Street South Street West Street\nApprove 33 54 42 26\nDisapprove 19 39 28 9\nTest, at the 5% significance level, whether the opinions of the residents are independent of the streets\non which they live. [7]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n........................................................................................................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2023-on-43-q02-06e5a8eff0.pdf"
    },
    {
      "id": "9231-2023-on-43-q03",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 3,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_43.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2023-on-43-q03-p01-672e659d9f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-43-q03",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "3 Scientists are studying the effects of exercise on LDL blood cholesterol levels. Over a three-month\nperiod, a large group of people exercised for 20 minutes each day. For a randomly chosen sample of 10\nof these people, the LDL blood cholesterol levels were measured at the beginning and the end of the\nthree-month period. The results, measured in suitable units, are as follows.\nPerson A B C D E F G H I J\nBeginning 72 84 120 90 102 135 64 75 80 88\nCholesterol\nlevel\nEnd 64 76 105 92 105 115 67 75 75 84\n(a) Test, at the 2.5% significance level, whether there is evidence that the population mean LDL blood\ncholesterol level has reduced by more than 2 units after the three-month period. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2023-on-43-q03-4bd7f5fab6.pdf"
    },
    {
      "id": "9231-2023-on-43-q04",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 4,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_43.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_43.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2023-on-43-q04-p01-f407e911ce.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2023-on-43-q04-p02-376b36f026.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-43-q04",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "4\nf(x)\nx\n0 2 6\nAs shown in the diagram, the continuous random variable X has probability density function f given by\nG G mx 0 x 2,\nk = f(x) G G +c 6, 2 x\n2 x *\n0 otherwise,\nwhere m, k and c are constants.\n1 1 G = = , show that and find the values of k and c. [4] P(X 2) m (a) Given that\n3 6\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the exact numerical value of th",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2023-on-43-q04-216773dbbf.pdf"
    },
    {
      "id": "9231-2023-on-43-q05",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 5,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_43.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_43.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2023-on-43-q05-p01-6dd3acf92d.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2023-on-43-q05-p02-4482d47d5f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-43-q05",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "5 The random variable X has the geometric distribution Geo(p).\npt\n= 1-p. , where q [3] (a) Show that the probability generating function of X is\n1-qt\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\nq\n= Var(X) . [5] (b) Use the probability generating function of X to show that\n2 p\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2023-on-43-q05-4dcff4b4c5.pdf"
    },
    {
      "id": "9231-2023-on-43-q06",
      "subject": "9231",
      "year": 2023,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 6,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2023-oct-nov/9231_w23_qp_43.pdf?download=true",
      "source_pages": [
        10,
        11,
        12
      ],
      "local_pdf": "_source-pdfs/2023-Oct-Nov/qp/9231_w23_qp_43.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2023-on-43-q06-p01-53316cd20c.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2023-on-43-q06-p02-fa3a9fa97e.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2023-on-43-q06-p03-3dafdf6388.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2023-on-43-q06",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "6 A school is conducting an experiment to see whether the distance that children can throw a ball increases\nin hot weather. On a cold day, all the children at the school were asked to throw a ball as far as possible.\nThe distances thrown were measured and recorded. The median distance thrown by a random sample\nof 25 of the children was 22.0 m. The children were asked to throw the ball again on a hot day. The\ndistances thrown by the same 25 children were measured and recorded and these distances, in m, are\nshown below.\n21.2 23.5 22.9 18.6 19.4\n22.1 26.5 20.2 25.7 20.6\n22.3 17.4 22.2 27.0 23.9\n28.2 22.6 27.2 23.0 23.7\n19.8 22.7 23.3 21.5 24.3\nThe teacher claims that on average the distances thrown will be further when it is hot.\nCarry out a Wilcoxon signed-rank test, at the 5% significance level, to test whether the data supports\nthe teacher’s claim. [10]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2023-on-43-q06-334c263de7.pdf"
    },
    {
      "id": "9231-2024-mj-11-q01",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 1,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_11.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_11.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2024-mj-11-q01-p01-b7ac0351a0.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2024-mj-11-q01-p02-4840c66680.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-11-q01",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "3 2 +x -px-5 = a, b, 2x 0, 1 The cubic equation where p is a positive constant, has roots c.\nab+bc+ca. [1] (a) State, in terms of p, the value of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 2 2 bc+ab c+abc . [2] (b) Find the value of a\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2024-mj-11-q01-4db0b6a3d7.pdf"
    },
    {
      "id": "9231-2024-mj-11-q02",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_11.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_11.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2024-mj-11-q02-p01-043c278d72.png",
        "9231-topic-17-proof-by-induction/assets/9231-2024-mj-11-q02-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-11-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "4n n +38 -2 6 2 Prove by mathematical induction that is divisible by 74 for all positive integers n. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2024-mj-11-q02-bd5e94b0cd.pdf"
    },
    {
      "id": "9231-2024-mj-11-q03",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 3,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_11.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_11.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2024-mj-11-q03-p01-ef85e601ae.png",
        "9231-topic-13-summation-of-series/assets/9231-2024-mj-11-q03-p02-9f6558e2c5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-11-q03",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "3 (a) Use standard results from the list of formulae (MF19) to show that\nN\n/\n1 r(r+1)(3r+4) = N(N+1)(N+2)(9N+19). [3]\n12\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2024-mj-11-q03-20f3517548.pdf"
    },
    {
      "id": "9231-2024-mj-11-q04",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_11.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_11.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2024-mj-11-q04-p01-dd8e313d84.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2024-mj-11-q04-p02-2069845c1e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-11-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "1 -1 3 14 0\n2 2 = M 4 The matrix M is given by .\n1 1 1o 0 pe 3 f\n2 2\n(a) The matrix M represents a sequence of two geometrical transformations in the x-y plane.\nGive full details of each transformation, and make clear the order in which they are applied. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1\nM as the product of two matrices, neither of which is I. [2] (b) Write\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2024-mj-11-q04-af11f516a1.pdf"
    },
    {
      "id": "9231-2024-mj-11-q05",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 5,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_11.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_11.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2024-mj-11-q05-p01-3e4f8b63a9.png",
        "9231-topic-16-vectors/assets/9231-2024-mj-11-q05-p02-d55dafe83c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-11-q05",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "5 The points A, B, C have position vectors\n2i+2j+4k, 2i+4j-k, -3i-3j+4k,\nrespectively, relative to the origin O.\nax+by+cz = d. [5] (a) Find the equation of the plane ABC, giving your answer in the form\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2024-mj-11-q05-21946c34e6.pdf"
    },
    {
      "id": "9231-2024-mj-11-q06",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_11.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_11.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2024-mj-11-q06-p01-85587e1f2c.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2024-mj-11-q06-p02-9d856af556.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-11-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 +ax+1 x 5 2 = y 6 The curve C has equation , where a .\nx+2 2\n(a) Find the equations of the asymptotes of C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Show that C has no stationary points. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2024-mj-11-q06-5907a91a05.pdf"
    },
    {
      "id": "9231-2024-mj-11-q07",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 7,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_11.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_11.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2024-mj-11-q07-p01-0f71030bde.png",
        "9231-topic-15-polar-coordinates/assets/9231-2024-mj-11-q07-p02-c97e5663ac.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-11-q07",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "-1 2 G G = (r-i)tan (r-i), r 7 The curve C has polar equation for r. 0 i\n(a) Sketch C and state the polar coordinates of the point of C furthest from the pole. [3]\n............................................................................................................................................................\n= r-i, u or otherwise, find the area of the region enclosed by C and the (b) Using the substitution\ninitial line. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2024-mj-11-q07-80aeaae933.pdf"
    },
    {
      "id": "9231-2024-mj-12-q01",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 1,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_12.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_12.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2024-mj-12-q01-p01-b7ac0351a0.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2024-mj-12-q01-p02-4840c66680.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-12-q01",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "3 2 +x -px-5 = a, b, 2x 0, 1 The cubic equation where p is a positive constant, has roots c.\nab+bc+ca. [1] (a) State, in terms of p, the value of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n2 2 2 bc+ab c+abc . [2] (b) Find the value of a\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.........................................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2024-mj-12-q01-828195603d.pdf"
    },
    {
      "id": "9231-2024-mj-12-q02",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_12.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_12.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2024-mj-12-q02-p01-043c278d72.png",
        "9231-topic-17-proof-by-induction/assets/9231-2024-mj-12-q02-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-12-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "4n n +38 -2 6 2 Prove by mathematical induction that is divisible by 74 for all positive integers n. [6]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2024-mj-12-q02-9f4a42d3ff.pdf"
    },
    {
      "id": "9231-2024-mj-12-q03",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 3,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_12.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_12.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2024-mj-12-q03-p01-ef85e601ae.png",
        "9231-topic-13-summation-of-series/assets/9231-2024-mj-12-q03-p02-9f6558e2c5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-12-q03",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "3 (a) Use standard results from the list of formulae (MF19) to show that\nN\n/\n1 r(r+1)(3r+4) = N(N+1)(N+2)(9N+19). [3]\n12\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2024-mj-12-q03-709c727f93.pdf"
    },
    {
      "id": "9231-2024-mj-12-q04",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_12.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_12.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2024-mj-12-q04-p01-dd8e313d84.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2024-mj-12-q04-p02-2069845c1e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-12-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "1 -1 3 14 0\n2 2 = M 4 The matrix M is given by .\n1 1 1o 0 pe 3 f\n2 2\n(a) The matrix M represents a sequence of two geometrical transformations in the x-y plane.\nGive full details of each transformation, and make clear the order in which they are applied. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n-1\nM as the product of two matrices, neither of which is I. [2] (b) Write\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...........................................................................................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2024-mj-12-q04-8f7ff5df76.pdf"
    },
    {
      "id": "9231-2024-mj-12-q05",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 5,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_12.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_12.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2024-mj-12-q05-p01-3e4f8b63a9.png",
        "9231-topic-16-vectors/assets/9231-2024-mj-12-q05-p02-d55dafe83c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-12-q05",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "5 The points A, B, C have position vectors\n2i+2j+4k, 2i+4j-k, -3i-3j+4k,\nrespectively, relative to the origin O.\nax+by+cz = d. [5] (a) Find the equation of the plane ABC, giving your answer in the form\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2024-mj-12-q05-14c0d28d87.pdf"
    },
    {
      "id": "9231-2024-mj-12-q06",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_12.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_12.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2024-mj-12-q06-p01-85587e1f2c.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2024-mj-12-q06-p02-9d856af556.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-12-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "2 +ax+1 x 5 2 = y 6 The curve C has equation , where a .\nx+2 2\n(a) Find the equations of the asymptotes of C. [3]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Show that C has no stationary points. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2024-mj-12-q06-254078cedc.pdf"
    },
    {
      "id": "9231-2024-mj-12-q07",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 7,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_12.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_12.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2024-mj-12-q07-p01-0f71030bde.png",
        "9231-topic-15-polar-coordinates/assets/9231-2024-mj-12-q07-p02-c97e5663ac.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-12-q07",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "-1 2 G G = (r-i)tan (r-i), r 7 The curve C has polar equation for r. 0 i\n(a) Sketch C and state the polar coordinates of the point of C furthest from the pole. [3]\n............................................................................................................................................................\n= r-i, u or otherwise, find the area of the region enclosed by C and the (b) Using the substitution\ninitial line. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2024-mj-12-q07-576bbec53e.pdf"
    },
    {
      "id": "9231-2024-mj-13-q01",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 1,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_13.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_13.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2024-mj-13-q01-p01-573e09d1c4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-13-q01",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:4)(cid:3)(cid:2)(cid:3)(cid:1)(cid:4)(cid:44)\n1 The matrix A is given by\nSIHT\nk 1 0 NI\nf p ETIRW A = 6 5 2 ,\n-1 -k 3\nTON\nwhere k is a real constant.\nOD\n(a) Show that A is non-singular. [3]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD -1 3 0\nf p A-1 , find the value of k. [2] 0 1 0 = (b) Given that\n23 1 3 -\n2 2\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n...................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2024-mj-13-q01-0dec9bafa4.pdf"
    },
    {
      "id": "9231-2024-mj-13-q02",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 2,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_13.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_13.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2024-mj-13-q02-p01-8bb03b1994.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2024-mj-13-q02-p02-c206d9745d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-13-q02",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:4)(cid:3)(cid:2)(cid:3)(cid:1)(cid:5)(cid:44)\nx3 +2xx2 +3 +1 = 0 2 The cubic equation has roots a, b, c. SIHT\nNI 2 2 2 +1, +1, +1. [3] (a) Find a cubic equation whose roots are a b c\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSI",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2024-mj-13-q02-361a715c1b.pdf"
    },
    {
      "id": "9231-2024-mj-13-q03",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 3,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_13.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_13.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2024-mj-13-q03-p01-6360da3ebe.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2024-mj-13-q03-p02-5e2a54a29f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-13-q03",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n1 2 7 0 SIHT\ne oe o. M = 3 The matrix M is given by\n0 1 0 1\nNI\nETIRW (a) The matrix M represents a sequence of two geometrical transformations in the x-y plane.\nGive full details of each transformation, and make clear the order in which they are applied. [4] TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n(b) Find the equations of the invariant lines, through the origin, of the transformation represented\nby M. [5]\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n...........................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2024-mj-13-q03-ca3d3c7468.pdf"
    },
    {
      "id": "9231-2024-mj-13-q04",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 4,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_13.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_13.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2024-mj-13-q04-p01-3c6f45abee.png",
        "9231-topic-13-summation-of-series/assets/9231-2024-mj-13-q04-p02-ab44380466.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-13-q04",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:4)(cid:3)(cid:2)(cid:3)(cid:1)(cid:9)(cid:44)\n4 (a) Prove by mathematical induction that, for all positive integers n,\nSIHT\n/n\n1n(n+1)(2n+1). r2 NI [5] =\n6 ETIRW\nr=1\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n....................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2024-mj-13-q04-226b0379e4.pdf"
    },
    {
      "id": "9231-2024-mj-13-q05",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 5,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_13.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_13.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2024-mj-13-q05-p01-245f27e4b6.png",
        "9231-topic-16-vectors/assets/9231-2024-mj-13-q05-p02-938e712b73.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-13-q05",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:4)(cid:3)(cid:2)(cid:3)(cid:2)(cid:1)(cid:44)\nl l r =-3i+4j+n(i+2j+k) 5 The lines and have equations r = i+4j-k+m(j-2k) and\nSIHT\n1 2\nrespectively.\nNI\nETIRW\n(a) Find the shortest distance between l l and . [5]\n1 2\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\nP l l contains and is parallel to . The plane\nNIGRAM 1 1 2\nP in the form s. [2] px+qy+rz = (b) Obtain an equation of\n1\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n.........",
      "pdf_path": "9231-topic-16-vectors/print/9231-2024-mj-13-q05-a50df10ee7.pdf"
    },
    {
      "id": "9231-2024-mj-13-q06",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_13.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_13.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2024-mj-13-q06-p01-d7c6d6b2df.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2024-mj-13-q06-p02-b970dcb511.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-13-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:4)(cid:3)(cid:2)(cid:4)(cid:2)(cid:3)(cid:44)\nx+1\nSIHT 6 The curve C has equation y = .\nx2 +3\nNI\n(a) Show that C has no vertical asymptotes and state the equation of the horizontal asymptote. [2]\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n(b) Find the coordinates of any stationary points on C. [4]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n.................................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2024-mj-13-q06-c7b68a4394.pdf"
    },
    {
      "id": "9231-2024-mj-13-q07",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 7,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_13.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_13.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2024-mj-13-q07-p01-6525114b00.png",
        "9231-topic-15-polar-coordinates/assets/9231-2024-mj-13-q07-p02-46dcca6aac.png",
        "9231-topic-15-polar-coordinates/assets/9231-2024-mj-13-q07-p03-d9c4801c6c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-13-q07",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:4)(cid:3)(cid:2)(cid:4)(cid:2)(cid:5)(cid:44)\nr2 = sin2icosi, 7 The curve C has polar equation for r. 0 G G i SIHT\nNI\n(a) Sketch C and state the equation of the line of symmetry. [3]\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\n............................................................................................................................................................\nNIGRAM\n(b) Find a Cartesian equation for C. [3]\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\n(c) Find the total area enclosed by C. [4]\nSIHT\nNI ..............",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2024-mj-13-q07-c317e74e38.pdf"
    },
    {
      "id": "9231-2024-mj-21-q01",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 1,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_21.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_21.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2024-mj-21-q01-p01-6798de5f61.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-21-q01",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "3 = -108 +108i, cosi+isini z 3 1 Find the roots of the equation giving your answers in the form , r\n` j\n2 1 1 where r 0 and 0 2r. [5] i\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...........................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2024-mj-21-q01-34c5aa27fe.pdf"
    },
    {
      "id": "9231-2024-mj-21-q02",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 2,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_21.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_21.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2024-mj-21-q02-p01-e68668b4ab.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-21-q02",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "2 1+x 1-x 2 +e x 2 Find the Maclaurin’s series for e up to and including the term in . [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n........................................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2024-mj-21-q02-f2db25a955.pdf"
    },
    {
      "id": "9231-2024-mj-21-q03",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 3,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_21.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_21.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2024-mj-21-q03-p01-4d091a987e.png",
        "9231-topic-23-differentiation/assets/9231-2024-mj-21-q03-p02-1620b0cb27.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-21-q03",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "3 It is given that\n-1 -1 1 G = = x sin t y tcos t, and for 0 t 1.\ndy\n-1 2 = -t+ 1-t cos t. [3] (a) Show that\ndx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2024-mj-21-q03-3cd89bb920.pdf"
    },
    {
      "id": "9231-2024-mj-21-q04",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 4,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_21.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-mj-21-q04-p01-406ce87d71.png",
        "9231-topic-24-integration/assets/9231-2024-mj-21-q04-p02-ad2fcd25a3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-21-q04",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "ln3\ny n H = 4 It is given that, for 0, I sech xdx. n\nn\n0\nH (a) Show that, for 2, n\nn-2\n3 4\nn-1 = + n-2 I I . [5]\nn-2 5 5 n ` j ` j b l b l\nd\n= -tanhxsech sech x x .] [You may use the result that\ndx` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-mj-21-q04-f20d355928.pdf"
    },
    {
      "id": "9231-2024-mj-21-q05",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 5,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_21.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-mj-21-q05-p01-4a09073847.png",
        "9231-topic-24-integration/assets/9231-2024-mj-21-q05-p02-1eb3eacafd.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-21-q05",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "5\ny\nx n-1 1 1 2 0\nn n n\n2 G G = 2x-x y n The diagram shows the curve with equation for 1, together with a set of 0 x\n1\nrectangles of width .\nn\n1\ny 2 1 2x-x dx U where (a) By considering the sum of the areas of these rectangles, show that ,\nn ` j\n0\n1 2 1\n= 1+ - U . [5]\nn le3 6no n b\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\ny 2 2x-x n, dx. L (b) Use a similar method to find, in terms of a lower bound for [4]\nn ` j\n0\n.............................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-mj-21-q05-a134e6dfaa.pdf"
    },
    {
      "id": "9231-2024-mj-21-q06",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_21.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2024-mj-21-q06-p01-cab402c2c0.png",
        "9231-topic-26-differential-equations/assets/9231-2024-mj-21-q06-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-21-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "1 1 x 2 coshx+sinhx = 6 (a) Show that . [2] e2\n` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the particular solution of the differential equation\n2 1 d y dy\n2, + +3y = coshx+sinhx 5\n2 dx ` j dx\ndy 4\n= = = given that, when x 0, y 1 and . [10]\n3 dx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2024-mj-21-q06-b5026c0590.pdf"
    },
    {
      "id": "9231-2024-mj-21-q07",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_21.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2024-mj-21-q07-p01-4ae142f6f1.png",
        "9231-topic-26-differential-equations/assets/9231-2024-mj-21-q07-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-21-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "2 = 1+x u 7 (a) Use the substitution to find\nx y\ndx. [2]\n2 1+x\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the solution of the differential equation\ndy\n-1 2 -y = x x sinh x,\ndx\n= = = y 1 when x 1. Give your answer in the form . [10] y f x given that\n` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2024-mj-21-q07-480b6e72df.pdf"
    },
    {
      "id": "9231-2024-mj-21-q08",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_21.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-mj-21-q08-p01-d07e27d669.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-mj-21-q08-p02-45c093b4b4.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-mj-21-q08-p03-de5c346739.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-21-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "a 8 (a) Find the set of values of for which the system of equations\n6x+ay = 3,\n2x- = y 1,\nx+5y+4z = 2\nhas a unique solution. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\na. [3] (b) Show that the system of equations in part (a) is consistent for all values of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2024-mj-21-q08-213fd6a4b1.pdf"
    },
    {
      "id": "9231-2024-mj-22-q01",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 1,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_22.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_22.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2024-mj-22-q01-p01-6798de5f61.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-22-q01",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "3 = -108 +108i, cosi+isini z 3 1 Find the roots of the equation giving your answers in the form , r\n` j\n2 1 1 where r 0 and 0 2r. [5] i\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n...........................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2024-mj-22-q01-2cbdb962c6.pdf"
    },
    {
      "id": "9231-2024-mj-22-q02",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 2,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_22.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_22.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2024-mj-22-q02-p01-e68668b4ab.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-22-q02",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "2 1+x 1-x 2 +e x 2 Find the Maclaurin’s series for e up to and including the term in . [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n........................................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2024-mj-22-q02-34ea5b1b4d.pdf"
    },
    {
      "id": "9231-2024-mj-22-q03",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 3,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_22.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_22.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2024-mj-22-q03-p01-4d091a987e.png",
        "9231-topic-23-differentiation/assets/9231-2024-mj-22-q03-p02-1620b0cb27.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-22-q03",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "3 It is given that\n-1 -1 1 G = = x sin t y tcos t, and for 0 t 1.\ndy\n-1 2 = -t+ 1-t cos t. [3] (a) Show that\ndx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..............................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2024-mj-22-q03-996796fb51.pdf"
    },
    {
      "id": "9231-2024-mj-22-q04",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 4,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_22.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-mj-22-q04-p01-6e1ac2002b.png",
        "9231-topic-24-integration/assets/9231-2024-mj-22-q04-p02-ad2fcd25a3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-22-q04",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "ln3\ny n H = 4 It is given that, for 0, I sech xdx. n\nn\n0\nH (a) Show that, for 2, n\nn-2\n3 4\nn-1 = + n-2 I I . [5]\nn-2 5 5 n ` j ` j b l b l\nd\n= -tanhxsech sech x x .] [You may use the result that\ndx` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..................................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-mj-22-q04-59a74641e8.pdf"
    },
    {
      "id": "9231-2024-mj-22-q05",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 5,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_22.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-mj-22-q05-p01-886ad1294e.png",
        "9231-topic-24-integration/assets/9231-2024-mj-22-q05-p02-5919f368f1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-22-q05",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "5\ny\nx n-1 1 1 2 0\nn n n\n2 G G = 2x-x y n The diagram shows the curve with equation for 1, together with a set of 0 x\n1\nrectangles of width .\nn\n1\ny 2 1 2x-x dx U where (a) By considering the sum of the areas of these rectangles, show that ,\nn ` j\n0\n1 2 1\n= 1+ - U . [5]\nn le3 6no n b\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n1\ny 2 2x-x n, dx. L (b) Use a similar method to find, in terms of a lower bound for [4]\nn ` j\n0\n.............................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-mj-22-q05-440512bff1.pdf"
    },
    {
      "id": "9231-2024-mj-22-q06",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_22.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2024-mj-22-q06-p01-cab402c2c0.png",
        "9231-topic-26-differential-equations/assets/9231-2024-mj-22-q06-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-22-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "1 1 x 2 coshx+sinhx = 6 (a) Show that . [2] e2\n` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the particular solution of the differential equation\n2 1 d y dy\n2, + +3y = coshx+sinhx 5\n2 dx ` j dx\ndy 4\n= = = given that, when x 0, y 1 and . [10]\n3 dx\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.....................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2024-mj-22-q06-b7e22eb81f.pdf"
    },
    {
      "id": "9231-2024-mj-22-q07",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_22.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2024-mj-22-q07-p01-98a2969846.png",
        "9231-topic-26-differential-equations/assets/9231-2024-mj-22-q07-p02-a9b884fa0d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-22-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "2 = 1+x u 7 (a) Use the substitution to find\nx y\ndx. [2]\n2 1+x\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Find the solution of the differential equation\ndy\n-1 2 -y = x x sinh x,\ndx\n= = = y 1 when x 1. Give your answer in the form . [10] y f x given that\n` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2024-mj-22-q07-cfdda23337.pdf"
    },
    {
      "id": "9231-2024-mj-22-q08",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_22.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-mj-22-q08-p01-d07e27d669.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-mj-22-q08-p02-45c093b4b4.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-mj-22-q08-p03-de5c346739.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-22-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "a 8 (a) Find the set of values of for which the system of equations\n6x+ay = 3,\n2x- = y 1,\nx+5y+4z = 2\nhas a unique solution. [2]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\na. [3] (b) Show that the system of equations in part (a) is consistent for all values of\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2024-mj-22-q08-64b4de4e6d.pdf"
    },
    {
      "id": "9231-2024-mj-23-q01",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 1,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_23.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-mj-23-q01-p01-c0a578dd57.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-23-q01",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n7 y SIHT 1\n2 dx. 1 Find the exact value of [5]\n4x-x2 -1 2 NI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n...................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-mj-23-q01-be483cfc06.pdf"
    },
    {
      "id": "9231-2024-mj-23-q02",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 2,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_23.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_23.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2024-mj-23-q02-p01-e7e77b1b1a.png",
        "9231-topic-21-hyperbolic-functions/assets/9231-2024-mj-23-q02-p02-2b891d4def.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-23-q02",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:6)(cid:4)(cid:8)(cid:7)(cid:1)(cid:5)(cid:44)\n2 The curve C has parametric equations\nSIHT\n3 NI 0 1 t G x = cosht y = sinht , , for .\n5 ETIRW\nThe length of C is denoted by s.\nTON\n3 y\n5 s = cosh2tdt. [4] (a) Show that\nOD\n0\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ..............................................................................................................................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2024-mj-23-q02-8b3c9ff0b1.pdf"
    },
    {
      "id": "9231-2024-mj-23-q03",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 3,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_23.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_23.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2024-mj-23-q03-p01-41e5d66106.png",
        "9231-topic-23-differentiation/assets/9231-2024-mj-23-q03-p02-858a78c3d3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-23-q03",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:6)(cid:4)(cid:8)(cid:7)(cid:1)(cid:7)(cid:44)\n3 The curve C has equation\nSIHT\nx3 +2xy+8y3 NI = -12.\nETIRW\ndy 1\nC, (-2,-1) = - on . [3] (a) Show that, at the point\n2 dx\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ...............",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2024-mj-23-q03-973596cce5.pdf"
    },
    {
      "id": "9231-2024-mj-23-q04",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 4,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_23.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-mj-23-q04-p01-b6aac73aeb.png",
        "9231-topic-24-integration/assets/9231-2024-mj-23-q04-p02-ff8ed9aa59.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-23-q04",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:6)(cid:4)(cid:8)(cid:7)(cid:1)(cid:9)(cid:44)\n4\nSIHT\ny\nNI\nETIRW\nTON\nOD\nx 0 N-1 1 2 3 4 N\nNIGRAM\nx-2 y = (N-2) The diagram shows the curve with equation for together with a set of 2 G x G N\nrectangles of unit width.\nSIHT\n(a) By considering the sum of the areas of these rectangles, show that\nNI\nETIRW /N\n1 3 1 1\n- + . [5] 2\nN 2 r2 N2\nTON r=1\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n...........",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-mj-23-q04-c2468657f4.pdf"
    },
    {
      "id": "9231-2024-mj-23-q05",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_23.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2024-mj-23-q05-p01-7c30b99f28.png",
        "9231-topic-26-differential-equations/assets/9231-2024-mj-23-q05-p02-fc6fedb8a2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-23-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:6)(cid:4)(cid:8)(cid:7)(cid:2)(cid:1)(cid:44)\n5 (a) Find the general solution of the differential equation\nSIHT\nd2x dx\nNI 338sint. +25x +10 = [7]\ndt dt2 ETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2024-mj-23-q05-c9c68c4bbc.pdf"
    },
    {
      "id": "9231-2024-mj-23-q06",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 6,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_23.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_23.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2024-mj-23-q06-p01-f7066eca62.png",
        "9231-topic-25-complex-numbers/assets/9231-2024-mj-23-q06-p02-fe2103a84e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-23-q06",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n/n\nz4n+2 -z2 SIHT\nz2 z-2. z4r ! 6 (a) Show that , for [2] =\nz2-z-2\nNI r=1\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\nz = cosi+isini, sin2i ! 0, show that, if (b) By letting\n/n TON cos2i-cos(4n+2)i\n. [5] sin(4ri) =\n2sin2i\nOD\nr=1\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ........................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2024-mj-23-q06-8ab6ff6b8f.pdf"
    },
    {
      "id": "9231-2024-mj-23-q07",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_23.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2024-mj-23-q07-p01-60b99393f6.png",
        "9231-topic-26-differential-equations/assets/9231-2024-mj-23-q07-p02-6a5d9eaff0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-23-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:6)(cid:4)(cid:8)(cid:8)(cid:2)(cid:5)(cid:44)\n7 (a) Show that\nSIHT\n9 d x x\nx2 cosh-1 x2 NI e o -9 - = -9. [3]\n2 2 3 dx\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n(b) Find the solution of the differential equation\nNI\ndy ETIRW\nx2 x2 x -y = -9,\ndx\nTON\ny = f(x). y = 1 when x = 3. Give your answer in the form [9] given that\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n...............................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2024-mj-23-q07-ddb71a8b37.pdf"
    },
    {
      "id": "9231-2024-mj-23-q08",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_23.pdf?download=true",
      "source_pages": [
        16,
        17,
        18
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-mj-23-q08-p01-4c4f2b9dab.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-mj-23-q08-p02-efb969d3e4.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-mj-23-q08-p03-9a325049fe.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-23-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:6)(cid:4)(cid:8)(cid:8)(cid:2)(cid:7)(cid:44)\nP P 8 The planes and do not intersect and are both perpendicular to i+2j+3k. The line l intersects\nSIHT\n1 2\nP P (1,6,0) (3, -6,0). at the point and intersects at the point\n1 2\nNI\nETIRW\nP P and . [3] (a) Find Cartesian equations of\n1 2\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\nx\nSIHT\nf p a+mb, where a and b are vectors to be y = (b) Express the vector equation of l in the form\nNI z\nETIRW\n1x+ 1 determined, and hence show that for points on l, and z = 0. [2] y = 1\n2 12\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n..............................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2024-mj-23-q08-a3f87e04f3.pdf"
    },
    {
      "id": "9231-2024-mj-31-q01",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 1,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_31.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_31.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2024-mj-31-q01-p01-3db378c826.png",
        "9231-topic-36-momentum/assets/9231-2024-mj-31-q01-p02-b4d61e0185.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-31-q01",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "1\nu\nA B\ni\nm 5m\nTwo smooth uniform spheres A and B of equal radii have masses m and 5m respectively. Sphere A is\nmoving on a smooth horizontal surface with speed u when it collides with sphere B which is at rest on\nthe surface. Immediately before the collision, A’s direction of motion makes an angle of with the line i\nof centres. After the collision, the kinetic energies of A and B are equal. The coefficient of restitution\n1 between the spheres is .\n2\ntani. [6] Find the value of\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2024-mj-31-q01-a09620a71a.pdf"
    },
    {
      "id": "9231-2024-mj-31-q02",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 2,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_31.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_31.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2024-mj-31-q02-p01-de50dec215.png",
        "9231-topic-34-hooke-s-law/assets/9231-2024-mj-31-q02-p02-14397ccec2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-31-q02",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "2 The points A and B are at the same horizontal level a distance 4a apart. The ends of a light elastic string,\nof natural length 4a and modulus of elasticity m, are attached to A and B. A particle P of mass m is\n3\nattached to the midpoint of the string. The system is in equilibrium with P at a distance a below M,\n2\nthe midpoint of AB.\n(a) Find in terms of m and g. [3] m\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2024-mj-31-q02-8078dacaf8.pdf"
    },
    {
      "id": "9231-2024-mj-31-q03",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 3,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_31.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_31.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-mj-31-q03-p01-8678b5bb4b.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-mj-31-q03-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-31-q03",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "-1 = u ms 3 At time t 0 seconds, a particle P is projected with speed at an angle 60° above the horizontal\nfrom a point O. In the subsequent motion P moves freely under gravity. The direction of motion of P\n= = when t 5 is perpendicular to its direction of motion when t 15.\nFind the value of u. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n............................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2024-mj-31-q03-94d36c9c29.pdf"
    },
    {
      "id": "9231-2024-mj-31-q04",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_31.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_31.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2024-mj-31-q04-p01-8a9dce96a6.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2024-mj-31-q04-p02-585fd5eb71.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-31-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "4\nP Q\na\nO\na\nA ring of weight W, with radius a and centre O, is at rest on a rough surface that is inclined to the\n1 = horizontal at an angle where . The plane of the ring is perpendicular to the inclined surface tana a\n2\nand parallel to a line of greatest slope of the surface. The point P on the circumference of the ring is\nsuch that OP is parallel to the surface.\nA light inextensible string is attached to P and to the point Q, which is on the surface, such that PQ is\nhorizontal (see diagram). The points O, P and Q are in the same vertical plane. The system is in limiting\nequilibrium and the coefficient of friction between the ring and the surface is n.\n(a) Find, in terms of W, the tension in the string PQ. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2024-mj-31-q04-b0c71ebc3c.pdf"
    },
    {
      "id": "9231-2024-mj-31-q05",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_31.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2024-mj-31-q05-p01-ca3a6a7eb7.png",
        "9231-topic-33-circular-motion/assets/9231-2024-mj-31-q05-p02-5eeed0af48.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-31-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "5 Two particles A and B of masses m and km respectively are connected by a light inextensible string of\nlength a. The particles are placed on a rough horizontal circular turntable with the string taut and lying\nalong a radius of the turntable. Particle A is at a distance a from the centre of the turntable and particle\nB is at a distance 2a from the centre of the turntable. The coefficient of friction between each particle\n1 and the turntable is .\n5\ng 2\n, the system is in limiting equilibrium. When the turntable is made to rotate with angular speed\na 5\n(a) Find the tension in the string, in terms of m and g. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2024-mj-31-q05-529c436d0d.pdf"
    },
    {
      "id": "9231-2024-mj-31-q06",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 6,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_31.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_31.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2024-mj-31-q06-p01-760193447c.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2024-mj-31-q06-p02-fb686d72d3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-31-q06",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "6 A particle P of mass 2 kg moving on a horizontal straight line has displacement x m from a fixed point\n-1\nO on the line and velocity v ms at time t s. The only horizontal force acting on P has magnitude\n-t 1 2 = = = (2v-1) and acts towards O. When t 0, x 1 and v 3. e N\n10\n(a) Find an expression for v in terms of t. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2024-mj-31-q06-b26d3ed8ee.pdf"
    },
    {
      "id": "9231-2024-mj-31-q07",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_31.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_31.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-mj-31-q07-p01-f5d0bbf40a.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-mj-31-q07-p02-6b84dfae3b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-31-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "7 A smooth sphere with centre O and of radius a is fixed to a horizontal plane. A particle P of mass m is\nprojected horizontally from the highest point of the sphere with speed u, so that it begins to move along\nthe surface of the sphere. The particle P loses contact with the sphere at the point Q on the sphere,\nwhere OQ makes an angle with the upward vertical through O. i\n2 +2ag u\n= cosi . [4] (a) Show that\n3ag\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2024-mj-31-q07-0261c589f5.pdf"
    },
    {
      "id": "9231-2024-mj-32-q01",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 1,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_32.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_32.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2024-mj-32-q01-p01-3db378c826.png",
        "9231-topic-36-momentum/assets/9231-2024-mj-32-q01-p02-b4d61e0185.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-32-q01",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "1\nu\nA B\ni\nm 5m\nTwo smooth uniform spheres A and B of equal radii have masses m and 5m respectively. Sphere A is\nmoving on a smooth horizontal surface with speed u when it collides with sphere B which is at rest on\nthe surface. Immediately before the collision, A’s direction of motion makes an angle of with the line i\nof centres. After the collision, the kinetic energies of A and B are equal. The coefficient of restitution\n1 between the spheres is .\n2\ntani. [6] Find the value of\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2024-mj-32-q01-5dd57c846d.pdf"
    },
    {
      "id": "9231-2024-mj-32-q02",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 2,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_32.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_32.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2024-mj-32-q02-p01-de50dec215.png",
        "9231-topic-34-hooke-s-law/assets/9231-2024-mj-32-q02-p02-14397ccec2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-32-q02",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "2 The points A and B are at the same horizontal level a distance 4a apart. The ends of a light elastic string,\nof natural length 4a and modulus of elasticity m, are attached to A and B. A particle P of mass m is\n3\nattached to the midpoint of the string. The system is in equilibrium with P at a distance a below M,\n2\nthe midpoint of AB.\n(a) Find in terms of m and g. [3] m\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...................................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2024-mj-32-q02-2ada14a331.pdf"
    },
    {
      "id": "9231-2024-mj-32-q03",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 3,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_32.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_32.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-mj-32-q03-p01-8678b5bb4b.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-mj-32-q03-p02-b0f14555e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-32-q03",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "-1 = u ms 3 At time t 0 seconds, a particle P is projected with speed at an angle 60° above the horizontal\nfrom a point O. In the subsequent motion P moves freely under gravity. The direction of motion of P\n= = when t 5 is perpendicular to its direction of motion when t 15.\nFind the value of u. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n............................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2024-mj-32-q03-48a06827d1.pdf"
    },
    {
      "id": "9231-2024-mj-32-q04",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_32.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_32.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2024-mj-32-q04-p01-8a9dce96a6.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2024-mj-32-q04-p02-585fd5eb71.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-32-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "4\nP Q\na\nO\na\nA ring of weight W, with radius a and centre O, is at rest on a rough surface that is inclined to the\n1 = horizontal at an angle where . The plane of the ring is perpendicular to the inclined surface tana a\n2\nand parallel to a line of greatest slope of the surface. The point P on the circumference of the ring is\nsuch that OP is parallel to the surface.\nA light inextensible string is attached to P and to the point Q, which is on the surface, such that PQ is\nhorizontal (see diagram). The points O, P and Q are in the same vertical plane. The system is in limiting\nequilibrium and the coefficient of friction between the ring and the surface is n.\n(a) Find, in terms of W, the tension in the string PQ. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n.................................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2024-mj-32-q04-191c7a368e.pdf"
    },
    {
      "id": "9231-2024-mj-32-q05",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_32.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2024-mj-32-q05-p01-ca3a6a7eb7.png",
        "9231-topic-33-circular-motion/assets/9231-2024-mj-32-q05-p02-5eeed0af48.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-32-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "5 Two particles A and B of masses m and km respectively are connected by a light inextensible string of\nlength a. The particles are placed on a rough horizontal circular turntable with the string taut and lying\nalong a radius of the turntable. Particle A is at a distance a from the centre of the turntable and particle\nB is at a distance 2a from the centre of the turntable. The coefficient of friction between each particle\n1 and the turntable is .\n5\ng 2\n, the system is in limiting equilibrium. When the turntable is made to rotate with angular speed\na 5\n(a) Find the tension in the string, in terms of m and g. [4]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n..........................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2024-mj-32-q05-d6889188b0.pdf"
    },
    {
      "id": "9231-2024-mj-32-q06",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 6,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_32.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_32.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2024-mj-32-q06-p01-760193447c.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2024-mj-32-q06-p02-fb686d72d3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-32-q06",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "6 A particle P of mass 2 kg moving on a horizontal straight line has displacement x m from a fixed point\n-1\nO on the line and velocity v ms at time t s. The only horizontal force acting on P has magnitude\n-t 1 2 = = = (2v-1) and acts towards O. When t 0, x 1 and v 3. e N\n10\n(a) Find an expression for v in terms of t. [5]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2024-mj-32-q06-d7c95e16d7.pdf"
    },
    {
      "id": "9231-2024-mj-32-q07",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_32.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_32.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-mj-32-q07-p01-f5d0bbf40a.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-mj-32-q07-p02-a545adc828.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-32-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "7 A smooth sphere with centre O and of radius a is fixed to a horizontal plane. A particle P of mass m is\nprojected horizontally from the highest point of the sphere with speed u, so that it begins to move along\nthe surface of the sphere. The particle P loses contact with the sphere at the point Q on the sphere,\nwhere OQ makes an angle with the upward vertical through O. i\n2 +2ag u\n= cosi . [4] (a) Show that\n3ag\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2024-mj-32-q07-6830e5eb7c.pdf"
    },
    {
      "id": "9231-2024-mj-33-q01",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 1,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_33.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_33.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2024-mj-33-q01-p01-784f69d785.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-33-q01",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:5)(cid:8)(cid:10)(cid:1)(cid:1)(cid:3)(cid:44)\n1\nSIHT\n1u\n2\nNI\nA B\nETIRW\nu i\nTON\nOD\nm 2m\nTwo smooth uniform spheres A and B of equal radii have masses m and 2m respectively . The two spheres\nNIGRAM\n1u respectively. Immediately are moving on a smooth horizontal surface when they collidew ith speeds u and\n2\nbefore the collision, A’s direction of motion is along the line of centres, and B’ s direction of motion makes\nwith the line of centres (see diagram). an angle i SIHT\nNI\n1u. The As a result of the collision, the direction of motion of A is reversed and its speed is reduced to\nETIRW 4\nwith the line of centres, but on the opposite side of the line direction of motion of B again makes an angle i\nof centres. The speed of B is unchanged.\nTON\nOD Find the value of the coefficient of restitution between the spheres. [4]\n...........................................................................................................................................................................\nNIGRAM ...........................................................................................................................................................................\n...........................................................................................................................................................................\nSIHT\n........................................................................................................................................................................... NI\nETIRW\n...........................................................................................................................................................................\nTON\n...........................................................................................................................................................................\nOD\n...........................................................................................................................................................................\n...........................................................................................................................................................................\nNIGRAM\n...........................................................................................................................................................................\nSIHT\n...........................................................................................................................................................................\nNI\nETIRW\n...........................................................................................................................................................................\nTON\n.................................................................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2024-mj-33-q01-1d5439f465.pdf"
    },
    {
      "id": "9231-2024-mj-33-q02",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 2,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_33.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2024-mj-33-q02-p01-423e9397bd.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-33-q02",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:5)(cid:8)(cid:10)(cid:1)(cid:1)(cid:4)(cid:44)\n2 A particle P of mass m is attached to one end of a light elastic string of natural length a and modulus of\nSIHT\nelasticity 2mg. A particle Q of mass km is attached to the other end of the string. Particle P lies on a smooth\nNI horizontal table. The string has part of its length in contact with the table and then passes through a small\nETIRW smooth hole H in the table.\n1ga. TON Particle Q Particle P moves in a horizontal circle on the surface of the table with constant speed\n2\n1a. HQ = hangs in equilibrium vertically below the hole with\nOD\n4\n(a) Find, in terms of a, the extension in the string. [4]\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n.............................................................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2024-mj-33-q02-78228431d6.pdf"
    },
    {
      "id": "9231-2024-mj-33-q03",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 3,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_33.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2024-mj-33-q03-p01-ac25a71916.png",
        "9231-topic-33-circular-motion/assets/9231-2024-mj-33-q03-p02-8dd1d2972f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-33-q03",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:5)(cid:8)(cid:10)(cid:1)(cid:1)(cid:5)(cid:44)\n3 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end of\nSIHT\nthe string is attached to a fixed point O. When the particle is hanging vertically below O, it is projected\nNI horizontally with speed u so that it begins to move along a circular path. When P is at the lowest point of its\nETIRW\nwith the upward vertical, the tension in motion, the tension in the string is T. When OP makes an angle i\nthe string is S.\nTON\nT-3mg(1+cosi ). = S [5] (a) Show that\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n.........................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2024-mj-33-q03-321fb41033.pdf"
    },
    {
      "id": "9231-2024-mj-33-q04",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 4,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_33.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_33.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2024-mj-33-q04-p01-d00b9bfd5c.png",
        "9231-topic-34-hooke-s-law/assets/9231-2024-mj-33-q04-p02-46f2bdd712.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-33-q04",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:5)(cid:8)(cid:10)(cid:1)(cid:1)(cid:7)(cid:44)\n4\nSIHT\nNI\nO 2a\nETIRW\nC 3 a\nTON 2 B 3 a\n4 OD A\ni\nA light spring of natural length a and modulus of elasticity kmg is attached to a fixed point O on a smooth\n3 sini = NIGRAM i, where . A particle of mass m is attached to the lower plane inclined to the horizontal at an angle\n4\nOA = 2a and OA is along a line of greatest end of the spring and is held at the point A on the plane, where\nslope of the plane (see diagram).\nSIHT\nThe particle is released from rest and is moving with speed V when it passes through the point B on the NI\n3 ETIRW 1V OB a. = The speed of the particle is when it passes through the point C on the plane, plane, where\n2 2\n3 OC = a. where\n4\nTON\nFind the value of k. [7]\nOD\n...........................................................................................................................................................................\n...........................................................................................................................................................................\nNIGRAM\n...........................................................................................................................................................................\nSIHT\n...........................................................................................................................................................................\nNI\nETIRW\n...........................................................................................................................................................................\nTON\n...........................................................................................................................................................................\nOD\n...........................................................................................................................................................................\n...........................................................................................................................................................................\nNIGRAM\n...........................................................................................................................................................................\nSIHT\n...........................................................................................................................................................................\nNI\nETIRW ...........................................................................................................................................................................\n........................................................................................................................................................................... TON\nOD\n.....................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2024-mj-33-q04-80ca7c98f5.pdf"
    },
    {
      "id": "9231-2024-mj-33-q05",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 5,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_33.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_33.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2024-mj-33-q05-p01-0374b72085.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2024-mj-33-q05-p02-1f0c32f98b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-33-q05",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:5)(cid:8)(cid:10)(cid:1)(cid:1)(cid:9)(cid:44)\n5\nSIHT\nC\nNI\nETIRW\nTON\nOD 18a\nNIGRAM\nO B x A\n24a\nSIHT\nNI OC = 18a, OB = 24a COB = 90°. A uniform lamina is in the form of a triangle OBC, with and angle The\nETIRW\nOA = x (see diagram). The triangle OAC is removed from the lamina. point A on OB is such that\nTON\n(a) Find, in terms of a and x, the distance of the centre of mass of the remaining object ABC from OC. [3]\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT ...................................................................................................................................................\nNI\n................................................................................................................................................... ETIRW\n...................................................................................................................................................\nTON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n...................................................................................................................................................\nSIHT\nNI ...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n.............................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2024-mj-33-q05-87fc09c3ae.pdf"
    },
    {
      "id": "9231-2024-mj-33-q06",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 6,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_33.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_33.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-mj-33-q06-p01-0d2cef611c.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-mj-33-q06-p02-00c368379b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-33-q06",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:5)(cid:8)(cid:10)(cid:2)(cid:2)(cid:1)(cid:44)\ns–1 6 A particle P is projected with speed u m at an angle above the horizontal from a point O and moves i\nSIHT\n3 u. freely under gravity. After 5 seconds the speed of P is\n4\nNI\nETIRW 7 u2 -100u sini+2500 = 0. [3] (a) Show that\n16\nTON ...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT ...................................................................................................................................................\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\nOD ...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM ...................................................................................................................................................\n...................................................................................................................................................\nSIHT\n................................................................................................................................................... NI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n.............................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2024-mj-33-q06-90e4f4490f.pdf"
    },
    {
      "id": "9231-2024-mj-33-q07",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 7,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_33.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_33.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2024-mj-33-q07-p01-643cb67fd7.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2024-mj-33-q07-p02-0da5ed14e4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-33-q07",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:5)(cid:8)(cid:10)(cid:2)(cid:2)(cid:3)(cid:44)\n7 A parachutist of mass m kg opens his parachute when he is moving vertically downwards with a speed of\nSIHT\ns–1. At time t s after opening his parachute, he has fallen a distance xm from the point where he opened 50 m\ns–1. NI The forces acting on him are his weight and a resistive force of his parachute, and his speed is v m\nETIRW\nmagnitude mv N.\nTON (a) Find an expression for v in terms of t. [6]\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n...................................................................................................................................................\nSIHT\nNI .......................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2024-mj-33-q07-697f3cbe08.pdf"
    },
    {
      "id": "9231-2024-mj-41-q01",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_41.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-mj-41-q01-p01-c97415a6d9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-41-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 The times taken by members of a large cycling club to complete a cross-country circuit have a normal\ndistribution with mean minutes. The times taken, x minutes, are recorded for a random sample of 14 n\nmembers of the club. The results are summarised as follows, where is the sample mean. x\n2 /(x-x) = = 941.5 x 42.8\nFind a 95% confidence interval for n. [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2024-mj-41-q01-2cabf83101.pdf"
    },
    {
      "id": "9231-2024-mj-41-q02",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 2,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_41.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_41.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2024-mj-41-q02-p01-6daf300f11.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-41-q02",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "2 A large number of students are taking a Physics course. They are assessed by a practical examination\nand a written examination. The marks out of 100 obtained by a random sample of 15 students in each of\nthe examinations are as follows.\nStudent A B C D E F G H I J K L M N O\nPractical\n66 63 24 52 59 76 88 51 48 36 91 72 68 67 60\nexamination\nWritten\n63 57 39 50 47 71 87 65 56 39 78 70 61 62 70\nexamination\nUse a sign test, at the 10% significance level, to test whether, on average, the practical examination\nmarks are higher than the written examination marks. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2024-mj-41-q02-7752c4d9e3.pdf"
    },
    {
      "id": "9231-2024-mj-41-q03",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 3,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_41.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_41.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2024-mj-41-q03-p01-d35dd183e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-41-q03",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "3 A factory produces metal discs. The manager claims that the diameters of these discs have a median\nof 22.0 mm. The diameters, in mm, of a random sample of 12 discs produced by this factory are as\nfollows.\n22.4 20.9 22.8 21.5 23.2 22.9 23.9 21.7 19.8 23.6 22.6 23.0\n(a) Carry out a Wilcoxon signed-rank test, at the 10% significance level, to test whether there is any\nevidence against the manager’s claim. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2024-mj-41-q03-38fb3ad4c7.pdf"
    },
    {
      "id": "9231-2024-mj-41-q04",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 4,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_41.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_41.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2024-mj-41-q04-p01-4987ad08b6.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2024-mj-41-q04-p02-668abdd72c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-41-q04",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "4 The random variable Y is the sum of two independent observations of the random variable X. The\nprobability generating function G (t) of Y is given by\nY\n2 t\n= G (t) .\n4 Y (4-3t)\nE(Y). [3] (a) Find\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Write down an expression for the probability generating function of X. [1]\n............................................................................................................................................................\n......................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2024-mj-41-q04-ac1bd5aed8.pdf"
    },
    {
      "id": "9231-2024-mj-41-q05",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 5,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_41.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_41.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2024-mj-41-q05-p01-f9e81ebc37.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2024-mj-41-q05-p02-b38ad10381.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-41-q05",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "5 Two companies, P and Q, produce a certain type of paint brush. An independent examiner rates the\nquality of the brushes produced as poor, satisfactory or good. He takes a random sample of brushes\nfrom each company. The examiner’s ratings are summarised in the table.\nCompany Poor Satisfactory Good\nP 18 43 64\nQ 22 22 31\n(a) Test, at the 5% significance level, whether quality of brushes is independent of company. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2024-mj-41-q05-cf6ca40919.pdf"
    },
    {
      "id": "9231-2024-mj-41-q06",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 6,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_41.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-mj-41-q06-p01-a63e318f6d.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-mj-41-q06-p02-fdd6c07422.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-41-q06",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "6 Jade is a swimming instructor at a sports college. She claims that, as a result of an intensive training\ncourse, the mean time taken by students to swim 50 metres has reduced by more than 1 second. She\nchooses a random sample of 10 students. The times taken, in seconds, before and after the training\ncourse are recorded in the table.\nStudent A B C D E F G H I J\nTime before\n54.2 47.4 52.1 59.0 55.3 51.0 48.9 52.2 58.4 51.4\ncourse\nTime after\n50.1 46.3 52.5 58.8 51.4 48.4 49.5 48.7 58.3 51.4\ncourse\n(a) Test, at the 10% significance level, whether Jade’s claim is justified. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2024-mj-41-q06-c7065f950d.pdf"
    },
    {
      "id": "9231-2024-mj-41-q07",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 7,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_41.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_41.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2024-mj-41-q07-p01-2fbec9e485.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2024-mj-41-q07-p02-adce2ca883.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-41-q07",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "7 The continuous random variable X has probability density function f given by\nx\n2 (4-x ) G G 0 x 2,\n4\n= f(x) *\n0 otherwise.\n(a) Find . [4] Var X\n` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2024-mj-41-q07-a7578ae446.pdf"
    },
    {
      "id": "9231-2024-mj-42-q01",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_42.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-mj-42-q01-p01-c97415a6d9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-42-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "1 The times taken by members of a large cycling club to complete a cross-country circuit have a normal\ndistribution with mean minutes. The times taken, x minutes, are recorded for a random sample of 14 n\nmembers of the club. The results are summarised as follows, where is the sample mean. x\n2 /(x-x) = = 941.5 x 42.8\nFind a 95% confidence interval for n. [4]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2024-mj-42-q01-19aafe203a.pdf"
    },
    {
      "id": "9231-2024-mj-42-q02",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 2,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_42.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_42.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2024-mj-42-q02-p01-6daf300f11.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-42-q02",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "2 A large number of students are taking a Physics course. They are assessed by a practical examination\nand a written examination. The marks out of 100 obtained by a random sample of 15 students in each of\nthe examinations are as follows.\nStudent A B C D E F G H I J K L M N O\nPractical\n66 63 24 52 59 76 88 51 48 36 91 72 68 67 60\nexamination\nWritten\n63 57 39 50 47 71 87 65 56 39 78 70 61 62 70\nexamination\nUse a sign test, at the 10% significance level, to test whether, on average, the practical examination\nmarks are higher than the written examination marks. [5]\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n....................................................................................................................................................................\n..........................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2024-mj-42-q02-ec098a4e32.pdf"
    },
    {
      "id": "9231-2024-mj-42-q03",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 3,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_42.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_42.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2024-mj-42-q03-p01-d35dd183e2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-42-q03",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "3 A factory produces metal discs. The manager claims that the diameters of these discs have a median\nof 22.0 mm. The diameters, in mm, of a random sample of 12 discs produced by this factory are as\nfollows.\n22.4 20.9 22.8 21.5 23.2 22.9 23.9 21.7 19.8 23.6 22.6 23.0\n(a) Carry out a Wilcoxon signed-rank test, at the 10% significance level, to test whether there is any\nevidence against the manager’s claim. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2024-mj-42-q03-160f000152.pdf"
    },
    {
      "id": "9231-2024-mj-42-q04",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 4,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_42.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_42.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2024-mj-42-q04-p01-4987ad08b6.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2024-mj-42-q04-p02-668abdd72c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-42-q04",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "4 The random variable Y is the sum of two independent observations of the random variable X. The\nprobability generating function G (t) of Y is given by\nY\n2 t\n= G (t) .\n4 Y (4-3t)\nE(Y). [3] (a) Find\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n(b) Write down an expression for the probability generating function of X. [1]\n............................................................................................................................................................\n......................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2024-mj-42-q04-7f81026b20.pdf"
    },
    {
      "id": "9231-2024-mj-42-q05",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 5,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_42.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_42.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2024-mj-42-q05-p01-f9e81ebc37.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2024-mj-42-q05-p02-b38ad10381.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-42-q05",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "5 Two companies, P and Q, produce a certain type of paint brush. An independent examiner rates the\nquality of the brushes produced as poor, satisfactory or good. He takes a random sample of brushes\nfrom each company. The examiner’s ratings are summarised in the table.\nCompany Poor Satisfactory Good\nP 18 43 64\nQ 22 22 31\n(a) Test, at the 5% significance level, whether quality of brushes is independent of company. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n....................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2024-mj-42-q05-8241267689.pdf"
    },
    {
      "id": "9231-2024-mj-42-q06",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 6,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_42.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-mj-42-q06-p01-a63e318f6d.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-mj-42-q06-p02-fdd6c07422.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-42-q06",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "6 Jade is a swimming instructor at a sports college. She claims that, as a result of an intensive training\ncourse, the mean time taken by students to swim 50 metres has reduced by more than 1 second. She\nchooses a random sample of 10 students. The times taken, in seconds, before and after the training\ncourse are recorded in the table.\nStudent A B C D E F G H I J\nTime before\n54.2 47.4 52.1 59.0 55.3 51.0 48.9 52.2 58.4 51.4\ncourse\nTime after\n50.1 46.3 52.5 58.8 51.4 48.4 49.5 48.7 58.3 51.4\ncourse\n(a) Test, at the 10% significance level, whether Jade’s claim is justified. [7]\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n...............................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2024-mj-42-q06-123b4b75c8.pdf"
    },
    {
      "id": "9231-2024-mj-42-q07",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 7,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_42.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_42.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2024-mj-42-q07-p01-2fbec9e485.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2024-mj-42-q07-p02-adce2ca883.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-42-q07",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "7 The continuous random variable X has probability density function f given by\nx\n2 (4-x ) G G 0 x 2,\n4\n= f(x) *\n0 otherwise.\n(a) Find . [4] Var X\n` j\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................\n........................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2024-mj-42-q07-7efe3a2dcc.pdf"
    },
    {
      "id": "9231-2024-mj-43-q01",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 1,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_43.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_43.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2024-mj-43-q01-p01-c742e1ac0a.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2024-mj-43-q01-p02-997a800533.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-43-q01",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:5)(cid:9)(cid:2)(cid:2)(cid:1)(cid:3)(cid:44)\n1 A college uses two assessments, X and Y, when interviewing applicants for research posts at the college.\nSIHT\nThese assessments have been used for a large number of applicants this year.\nNI\nETIRW The scores for a random sample of 9 applicants who took assessment X are as follows.\nTON 21.4 24.6 25.3 22.7 20.8 21.5 22.9 21.3 22.3\nOD\nThe scores for a random sample of 10 applicants who took assessment Y are as follows.\n20.9 23.5 24.8 21.9 23.4 24.0 23.8 24.1 25.1 25.8\nNIGRAM\nThe interviewer believes that the population median score from assessment X is lower than the\npopulation median score from assessment Y.\nSIHT\nCarry out a Wilcoxon rank-sum test, at the 1% significance level, to test whether the interviewer’s\nNI\nbelief is supported by the data. [7]\nETIRW\n..........................................................................................................................................................\nTON\nOD ..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM ..........................................................................................................................................................\n..........................................................................................................................................................\nSIHT\n.......................................................................................................................................................... NI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW\n..........................................................................................................................................................\nTON\n...........................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2024-mj-43-q01-4d5a74adb5.pdf"
    },
    {
      "id": "9231-2024-mj-43-q02",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 2,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_43.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-mj-43-q02-p01-bd21691952.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-43-q02",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:5)(cid:9)(cid:2)(cid:2)(cid:1)(cid:5)(cid:44)\n2 A rowing club has a lar ge number of members. A random sample of 12 of these members is taken\nSIHT\nand the pulse rate, x beats per minute (bpm), of each is measured after a 30-minute training session. A\nNI 98% confidence interval for the population mean pulse rate, bpm, is calculated from the sample as n\nETIRW 64.22 1 1 68.66. n\nTON /x2. /x and [6] (a) Find the values of\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT ...................................................................................................................................................\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\nOD ...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM ...................................................................................................................................................\n...................................................................................................................................................\nSIHT\n................................................................................................................................................... NI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n(b) State an assumption that is necessary for the confidence interval to be valid. [1]\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n..................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2024-mj-43-q02-65cd68e69a.pdf"
    },
    {
      "id": "9231-2024-mj-43-q03",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 3,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_43.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_43.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2024-mj-43-q03-p01-27eccab20f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-43-q03",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:5)(cid:9)(cid:2)(cid:2)(cid:1)(cid:6)(cid:44)\n3 There are three bus companies in a city. The council is investigating whether the buses reliably arrive at\nSIHT\ntheir destination on time. The results from random samples of buses from each company are summarised\nNI in the following table.\nETIRW\nBus company\nTON\nA B C Total\nOD\nEarly 22 22 10 54\nArrival On time 30 52 42 124\nNIGRAM\nLate 28 26 18 72\nTotal 80 100 70 250\nSIHT\nTest, at the 5% significance level, whether the reliability of buses is independent of bus company. [7] NI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW ..........................................................................................................................................................\n.......................................................................................................................................................... TON\nOD\n..........................................................................................................................................................\n........",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2024-mj-43-q03-2c7d88fd19.pdf"
    },
    {
      "id": "9231-2024-mj-43-q04",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 4,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_43.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_43.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2024-mj-43-q04-p01-f2abc7a1d7.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2024-mj-43-q04-p02-d8b6727f32.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-43-q04",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:5)(cid:9)(cid:2)(cid:2)(cid:1)(cid:7)(cid:44)\nG (t) 4 The random variable X has probability generating function given by\nSIHT\nX\nct(1+t)5, NI G (t) =\nX ETIRW\nwhere c is a constant.\nTON\n(a) Find the value of c. [1]\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\nE(X). [2] (b) Find the value of\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\n.....................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2024-mj-43-q04-e7ac27f9da.pdf"
    },
    {
      "id": "9231-2024-mj-43-q05",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 5,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_43.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_43.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2024-mj-43-q05-p01-b052b916fe.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2024-mj-43-q05-p02-513ce24cc0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-43-q05",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:5)(cid:9)(cid:2)(cid:3)(cid:1)(cid:9)(cid:44)\n5 The continuous random variable X has cumulative distribution function F given by\nSIHT\nZ 2, x 1 0 NI\n] ETIRW\n(x-2)2 ]\n] 2 4, G 1 x\n12\nTON F(x) = [\n(8-x)2\n]\n1- 4 8, G x G OD\n] 24\n]\n8. 1 x 2\n\\\n(a) Sketch the graph of the probability density function of X. [3]\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\n(b) Find E(X). [3]\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI ...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\n................................................................................................................................................... OD\n...................................................................................................................................................\n................................................................................................................................................... NIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\nNIGRAM\n(c) Find the exact value of the interquartile range of X. [4]\nSIHT\nNI ...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...........................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2024-mj-43-q05-64e07a5295.pdf"
    },
    {
      "id": "9231-2024-mj-43-q06",
      "subject": "9231",
      "year": 2024,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 6,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-may-june/9231_s24_qp_43.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-May-June/qp/9231_s24_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-mj-43-q06-p01-eb73dff76e.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-mj-43-q06-p02-e8ad4bc7d6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-mj-43-q06",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:2)(cid:10)(cid:7)(cid:6)(cid:6)(cid:5)(cid:9)(cid:2)(cid:3)(cid:2)(cid:1)(cid:44)\n6 Seva is investigating the lengths of the tails of adult wallabies in two regions of Australia, X and Y. He\nSIHT\nchooses a random sample of 50 adult wallabies from region X and records the lengths, x cm, of their\nNI tails. He also chooses a random sample of 40 adult wallabies from region Y and records the lengths,\nETIRW y cm, of their tails. His results are summarised as follows.\n/x2 /y2 TON /x /y = 23480 = 22220 = 1080 = 940\nOD\nIt cannot be assumed that the population variances of the two distributions are the same.\n(a) Find a 90% confidence interval for the difference between the population mean lengths of the tails\nof adult wallabies in regions X and Y. [6]\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI .......................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2024-mj-43-q06-298c84942f.pdf"
    },
    {
      "id": "9231-2024-on-11-q01",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 1,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_11.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_11.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2024-on-11-q01-p01-2b4c04fb9d.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2024-on-11-q01-p02-48ed56c9d7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-11-q01",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 The matrix M represents the sequence of two transformations in the x-y plane given by a stretch parallel\nSIHT\nto the x-axis, scale factor k (k ! 0), followed by a shear, x-axis fixed, with (0, 1) mapped to (k, 1).\nNI\nk k\nETIRW e o. M = [4] (a) Show that\n0 1\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n(b) The transformation represented by M has a line of invariant points.\nSIHT\nFind, in terms of k, the equation of this line. [3]\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n.......................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2024-on-11-q01-18bf2978d3.pdf"
    },
    {
      "id": "9231-2024-on-11-q02",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_11.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_11.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2024-on-11-q02-p01-06b299f062.png",
        "9231-topic-17-proof-by-induction/assets/9231-2024-on-11-q02-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-11-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 Prove by mathematical induction that, for all positive integers n,\nSIHT\ndn -n\n`tan-1xj (x)`1+x2j = P , NI\ndxn n ETIRW\nP (x) is a polynomial of degree n-1. [6] where\nn\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n......",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2024-on-11-q02-b13900dcfc.pdf"
    },
    {
      "id": "9231-2024-on-11-q03",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 3,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_11.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_11.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2024-on-11-q03-p01-e81e7720e0.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2024-on-11-q03-p02-3e3792a2b3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-11-q03",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\nx4 +2x3 a, b, c, -1 = 0 3 The quartic equation has roots d. SIHT\nNI 4, 4, 4, 4 4 4 4 4. +b +c +d and state the value of (a) Find a quartic equation whose roots are a b c d a\nETIRW\n[5]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n.......................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2024-on-11-q03-01f06d2672.pdf"
    },
    {
      "id": "9231-2024-on-11-q04",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 4,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_11.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_11.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2024-on-11-q04-p01-64a76df20b.png",
        "9231-topic-13-summation-of-series/assets/9231-2024-on-11-q04-p02-fcb7ecbc02.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-11-q04",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n/n\n5k SIHT\n4 (a) Use the method of dif ferences to find in terms of n and k, where k is a\n(5r+k)(5r+5+k)\nNI r=1\npositive constant. [4]\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2024-on-11-q04-f502a1c8a8.pdf"
    },
    {
      "id": "9231-2024-on-11-q05",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_11.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_11.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2024-on-11-q05-p01-e42ab27bc6.png",
        "9231-topic-15-polar-coordinates/assets/9231-2024-on-11-q05-p02-abaaca4118.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-11-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 (a) Show that the curve with Cartesian equation\nSIHT\n2\n`x2 +y2j = 6xy NI\nETIRW\nr2 = 3sin2i. has polar equation [2]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n1 r2 = 3sin2i, for r. 0 G G The curve C has polar equation i\n2\nOD\n(b) Sketch C and state the maximum distance of a point on C from the pole. [3]\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\n(c) Find the area of the region enclosed by C. [2]\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n(d) Find the maximum distance of a point on C from the initial line. [6]\nTON\n.......................................................................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2024-on-11-q05-b06b2a4f1a.pdf"
    },
    {
      "id": "9231-2024-on-11-q06",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_11.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_11.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2024-on-11-q06-p01-af0ae3df9a.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2024-on-11-q06-p02-05e5b43026.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-11-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\n4x2 +x+1 SIHT\ny = 6 The curve C has equation .\n2x2 -7x+3\nNI\n(a) Find the equations of the asymptotes of C. [2] ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n(b) Find the coordinates of any stationary points on C. [4]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ...............................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2024-on-11-q06-1671031c24.pdf"
    },
    {
      "id": "9231-2024-on-11-q07",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 7,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_11.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_11.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2024-on-11-q07-p01-2e553e28d8.png",
        "9231-topic-16-vectors/assets/9231-2024-on-11-q07-p02-32d32aefd7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-11-q07",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\nl l r = i-2j+9k+n(i-4jk+2 ) 7 The lines and have equations r = i+3j-2k+m(2i+jk+ ) and\nSIHT\n1 2\nrespectively. The plane P l l contains and is parallel to .\n1 1 2\nNI\nETIRW\nP ax+by+cz = d. , giving your answer in the form [4] (a) Find the equation of\n1\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nP l (2, -1, 7). contains and the point with coordinates The plane NI\n2 2\nETIRW\nP P and . [4] (b) Find the acute angle between\n1 2\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n....................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2024-on-11-q07-9575033c4f.pdf"
    },
    {
      "id": "9231-2024-on-12-q01",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 1,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_12.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_12.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2024-on-12-q01-p01-4d9e86b1e9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-12-q01",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\nu u u 1 The sequence , , , … is such that u = 4 and u = 3u -2 for 1. n H\nSIHT 1 2 3 1 n+1 n\n3n NI for all positive integers n. [5] u = +1 Prove by induction that\nn ETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW ....................................................................................................................................................................\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n.....",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2024-on-12-q01-11252c20fd.pdf"
    },
    {
      "id": "9231-2024-on-12-q02",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 2,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_12.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_12.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2024-on-12-q02-p01-43c87fb7d4.png",
        "9231-topic-16-vectors/assets/9231-2024-on-12-q02-p02-d299d8573c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-12-q02",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\nl 2 The line has equation r = i+3j-k+m(i-j-4k).\nSIHT\n1\nNI l contains and is parallel to the vector 2i+5jk-4 . The plane P\n1 ETIRW\nax+by+cz = d. P, giving your answer in the form [4] (a) Find the equation of\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2024-on-12-q02-2997d296f3.pdf"
    },
    {
      "id": "9231-2024-on-12-q03",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 3,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_12.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_12.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2024-on-12-q03-p01-95d972f305.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2024-on-12-q03-p02-e78a400fd0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-12-q03",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 It is given that\nSIHT\n+ + + = 2, a b c d\nNI\nETIRW 2 2 2 2 +b +c +d = 3, a\n3 3 3 3 +b +c +d = 4. a\nTON\nOD\nab+ac+ad+bc+bd+cd. (a) Find the value of [2]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n2 2 2 2 2 2 2 2 2 2 2 2 b+a c+a d+b a+b c+b d+c a+c b+c d+d a+d b+d c. [3] (b) Find the value of a\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON .........................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2024-on-12-q03-78cfe4e7cd.pdf"
    },
    {
      "id": "9231-2024-on-12-q04",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_12.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_12.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2024-on-12-q04-p01-7072aba58a.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2024-on-12-q04-p02-27d218a51a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-12-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 The matrices A, B and C are given by\nSIHT\n-2 1 2 3 0 NI\n-2 -1 1\nf p f p ETIRW e o. = C = -1 A = 2 1 3 B 3 , and\n1 1 3\n3 2 5 0 0\nTON\n-7 3 OD\ne o. CAB = [3] (a) Show that\n-9 3\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n(b) Find the equations of the invariant lines, through the origin, of the transformation in the x-y plane\nrepresented by CAB. [5]\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ...............................................................................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2024-on-12-q04-cc648342c6.pdf"
    },
    {
      "id": "9231-2024-on-12-q05",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 5,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_12.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_12.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2024-on-12-q05-p01-9026de497c.png",
        "9231-topic-13-summation-of-series/assets/9231-2024-on-12-q05-p02-856c811a6e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-12-q05",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\nn\nxf(r) -xf(r+1) / SIHT 5 It is given that , where u = and x 2 0. S = u\nr n r\nr=1\nNI\nS in terms of n, x and the function f. [2] (a) Find\nETIRW\nn\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\nf(r) = lnr, find the set of values of x for which the infinite series (b) Given that\nNIGRAM\nu +u +u +…\n1 2 3\nSIHT\nis conver gent and give the sum to infinity when this exists. [3]\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n...........................................................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2024-on-12-q05-6d34f222db.pdf"
    },
    {
      "id": "9231-2024-on-12-q06",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_12.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_12.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2024-on-12-q06-p01-a839ae34a7.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2024-on-12-q06-p02-4d433beaeb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-12-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\nx2 +3 SIHT\ny = 6 The curve C has equation .\nx2 +1\nNI\n(a) Show that C has no vertical asymptotes and state the equation of the horizontal asymptote. [2] ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n1 1 y G 3 for all real values of x. [4] (b) Show that\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n(c) Find the coordinates of any stationary points on C. [2]\nSIHT\n............................................................................................................................................................\nNI\nETIRW .............................................................................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2024-on-12-q06-8d726784b2.pdf"
    },
    {
      "id": "9231-2024-on-12-q07",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 7,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_12.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_12.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2024-on-12-q07-p01-7546df0b8f.png",
        "9231-topic-15-polar-coordinates/assets/9231-2024-on-12-q07-p02-d1c8094ac1.png",
        "9231-topic-15-polar-coordinates/assets/9231-2024-on-12-q07-p03-c070135150.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-12-q07",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n3 -1 G r = a(cosi+sini) 7 The curve C has polar equation for r, where a is a positive constant. r i G SIHT\n4 4 1\nNI\nC and show that it represents a circle, stating its radius and the (a) Find a Cartesian equation for\nETIRW 1\nCartesian coordinates of its centre. [4]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI C and state the greatest distance of a point on C from the pole. [3] (b) Sketch\n1 1 ETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\nThe curve C with polar equation r = ai intersects C at the pole and the point with polar coordinates\nSIHT 2 1\n(az, z).\nNI\nETIRW\n1.25 1 1.26. [2] (c) Verify that z 1\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n..........................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2024-on-12-q07-f35b0382f6.pdf"
    },
    {
      "id": "9231-2024-on-13-q01",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 1,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_13.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_13.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2024-on-13-q01-p01-2b4c04fb9d.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2024-on-13-q01-p02-48ed56c9d7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-13-q01",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 The matrix M represents the sequence of two transformations in the x-y plane given by a stretch parallel\nSIHT\nto the x-axis, scale factor k (k ! 0), followed by a shear, x-axis fixed, with (0, 1) mapped to (k, 1).\nNI\nk k\nETIRW e o. M = [4] (a) Show that\n0 1\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n(b) The transformation represented by M has a line of invariant points.\nSIHT\nFind, in terms of k, the equation of this line. [3]\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n.......................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2024-on-13-q01-8b5cd81967.pdf"
    },
    {
      "id": "9231-2024-on-13-q02",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_13.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_13.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2024-on-13-q02-p01-06b299f062.png",
        "9231-topic-17-proof-by-induction/assets/9231-2024-on-13-q02-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-13-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 Prove by mathematical induction that, for all positive integers n,\nSIHT\ndn -n\n`tan-1xj (x)`1+x2j = P , NI\ndxn n ETIRW\nP (x) is a polynomial of degree n-1. [6] where\nn\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n......",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2024-on-13-q02-3d0fde024a.pdf"
    },
    {
      "id": "9231-2024-on-13-q03",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 3,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_13.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_13.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2024-on-13-q03-p01-e81e7720e0.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2024-on-13-q03-p02-3e3792a2b3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-13-q03",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\nx4 +2x3 a, b, c, -1 = 0 3 The quartic equation has roots d. SIHT\nNI 4, 4, 4, 4 4 4 4 4. +b +c +d and state the value of (a) Find a quartic equation whose roots are a b c d a\nETIRW\n[5]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n.......................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2024-on-13-q03-277b1bea79.pdf"
    },
    {
      "id": "9231-2024-on-13-q04",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 4,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_13.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_13.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2024-on-13-q04-p01-64a76df20b.png",
        "9231-topic-13-summation-of-series/assets/9231-2024-on-13-q04-p02-fcb7ecbc02.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-13-q04",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n/n\n5k SIHT\n4 (a) Use the method of dif ferences to find in terms of n and k, where k is a\n(5r+k)(5r+5+k)\nNI r=1\npositive constant. [4]\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2024-on-13-q04-dade361952.pdf"
    },
    {
      "id": "9231-2024-on-13-q05",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_13.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_13.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2024-on-13-q05-p01-e42ab27bc6.png",
        "9231-topic-15-polar-coordinates/assets/9231-2024-on-13-q05-p02-abaaca4118.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-13-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 (a) Show that the curve with Cartesian equation\nSIHT\n2\n`x2 +y2j = 6xy NI\nETIRW\nr2 = 3sin2i. has polar equation [2]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n1 r2 = 3sin2i, for r. 0 G G The curve C has polar equation i\n2\nOD\n(b) Sketch C and state the maximum distance of a point on C from the pole. [3]\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\n(c) Find the area of the region enclosed by C. [2]\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n(d) Find the maximum distance of a point on C from the initial line. [6]\nTON\n.......................................................................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2024-on-13-q05-e66f052874.pdf"
    },
    {
      "id": "9231-2024-on-13-q06",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_13.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_13.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2024-on-13-q06-p01-af0ae3df9a.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2024-on-13-q06-p02-05e5b43026.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-13-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\n4x2 +x+1 SIHT\ny = 6 The curve C has equation .\n2x2 -7x+3\nNI\n(a) Find the equations of the asymptotes of C. [2] ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n(b) Find the coordinates of any stationary points on C. [4]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ...............................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2024-on-13-q06-b5cd4ca974.pdf"
    },
    {
      "id": "9231-2024-on-13-q07",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 7,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_13.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_13.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2024-on-13-q07-p01-2e553e28d8.png",
        "9231-topic-16-vectors/assets/9231-2024-on-13-q07-p02-32d32aefd7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-13-q07",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\nl l r = i-2j+9k+n(i-4jk+2 ) 7 The lines and have equations r = i+3j-2k+m(2i+jk+ ) and\nSIHT\n1 2\nrespectively. The plane P l l contains and is parallel to .\n1 1 2\nNI\nETIRW\nP ax+by+cz = d. , giving your answer in the form [4] (a) Find the equation of\n1\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nP l (2, -1, 7). contains and the point with coordinates The plane NI\n2 2\nETIRW\nP P and . [4] (b) Find the acute angle between\n1 2\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n....................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2024-on-13-q07-63fb0ae895.pdf"
    },
    {
      "id": "9231-2024-on-21-q01",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 1,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_21.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-on-21-q01-p01-6dc647fcb1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-21-q01",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 Find the set of values of k for which the system of equations\nSIHT\nx+5yz+6 = 1,\nNI\nETIRW kx+2y+2z = 2,\n-3x+48yz+ 3= ,\nTON\nhas a unique solution and interpret this situation geometrically. [4]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n......................................................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2024-on-21-q01-aed54b8833.pdf"
    },
    {
      "id": "9231-2024-on-21-q02",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 2,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_21.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_21.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2024-on-21-q02-p01-0773d756a7.png",
        "9231-topic-23-differentiation/assets/9231-2024-on-21-q02-p02-53f51023e6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-21-q02",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 It is given that\nSIHT\n1\ncos-1t = 1+ y = x and for 0 1 t 1 1. NI\nt\nETIRW\ndy t2\nTON (a) Show that = . [2]\ndx 1-t2\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n......................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2024-on-21-q02-9b392c67db.pdf"
    },
    {
      "id": "9231-2024-on-21-q03",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 3,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_21.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-on-21-q03-p01-489c856ac8.png",
        "9231-topic-24-integration/assets/9231-2024-on-21-q03-p02-c070135150.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-21-q03",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 12. ex 3 A curve has equation for The area of the surface generated when the curve is y = ln ln G x G SIHT\n5 3\nrotated through 2r radians about the x-axis is denoted by A.\nNI\nETIRW\nex to show that u = (a) Use the substitution\n12 y TON 5 1+u2du. A = 2r [2]\n4\nOD\n3\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\nu = sinhv to show that (b) Use the substitution\nTON\nrb904 5l. A = +ln [6]\nOD 225 3\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-on-21-q03-03732f0a0c.pdf"
    },
    {
      "id": "9231-2024-on-21-q04",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 4,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_21.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-on-21-q04-p01-61754f2745.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-on-21-q04-p02-3b4e45da21.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-21-q04",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 The matrix A is given by\nSIHT\n-11 1 8 NI\nf p ETIRW A = 0 -2 0 .\n-16 1 13\nTON\nOD\n1\nf p\n(a) Show that is an eigenvector of A and state the corresponding eigenvalue. [2] 1\n1\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n3 -19m-30 = 0 and hence find the other (b) Show that the characteristic equation of A is ETIRW m\neigenvalues of A. [3]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n...",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2024-on-21-q04-fc55418261.pdf"
    },
    {
      "id": "9231-2024-on-21-q05",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_21.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2024-on-21-q05-p01-f2c10ca233.png",
        "9231-topic-26-differential-equations/assets/9231-2024-on-21-q05-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-21-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 Find the particular solution of the differential equation\nSIHT\nNI\nd2x dx\nETIRW t2 6 -5 +x = +t+1,\ndt dt2\nTON\ndx\ngiven that, when = -6. t = 0, x = 12 and [10]\nOD dt\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n..",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2024-on-21-q05-ddc74048eb.pdf"
    },
    {
      "id": "9231-2024-on-21-q06",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 6,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_21.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-on-21-q06-p01-f921b8b67e.png",
        "9231-topic-24-integration/assets/9231-2024-on-21-q06-p02-0a6507155c.png",
        "9231-topic-24-integration/assets/9231-2024-on-21-q06-p03-ea5ad0e219.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-21-q06",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6\nSIHT\ny\nNI\n1 ETIRW\nTON\nOD\nNIGRAM\nx 0\n1 2 N-1 1\nN N N\nSIHT\nx NI b1l The diagram shows the curve with equation for 1, together with a set of N rectangles 0 G x G y =\nETIRW 2\n1\neach of width .\nN\nTON\nx y 1\nb1l dx 2 L , where (a) By considering the sum of the areas of these rectangles, show that\n2 N OD\n0\n1\n= L . [4]\nN 1 ` j 2N 2 -1\nN\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n.........................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-on-21-q06-8369a2ac2a.pdf"
    },
    {
      "id": "9231-2024-on-21-q07",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_21.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2024-on-21-q07-p01-6342a75766.png",
        "9231-topic-26-differential-equations/assets/9231-2024-on-21-q07-p02-02b7aff01d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-21-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "7 (a) Show that an appropriate integrating factor for\nNIGRAM\ndy\nx2 x2 +16 +y = x +16\ndx\nSIHT\nNI\n1x+ 1 x2 is +16. [4]\nETIRW 4 4\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\n(b) Hence find the solution of the differential equation\nSIHT\ndy\nx2 x2 NI +16 +y = x +16\ndx\nETIRW\nfor which y = 6 when x = 3. [6]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n......................................................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2024-on-21-q07-29ae623233.pdf"
    },
    {
      "id": "9231-2024-on-21-q08",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 8,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_21.pdf?download=true",
      "source_pages": [
        17,
        18,
        19
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-on-21-q08-p01-f5f102b93c.png",
        "9231-topic-24-integration/assets/9231-2024-on-21-q08-p02-51af62e2cf.png",
        "9231-topic-24-integration/assets/9231-2024-on-21-q08-p03-08011c493d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-21-q08",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n7 1 SIHT bz+ l z = cosi+isini, 8 (a) By considering the binomial expansion of , where use de Moivre’ s\nz\ntheorem to show that\nNI\nETIRW\ncos7 acos7i+bcos5i+ccos3i+dcosi, = i\nTON\nwhere a, b, c and d are constants to be determined. [5]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-on-21-q08-d506afdf6e.pdf"
    },
    {
      "id": "9231-2024-on-22-q01",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 1,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_22.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-on-22-q01-p01-aeada6c5f1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-22-q01",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\ny 1 7 SIHT\n1 Find the value of dx, giving your answer in the form b), where a and b are ln(a+\n(x-5)2 -1\nNI 6\nintegers to be determined. [4] ETIRW\n.......................................................................................................................................................... TON\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\n..........................................................................................................................................................\nSIHT\nNI ..........................................................................................................................................................\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n.........................................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-on-22-q01-b73e311489.pdf"
    },
    {
      "id": "9231-2024-on-22-q02",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 2,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_22.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_22.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2024-on-22-q02-p01-4c671df7f8.png",
        "9231-topic-23-differentiation/assets/9231-2024-on-22-q02-p02-47c71783ce.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-22-q02",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 The curve C has equation\nSIHT\n4y2 +4ln(xy) = 1. NI\nETIRW\ndy 1\n1l TON b2, (a) Show that, at the point = - on C, . [3]\n6 dx 2\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n..........",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2024-on-22-q02-a2d7967212.pdf"
    },
    {
      "id": "9231-2024-on-22-q03",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 3,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_22.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-on-22-q03-p01-30a6ce1820.png",
        "9231-topic-24-integration/assets/9231-2024-on-22-q03-p02-5e32caa22a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-22-q03",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 The curve C has parametric equations\nSIHT\n1 1 1\n2et(t-1), e2t t3 x = - - , for 1. 0 G t G y = NI\n2 3 2\nETIRW\nFind the exact length of C. [7]\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW ..........................................................................................................................................................\n.......................................................................................................................................................... TON\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\n..........................................................................................................................................................\nSIHT\nNI .............................",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-on-22-q03-871b443f7c.pdf"
    },
    {
      "id": "9231-2024-on-22-q04",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 4,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_22.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_22.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2024-on-22-q04-p01-8884ec1a0b.png",
        "9231-topic-25-complex-numbers/assets/9231-2024-on-22-q04-p02-f01eb618d3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-22-q04",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 (a) Use de Moivre’s theorem to show that\nSIHT\ncot6 i-15cot4 i+-15cot2 1 i\ncot6i = . [6] NI\n6cot5 i-20cot3 i+6coti ETIRW\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n...................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2024-on-22-q04-930026ce0c.pdf"
    },
    {
      "id": "9231-2024-on-22-q05",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_22.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2024-on-22-q05-p01-b12606d420.png",
        "9231-topic-26-differential-equations/assets/9231-2024-on-22-q05-p02-5e32caa22a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-22-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 Find the particular solution of the differential equation\nSIHT\nd2y dy\nx2, +y 3 +2 = NI\ndx dx2 ETIRW\ndy\ny = = 0. x = 0, [10] given that, when\ndx\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW ..........................................................................................................................................................\n.......................................................................................................................................................... TON\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\n..........................................................................................................................................................\nSIHT\nNI ..........................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2024-on-22-q05-e4c0d1027d.pdf"
    },
    {
      "id": "9231-2024-on-22-q06",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 6,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_22.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-on-22-q06-p01-21657cbdd0.png",
        "9231-topic-24-integration/assets/9231-2024-on-22-q06-p02-27d5419eb6.png",
        "9231-topic-24-integration/assets/9231-2024-on-22-q06-p03-22255a9f5b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-22-q06",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6\nSIHT\ny\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON x 0\n1 2 n-1 1\nOD n n n\ne1-x y = The diagram shows the curve with equation for 1, together with a set of n rectangles 0 G x G\n1\nof width .\nn NIGRAM\ny 1\ne1-xdx 1 U , where (a) By considering the sum of the areas of these rectangles, show that\nn\n0 SIHT\ne-1\n[4] U = .\nn 1 NI n(1-e- ) n\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\ny 1\nSIHT e1-xdx. L (b) Use a similar method to find, in terms of n, a lower bound for [4]\nn\n0\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n...............",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-on-22-q06-a4c6ee09b1.pdf"
    },
    {
      "id": "9231-2024-on-22-q07",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_22.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2024-on-22-q07-p01-bc04d19af6.png",
        "9231-topic-26-differential-equations/assets/9231-2024-on-22-q07-p02-ceb472fa9e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-22-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "SIHT\nd NI\n(ln(tanhx)) 2cosech2x. = [3] 7 (a) Show that\ndx ETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nNIGRAM\n(b) Find the solution of the differential equation\nSIHT\ndy\nNI +2y sinh2x sinh2x =\ndx\nETIRW\ny = 5 x = ln2. for which when Give your answer in an exact form. [7]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n..........................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2024-on-22-q07-90346917db.pdf"
    },
    {
      "id": "9231-2024-on-22-q08",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_22.pdf?download=true",
      "source_pages": [
        16,
        17,
        18
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-on-22-q08-p01-aa106fb0f7.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-on-22-q08-p02-9e30d996b6.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-on-22-q08-p03-ac6c9e3217.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-22-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:7)(cid:44)\n8 The matrix A is given by\nSIHT\n-2 0 0 NI\nf p ETIRW A = 0 7 9 .\n4 1 7\nTON\n3 2 (a) Show that the characteristic equation of A is -12mm +12 +80 = 0 and find the eigenvalues of m\nOD\nA. [4]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n..............................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2024-on-22-q08-35ee25843b.pdf"
    },
    {
      "id": "9231-2024-on-23-q01",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 1,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_23.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-on-23-q01-p01-6dc647fcb1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-23-q01",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 Find the set of values of k for which the system of equations\nSIHT\nx+5yz+6 = 1,\nNI\nETIRW kx+2y+2z = 2,\n-3x+48yz+ 3= ,\nTON\nhas a unique solution and interpret this situation geometrically. [4]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n......................................................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2024-on-23-q01-c582a44ea9.pdf"
    },
    {
      "id": "9231-2024-on-23-q02",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 2,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_23.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_23.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2024-on-23-q02-p01-0773d756a7.png",
        "9231-topic-23-differentiation/assets/9231-2024-on-23-q02-p02-53f51023e6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-23-q02",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 It is given that\nSIHT\n1\ncos-1t = 1+ y = x and for 0 1 t 1 1. NI\nt\nETIRW\ndy t2\nTON (a) Show that = . [2]\ndx 1-t2\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n......................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2024-on-23-q02-bfd1c6e970.pdf"
    },
    {
      "id": "9231-2024-on-23-q03",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 3,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_23.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-on-23-q03-p01-489c856ac8.png",
        "9231-topic-24-integration/assets/9231-2024-on-23-q03-p02-c070135150.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-23-q03",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 12. ex 3 A curve has equation for The area of the surface generated when the curve is y = ln ln G x G SIHT\n5 3\nrotated through 2r radians about the x-axis is denoted by A.\nNI\nETIRW\nex to show that u = (a) Use the substitution\n12 y TON 5 1+u2du. A = 2r [2]\n4\nOD\n3\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\nu = sinhv to show that (b) Use the substitution\nTON\nrb904 5l. A = +ln [6]\nOD 225 3\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-on-23-q03-aa1302e275.pdf"
    },
    {
      "id": "9231-2024-on-23-q04",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 4,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_23.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-on-23-q04-p01-61754f2745.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2024-on-23-q04-p02-3b4e45da21.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-23-q04",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 The matrix A is given by\nSIHT\n-11 1 8 NI\nf p ETIRW A = 0 -2 0 .\n-16 1 13\nTON\nOD\n1\nf p\n(a) Show that is an eigenvector of A and state the corresponding eigenvalue. [2] 1\n1\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n3 -19m-30 = 0 and hence find the other (b) Show that the characteristic equation of A is ETIRW m\neigenvalues of A. [3]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n...",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2024-on-23-q04-13371a571d.pdf"
    },
    {
      "id": "9231-2024-on-23-q05",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_23.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2024-on-23-q05-p01-f2c10ca233.png",
        "9231-topic-26-differential-equations/assets/9231-2024-on-23-q05-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-23-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 Find the particular solution of the differential equation\nSIHT\nNI\nd2x dx\nETIRW t2 6 -5 +x = +t+1,\ndt dt2\nTON\ndx\ngiven that, when = -6. t = 0, x = 12 and [10]\nOD dt\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n..",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2024-on-23-q05-6ad4cb9c3f.pdf"
    },
    {
      "id": "9231-2024-on-23-q06",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 6,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_23.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-on-23-q06-p01-f921b8b67e.png",
        "9231-topic-24-integration/assets/9231-2024-on-23-q06-p02-0a6507155c.png",
        "9231-topic-24-integration/assets/9231-2024-on-23-q06-p03-ea5ad0e219.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-23-q06",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6\nSIHT\ny\nNI\n1 ETIRW\nTON\nOD\nNIGRAM\nx 0\n1 2 N-1 1\nN N N\nSIHT\nx NI b1l The diagram shows the curve with equation for 1, together with a set of N rectangles 0 G x G y =\nETIRW 2\n1\neach of width .\nN\nTON\nx y 1\nb1l dx 2 L , where (a) By considering the sum of the areas of these rectangles, show that\n2 N OD\n0\n1\n= L . [4]\nN 1 ` j 2N 2 -1\nN\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n.........................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-on-23-q06-a3285243c6.pdf"
    },
    {
      "id": "9231-2024-on-23-q07",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_23.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2024-on-23-q07-p01-6342a75766.png",
        "9231-topic-26-differential-equations/assets/9231-2024-on-23-q07-p02-02b7aff01d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-23-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "7 (a) Show that an appropriate integrating factor for\nNIGRAM\ndy\nx2 x2 +16 +y = x +16\ndx\nSIHT\nNI\n1x+ 1 x2 is +16. [4]\nETIRW 4 4\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\n(b) Hence find the solution of the differential equation\nSIHT\ndy\nx2 x2 NI +16 +y = x +16\ndx\nETIRW\nfor which y = 6 when x = 3. [6]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n......................................................................................................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2024-on-23-q07-20d83dafa5.pdf"
    },
    {
      "id": "9231-2024-on-23-q08",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 8,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_23.pdf?download=true",
      "source_pages": [
        17,
        18,
        19
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2024-on-23-q08-p01-f5f102b93c.png",
        "9231-topic-24-integration/assets/9231-2024-on-23-q08-p02-51af62e2cf.png",
        "9231-topic-24-integration/assets/9231-2024-on-23-q08-p03-08011c493d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-23-q08",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n7 1 SIHT bz+ l z = cosi+isini, 8 (a) By considering the binomial expansion of , where use de Moivre’ s\nz\ntheorem to show that\nNI\nETIRW\ncos7 acos7i+bcos5i+ccos3i+dcosi, = i\nTON\nwhere a, b, c and d are constants to be determined. [5]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............",
      "pdf_path": "9231-topic-24-integration/print/9231-2024-on-23-q08-391a2310a0.pdf"
    },
    {
      "id": "9231-2024-on-31-q01",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 1,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_31.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_31.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-on-31-q01-p01-8669c3e0a5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-31-q01",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\nums-1 tan-12 1 A particle P is projected with speed at an angle above the horizontal from a point O on SIHT\na horizontal plane and moves freely under gravity. When P has travelled a distance 56 m horizontally\nNI\nfrom O, it is at a vertical height H m above the plane. When P has travelled a distance 84 m horizontally\nETIRW\n1H from O, it is at a vertical height m above the plane.\n2\nTON\nFind, in either order, the value of u and the value of H. [5]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n...................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2024-on-31-q01-eb5ab089f1.pdf"
    },
    {
      "id": "9231-2024-on-31-q02",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 2,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_31.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2024-on-31-q02-p01-119bb52dc1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-31-q02",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n2 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end\nSIHT\nof the string is attached to a fixed point O. The particle P is held at the point A with the string taut. It is\n3 NI given that OA makes an angle with the downward vertical through O, where tani = . The particle P i\n4 ETIRW is projected perpendicular to OA in an upwards direction with speed 5ag, and it starts to move along a\ncircular path in a vertical plane. When P is at the point B, where angle AOB is a right angle, the tension\nTON in the string is T.\nOD\nFind T in terms of m and g. [5]\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2024-on-31-q02-01013f4fe4.pdf"
    },
    {
      "id": "9231-2024-on-31-q03",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 3,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_31.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_31.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2024-on-31-q03-p01-54b4ae322e.png",
        "9231-topic-34-hooke-s-law/assets/9231-2024-on-31-q03-p02-7f14540125.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-31-q03",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 A particle P of mass m kg is attached to one end of a light elastic string of natural length 2 m and\nSIHT\nmodulus of elasticity 2mg N. The other end of the string is attached to a fixed point O. The particle P\nhangs in equilibrium vertically below O. The particle P is pulled down vertically a distance d m below NI\nETIRW its equilibrium position and released from rest.\nTON (a) Given that the particle just reaches O in the subsequent motion, find the value of d. [6]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\n",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2024-on-31-q03-bb70a20a01.pdf"
    },
    {
      "id": "9231-2024-on-31-q04",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_31.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_31.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2024-on-31-q04-p01-62ab1564b1.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2024-on-31-q04-p02-9b4db6a532.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-31-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4\nSIHT\nNI\nETIRW\n2 A B a\n3 TON\nOD\nkh\nh\nNIGRAM\nSIHT\nNI a\nETIRW\nTON\nOD 2a (k 1 1) An object is formed by removing a cylinder of radius and height kh from a uniform solid\n3\ncylinder of radius a and height h. The vertical axes of symmetry of the two cy linders coincide. The\nupper faces of the two cylinders are in the same plane as each other. The points A and B are the opposite\nends of a diameter of the upper face of the object (see diagram).\nNIGRAM\n(a) Find, in terms of h and k, the distance of the centre of mass of the object from AB. [4]\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n...........................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2024-on-31-q04-2252cbcee8.pdf"
    },
    {
      "id": "9231-2024-on-31-q05",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 5,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_31.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_31.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2024-on-31-q05-p01-d415a84c1d.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2024-on-31-q05-p02-9672a61f91.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-31-q05",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 A particle P of mass 2 kg moving on a horizontal straight line has displacement x m from a fixed point\nSIHT\nms-1 O on the line and velocity v at time t s. The only horizontal force acting on P is a variable force\nF N which can be expressed as a function of t. It is given that NI\nETIRW\n3-t v\n=\nx 1+t\nTON\nand when t = 0, x = 5.\nOD\n(a) Find an expression for x in terms of t. [4]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ...............................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2024-on-31-q05-893e0556ab.pdf"
    },
    {
      "id": "9231-2024-on-31-q06",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 6,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_31.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2024-on-31-q06-p01-8286623d4e.png",
        "9231-topic-33-circular-motion/assets/9231-2024-on-31-q06-p02-216300a5b7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-31-q06",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6\nSIHT\nO\nNI\na ETIRW\n1 m\nTON\nP\n0.8 m OD\nb\nQ\nNIGRAM\n1.4 m\nSIHT\nparticle P of mass 0.05 kg is attached to one end of a light inextensible string of length 1m . The other A\nNI\nend of the string is attached to a fixed point O. A particle Q of mass 0.04 kg is attached to one end of a ETIRW\nsecond light inextensible string. The other end of this string is attached to P.\nTON\ns-1. The particle P moves in a horizontal circle of radius 0.8 m with angular speed rad The particle Q ~\nOD s-1. moves in a horizontal circle of radius 1.4 m also with angular speed rad The centres of the circles ~\nare vertically below O, and O, P and Q are always in the same vertical plane. The strings OP and PQ\nremain at constant angles and respectively to the vertical (see diagram). a b\nNIGRAM\n(a) Find the tension in the string OP. [3]\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n....................................................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2024-on-31-q06-54a7949c7f.pdf"
    },
    {
      "id": "9231-2024-on-31-q07",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 7,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_31.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_31.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2024-on-31-q07-p01-b7b510f4aa.png",
        "9231-topic-36-momentum/assets/9231-2024-on-31-q07-p02-4bd135dcbc.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-31-q07",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\ntan-1b4l SIHT 7 A particle P is projected with speed u at an angle above the horizontal from a point O on a\n3\nhorizontal plane and moves freely under gravity. When P is moving horizontally , it strikes a smooth NI\nETIRW inclined plane at the point A. This plane is inclined to the horizontal at an angle a, and the line of\ngreatest slope through A lies in the vertical plane through O and A.\nTON\nAs a result of the impact, P moves vertically upwards. The coefficient of restitution between P and the\nOD\ninclined plane is e.\netan2 (a) Show that = 1. [4] a\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ......",
      "pdf_path": "9231-topic-36-momentum/print/9231-2024-on-31-q07-eced0cb5ea.pdf"
    },
    {
      "id": "9231-2024-on-32-q01",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 1,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 3,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_32.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2024-on-32-q01-p01-345716060a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-32-q01",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 A particle of mass 2k g is attached to one end of a light elastic string of natural length 0.8m and modulus\nSIHT\nof elasticity 100 N. The other end of the string is attached to a fixed point O on a smooth horizontal\nsurface. The particle is moving in a horizontal circle about O with the string taut and with constant NI\nETIRW angular speed 5 radians per second.\nTON Find the extension of the string. [3]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n.............................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2024-on-32-q01-18f468c7de.pdf"
    },
    {
      "id": "9231-2024-on-32-q02",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 2,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_32.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_32.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2024-on-32-q02-p01-9eb600472b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-32-q02",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n2 A particle P of mass m is attached to one end of a light elastic spring of natural length a and modulus\nSIHT\nof elasticity 5mg. The other end of the spring is attached to a fixed point O. The spring hangs vertically\nNI with P below O. The particle P is pulled down vertically and released from rest when the length of the\nETIRW\n3 spring is a.\n2\nTON\nFind the distance of P below O when P first comes to instantaneous rest. [4]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n........................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2024-on-32-q02-acc939ef48.pdf"
    },
    {
      "id": "9231-2024-on-32-q03",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 3,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_32.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_32.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2024-on-32-q03-p01-d3d88ae889.png",
        "9231-topic-36-momentum/assets/9231-2024-on-32-q03-p02-14e02b5398.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-32-q03",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3\nSIHT\n3u\n2u NI\nETIRW\nA B\nθ\nTON\nOD\nm m\nThe diagram shows two identical smooth uniform spheres A and B of equal radii and each of mass m.\nNIGRAM The two spheres are moving on a smooth horizontal surface when they collide with speeds 2u and\n3u respectively. Immediately before the collision, A’ s direction of motion makes an angle with the i\nline of centres and B’ s direction of motion is perpendicular to that of A. After the collision, B moves\nSIHT\n1. perpendicular to the line of centres. The coefficient of restitution between the spheres is\n3\nNI\nETIRW\ntani. [3] (a) Find the value of\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n...................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2024-on-32-q03-9d6406ec50.pdf"
    },
    {
      "id": "9231-2024-on-32-q04",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_32.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_32.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2024-on-32-q04-p01-fed990fdf4.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2024-on-32-q04-p02-724b6b46ed.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-32-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4\nSIHT\nB\nNI\nETIRW\nD\nTON\nE 3a\nOD\nC\nNIGRAM\nka\n3a i\nSIHT\nA\nNI\nETIRW\nTON\nOD The end A of a uniform rod AB of length 6a and weight W is in contact with a rough vertical wall. One\nend of a light inextensible string of length 3a is attached to the midpoint C of the rod. The other end of\nthe string is attached to a point D on the wall, vertically above A. The rod is in equilibrium when the\n3 angle between the rod and the wall is i, where tani = . A particle of weight W is attached to the point\n2 NIGRAM\nE on the rod, where the distance AE is equal to ka (3 1 k 1 6) (see diagram). The rod and the string are\nin a vertical plane perpendicular to the wall. The coefficient of friction between the rod and the wall\n1. is The rod is about to slip down the wall. SIHT\n3\nNI\n(a) Find the value of k. [5]\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n.................................................................................................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2024-on-32-q04-469eacc7dd.pdf"
    },
    {
      "id": "9231-2024-on-32-q05",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 5,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_32.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_32.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-on-32-q05-p01-07a3e7205b.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-on-32-q05-p02-b0eaf21fa4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-32-q05",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 A particle P is projected from a point O on horizontal ground with speed u at an angle above the i\nSIHT\n1. horizontal, where The particle P moves freely under gravity and passes through the point tani =\n3 NI 4a) with coordinates (3a, relative to horizontal and vertical axes through O in the plane of the motion.\nETIRW 5\nu2 = 25ag. [2] (a) Use the equation of the trajectory to show that\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n......................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2024-on-32-q05-c082781749.pdf"
    },
    {
      "id": "9231-2024-on-32-q06",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 6,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_32.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2024-on-32-q06-p01-c8bdb4ce3a.png",
        "9231-topic-33-circular-motion/assets/9231-2024-on-32-q06-p02-29242d2a82.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-32-q06",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6\nSIHT\nO\nNI\nθ ETIRW\na\nTON\nθ\nOD\nA\n1 10ag\n3\nNIGRAM\nSIHT A particle P of mass m is attached to one end of a light inextensible string of length a. The other end\nof the string is attached to a fixed point O. The particle P is held with the string taut and the string\nNI\nmakes an angle with the downward vertical through O. The particle P is projected at right angles to i ETIRW\n1 the string with speed and begins to move downwards along a circular path. When the string is 10ag\n3\nvertical, it strikes a small smooth peg at the point A which is vertically below O. The circular path and TON\nthe point A are in the same vertical plane. After the string strikes the peg, the particle P begins to move\nOD\nin a vertical circle with centre A. When the string makes an angle with the upward vertical through A i\n5 the string becomes slack (see diagram). The distance of A below O is a.\n9\ncosi. [6] (a) Find the value of\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n.............................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2024-on-32-q06-c69ef70752.pdf"
    },
    {
      "id": "9231-2024-on-32-q07",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 7,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_32.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_32.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2024-on-32-q07-p01-0c417e0b82.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2024-on-32-q07-p02-76fdcd96d3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-32-q07",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n7 A particle P of mass m kg is held at rest at a point O and released so that it moves vertically under\nSIHT\n0.1mv2 gravity against a resistive force of magnitude N, where v m s-1 is the velocity of P at time t s.\nNI\nETIRW (a) Find an expression for v in terms of t. [6]\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2024-on-32-q07-50dfe8ae13.pdf"
    },
    {
      "id": "9231-2024-on-33-q01",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 1,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_33.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_33.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2024-on-33-q01-p01-8669c3e0a5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-33-q01",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\nums-1 tan-12 1 A particle P is projected with speed at an angle above the horizontal from a point O on SIHT\na horizontal plane and moves freely under gravity. When P has travelled a distance 56 m horizontally\nNI\nfrom O, it is at a vertical height H m above the plane. When P has travelled a distance 84 m horizontally\nETIRW\n1H from O, it is at a vertical height m above the plane.\n2\nTON\nFind, in either order, the value of u and the value of H. [5]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n...................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2024-on-33-q01-f79cb1008e.pdf"
    },
    {
      "id": "9231-2024-on-33-q02",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 2,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_33.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2024-on-33-q02-p01-119bb52dc1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-33-q02",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n2 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end\nSIHT\nof the string is attached to a fixed point O. The particle P is held at the point A with the string taut. It is\n3 NI given that OA makes an angle with the downward vertical through O, where tani = . The particle P i\n4 ETIRW is projected perpendicular to OA in an upwards direction with speed 5ag, and it starts to move along a\ncircular path in a vertical plane. When P is at the point B, where angle AOB is a right angle, the tension\nTON in the string is T.\nOD\nFind T in terms of m and g. [5]\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2024-on-33-q02-d779ecc530.pdf"
    },
    {
      "id": "9231-2024-on-33-q03",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 3,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_33.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_33.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2024-on-33-q03-p01-54b4ae322e.png",
        "9231-topic-34-hooke-s-law/assets/9231-2024-on-33-q03-p02-7f14540125.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-33-q03",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 A particle P of mass m kg is attached to one end of a light elastic string of natural length 2 m and\nSIHT\nmodulus of elasticity 2mg N. The other end of the string is attached to a fixed point O. The particle P\nhangs in equilibrium vertically below O. The particle P is pulled down vertically a distance d m below NI\nETIRW its equilibrium position and released from rest.\nTON (a) Given that the particle just reaches O in the subsequent motion, find the value of d. [6]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\n",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2024-on-33-q03-d4f51e3e36.pdf"
    },
    {
      "id": "9231-2024-on-33-q04",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_33.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_33.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2024-on-33-q04-p01-62ab1564b1.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2024-on-33-q04-p02-9b4db6a532.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-33-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4\nSIHT\nNI\nETIRW\n2 A B a\n3 TON\nOD\nkh\nh\nNIGRAM\nSIHT\nNI a\nETIRW\nTON\nOD 2a (k 1 1) An object is formed by removing a cylinder of radius and height kh from a uniform solid\n3\ncylinder of radius a and height h. The vertical axes of symmetry of the two cy linders coincide. The\nupper faces of the two cylinders are in the same plane as each other. The points A and B are the opposite\nends of a diameter of the upper face of the object (see diagram).\nNIGRAM\n(a) Find, in terms of h and k, the distance of the centre of mass of the object from AB. [4]\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n...........................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2024-on-33-q04-958e50f784.pdf"
    },
    {
      "id": "9231-2024-on-33-q05",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 5,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_33.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_33.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2024-on-33-q05-p01-d415a84c1d.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2024-on-33-q05-p02-9672a61f91.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-33-q05",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 A particle P of mass 2 kg moving on a horizontal straight line has displacement x m from a fixed point\nSIHT\nms-1 O on the line and velocity v at time t s. The only horizontal force acting on P is a variable force\nF N which can be expressed as a function of t. It is given that NI\nETIRW\n3-t v\n=\nx 1+t\nTON\nand when t = 0, x = 5.\nOD\n(a) Find an expression for x in terms of t. [4]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ...............................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2024-on-33-q05-8439aeb9ff.pdf"
    },
    {
      "id": "9231-2024-on-33-q06",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 6,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_33.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2024-on-33-q06-p01-8286623d4e.png",
        "9231-topic-33-circular-motion/assets/9231-2024-on-33-q06-p02-216300a5b7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-33-q06",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6\nSIHT\nO\nNI\na ETIRW\n1 m\nTON\nP\n0.8 m OD\nb\nQ\nNIGRAM\n1.4 m\nSIHT\nparticle P of mass 0.05 kg is attached to one end of a light inextensible string of length 1m . The other A\nNI\nend of the string is attached to a fixed point O. A particle Q of mass 0.04 kg is attached to one end of a ETIRW\nsecond light inextensible string. The other end of this string is attached to P.\nTON\ns-1. The particle P moves in a horizontal circle of radius 0.8 m with angular speed rad The particle Q ~\nOD s-1. moves in a horizontal circle of radius 1.4 m also with angular speed rad The centres of the circles ~\nare vertically below O, and O, P and Q are always in the same vertical plane. The strings OP and PQ\nremain at constant angles and respectively to the vertical (see diagram). a b\nNIGRAM\n(a) Find the tension in the string OP. [3]\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n....................................................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2024-on-33-q06-48447ac245.pdf"
    },
    {
      "id": "9231-2024-on-33-q07",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 7,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_33.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_33.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2024-on-33-q07-p01-b7b510f4aa.png",
        "9231-topic-36-momentum/assets/9231-2024-on-33-q07-p02-4bd135dcbc.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-33-q07",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\ntan-1b4l SIHT 7 A particle P is projected with speed u at an angle above the horizontal from a point O on a\n3\nhorizontal plane and moves freely under gravity. When P is moving horizontally , it strikes a smooth NI\nETIRW inclined plane at the point A. This plane is inclined to the horizontal at an angle a, and the line of\ngreatest slope through A lies in the vertical plane through O and A.\nTON\nAs a result of the impact, P moves vertically upwards. The coefficient of restitution between P and the\nOD\ninclined plane is e.\netan2 (a) Show that = 1. [4] a\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ......",
      "pdf_path": "9231-topic-36-momentum/print/9231-2024-on-33-q07-5366658653.pdf"
    },
    {
      "id": "9231-2024-on-41-q01",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_41.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-on-41-q01-p01-32d0179186.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-41-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 Ellie is investigating the heights of two types of beech tree, A and B, in a certain region. She has\nSIHT\nchosen a random sample of 60 beech trees of type A in the region, recorded their heights, x m, and\n4.95m2 NI calculated unbiased estimates for the population mean and population variance as 35.6 m and\nETIRW respectively.\nTON Ellie also chooses a random sample of 50 beech trees of type B in the region and records their heights,\ny m. Her results are summarised as follows.\nOD\n/y2 /y = 1654 = 54850\nFind a 95% confidence interval for the difference between the population mean heights of type A and\nNIGRAM\ntype B beech trees in the region. [6]\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n............................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2024-on-41-q01-ada7662ce7.pdf"
    },
    {
      "id": "9231-2024-on-41-q02",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 2,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_41.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_41.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2024-on-41-q02-p01-7c205cffe1.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2024-on-41-q02-p02-5980832827.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-41-q02",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 A school with a lar ge number of students is updating its logo. Each student has designed a new logo\nSIHT\nand two teachers have each awarded a mark out of 50 for each logo. The marks awarded to a random\nsample of 12 students are shown in the following table. NI\nETIRW\nStudent A B C D E F G H I J K L\nTON\nTeacher 1 36 38 40 36 22 34 45 44 48 35 28 30\nOD\nTeacher 2 38 42 32 41 32 41 42 50 36 44 42 41\nOne of the students claims that Teacher 2 is awarding higher marks than Teacher 1.\nNIGRAM\n(a) Carry out a Wilcoxon matched-pairs signed-rank test, at the 5% significance level, to test whether\nthe data supports the claim. [7] SIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2024-on-41-q02-e0f051f077.pdf"
    },
    {
      "id": "9231-2024-on-41-q03",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 3,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_41.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_41.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2024-on-41-q03-p01-c375c8d95a.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2024-on-41-q03-p02-c070135150.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-41-q03",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 A statistician believes that the number of telephone calls received by an advice centre in a 10-minute\nSIHT\ninterval can be modelled by theP oisson distribution Po(1.9). The number of calls received in ar andomly\nchosen 10-minute interval was recorded on each of 100 days. The results are summarised in the table, NI\nETIRW together with some of the expected frequencies corresponding to the distribution Po(1.9).\nTON\n6 or\nNumber of calls 0 1 2 3 4 5\nmore OD\nObserved\n10 18 35 21 11 4 1\nfrequency\nNIGRAM Expected\n14.957 28.418 26.997 1.322\nfrequency\nSIHT\n(a) Complete the table. [2]\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD (b) Carry out a goodness of fit test, at the 10% significance level, to determinwe hether the statistician’s\nbelief is reasonable. [6]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n.....................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2024-on-41-q03-a8696ead57.pdf"
    },
    {
      "id": "9231-2024-on-41-q04",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 4,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_41.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_41.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2024-on-41-q04-p01-fc3b2850fe.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2024-on-41-q04-p02-a4888a4696.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-41-q04",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 The continuous random variable X has probability density function f given by\nSIHT\nNI kx3 G 1 0 1, x\nETIRW\n*\n= k(5-x) G G 1 5, f(x) x\n0 otherwise, TON\nOD where k is a constant.\n(a) Sketch the graph of f. [1]\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\n4 (b) Show that . [2] k =\n33\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\n(c) Find the cumulative distribution function of X. [3]\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n...................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2024-on-41-q04-2176cfddcf.pdf"
    },
    {
      "id": "9231-2024-on-41-q05",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 5,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_41.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_41.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2024-on-41-q05-p01-0a42b06311.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2024-on-41-q05-p02-2529c26a43.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-41-q05",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is\nSIHT\n1. 1 and the third coin is biased so that the probability of obtaining a head is The random variable X is\n5 3 NI\nthe number of heads that Nikita obtains when he throws all three coins at the same time.\nETIRW\n(a) Find the probability generating function of X. [3]\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\nRajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable Y is the\nnumber of 4s that Rajesh obtains when he throws the two dice. NIGRAM\nThe random variable Z is the sum of the number of heads obtained by Nikita ",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2024-on-41-q05-4f38ed35f1.pdf"
    },
    {
      "id": "9231-2024-on-41-q06",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 6,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_41.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-on-41-q06-p01-6eee1f7b8b.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-on-41-q06-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-41-q06",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 Ansal is investigating the wingspans of Monarch butterflies in two different regions, X and Y. He takes\nSIHT\na random sample of 8 Monarch butterflies from region X and records their wingspans, x cm. His results\nare as follows. NI\nETIRW\n8.2 7.0 7.3 8.8 7.8 8.5 9.2 7.4\nTON\nAnsal also takes a random sample of 9 Monarch butterflies from region Y and records their wingspans,\nOD\ny cm. His results are summarised as follows.\n/y2 /y = 71.10 = 567.13\nNIGRAM\nAnsal suspects that the mean wingspan of Monarch butterflies from region X is greater than the mean\nwingspan of Monarch butterflies from region Y. It is known that the wingspans of Monarch butterflies\nin regions X and Y are normally distributed with equal population variances. SIHT\nNI\nTest, at the 10% significance level, whether Ansal’s suspicion is supported by the data. [8]\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n..........",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2024-on-41-q06-22add6dfff.pdf"
    },
    {
      "id": "9231-2024-on-42-q01",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_42.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-on-42-q01-p01-7b99aa5737.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-42-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 A scientist is investigating the lengths of the leaves of a certain type of plant. The scientist assumes that\nSIHT\nthe lengths of the leaves of this type of plant are normally distributed. He measures the lengths, x cm, of\nNI the leaves of a random sample of 8 plants of this type. His results are as follows.\nETIRW\n3.5 4.2 3.8 5.2 2.9 3.7 4.1 3.2\nTON\nFind a 90% confidence interval for the population mean length of leaves of this type of plant. [4]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT ....................................................................................................................................................................\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n...............................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2024-on-42-q01-857722f14f.pdf"
    },
    {
      "id": "9231-2024-on-42-q02",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 2,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_42.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_42.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2024-on-42-q02-p01-a1ce99c8d9.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2024-on-42-q02-p02-e2b589d933.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-42-q02",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\nG (t) 2 The random variable X has probability generating function given by\nSIHT X\n1 +pt+qt2, NI G (t) =\n5 X ETIRW\nwhere p and q are constants.\nTON\nE(X) = 1.1, find the numerical value of Var(X ). [4] (a) Given that\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI ....................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2024-on-42-q02-bdbd93c981.pdf"
    },
    {
      "id": "9231-2024-on-42-q03",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 3,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_42.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_42.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2024-on-42-q03-p01-b6b4be07ab.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2024-on-42-q03-p02-c070135150.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-42-q03",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 Rosie sows 5 seeds in each of 150 plant pots. The number of seeds that germinate is recorded for each\nSIHT\npot. The results are summarised in the following table.\nNI\nETIRW\nNumber of seeds\n0 1 2 3 4 5\nthat germinate\nTON\nOD Number of pots 12 40 43 35 16 4\nRosie suggests that the number of seeds that germinate follows the binomial distribution B(5, p).\nNIGRAM\np = 0.42. [1] (a) Use Rosie’s results to show that\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n(b) Carry out a goodness of fit test, at the 10% significan ce level, to test whether the distribution\nOD\nB(5, 0.42) is a good fit for the data. [9]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ....................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2024-on-42-q03-16c291ae32.pdf"
    },
    {
      "id": "9231-2024-on-42-q04",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 4,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_42.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_42.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2024-on-42-q04-p01-2fc0925422.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2024-on-42-q04-p02-b6777c6e71.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-42-q04",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 The random variable X has probability density function f given by\nSIHT\nNI\n1 (x-1)2 x 5, 2 G G\nETIRW * f(x) = 21\notherwise. 0\nTON\n(a) Find the cumulative distribution function of X. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n(X-1)4. Y = The random variable Y is defined by\nNIGRAM\n(b) Find the probability density function of Y. [3]\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n..",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2024-on-42-q04-4a3cd6fa41.pdf"
    },
    {
      "id": "9231-2024-on-42-q05",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 5,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_42.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-on-42-q05-p01-376d1b2c42.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-on-42-q05-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-42-q05",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 Dev owns a small company which produces bottles of juice. He uses two machines, X and Y, to fill\nSIHT\nempty bottles with juice. Dev is investigating the volumes of juice in the bottles. He chooses a random\nsample of 35 bottles filled by machine X and a random sample of 60 bottles filled by machine Y . The NI\nETIRW volumes of juice, x and y respectively, measured in suitable units, are summarised by\n/x2 /y2 /x /y TON = 30.8, = 29.0, = 62.4, = 76.8.\nOD\nDev claims that the mean volume of juice in bottles filled by machine Y is greater than the mean\nvolume of juice in bottles filled by machine X. A test at the a% significance level suggests that there is\nsufficient evidence to support Dev’s claim.\nNIGRAM\nFind the set of possible values of a. [9]\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n...............................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2024-on-42-q05-e87ced6480.pdf"
    },
    {
      "id": "9231-2024-on-42-q06",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 6,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_42.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_42.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2024-on-42-q06-p01-93fd987357.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2024-on-42-q06-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-42-q06",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 A sports college keeps records of the times taken by students to run one lap of a running track. The\nSIHT\npopulation median time taken is 51.0 seconds. After a month of intensive training, a random sample of\n22 new students run one lap of the track, giving times, in seconds, as follows. NI\nETIRW\n51.3 52.0 53.4 49.2 49.3 51.1 52.2 47.2\nTON\n53.0 48.5 49.4 50.3 50.8 51.6 49.1 52.3\nOD\n51.8 52.4 47.9 48.9 50.6 51.9\nIt is claimed that the intensive training has led to a decrease in the median time taken to run one lap of\nNIGRAM\nthe track.\nCarry out a Wilcoxon signed-rank test, at the 5% significance level, to test whether there is suf ficient SIHT\nevidence to support the claim. [9]\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ...............................................................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2024-on-42-q06-cefe117e7d.pdf"
    },
    {
      "id": "9231-2024-on-43-q01",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_43.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-on-43-q01-p01-32d0179186.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-43-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 Ellie is investigating the heights of two types of beech tree, A and B, in a certain region. She has\nSIHT\nchosen a random sample of 60 beech trees of type A in the region, recorded their heights, x m, and\n4.95m2 NI calculated unbiased estimates for the population mean and population variance as 35.6 m and\nETIRW respectively.\nTON Ellie also chooses a random sample of 50 beech trees of type B in the region and records their heights,\ny m. Her results are summarised as follows.\nOD\n/y2 /y = 1654 = 54850\nFind a 95% confidence interval for the difference between the population mean heights of type A and\nNIGRAM\ntype B beech trees in the region. [6]\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n............................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2024-on-43-q01-6b7845b32d.pdf"
    },
    {
      "id": "9231-2024-on-43-q02",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 2,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_43.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_43.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2024-on-43-q02-p01-7c205cffe1.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2024-on-43-q02-p02-5980832827.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-43-q02",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 A school with a lar ge number of students is updating its logo. Each student has designed a new logo\nSIHT\nand two teachers have each awarded a mark out of 50 for each logo. The marks awarded to a random\nsample of 12 students are shown in the following table. NI\nETIRW\nStudent A B C D E F G H I J K L\nTON\nTeacher 1 36 38 40 36 22 34 45 44 48 35 28 30\nOD\nTeacher 2 38 42 32 41 32 41 42 50 36 44 42 41\nOne of the students claims that Teacher 2 is awarding higher marks than Teacher 1.\nNIGRAM\n(a) Carry out a Wilcoxon matched-pairs signed-rank test, at the 5% significance level, to test whether\nthe data supports the claim. [7] SIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2024-on-43-q02-3a4cc2b84a.pdf"
    },
    {
      "id": "9231-2024-on-43-q03",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 3,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_43.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_43.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2024-on-43-q03-p01-c375c8d95a.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2024-on-43-q03-p02-c070135150.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-43-q03",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 A statistician believes that the number of telephone calls received by an advice centre in a 10-minute\nSIHT\ninterval can be modelled by theP oisson distribution Po(1.9). The number of calls received in ar andomly\nchosen 10-minute interval was recorded on each of 100 days. The results are summarised in the table, NI\nETIRW together with some of the expected frequencies corresponding to the distribution Po(1.9).\nTON\n6 or\nNumber of calls 0 1 2 3 4 5\nmore OD\nObserved\n10 18 35 21 11 4 1\nfrequency\nNIGRAM Expected\n14.957 28.418 26.997 1.322\nfrequency\nSIHT\n(a) Complete the table. [2]\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD (b) Carry out a goodness of fit test, at the 10% significance level, to determinwe hether the statistician’s\nbelief is reasonable. [6]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n.....................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2024-on-43-q03-7a239f4703.pdf"
    },
    {
      "id": "9231-2024-on-43-q04",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 4,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_43.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_43.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2024-on-43-q04-p01-fc3b2850fe.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2024-on-43-q04-p02-a4888a4696.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-43-q04",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 The continuous random variable X has probability density function f given by\nSIHT\nNI kx3 G 1 0 1, x\nETIRW\n*\n= k(5-x) G G 1 5, f(x) x\n0 otherwise, TON\nOD where k is a constant.\n(a) Sketch the graph of f. [1]\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\n4 (b) Show that . [2] k =\n33\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\n(c) Find the cumulative distribution function of X. [3]\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n...................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2024-on-43-q04-7cd2c0aebd.pdf"
    },
    {
      "id": "9231-2024-on-43-q05",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 5,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_43.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_43.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2024-on-43-q05-p01-0a42b06311.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2024-on-43-q05-p02-2529c26a43.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-43-q05",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 Nikita has three coins. One coin is fair, one coin is biased so that the probability of obtaining a head is\nSIHT\n1. 1 and the third coin is biased so that the probability of obtaining a head is The random variable X is\n5 3 NI\nthe number of heads that Nikita obtains when he throws all three coins at the same time.\nETIRW\n(a) Find the probability generating function of X. [3]\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\nRajesh has two fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6. The random variable Y is the\nnumber of 4s that Rajesh obtains when he throws the two dice. NIGRAM\nThe random variable Z is the sum of the number of heads obtained by Nikita ",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2024-on-43-q05-cd706bd2aa.pdf"
    },
    {
      "id": "9231-2024-on-43-q06",
      "subject": "9231",
      "year": 2024,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 6,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2024-oct-nov/9231_w24_qp_43.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2024-Oct-Nov/qp/9231_w24_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-on-43-q06-p01-6eee1f7b8b.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2024-on-43-q06-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2024-on-43-q06",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 Ansal is investigating the wingspans of Monarch butterflies in two different regions, X and Y. He takes\nSIHT\na random sample of 8 Monarch butterflies from region X and records their wingspans, x cm. His results\nare as follows. NI\nETIRW\n8.2 7.0 7.3 8.8 7.8 8.5 9.2 7.4\nTON\nAnsal also takes a random sample of 9 Monarch butterflies from region Y and records their wingspans,\nOD\ny cm. His results are summarised as follows.\n/y2 /y = 71.10 = 567.13\nNIGRAM\nAnsal suspects that the mean wingspan of Monarch butterflies from region X is greater than the mean\nwingspan of Monarch butterflies from region Y. It is known that the wingspans of Monarch butterflies\nin regions X and Y are normally distributed with equal population variances. SIHT\nNI\nTest, at the 10% significance level, whether Ansal’s suspicion is supported by the data. [8]\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n..........",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2024-on-43-q06-bb0e73d0d9.pdf"
    },
    {
      "id": "9231-2025-mj-11-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_11.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_11.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2025-mj-11-q01-p01-55e9d04da4.png",
        "9231-topic-13-summation-of-series/assets/9231-2025-mj-11-q01-p02-9488f11714.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-11-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 (a) Use standard results from the list of formulae (MF19) to show that\nSIHT\nNI /n\nETIRW an3 +bn2 (2-3r)(5-3r) = +cn,\nr=1\nTON\nwhere a, b and c are integers to be determined. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ........................................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2025-mj-11-q01-38a2fcfc7b.pdf"
    },
    {
      "id": "9231-2025-mj-11-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 2,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_11.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_11.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-mj-11-q02-p01-636fa3f219.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-mj-11-q02-p02-fe5262cec7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-11-q02",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\nx3 +2x+1 = 0 2 The cubic equation has roots a, b, c. SIHT\nNI 3 3 3 -1. -1, -1, [3] (a) Find a cubic equation whose roots are c a b\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ....",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2025-mj-11-q02-daed8b492a.pdf"
    },
    {
      "id": "9231-2025-mj-11-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 3,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_11.pdf?download=true",
      "source_pages": [
        6
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_11.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2025-mj-11-q03-p01-8b396d8233.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-11-q03",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\nu u u u = 5 3 The sequence , , , … is such that and u = 6u +5 for n H 1.\nSIHT 1 2 3 1 n+1 n\n6n NI for all positive integers n. [5] = -1 u (a) Prove by induction that\nn ETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n........................................................................................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2025-mj-11-q03-c4e8b4304d.pdf"
    },
    {
      "id": "9231-2025-mj-11-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_11.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_11.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-mj-11-q04-p01-c04058fcfc.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-mj-11-q04-p02-c070135150.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-11-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n-sini 1 2 cosi SIHT\ne oe o, = 0 1 1 2r. 4 The matrix M is given by where M i\n0 1 sini cosi\nNI\nETIRW\n(a) The matrix M represents a sequence of two geometrical transformations in the x–y plane.\nTON\nState the type of each transformation, and make clear the order in which they are applied. [2]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nfor which the transformation represented by M has a line of invariant points. [7] (b) Find the value of i OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n....",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2025-mj-11-q04-a8813fcd4a.pdf"
    },
    {
      "id": "9231-2025-mj-11-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_11.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_11.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2025-mj-11-q05-p01-4fd47c79f9.png",
        "9231-topic-15-polar-coordinates/assets/9231-2025-mj-11-q05-p02-fcaf2e489d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-11-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n1 i, SIHT 5 The curve C has polar equation for 0 G G 2r. r = ie i 8\nNI\n(a) Sketch C. [2]\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\n(b) Find the area of the region bounded by C and the initial line, giving your answer in the form\n1 2 `pr +qr+rje r +s, where p, q, r and s are integers to be determined. [6]\nNIGRAM 2\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n..................................................................................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2025-mj-11-q05-56b190a845.pdf"
    },
    {
      "id": "9231-2025-mj-11-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 6,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_11.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_11.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2025-mj-11-q06-p01-521cc0f6fb.png",
        "9231-topic-16-vectors/assets/9231-2025-mj-11-q06-p02-5a9eeda9a0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-11-q06",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 The points A, B, C have position vectors\nSIHT\ni-2k, i+2jk+2 ,i 2 -j-k, NI\nETIRW\nrespectively .\nTON\nax+by+cz = d . [5] (a) Find the equation of the plane ABC, giving your answer in the form\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\ni+tk, where t !-2. A point D has position vector TON\nOD\n(b) Find the acute angle between the planes ABC and ABD. [4]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n.....................................................................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2025-mj-11-q06-4a1b98c703.pdf"
    },
    {
      "id": "9231-2025-mj-11-q07",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_11.pdf?download=true",
      "source_pages": [
        14,
        15,
        16,
        17
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_11.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-11-q07-p01-3c41d8bdb0.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-11-q07-p02-4ed6a883d8.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-11-q07-p03-d6e90ded7f.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-11-q07-p04-334c0562ba.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-11-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\n2x2 -5x SIHT\n= y 7 The curve C has equation .\n2x2 -7x-4\nNI\nETIRW\n(a) Find the equations of the asymptotes of C. [2]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD (b) Find the coordinates of any stationary points on C. [4]\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2025-mj-11-q07-33376aba26.pdf"
    },
    {
      "id": "9231-2025-mj-12-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_12.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_12.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2025-mj-12-q01-p01-55e9d04da4.png",
        "9231-topic-13-summation-of-series/assets/9231-2025-mj-12-q01-p02-9488f11714.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-12-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 (a) Use standard results from the list of formulae (MF19) to show that\nSIHT\nNI /n\nETIRW an3 +bn2 (2-3r)(5-3r) = +cn,\nr=1\nTON\nwhere a, b and c are integers to be determined. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ........................................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2025-mj-12-q01-e724746acb.pdf"
    },
    {
      "id": "9231-2025-mj-12-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 2,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_12.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_12.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-mj-12-q02-p01-636fa3f219.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-mj-12-q02-p02-fe5262cec7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-12-q02",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\nx3 +2x+1 = 0 2 The cubic equation has roots a, b, c. SIHT\nNI 3 3 3 -1. -1, -1, [3] (a) Find a cubic equation whose roots are c a b\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ....",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2025-mj-12-q02-cfc63a87e9.pdf"
    },
    {
      "id": "9231-2025-mj-12-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 3,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_12.pdf?download=true",
      "source_pages": [
        6
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_12.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2025-mj-12-q03-p01-8b396d8233.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-12-q03",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\nu u u u = 5 3 The sequence , , , … is such that and u = 6u +5 for n H 1.\nSIHT 1 2 3 1 n+1 n\n6n NI for all positive integers n. [5] = -1 u (a) Prove by induction that\nn ETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n........................................................................................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2025-mj-12-q03-206d9ad21c.pdf"
    },
    {
      "id": "9231-2025-mj-12-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_12.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_12.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-mj-12-q04-p01-c04058fcfc.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-mj-12-q04-p02-c070135150.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-12-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n-sini 1 2 cosi SIHT\ne oe o, = 0 1 1 2r. 4 The matrix M is given by where M i\n0 1 sini cosi\nNI\nETIRW\n(a) The matrix M represents a sequence of two geometrical transformations in the x–y plane.\nTON\nState the type of each transformation, and make clear the order in which they are applied. [2]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nfor which the transformation represented by M has a line of invariant points. [7] (b) Find the value of i OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n....",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2025-mj-12-q04-04e3ef059c.pdf"
    },
    {
      "id": "9231-2025-mj-12-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_12.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_12.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2025-mj-12-q05-p01-4fd47c79f9.png",
        "9231-topic-15-polar-coordinates/assets/9231-2025-mj-12-q05-p02-fcaf2e489d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-12-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n1 i, SIHT 5 The curve C has polar equation for 0 G G 2r. r = ie i 8\nNI\n(a) Sketch C. [2]\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\n(b) Find the area of the region bounded by C and the initial line, giving your answer in the form\n1 2 `pr +qr+rje r +s, where p, q, r and s are integers to be determined. [6]\nNIGRAM 2\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n..................................................................................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2025-mj-12-q05-f62dbc22f2.pdf"
    },
    {
      "id": "9231-2025-mj-12-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 6,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_12.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_12.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2025-mj-12-q06-p01-521cc0f6fb.png",
        "9231-topic-16-vectors/assets/9231-2025-mj-12-q06-p02-5a9eeda9a0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-12-q06",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 The points A, B, C have position vectors\nSIHT\ni-2k, i+2jk+2 ,i 2 -j-k, NI\nETIRW\nrespectively .\nTON\nax+by+cz = d . [5] (a) Find the equation of the plane ABC, giving your answer in the form\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\ni+tk, where t !-2. A point D has position vector TON\nOD\n(b) Find the acute angle between the planes ABC and ABD. [4]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n.....................................................................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2025-mj-12-q06-abfb774fea.pdf"
    },
    {
      "id": "9231-2025-mj-12-q07",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_12.pdf?download=true",
      "source_pages": [
        14,
        15,
        16,
        17
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_12.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-12-q07-p01-3c41d8bdb0.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-12-q07-p02-4ed6a883d8.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-12-q07-p03-d6e90ded7f.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-12-q07-p04-334c0562ba.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-12-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\n2x2 -5x SIHT\n= y 7 The curve C has equation .\n2x2 -7x-4\nNI\nETIRW\n(a) Find the equations of the asymptotes of C. [2]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD (b) Find the coordinates of any stationary points on C. [4]\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2025-mj-12-q07-332ce5a056.pdf"
    },
    {
      "id": "9231-2025-mj-13-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 1,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_13.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_13.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-mj-13-q01-p01-0133896659.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-mj-13-q01-p02-3ca23a2b19.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-13-q01",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 The matrix M represents the sequence of two transformations in the x-y plane given by a stretch parallel\nSIHT\n1 to the x-axis, scale factor 14, followed by a rotation anticlockwise about the origin through angle r.\n3\nNI\n14 - 3\nETIRW e o. 2M = [4] (a) Show that\n14 3 1\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n(b) Find the equations of the invariant lines, through the origin, of the transformation in the x-y plane\nSIHT\nrepresented by M. [5]\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n..............................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2025-mj-13-q01-8d4cd1b73b.pdf"
    },
    {
      "id": "9231-2025-mj-13-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_13.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_13.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2025-mj-13-q02-p01-ed8f083e8d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-13-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2025n +47n 2 Prove by mathematical induction that is divisible by 46 for all positive integers n. [6] -2\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW ....................................................................................................................................................................\nTON ....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n...............................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2025-mj-13-q02-871a755774.pdf"
    },
    {
      "id": "9231-2025-mj-13-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 3,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_13.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_13.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-mj-13-q03-p01-80b8431b63.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-mj-13-q03-p02-09a770378d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-13-q03",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\nx4 +7xx2 a, b, c,d. +3 +22 = 0 3 The quartic equation has roots SIHT\nNI 2 2 2 2. +b +c +d [2] (a) Find the value of a\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n4 4 4 4. +b +c +d [2] (b) Find the value of a\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ................................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2025-mj-13-q03-5ed2899a7d.pdf"
    },
    {
      "id": "9231-2025-mj-13-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 4,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_13.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_13.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2025-mj-13-q04-p01-935e518a98.png",
        "9231-topic-13-summation-of-series/assets/9231-2025-mj-13-q04-p02-4b4bdac246.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-13-q04",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\nw = r(r+1)(r+2)f(r+9). 4 Let\nSIHT\nr\nNI (a) Show that\nETIRW\nw -w = 10(r+1)(r+2)f(r+9). [2]\nr+1 r\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n/n\nSIHT\nu u = (r+1)(r+2)f(r+9), find in terms of n. [3] (b) Given that\nr r\nNI\nr=1\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI .............................................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2025-mj-13-q04-905561b0a5.pdf"
    },
    {
      "id": "9231-2025-mj-13-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 5,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_13.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_13.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2025-mj-13-q05-p01-f9fa935aea.png",
        "9231-topic-16-vectors/assets/9231-2025-mj-13-q05-p02-4405dc6f84.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-13-q05",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 The plane has equation r = 2i+3j-2k+m(i-2j-k) +n(3i+2j-2k). P\nSIHT\nNI ax+by+cz = d. P, giving your answer in the form [4] (a) Find a Cartesian equation of\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW 4i+2j+9k. The point P has position vector\nP. [4] (b) Find the position vector of the foot of the perpendicular from P to TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n...........",
      "pdf_path": "9231-topic-16-vectors/print/9231-2025-mj-13-q05-0dc329d333.pdf"
    },
    {
      "id": "9231-2025-mj-13-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_13.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_13.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-13-q06-p01-fd4d9ff14a.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-13-q06-p02-222cea3823.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-13-q06-p03-e301492354.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-13-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\nx2 +a SIHT\ny = 6 The curve C has equation , where a is a positive constant.\nx+a\nNI\n(a) Find the equations of the asymptotes of C. [3] ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n(b) Find, in terms of a, the x-coordinates of the stationary points on C. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ......................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2025-mj-13-q06-f5a14ed650.pdf"
    },
    {
      "id": "9231-2025-mj-13-q07",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 7,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_13.pdf?download=true",
      "source_pages": [
        16,
        17
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_13.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2025-mj-13-q07-p01-4cc83df14a.png",
        "9231-topic-15-polar-coordinates/assets/9231-2025-mj-13-q07-p02-74d412e880.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-13-q07",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:7)(cid:44)\n-1 1 r2 esinicosi, 7 The curve C has polar equation = for r. G G r i SIHT\n2 2\nNI\n(a) Find the polar coordinates of the point on C that is furthest from the pole, giving your answers\nETIRW\ncorrect to 3 decimal places. [5]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n1 r, giving your = (b) Find the polar coordinates of the point on C that is furthest from the half-line i\n2\nTON answers correct to 3 decimal places. [5]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT .....................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2025-mj-13-q07-7fac740e99.pdf"
    },
    {
      "id": "9231-2025-mj-14-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_14.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_14.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2025-mj-14-q01-p01-e3e99a848a.png",
        "9231-topic-13-summation-of-series/assets/9231-2025-mj-14-q01-p02-d145d6fa1d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-14-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n/n\nSIHT\n1 (a) Use the List of formulae (MF19) to find in terms of n, simplifying your answer. [2] (2r+1)\nNI r=1\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n2r+1 1 1\n(b) Show that = - . [1]\n(r2 +1)(r2 r2 r2 +2r+2) +1 +2r+2\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n.................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2025-mj-14-q01-55feb8a0bf.pdf"
    },
    {
      "id": "9231-2025-mj-14-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_14.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_14.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2025-mj-14-q02-p01-43a7555baf.png",
        "9231-topic-17-proof-by-induction/assets/9231-2025-mj-14-q02-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-14-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 Prove by mathematical induction that, for every integer 2, n H\nSIHT\ndn\n(-1)n(n-2)!x1-n. NI [6] (xlnx) =\ndxn ETIRW\nTON ....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT ....................................................................................................................................................................\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\nOD ....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM ....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\n.................................................................................................................................................................... NI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n...........................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2025-mj-14-q02-477764b2ce.pdf"
    },
    {
      "id": "9231-2025-mj-14-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 3,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_14.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_14.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2025-mj-14-q03-p01-04fdfbb378.png",
        "9231-topic-16-vectors/assets/9231-2025-mj-14-q03-p02-3679961489.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-14-q03",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 The points A, B and C have position vectors\nSIHT\n2j+3k, -5i+3j+k and i+2j+5k NI\nETIRW\nrespectively, relative to the origin O.\nTON\nax+by+cz = d. [5] (a) Find the equation of the plane ABC, giving your answer in the form\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ..............................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2025-mj-14-q03-2a1a700f1b.pdf"
    },
    {
      "id": "9231-2025-mj-14-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 4,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_14.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_14.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-mj-14-q04-p01-0d4e0c4440.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-mj-14-q04-p02-ae12440acb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-14-q04",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\nx3 +bx2 +cx-1 = 0, 4 The cubic equation where b and c are constants, has roots a, b, c. SIHT\nNI 1 a b\nETIRW f p\nIt is given that the matrix 1 is singular. a c\n1 b c\nTON\n2 2 2 (a) Show that +b +c = 3. [4] a OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n.............................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2025-mj-14-q04-fdaa9985c7.pdf"
    },
    {
      "id": "9231-2025-mj-14-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 5,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_14.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_14.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-mj-14-q05-p01-31beb9953c.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-mj-14-q05-p02-52af6d880c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-14-q05",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 The matrix M represents a sequence of two transformations in the x-y plane given by a one-way stretch\nSIHT\nin the x-direction, scale factor 3, followed by a reflection in the line y = x.\nNI\nETIRW (a) Find M. [3]\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nM-1. [3] (b) Give full details of the geometrical transformation in the x-y plane represented by OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON 1 2\ne o. = MN The matrix N is such that\n3 2 OD\n(c) Find N. [3]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n...........................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2025-mj-14-q05-aebd06fb45.pdf"
    },
    {
      "id": "9231-2025-mj-14-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_14.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_14.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2025-mj-14-q06-p01-0cfe1da378.png",
        "9231-topic-15-polar-coordinates/assets/9231-2025-mj-14-q06-p02-9585e63bba.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-14-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\natanb1 SIHT il, 6 The curve C has polar equation where a is a positive constant and 2r. 0 G G r = i\n8\nNI\n(a) Sketch C and state, in terms of a, the greatest distance of a point on C from the pole. [3] ETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\n............................................................................................................................................................\nNIGRAM\n(b) Find, in terms of a, the area of the region bounded by C and the initial line. [4]\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\n(c) Show that, at the point on C furthest from the initial line,",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2025-mj-14-q06-c12ca27084.pdf"
    },
    {
      "id": "9231-2025-mj-14-q07",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_14.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_14.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-14-q07-p01-480f617471.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-14-q07-p02-70029f662c.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-mj-14-q07-p03-398fb360ec.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-14-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\nx2 +x-4 SIHT\ny = 7 The curve C has equation .\nx2 +x+2\nNI\nETIRW\n(a) State the equation of the asymptote of C. [1]\nTON\n............................................................................................................................................................\nOD\n17 (b) Show that, for all real values of x, - G y 1 1. [4]\n7\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n(c) Find the coordinates of any stationary points of C. [3] OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n..",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2025-mj-14-q07-d661fe3f48.pdf"
    },
    {
      "id": "9231-2025-mj-21-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 1,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_21.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_21.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2025-mj-21-q01-p01-ccd47ef440.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-21-q01",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\nz3 reii, r 2 0 = 27-27i, 1 Find the roots of the equation giving your answers in the form where and SIHT\n-r G 1 r. [5] i\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2025-mj-21-q01-377a6c7335.pdf"
    },
    {
      "id": "9231-2025-mj-21-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 2,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_21.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-mj-21-q02-p01-43d44a83d1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-21-q02",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\ny 1 n\n`1-xj I = sinhxdx, 2 Let where n is a non-negative integer. SIHT\nn\n0\nNI\nI =-1+n(n-1)I n H 2, . [4] (a) Show that, for\nETIRW n n-2\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\nI . [3] (b) Find the exact value of\n2\nNIGRAM\n............................................................................................................................................................\n.................................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-mj-21-q02-691f387b3f.pdf"
    },
    {
      "id": "9231-2025-mj-21-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 3,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_21.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_21.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2025-mj-21-q03-p01-dddcdd9ea6.png",
        "9231-topic-25-complex-numbers/assets/9231-2025-mj-21-q03-p02-994fae2397.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-21-q03",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n5 1 SIHT bz- l z = cosi+isini, 3 By considering the binomial expansion of , where use de Moivre’s theorem\nz\nto show that\nNI\nETIRW\na\ncosec5 = , i TON\nsin5i+bsin3i+csini\nOD\nwhere a, b and c are integers to be determined. [6]\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON ....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT ....................................................................................................................................................................\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\nOD ....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM ....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\n.................................................................................................................................................................... NI\nETIRW\n.....................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2025-mj-21-q03-b3ea515f4d.pdf"
    },
    {
      "id": "9231-2025-mj-21-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 4,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_21.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-mj-21-q04-p01-071bc22515.png",
        "9231-topic-24-integration/assets/9231-2025-mj-21-q04-p02-633aeda43a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-21-q04",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4\nSIHT\ny\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nx\n0 1 2 3 n-1 n\nNI\nETIRW\n1\nx The diagram shows the curve with equation y = e for x H 1, together with a set of n-1 rectangles\nx\nof unit width. TON\nOD\n(a) By considering the sum of the areas of these rectangles, show that\n/n\n1 1\nr n e 1 e2+ oe -2e. [5]\nNIGRAM r n\nr=1\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n/n\n1\nr. e [4] (b) Use a similar method to find, in terms of n, a lower bound for\nr\nSIHT\nr=1\n............................................................................................................................................................\nNI\nETIRW\n................",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-mj-21-q04-36c767a765.pdf"
    },
    {
      "id": "9231-2025-mj-21-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_21.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-mj-21-q05-p01-b38aa9cc82.png",
        "9231-topic-26-differential-equations/assets/9231-2025-mj-21-q05-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-21-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 Find the particular solution of the differential equation\nSIHT\nNI\nd2x dx\nETIRW e-t, 6 +3 +6x =\ndt dt2\nTON\ndx\n= = given that, when x 0. t = 0, [10]\nOD dt\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ...................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-mj-21-q05-bc11fbd796.pdf"
    },
    {
      "id": "9231-2025-mj-21-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 6,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_21.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_21.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2025-mj-21-q06-p01-1462d9f833.png",
        "9231-topic-21-hyperbolic-functions/assets/9231-2025-mj-21-q06-p02-387a4bfd0e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-21-q06",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6 (a) Starting from the definitions of tanh and sech in terms of exponentials, prove that\nSIHT\n1-tanh2u sech2u. = [3] NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\nd 1\n(sech-1t) =- . [4] (b) Show that\ndt 1-t2 t\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI .........................................................................................................................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2025-mj-21-q06-3b9ecda6e0.pdf"
    },
    {
      "id": "9231-2025-mj-21-q07",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_21.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-mj-21-q07-p01-563413f81b.png",
        "9231-topic-26-differential-equations/assets/9231-2025-mj-21-q07-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-21-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n7 Find the solution of the differential equation\nSIHT\nNI\ndy x+5\nETIRW = 1, y -\ndx x2 +10x+61\nTON\ngiven that y = 0 when x = 3. Give your answer in an exact form. [10]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI .....",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-mj-21-q07-9d511eb606.pdf"
    },
    {
      "id": "9231-2025-mj-21-q08",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_21.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-mj-21-q08-p01-5d0e232fd4.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-mj-21-q08-p02-aad4398852.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-21-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n8 (a) It is given that is an eigenvalue of the non-singular square matrix A, with corresponding m\nSIHT\neigenvector e.\nNI\nETIRW 3. A3 Show that e is an eigenvector of with corresponding eigenvalue [2] m\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\nThe matrix A is given by\n-1 4 3 NIGRAM\nf p\n= 0 1 A 0 .\n5 -2 0\nSIHT\nNI\nETIRW -1, 1 and 5. [2] (b) Show that the eigenvalues of A are\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n.................................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2025-mj-21-q08-0e0c606075.pdf"
    },
    {
      "id": "9231-2025-mj-22-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 1,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_22.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_22.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2025-mj-22-q01-p01-ccd47ef440.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-22-q01",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\nz3 reii, r 2 0 = 27-27i, 1 Find the roots of the equation giving your answers in the form where and SIHT\n-r G 1 r. [5] i\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2025-mj-22-q01-ac9c51c42d.pdf"
    },
    {
      "id": "9231-2025-mj-22-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 2,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_22.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-mj-22-q02-p01-43d44a83d1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-22-q02",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\ny 1 n\n`1-xj I = sinhxdx, 2 Let where n is a non-negative integer. SIHT\nn\n0\nNI\nI =-1+n(n-1)I n H 2, . [4] (a) Show that, for\nETIRW n n-2\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\nI . [3] (b) Find the exact value of\n2\nNIGRAM\n............................................................................................................................................................\n.................................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-mj-22-q02-8d118ec29b.pdf"
    },
    {
      "id": "9231-2025-mj-22-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 3,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_22.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_22.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2025-mj-22-q03-p01-dddcdd9ea6.png",
        "9231-topic-25-complex-numbers/assets/9231-2025-mj-22-q03-p02-994fae2397.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-22-q03",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n5 1 SIHT bz- l z = cosi+isini, 3 By considering the binomial expansion of , where use de Moivre’s theorem\nz\nto show that\nNI\nETIRW\na\ncosec5 = , i TON\nsin5i+bsin3i+csini\nOD\nwhere a, b and c are integers to be determined. [6]\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON ....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT ....................................................................................................................................................................\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\nOD ....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM ....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\n.................................................................................................................................................................... NI\nETIRW\n.....................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2025-mj-22-q03-90706719fc.pdf"
    },
    {
      "id": "9231-2025-mj-22-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 4,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_22.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-mj-22-q04-p01-071bc22515.png",
        "9231-topic-24-integration/assets/9231-2025-mj-22-q04-p02-633aeda43a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-22-q04",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4\nSIHT\ny\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nx\n0 1 2 3 n-1 n\nNI\nETIRW\n1\nx The diagram shows the curve with equation y = e for x H 1, together with a set of n-1 rectangles\nx\nof unit width. TON\nOD\n(a) By considering the sum of the areas of these rectangles, show that\n/n\n1 1\nr n e 1 e2+ oe -2e. [5]\nNIGRAM r n\nr=1\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n/n\n1\nr. e [4] (b) Use a similar method to find, in terms of n, a lower bound for\nr\nSIHT\nr=1\n............................................................................................................................................................\nNI\nETIRW\n................",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-mj-22-q04-c1fe05c90c.pdf"
    },
    {
      "id": "9231-2025-mj-22-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_22.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-mj-22-q05-p01-b38aa9cc82.png",
        "9231-topic-26-differential-equations/assets/9231-2025-mj-22-q05-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-22-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 Find the particular solution of the differential equation\nSIHT\nNI\nd2x dx\nETIRW e-t, 6 +3 +6x =\ndt dt2\nTON\ndx\n= = given that, when x 0. t = 0, [10]\nOD dt\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ...................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-mj-22-q05-1dafc5f746.pdf"
    },
    {
      "id": "9231-2025-mj-22-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 6,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_22.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_22.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2025-mj-22-q06-p01-1462d9f833.png",
        "9231-topic-21-hyperbolic-functions/assets/9231-2025-mj-22-q06-p02-387a4bfd0e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-22-q06",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6 (a) Starting from the definitions of tanh and sech in terms of exponentials, prove that\nSIHT\n1-tanh2u sech2u. = [3] NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\nd 1\n(sech-1t) =- . [4] (b) Show that\ndt 1-t2 t\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI .........................................................................................................................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2025-mj-22-q06-7956eb9c82.pdf"
    },
    {
      "id": "9231-2025-mj-22-q07",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_22.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-mj-22-q07-p01-563413f81b.png",
        "9231-topic-26-differential-equations/assets/9231-2025-mj-22-q07-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-22-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n7 Find the solution of the differential equation\nSIHT\nNI\ndy x+5\nETIRW = 1, y -\ndx x2 +10x+61\nTON\ngiven that y = 0 when x = 3. Give your answer in an exact form. [10]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI .....",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-mj-22-q07-a72edba816.pdf"
    },
    {
      "id": "9231-2025-mj-22-q08",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_22.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-mj-22-q08-p01-5d0e232fd4.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-mj-22-q08-p02-aad4398852.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-22-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n8 (a) It is given that is an eigenvalue of the non-singular square matrix A, with corresponding m\nSIHT\neigenvector e.\nNI\nETIRW 3. A3 Show that e is an eigenvector of with corresponding eigenvalue [2] m\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\nThe matrix A is given by\n-1 4 3 NIGRAM\nf p\n= 0 1 A 0 .\n5 -2 0\nSIHT\nNI\nETIRW -1, 1 and 5. [2] (b) Show that the eigenvalues of A are\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n.................................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2025-mj-22-q08-3c0ecb6d60.pdf"
    },
    {
      "id": "9231-2025-mj-23-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 1,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_23.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_23.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2025-mj-23-q01-p01-239080e999.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-23-q01",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "NIGRAM\n1 SIHT eb l x2. 1 Find the Maclaurin’s series for up to and including the term in [5]\nx+2\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD ....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM ....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n...................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2025-mj-23-q01-f6c61bcf23.pdf"
    },
    {
      "id": "9231-2025-mj-23-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 2,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_23.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_23.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2025-mj-23-q02-p01-118d905a4d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-23-q02",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n2 (a) Starting from the definitions of tanh and sech in terms of exponentials, prove that\nSIHT\ntanh2t+sech2t NI = 1. [3]\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n(b) The curve C has parametric equations\ntan-1(sinht), x = ln(cosht), y = for 0 G t G 1.\nNIGRAM\nFind the length of C. [5]\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n...............................................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2025-mj-23-q02-ae514d8b29.pdf"
    },
    {
      "id": "9231-2025-mj-23-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 3,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_23.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_23.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2025-mj-23-q03-p01-17ae03fdf5.png",
        "9231-topic-23-differentiation/assets/9231-2025-mj-23-q03-p02-1b150c9f2a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-23-q03",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 The curve C has equation\nSIHT\n9y2 -3sinh-1(xy) = 1-3ln3. NI\nETIRW\ndy 1\n1) TON (a) Show that, at the point =- on C, . [4] (4,\n2 dx 3\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n..",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2025-mj-23-q03-e7fe9109b3.pdf"
    },
    {
      "id": "9231-2025-mj-23-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 4,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_23.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-mj-23-q04-p01-a0ae3f2ded.png",
        "9231-topic-26-differential-equations/assets/9231-2025-mj-23-q04-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-23-q04",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 Find the particular solution of the differential equation\nSIHT\nNI\nd2x dx\nETIRW 2t2 + -2x = +t-1,\ndt dt2\nTON\ndx\n= = given that, when x 0. t = 0, [10]\nOD dt\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n...........",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-mj-23-q04-dc4c4de006.pdf"
    },
    {
      "id": "9231-2025-mj-23-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 5,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_23.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_23.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2025-mj-23-q05-p01-e21eba0db7.png",
        "9231-topic-25-complex-numbers/assets/9231-2025-mj-23-q05-p02-6e850391ed.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-23-q05",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 (a) Use de Moivre’s theorem to show that\nSIHT\nNI\nsec5 i ETIRW sec5i = . [6]\n5sec4 i-20sec2 i+16\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ........................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2025-mj-23-q05-1c56078b89.pdf"
    },
    {
      "id": "9231-2025-mj-23-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 6,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_23.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-mj-23-q06-p01-ddf52fedd8.png",
        "9231-topic-24-integration/assets/9231-2025-mj-23-q06-p02-f0cb793411.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-23-q06",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6\nSIHT\ny\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nx 0 OD 1 n-1 1 2\nn n n\n1\nThe diagram shows the curve with equation y = for 0 G x G 1, together with a set of n rectangles\nx2 NIGRAM +1 1\nof width .\nn\nSIHT (a) By considering the sum of the areas of these rectangles, show that\nNI\n/n ETIRW 1 n\n1 r. [5]\n4 2 2 +r n\nr=1 TON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n/n\nTON n\n. Give your answer in terms of n (b) Use a similar method to find a lower bound for\n2 +r2 n\nOD\nr=1\nr. [4] and\n...............................................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-mj-23-q06-c187f605c8.pdf"
    },
    {
      "id": "9231-2025-mj-23-q07",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_23.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-mj-23-q07-p01-c69b8e8f75.png",
        "9231-topic-26-differential-equations/assets/9231-2025-mj-23-q07-p02-994fae2397.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-23-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n7 Find the solution of the differential equation\nSIHT\nNI\ndy 2x+6\nETIRW = y 4, -\ndx x2 +6x+5\nTON\ngiven that y = 0 when x = 0. Give your answer in an exact form. [9]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI .......",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-mj-23-q07-7827b5c865.pdf"
    },
    {
      "id": "9231-2025-mj-23-q08",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_23.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-mj-23-q08-p01-c7f5ebbef2.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-mj-23-q08-p02-c097cf76c6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-23-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n8 (a) Find the values of a for which the system of equations\nSIHT\n3 x+3y+8z 1, =\nNI 2\nETIRW ax+3y+4z 2, =\nay- z = 3,\nTON\ndoes not have a unique solution. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\nThe matrix A is given by\nSIHT 3 3 8\n2 f p\nA 0 = . 3 4\nNI\nETIRW 0 0 -1\nA-1, B2 (b) Given that B = = pI+qA, use the characteristic equation of A to show that where p and TON\nq are constants to be determined. [4]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n.............................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2025-mj-23-q08-0e9e5014c1.pdf"
    },
    {
      "id": "9231-2025-mj-24-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 1,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_24.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_24.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-mj-24-q01-p01-9704ddfe37.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-24-q01",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 (a) Find the values of k for which the system of equations\nSIHT\nx+ 2y+3z = 1,\nNI\nETIRW kx+ 5y+6z = 2,\n7x+2ky+9z = 3,\nTON\ndoes not have a unique solution. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\nk = 1, show that the system of equations in part (a) is consistent. Interpret this situation (b) Given that\ngeometrically. [3]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ..",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2025-mj-24-q01-c6e284ef0e.pdf"
    },
    {
      "id": "9231-2025-mj-24-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 2,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_24.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_24.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-mj-24-q02-p01-e9e1e650e0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-24-q02",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n5 y SIHT 1\n2 dx, 2 Find the exact value of giving your answer in logarithmic form. [6]\nx2 -2x+5 1 NI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON ....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n...........",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-mj-24-q02-eb7881425b.pdf"
    },
    {
      "id": "9231-2025-mj-24-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 3,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_24.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_24.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-mj-24-q03-p01-21d76b4e5b.png",
        "9231-topic-26-differential-equations/assets/9231-2025-mj-24-q03-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-24-q03",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 Find the particular solution of the differential equation\nSIHT\nd2y dy\n13e3x +4 +5y = NI\ndx dx2 ETIRW\ndy\nTON = 0 given that y = 1 and when x = 0. [10]\ndx\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-mj-24-q03-fb3b0feccb.pdf"
    },
    {
      "id": "9231-2025-mj-24-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 4,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_24.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_24.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2025-mj-24-q04-p01-3db4326785.png",
        "9231-topic-23-differentiation/assets/9231-2025-mj-24-q04-p02-e64e633979.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-24-q04",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 A curve has parametric equations\nSIHT\nt3 -t2 tet. x = +t-1 and y = NI\nETIRW\nx = 0. [1] (a) Show that 1 is the only real value of t for which\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM (t+1)et dy\n= . [3] (b) Show that\ndx 3t2 -2t+1\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n....................................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2025-mj-24-q04-732ef8244a.pdf"
    },
    {
      "id": "9231-2025-mj-24-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 5,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_24.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_24.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2025-mj-24-q05-p01-7e6c64ca4e.png",
        "9231-topic-25-complex-numbers/assets/9231-2025-mj-24-q05-p02-29c0d74a2d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-24-q05",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 (a) Use de Moivre’s theorem to show that\nSIHT\n=-64sin7 i+112sin5 i-56sin3 sin7i i+7sini. [5] NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n..........................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2025-mj-24-q05-2c21271e5f.pdf"
    },
    {
      "id": "9231-2025-mj-24-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_24.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_24.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-mj-24-q06-p01-32291f18fd.png",
        "9231-topic-26-differential-equations/assets/9231-2025-mj-24-q06-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-24-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6 Find the solution of the differential equation\nSIHT\ndy\n2x2tan-1x x -y = NI\ndx\nETIRW\n1 y = f(x). for which when x = 1. Give your answer in the form [9] y = r\n2\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n..............",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-mj-24-q06-bd1e5ad8f8.pdf"
    },
    {
      "id": "9231-2025-mj-24-q07",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 7,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_24.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_24.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-mj-24-q07-p01-cb67694ab9.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-mj-24-q07-p02-b194b6869b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-24-q07",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n7 The matrix A is given by\nSIHT\n1 7 11 NI\nf p ETIRW A = 0 2 5 .\n0 0 -3\nTON\nA6 PDP-1. (a) Find a matrix P and a diagonal matrix D such that = [7]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ..........................................................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2025-mj-24-q07-adb1ab0f2b.pdf"
    },
    {
      "id": "9231-2025-mj-24-q08",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 8,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_24.pdf?download=true",
      "source_pages": [
        14,
        15,
        16,
        17
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_24.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-mj-24-q08-p01-3ece0df29e.png",
        "9231-topic-24-integration/assets/9231-2025-mj-24-q08-p02-c2eb4c7634.png",
        "9231-topic-24-integration/assets/9231-2025-mj-24-q08-p03-cefb8d2a0c.png",
        "9231-topic-24-integration/assets/9231-2025-mj-24-q08-p04-168aea64e9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-24-q08",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\ny = tanhx 8 The curve C has equation for 0. x H\nSIHT\nNI (a) Sketch C and state the equation of the asymptote. [2]\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\n............................................................................................................................................................\n(b) By considering a suitable set of N rectangles of unit width, use your sketch to show that\nNIGRAM /N\ntanhr 2 ln(coshN). [3]\nr=1\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nNIGRAM\n1 (c) ",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-mj-24-q08-f2ca10d984.pdf"
    },
    {
      "id": "9231-2025-mj-31-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 1,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_31.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_31.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2025-mj-31-q01-p01-d17783117d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-31-q01",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 A particle P of mass 8k g is moving in a straight horizontal line.A t time t s, P has displacement x m from\nSIHT\ns-1. a fixed point O on the line and velocity v m The only horizontal force acting on P has magnitude\n(x3 NI + 4x) N and acts in the direction OP. When t = 0, x = 0 and v = 1.\nETIRW\nax2 v = +b, where a and b (a) Find an expression for v in terms of x, giving your answer in the form\nTON are constants to be determined. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n(b) Find an expression for",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2025-mj-31-q01-5a57ebed15.pdf"
    },
    {
      "id": "9231-2025-mj-31-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 2,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_31.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_31.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-mj-31-q02-p01-e15cff516c.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-mj-31-q02-p02-3454143b54.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-31-q02",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\ns-1 24 m 2 A particle P is projected with speed at an angle i° above the horizontal from a point O on SIHT\na horizontal plane, and moves freely under gravity . At a horizontal distance 35 m from O, there is a\nNI\nvertical wall of height 10 m which is perpendicular to the vertical plane of motion of P.\nETIRW\n(a) Determine the two values of for which P just clears the wall. [4] i\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2025-mj-31-q02-d50777b9c5.pdf"
    },
    {
      "id": "9231-2025-mj-31-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 3,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_31.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-mj-31-q03-p01-79ee24f2c2.png",
        "9231-topic-33-circular-motion/assets/9231-2025-mj-31-q03-p02-23fe940a2b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-31-q03",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3\nSIHT\nA\nNI\nETIRW\na\nTON\nOD\nO 1.5 P m\nNIGRAM\ns-1. rad A rough horizontal disc, centre O, rotates with constant angular speed A particle P of mass ~\n1.6 kg lies on the disc at a distance 1.5 m from O, and is attached to a point A vertically above O by a SIHT\nlight elastic string. The string has natural length 2 m, modulus of elasticity 32 N and makes an angle a\nNI\nwith the vertical OA (see diagram). Particle P moves in a horizontal circle also at a constant angular\nETIRW\ns-1. speed rad Particle P is on the point of slipping in the direction OP. The coefficient of friction ~\nbetween the particle and the disc is 0.5.\nTON\n(a) Given that the tension in the string is 8 N, show that sina = 0.6. [2] OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n.....................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-mj-31-q03-49a801c8e2.pdf"
    },
    {
      "id": "9231-2025-mj-31-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_31.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_31.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-mj-31-q04-p01-a48251aa42.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-mj-31-q04-p02-f628785147.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-31-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4\nSIHT\nNI\nETIRW\nθ\nTON\nOD\nNIGRAM\nSIHT\nNI\n2r 3r ETIRW\nTON An object is formed by removing a solid hemisphere, radius 2r, from a uniform solid cone, radius 3r\n1. and semi-vertical angle i, where The axes of symmetry of the cone and the hemisphere tani =\nOD 2\ncoincide. The base of the cone and the base of the hemisphere are in the same plane as each other (see\ndiagram).\n(a) Find, in terms of r, the distance of the centre of mass of the object from its base. [4]\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ..........................................................................................................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2025-mj-31-q04-6a25e8845c.pdf"
    },
    {
      "id": "9231-2025-mj-31-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 5,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_31.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_31.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2025-mj-31-q05-p01-9df19394ad.png",
        "9231-topic-34-hooke-s-law/assets/9231-2025-mj-31-q05-p02-994fae2397.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-31-q05",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 14 N is attached to\nSIHT\n7 a fixed point A on a smooth plane. The plane makes an angle tana = . to the horizontal, where a\n24\nA particle P of mass 2 kg is attached to the other end of the string. The string lies along a line of greatest NI\nETIRW slope of the plane. The particle P is initially held on the plane above the level of A, where AP = 0.8 m.\nThe particle P is then released from rest.\nTON\nFind the maximum velocity of P during the subsequent motion. [6]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n...................................................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2025-mj-31-q05-47eb385d1f.pdf"
    },
    {
      "id": "9231-2025-mj-31-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 6,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_31.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_31.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2025-mj-31-q06-p01-2eaed03d2d.png",
        "9231-topic-36-momentum/assets/9231-2025-mj-31-q06-p02-c070135150.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-31-q06",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 Two uniform smooth spheres A and B of equal radii have masses 2m and m respectively . Sphere A is\nSIHT\nmoving in a straight horizontal line with speed u, and sphere B is stationary. Sphere A collides directly\nwith B, and they both then move in the same direction with speeds v v and respectively. After the NI\nA B ETIRW 9 collision, the kinetic energy of B is times the kinetic energy of A.\n2\n6 TON v = u. [3] (a) Show that\n5 B\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\nSphere B then collides with a fixed vertical barrier . Immediately before the collision, the direction of\nSIHT\nwith the barrier. Immediately after the collision, the direction of motion motion of B makes an angle a\n4. of B makes an angle with the barrier. The coefficient of restitution between B and the barrier is As b NI\n5\n12 a result of the collision, the velocity of B is reduced to 5u. ETIRW\n25\nsin(a+b). [6] (b) Find the value of TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n..................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2025-mj-31-q06-efff5a98ba.pdf"
    },
    {
      "id": "9231-2025-mj-31-q07",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 7,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_31.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-mj-31-q07-p01-3771145591.png",
        "9231-topic-33-circular-motion/assets/9231-2025-mj-31-q07-p02-453fefc255.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-31-q07",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n7\nSIHT\nNI C B\nETIRW\nθ\nka a\nu\nTON\nO A\nOD\nNIGRAM\nSIHT\nNI\nA fixed hollow sphere has radius a and centre O. The points A, B and C lie on the inner surface of\nETIRW\nthe sphere with OA and BC horizontal. A portion of the sphere has been removed by a horizontal cut\nthrough points B and C at a vertical distance ka above the centre of the sphere, where k is a positive\nTON\nconstant and k 1 1. The points O, A, B and C all lie in the same vertical plane. OB makes an angle i\nOD with the upward vertical through O (see diagram).\nA particle P of mass m is free to move on the smooth inner surface of the sphere. The particle P is\nprojected vertically downwards from A with speed u and begins to move in a vertical circle.\nNIGRAM\n6 (a) In the case where u = ga, the reaction on P at B is half the reaction on P at A.\n5\nSIHT\nFind the value of k. [5]\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n...........................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-mj-31-q07-cc5375c213.pdf"
    },
    {
      "id": "9231-2025-mj-32-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 1,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_32.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_32.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2025-mj-32-q01-p01-d17783117d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-32-q01",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 A particle P of mass 8k g is moving in a straight horizontal line.A t time t s, P has displacement x m from\nSIHT\ns-1. a fixed point O on the line and velocity v m The only horizontal force acting on P has magnitude\n(x3 NI + 4x) N and acts in the direction OP. When t = 0, x = 0 and v = 1.\nETIRW\nax2 v = +b, where a and b (a) Find an expression for v in terms of x, giving your answer in the form\nTON are constants to be determined. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n(b) Find an expression for",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2025-mj-32-q01-3674294922.pdf"
    },
    {
      "id": "9231-2025-mj-32-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 2,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_32.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_32.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-mj-32-q02-p01-e15cff516c.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-mj-32-q02-p02-3454143b54.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-32-q02",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\ns-1 24 m 2 A particle P is projected with speed at an angle i° above the horizontal from a point O on SIHT\na horizontal plane, and moves freely under gravity . At a horizontal distance 35 m from O, there is a\nNI\nvertical wall of height 10 m which is perpendicular to the vertical plane of motion of P.\nETIRW\n(a) Determine the two values of for which P just clears the wall. [4] i\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2025-mj-32-q02-fcebdc7a37.pdf"
    },
    {
      "id": "9231-2025-mj-32-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 3,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_32.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-mj-32-q03-p01-79ee24f2c2.png",
        "9231-topic-33-circular-motion/assets/9231-2025-mj-32-q03-p02-23fe940a2b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-32-q03",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3\nSIHT\nA\nNI\nETIRW\na\nTON\nOD\nO 1.5 P m\nNIGRAM\ns-1. rad A rough horizontal disc, centre O, rotates with constant angular speed A particle P of mass ~\n1.6 kg lies on the disc at a distance 1.5 m from O, and is attached to a point A vertically above O by a SIHT\nlight elastic string. The string has natural length 2 m, modulus of elasticity 32 N and makes an angle a\nNI\nwith the vertical OA (see diagram). Particle P moves in a horizontal circle also at a constant angular\nETIRW\ns-1. speed rad Particle P is on the point of slipping in the direction OP. The coefficient of friction ~\nbetween the particle and the disc is 0.5.\nTON\n(a) Given that the tension in the string is 8 N, show that sina = 0.6. [2] OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n.....................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-mj-32-q03-a5070973bf.pdf"
    },
    {
      "id": "9231-2025-mj-32-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_32.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_32.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-mj-32-q04-p01-a48251aa42.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-mj-32-q04-p02-f628785147.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-32-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4\nSIHT\nNI\nETIRW\nθ\nTON\nOD\nNIGRAM\nSIHT\nNI\n2r 3r ETIRW\nTON An object is formed by removing a solid hemisphere, radius 2r, from a uniform solid cone, radius 3r\n1. and semi-vertical angle i, where The axes of symmetry of the cone and the hemisphere tani =\nOD 2\ncoincide. The base of the cone and the base of the hemisphere are in the same plane as each other (see\ndiagram).\n(a) Find, in terms of r, the distance of the centre of mass of the object from its base. [4]\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ..........................................................................................................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2025-mj-32-q04-9b5413fbf4.pdf"
    },
    {
      "id": "9231-2025-mj-32-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 5,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_32.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_32.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2025-mj-32-q05-p01-9df19394ad.png",
        "9231-topic-34-hooke-s-law/assets/9231-2025-mj-32-q05-p02-994fae2397.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-32-q05",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 14 N is attached to\nSIHT\n7 a fixed point A on a smooth plane. The plane makes an angle tana = . to the horizontal, where a\n24\nA particle P of mass 2 kg is attached to the other end of the string. The string lies along a line of greatest NI\nETIRW slope of the plane. The particle P is initially held on the plane above the level of A, where AP = 0.8 m.\nThe particle P is then released from rest.\nTON\nFind the maximum velocity of P during the subsequent motion. [6]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n...................................................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2025-mj-32-q05-c2242d94bd.pdf"
    },
    {
      "id": "9231-2025-mj-32-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 6,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_32.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_32.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2025-mj-32-q06-p01-2eaed03d2d.png",
        "9231-topic-36-momentum/assets/9231-2025-mj-32-q06-p02-c070135150.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-32-q06",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 Two uniform smooth spheres A and B of equal radii have masses 2m and m respectively . Sphere A is\nSIHT\nmoving in a straight horizontal line with speed u, and sphere B is stationary. Sphere A collides directly\nwith B, and they both then move in the same direction with speeds v v and respectively. After the NI\nA B ETIRW 9 collision, the kinetic energy of B is times the kinetic energy of A.\n2\n6 TON v = u. [3] (a) Show that\n5 B\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\nSphere B then collides with a fixed vertical barrier . Immediately before the collision, the direction of\nSIHT\nwith the barrier. Immediately after the collision, the direction of motion motion of B makes an angle a\n4. of B makes an angle with the barrier. The coefficient of restitution between B and the barrier is As b NI\n5\n12 a result of the collision, the velocity of B is reduced to 5u. ETIRW\n25\nsin(a+b). [6] (b) Find the value of TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n..................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2025-mj-32-q06-fb1195b499.pdf"
    },
    {
      "id": "9231-2025-mj-32-q07",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 7,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_32.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-mj-32-q07-p01-3771145591.png",
        "9231-topic-33-circular-motion/assets/9231-2025-mj-32-q07-p02-453fefc255.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-32-q07",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n7\nSIHT\nNI C B\nETIRW\nθ\nka a\nu\nTON\nO A\nOD\nNIGRAM\nSIHT\nNI\nA fixed hollow sphere has radius a and centre O. The points A, B and C lie on the inner surface of\nETIRW\nthe sphere with OA and BC horizontal. A portion of the sphere has been removed by a horizontal cut\nthrough points B and C at a vertical distance ka above the centre of the sphere, where k is a positive\nTON\nconstant and k 1 1. The points O, A, B and C all lie in the same vertical plane. OB makes an angle i\nOD with the upward vertical through O (see diagram).\nA particle P of mass m is free to move on the smooth inner surface of the sphere. The particle P is\nprojected vertically downwards from A with speed u and begins to move in a vertical circle.\nNIGRAM\n6 (a) In the case where u = ga, the reaction on P at B is half the reaction on P at A.\n5\nSIHT\nFind the value of k. [5]\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n...........................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-mj-32-q07-289b16df90.pdf"
    },
    {
      "id": "9231-2025-mj-33-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 1,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 3,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_33.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-mj-33-q01-p01-7137e17f36.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-33-q01",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end of\nSIHT\nthe string is attached to a fixed point O. The particle moves in a horizontal circle with constant angular\nNI speed and with the string inclined at an angle of to the downward vertical. i ~\nETIRW\n4 , find in terms of a and g. [3] tani = Given that ~\n3\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW ..........................................................................................................................................................\n.......................................................................................................................................................... TON\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n............................................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-mj-33-q01-230972d34b.pdf"
    },
    {
      "id": "9231-2025-mj-33-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 2,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_33.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_33.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2025-mj-33-q02-p01-f67f873e24.png",
        "9231-topic-34-hooke-s-law/assets/9231-2025-mj-33-q02-p02-c507962b4e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-33-q02",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 A particle P of mass m is attached to one end of a light elastic string of natural length a and modulus\nSIHT\nof elasticity mg. The other end of the string is attached to a fixed point O on a rough plane inclined at\nan angle of 30° to the horizontal. The particle P is held at rest at point O before being released. The NI\nETIRW 11mg. frictional force acting on P as it slides down the plane is\n30\nTON (a) Find, in terms of a, the distance that P moves down the plane before coming to rest. [5]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n....................................................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2025-mj-33-q02-fcde687842.pdf"
    },
    {
      "id": "9231-2025-mj-33-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 3,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_33.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_33.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2025-mj-33-q03-p01-f6b2976092.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2025-mj-33-q03-p02-b42af7eec6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-33-q03",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\ns-1 U m 3 A ball of mass m kg is projected vertically upwards with initial speed and moves under gravity. SIHT\ns-1. At time t s after projection, the ball has travelled a distance x m and its speed is v m There is a\nNI mkv2 resistive force of magnitude N, where k is a positive constant.\nETIRW\ng+kU2\n1 TON lnf p. (a) Show that the distance travelled by the ball when it is moving upwards is = x [4]\n2k g+kv2\nOD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ...................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2025-mj-33-q03-3def14b7f3.pdf"
    },
    {
      "id": "9231-2025-mj-33-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_33.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_33.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-mj-33-q04-p01-509818b76a.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-mj-33-q04-p02-286c9c805c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-33-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4\nSIHT\nB C\nNI\nETIRW\na\nE\nTON\nOD\n2a\nF\n2a\nNIGRAM\nA D\nSIHT\na\nNI\nETIRW\nAn object consists of a uniform lamina with a particle attached. The uniform lamina ABCEFD of\nmass m is formed from a rectangle ABCD and an isosceles triangle CEF, where F is the midpoint\nTON\nof CD. The rectangle has sides AB = 2a and AD = a. The triangle CEF has base a and height 2a. The\nparticle of mass km is attached to the lamina at E. The object rests in a vertical plane with its edge AD OD\non horizontal ground (see diagram).\nGiven that the object is on the point of toppling in its vertical plane about the vertex D, find the\nvalue of k. [4] NIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW ..........................................................................................................................................................\nTON ..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\n.......................................................................................................................................................... SIHT\nNI\n..........................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2025-mj-33-q04-d480e570fb.pdf"
    },
    {
      "id": "9231-2025-mj-33-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_33.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-mj-33-q05-p01-bb73a360f6.png",
        "9231-topic-33-circular-motion/assets/9231-2025-mj-33-q05-p02-ac0088d052.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-33-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5\nSIHT\nNI\nB\nETIRW\na\nA TON 3 rg\nr O 2\nOD\nNIGRAM\nA hollow cylinder of radius r is fixed with its axis horizontal. Points A, B and O are in the same vertical\nplane perpendicular to the axis of the cylinder, with A and B on the smooth inner surface and O on the SIHT\naxis. OA and OB make angles 90° and respectively with the upward vertical through O, with A and B a\nNI\non opposite sides of the vertical. A particle of mass m is projected vertically downwards from point A\nETIRW\n3 with speed rg and moves in a vertical circle inside the cylinder (see diagram). The particle loses\n2\ncontact with the cylinder at point B. TON\nOD\n(a) Find the value of a. [4]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n........................................................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-mj-33-q05-5e2dae807e.pdf"
    },
    {
      "id": "9231-2025-mj-33-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 6,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_33.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_33.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2025-mj-33-q06-p01-8ccac1a3a7.png",
        "9231-topic-36-momentum/assets/9231-2025-mj-33-q06-p02-8f98eaf9ca.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-33-q06",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6\nSIHT\nA B\nNI\nu ETIRW\nθ\nTON\nθ\nOD\n2u\nm m\nNIGRAM Two identical uniform smooth spheres A and B, each with mass m, are moving on a horizontal surface\nwith speeds 2u and u respectively when they collide. Immediately before the collision, the spheres are\nmoving parallel to each other in opposite directions such that their directions of motion each make an\nSIHT\nangle with the line of centres (see diagram). As a result of the collision, B moves in a direction which i\nNI is perpendicular to its initial direction of motion. The coefficient of restitution between the spheres is e.\nETIRW\n(a) Find an expression for tani in terms of e. [6]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n...................................................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2025-mj-33-q06-a035808044.pdf"
    },
    {
      "id": "9231-2025-mj-33-q07",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_33.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_33.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-mj-33-q07-p01-9bcf410498.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-mj-33-q07-p02-c5ef21d139.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-33-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n7 A particle P is projected from a point O with speed U at an angle 45° above the horizontal and moves\nSIHT\nfreely under gravity.\nNI\nETIRW (a) State the vertical and horizontal components of velocity at time t. [1]\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nAt time T, particle P is moving at an angle of 60° below the horizontal. OD\nU\n` 6j. T = 2 + [3] (b) Show that\n2g\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n..................................................................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2025-mj-33-q07-3ca5af62d1.pdf"
    },
    {
      "id": "9231-2025-mj-34-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 1,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_34.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_34.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2025-mj-34-q01-p01-c2f4ff4d2f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-34-q01",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 A light spring of natural length a and modulus of elasticity 20mg is placed so that it stands verticallyo n\nSIHT\na horizontal plane. The lower end of the spring is fixed to the plane. A particle of mass m is attached to\nthe upper end of the spring. NI\nETIRW\n3 The particle is pushed vertically downwards until the length of the spring is a. The system is then\n5\nTON released from rest.\nOD\nFind the maximum extension of the spring in the subsequent motion. [5]\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\n.......................................................................................................................................................... SIHT\nNI\n..........................................................................................................................................................\nETIRW\n..........................................................................................................................................................\nTON\nOD ..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM ..........................................................................................................................................................\n..........................................................................................................................................................\nSIHT\n.......................................................................................................................................................... NI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..............................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2025-mj-34-q01-def88ef0b4.pdf"
    },
    {
      "id": "9231-2025-mj-34-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 2,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_34.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_34.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2025-mj-34-q02-p01-2516f6f3d5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-34-q02",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n2 A particle P of mass m kg moves along a horizontal straight line against a resistive force of magnitude\nSIHT\n2mv3N, vms-1 where is the velocity of P at time t s. When t = 0, v = 1.\nNI\nETIRW (a) Find an expression for v in terms of t. [4]\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\nt = 6. [3] (b) Find the displacement of P from its initial position when\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n...................................................................................................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2025-mj-34-q02-ad13a5ea9e.pdf"
    },
    {
      "id": "9231-2025-mj-34-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 3,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_34.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_34.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-mj-34-q03-p01-9f6717f5c4.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-mj-34-q03-p02-1e33e909d3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-34-q03",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 A particle P is projected with speed u at an angle above the horizontal from a point O on a horizontal a\nSIHT\nplane and moves freely under gravity . The horizontal and vertical displacements of P from O at any\nNI subsequent time t are denoted by x and y respectively.\nETIRW\n(a) Derive the equation of the trajectory of P in the form\nTON\ngx2\nsec2 y = xtana- a. [4]\nOD 2u2\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n..............................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2025-mj-34-q03-7bf67ce313.pdf"
    },
    {
      "id": "9231-2025-mj-34-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 4,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_34.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_34.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-mj-34-q04-p01-1cb07a9c70.png",
        "9231-topic-33-circular-motion/assets/9231-2025-mj-34-q04-p02-4c0738819e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-34-q04",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4\nSIHT\nNI\nETIRW A\nTON\nh OD\na\nO\nNIGRAM\nB\nSIHT\nNI A hollow cone with a smooth inner surface is fixed with its vertex O downwards. The semi-vertical\nETIRW 3 angle of the cone is tana = a, where . A light inextensible string has a particle A of mass m attached\n4\nto one end and a particle B of mass m attached to the other end. The string passes through a small hole\nTON in the cone at O. Particle B hangs in equilibrium below O. Particle A is on the inner surface of the cone\nat a height h above the level of O and moves in horizontal circles with constant angular speed (see ~ OD\ndiagram).\nFind in terms of g and h. [6] ~\nNIGRAM\n....................................................................................................................................................................\nSIHT ....................................................................................................................................................................\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\nOD ....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM ....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\n.................................................................................................................................................................... NI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n.........................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-mj-34-q04-f7bbe20204.pdf"
    },
    {
      "id": "9231-2025-mj-34-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 5,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_34.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_34.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-mj-34-q05-p01-a5704d08ac.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-mj-34-q05-p02-366854adda.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-34-q05",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5\nSIHT\nh D E C\nNI\nETIRW\nTON\nOD 6a\nNIGRAM\nA F B h\nSIHT\nABCD is a uniform square lamina of side 6a. Points E and F are on DC and AB respectively and are NI\nETIRW such that DE = FB = h. The quadrilateral BCEF is removed from the square lamina (see diagram).\n(a) Show that the distance of the centre of mass of the resulting lamina AFED from AD is TON\nh2 -6ah+36a2\nand find a corresponding expression for the distance of the centre of mass from AB. OD\n18a\n[5]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n......",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2025-mj-34-q05-ed5365e360.pdf"
    },
    {
      "id": "9231-2025-mj-34-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 6,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_34.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_34.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2025-mj-34-q06-p01-24350e0017.png",
        "9231-topic-36-momentum/assets/9231-2025-mj-34-q06-p02-fe0035b553.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-34-q06",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6\nSIHT\nu\nNI\nA B ETIRW\ni\nTON\nOD\n5m 4m\nTwo uniform smooth spheres A and B of equal radii have masses 5m and 4m respectively . Sphere A is\nNIGRAM\nmoving with speed u on a horizontal surface when it collides with sphere B which is at rest. Immediately\nbefore the collision, A’s direction of motion makes an angle with the line of centres (see diagram). i\nThe coefficient of restitution between the spheres is e. SIHT\nNI 5 (a) Show that the speed of B after the collision is u(1+e)cosi. [3]\nETIRW 9\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n..................................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2025-mj-34-q06-695cb066e0.pdf"
    },
    {
      "id": "9231-2025-mj-34-q07",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 7,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_34.pdf?download=true",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_34.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-mj-34-q07-p01-ed593ccea7.png",
        "9231-topic-33-circular-motion/assets/9231-2025-mj-34-q07-p02-473dcb251b.png",
        "9231-topic-33-circular-motion/assets/9231-2025-mj-34-q07-p03-83248a0c80.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-34-q07",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n7 A particle P of mass m is attached to one end of a light inextensible string of length a. The other end of\nSIHT\nthe string is attached to a fixed point O. The particle moves in complete vertical circles with centre O,\nwith the string taut. When the string makes an angle with the downward vertical through O the speed NI i\nETIRW of P is 4ag. The ratio of the greatest and least tensions in the string during the motion is 11 : 1.\nTON cosi. [8] Find the value of\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-mj-34-q07-1cc5a0a0db.pdf"
    },
    {
      "id": "9231-2025-mj-41-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_41.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-mj-41-q01-p01-ccb0be6c9f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-41-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 The manager of a hardware store is interested in whether there is a dif ference in the amount spent\nSIHT\nper customer on weekdays ($x) compared to weekends ($y). Random samples of 120 customers on\nNI weekdays and 80 customers on weekends are taken and the amount spent by each customer is recorded.\nETIRW The results are summarised as follows.\n/(x-x)2 /(y-y)2 /x /y TON = 10 470 = 12 283 = 6560 = 13520\nOD\nTest at the 1% significance level whether there is a difference in the mean amount spent per customer on\nweekdays compared to weekends. You should not assume that the population variances of the amounts\nspent on weekdays and weekends are equal. [7]\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-mj-41-q01-e8a79f94e7.pdf"
    },
    {
      "id": "9231-2025-mj-41-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 2,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_41.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_41.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2025-mj-41-q02-p01-2ae31448cf.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2025-mj-41-q02-p02-7db3ca2de8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-41-q02",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2\nSIHT\nf(x)\nNI\nETIRW\nTON\nOD\nx 0\n5 8\nNIGRAM\nAs shown in the diagram, the continuous random variable X has probability density function f given by\nSIHT\na 0 G G 5, x\nNI\n* f(x) = 5 8, b-cx G x G ETIRW\n0 otherwise,\nTON\nwhere a, b and c are constants.\nOD\n2 and find the values of b and c. [3] a = (a) Show that\n13\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n.......................................................................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2025-mj-41-q02-1b207cd3a0.pdf"
    },
    {
      "id": "9231-2025-mj-41-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 3,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_41.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_41.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2025-mj-41-q03-p01-fc16b80409.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2025-mj-41-q03-p02-ae14cbe94b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-41-q03",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 Eggs in a supermarket are sold in boxes of six. A supermarket manager wishes to model the number of\nSIHT\nbroken eggs in the boxes sold in the store. A random sample of 2000 boxes is taken and the number of\nbroken eggs recorded. The observed frequencies are shown in the table below. NI\nETIRW\nNumber of broken eggs 0 1 2 3 4 5 6\nTON\nObserved frequency 1844 143 11 0 1 0 1 OD\n(a) Use the data to estimate the probability that an egg is broken. Give your answer correct to\n4 significant figures. [1]\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\nIt is decided to carry out a goodness of fit test at the 0.5% significance level to determine whether a\nbinomial distribution fits the data.\nSIHT\nThe observed frequencies and the expected frequencies are given in the following table.\nNI\nETIRW\nNumber of broken eggs 0 1 2 3 4 5 6\nTON\nObserved frequency 1844 143 11 0 1 0 1\nOD\nExpected frequency 1831.3 a 6.016 0.119 0.001 0.000 0.000\na = 162.6 correct to 1 decimal place. [1] (b) Show that\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\n",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2025-mj-41-q03-d385e43957.pdf"
    },
    {
      "id": "9231-2025-mj-41-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 4,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_41.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-mj-41-q04-p01-af00695f92.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-mj-41-q04-p02-6026ce9c2d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-41-q04",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 A researcher is interested in whether there is a difference between two schools in students’ aptitude for\nSIHT\nEnglish. She randomly chooses 10 students from school X and 8 students of a similar age from school\nY to take a written English test. The scores for the students from school X (x) and school Y (y) are NI\nETIRW summarised as follows.\n/x2 /y2 /x /y TON = 612 = 40104 = 444 = 27 460\nOD\nYou should assume that the two distributions are normal and have the same population variance.\n(a) Find a 95% confidence interval for the dif ference in the mean scores for students from school X\nand students from school Y in the written English test. [6]\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n.................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-mj-41-q04-46a1d638c0.pdf"
    },
    {
      "id": "9231-2025-mj-41-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 5,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_41.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_41.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2025-mj-41-q05-p01-9e06bc79fb.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2025-mj-41-q05-p02-74d93e4432.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-41-q05",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 Two jigsaw puzzles have the same number of pieces with identical shapes but have dif ferent pictures\nSIHT\nprinted on them. One puzzle has a seaside picture and the other has a cartoon picture. A researcher\nbelieves that children will complete the cartoon puzzle more quickly. To test this belief, 10 children are NI\nETIRW randomly selected. The time taken in seconds for each child to complete each puzzle is recorded below.\nTON\nChild A B C D E F G H I J\nOD\nSeaside 182 130 193 181 192 204 184 192 180 189\nCartoon 161 111 195 159 202 200 168 165 145 160\nNIGRAM\n(a) Carry out a Wilcoxon matched-pairs signed-rank test at the 5% significance level to test the\nresearcher’s belief. [6]\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n...............................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2025-mj-41-q05-35ee7cfb64.pdf"
    },
    {
      "id": "9231-2025-mj-41-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 6,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_41.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_41.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2025-mj-41-q06-p01-5c4eb2e9e1.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2025-mj-41-q06-p02-0618a6d635.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-41-q06",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 Y is a discrete random variable which takes the values 0, 2, 4, … . The probability generating function\nSIHT\nof Y is given by\nNI\nk\nETIRW = G (t) .\n1-at2 Y\nTON (a) Find k in terms of a. [1]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\na2. OD P(Y 2 2) = [3] (b) Show that\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n.....................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2025-mj-41-q06-9e69a5a23a.pdf"
    },
    {
      "id": "9231-2025-mj-42-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_42.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-mj-42-q01-p01-ccb0be6c9f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-42-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 The manager of a hardware store is interested in whether there is a dif ference in the amount spent\nSIHT\nper customer on weekdays ($x) compared to weekends ($y). Random samples of 120 customers on\nNI weekdays and 80 customers on weekends are taken and the amount spent by each customer is recorded.\nETIRW The results are summarised as follows.\n/(x-x)2 /(y-y)2 /x /y TON = 10 470 = 12 283 = 6560 = 13520\nOD\nTest at the 1% significance level whether there is a difference in the mean amount spent per customer on\nweekdays compared to weekends. You should not assume that the population variances of the amounts\nspent on weekdays and weekends are equal. [7]\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-mj-42-q01-ee1821560e.pdf"
    },
    {
      "id": "9231-2025-mj-42-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 2,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_42.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_42.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2025-mj-42-q02-p01-2ae31448cf.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2025-mj-42-q02-p02-7db3ca2de8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-42-q02",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2\nSIHT\nf(x)\nNI\nETIRW\nTON\nOD\nx 0\n5 8\nNIGRAM\nAs shown in the diagram, the continuous random variable X has probability density function f given by\nSIHT\na 0 G G 5, x\nNI\n* f(x) = 5 8, b-cx G x G ETIRW\n0 otherwise,\nTON\nwhere a, b and c are constants.\nOD\n2 and find the values of b and c. [3] a = (a) Show that\n13\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n.......................................................................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2025-mj-42-q02-429e8d9620.pdf"
    },
    {
      "id": "9231-2025-mj-42-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 3,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_42.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_42.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2025-mj-42-q03-p01-fc16b80409.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2025-mj-42-q03-p02-ae14cbe94b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-42-q03",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 Eggs in a supermarket are sold in boxes of six. A supermarket manager wishes to model the number of\nSIHT\nbroken eggs in the boxes sold in the store. A random sample of 2000 boxes is taken and the number of\nbroken eggs recorded. The observed frequencies are shown in the table below. NI\nETIRW\nNumber of broken eggs 0 1 2 3 4 5 6\nTON\nObserved frequency 1844 143 11 0 1 0 1 OD\n(a) Use the data to estimate the probability that an egg is broken. Give your answer correct to\n4 significant figures. [1]\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\nIt is decided to carry out a goodness of fit test at the 0.5% significance level to determine whether a\nbinomial distribution fits the data.\nSIHT\nThe observed frequencies and the expected frequencies are given in the following table.\nNI\nETIRW\nNumber of broken eggs 0 1 2 3 4 5 6\nTON\nObserved frequency 1844 143 11 0 1 0 1\nOD\nExpected frequency 1831.3 a 6.016 0.119 0.001 0.000 0.000\na = 162.6 correct to 1 decimal place. [1] (b) Show that\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\n",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2025-mj-42-q03-9b26e53ce2.pdf"
    },
    {
      "id": "9231-2025-mj-42-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 4,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_42.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-mj-42-q04-p01-af00695f92.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-mj-42-q04-p02-6026ce9c2d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-42-q04",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 A researcher is interested in whether there is a difference between two schools in students’ aptitude for\nSIHT\nEnglish. She randomly chooses 10 students from school X and 8 students of a similar age from school\nY to take a written English test. The scores for the students from school X (x) and school Y (y) are NI\nETIRW summarised as follows.\n/x2 /y2 /x /y TON = 612 = 40104 = 444 = 27 460\nOD\nYou should assume that the two distributions are normal and have the same population variance.\n(a) Find a 95% confidence interval for the dif ference in the mean scores for students from school X\nand students from school Y in the written English test. [6]\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n.................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-mj-42-q04-d01f258b8b.pdf"
    },
    {
      "id": "9231-2025-mj-42-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 5,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_42.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_42.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2025-mj-42-q05-p01-9e06bc79fb.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2025-mj-42-q05-p02-74d93e4432.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-42-q05",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 Two jigsaw puzzles have the same number of pieces with identical shapes but have dif ferent pictures\nSIHT\nprinted on them. One puzzle has a seaside picture and the other has a cartoon picture. A researcher\nbelieves that children will complete the cartoon puzzle more quickly. To test this belief, 10 children are NI\nETIRW randomly selected. The time taken in seconds for each child to complete each puzzle is recorded below.\nTON\nChild A B C D E F G H I J\nOD\nSeaside 182 130 193 181 192 204 184 192 180 189\nCartoon 161 111 195 159 202 200 168 165 145 160\nNIGRAM\n(a) Carry out a Wilcoxon matched-pairs signed-rank test at the 5% significance level to test the\nresearcher’s belief. [6]\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n...............................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2025-mj-42-q05-633e7ec0d5.pdf"
    },
    {
      "id": "9231-2025-mj-42-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 6,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_42.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_42.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2025-mj-42-q06-p01-5c4eb2e9e1.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2025-mj-42-q06-p02-0618a6d635.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-42-q06",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 Y is a discrete random variable which takes the values 0, 2, 4, … . The probability generating function\nSIHT\nof Y is given by\nNI\nk\nETIRW = G (t) .\n1-at2 Y\nTON (a) Find k in terms of a. [1]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\na2. OD P(Y 2 2) = [3] (b) Show that\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n.....................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2025-mj-42-q06-fb67ce5f30.pdf"
    },
    {
      "id": "9231-2025-mj-43-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 1,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_43.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_43.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2025-mj-43-q01-p01-2e6d184876.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-43-q01",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 A person’s eye colour may be categorised as “brown”, “blue” or “other”. A scientist claims that these\nSIHT\neye colours are uniformly distributed and hence are equally likely to occur in the population. A survey\nof 120 people from this population found that 38 people had brown eyes, 52 people had blue eyes and NI\nETIRW 30 people had eyes which were neither brown nor blue.\nTON Use the data to carry out a goodness of fit test at the 5% significance level to test the scientist’ s claim.\n[6]\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW ..........................................................................................................................................................\n.......................................................................................................................................................... TON\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\n",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2025-mj-43-q01-6c5a40654e.pdf"
    },
    {
      "id": "9231-2025-mj-43-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 2,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_43.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-mj-43-q02-p01-be2af92bf8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-43-q02",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n2 A farmer is investigating whether using a new fertiliser will increase the y ield of tomato plants. The\nSIHT\nfarmer selects 40 tomato plants at random and gives them the new fertiliser . The crop mass, x kg, of\nNI each of these 40 plants is recorded. The farmer selects a further 60 tomato plants at random and gives\nETIRW them a standard fertiliser. The crop mass, y kg, of each of these 60 plants is recorded. The results are\nsummarised as follows.\nTON\n/x / /x2 y2 / = 168 y = 228 720 900 = =\nOD\nFind a 90% confidence interval for the dif ference in mean crop mass associated with each type of\nfertiliser. [7]\nNIGRAM\n..........................................................................................................................................................\n.......................................................................................................................................................... SIHT\nNI\n..........................................................................................................................................................\nETIRW\n..........................................................................................................................................................\nTON\nOD ..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM ..........................................................................................................................................................\n..........................................................................................................................................................\nSIHT\n.......................................................................................................................................................... NI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n.................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-mj-43-q02-056af63c37.pdf"
    },
    {
      "id": "9231-2025-mj-43-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 3,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_43.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_43.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2025-mj-43-q03-p01-67a5012dbc.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2025-mj-43-q03-p02-a46f8bad4b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-43-q03",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 A continuous random variable X has probability density function f given by\nSIHT\nZ\nNI ]kx 0 G x 11,\nETIRW\n]]\nf(x) = [k(8-x) 1 G x G 8,\n] TON\n]\n0 otherwise,\n\\ OD\nwhere k is a constant.\n1 . [2] k = (a) Show that\nNIGRAM\n25\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n(b) Find the median value of X. [3]\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\nTON ...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\n................................................................................................................................................... OD\n...................................................................................................................................................\n...................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2025-mj-43-q03-bae17d761e.pdf"
    },
    {
      "id": "9231-2025-mj-43-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 4,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_43.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_43.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2025-mj-43-q04-p01-d3cd2f4d29.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2025-mj-43-q04-p02-5e32caa22a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-43-q04",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 A researcher claims that older people take longer to react to a sudden loud noise than younger people.\nSIHT\nTo investigate this, the researcher randomly selects 6 people over 50 years old and 8 people under 25\nyears old and records their reaction times, in milliseconds, to a sudden loud noise. The reaction times NI\nETIRW are as follows.\nTON Over 50 198 212 217 229 235 242\nOD\nUnder 25 178 181 183 192 203 209 223 231\nCarry out a Wilcoxon rank-sum test at the 5% significance level to test the researcher’s claim. [8]\nNIGRAM\n..........................................................................................................................................................\n.......................................................................................................................................................... SIHT\nNI\n..........................................................................................................................................................\nETIRW\n..........................................................................................................................................................\nTON\nOD ..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM ..........................................................................................................................................................\n..........................................................................................................................................................\nSIHT\n.......................................................................................................................................................... NI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..............................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2025-mj-43-q04-db743240e6.pdf"
    },
    {
      "id": "9231-2025-mj-43-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 5,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_43.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-mj-43-q05-p01-dff7d54838.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-mj-43-q05-p02-07d4aec870.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-43-q05",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 A doctor is investigating the concentration of blood glucose in patients at risk of developing type 2\nSIHT\ndiabetes, where blood glucose is measured in appropriate units. The doctor claims that a particular\nintervention reduces the concentration by more than k units on average. A group of 8 at risk patients NI\nETIRW is selected at random and each patient follows the intervention for six months. The blood glucose\nconcentrations before and after the intervention are given in the following table.\nTON\nPatient A B C D E F G H OD\nBefore 183 165 172 165 143 176 161 153\nAfter 164 148 164 149 134 153 155 148\nNIGRAM\nSIHT\nNI\n(a) Use a t-test at the 5% significance level to find the range of values of k for which the result of the\nETIRW\ntest is to reject the null hypothesis. [7]\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\n.........................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-mj-43-q05-0992cd8bf2.pdf"
    },
    {
      "id": "9231-2025-mj-43-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 6,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_43.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_43.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2025-mj-43-q06-p01-10cac348be.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2025-mj-43-q06-p02-3ed6290066.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-43-q06",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6 A bag contains 7 red balls and 3 blue balls. Kieran selects 2 balls at random, without replacement. The\nSIHT\nnumber of red balls selected by Kieran is denoted by X, and the number of different colours present in\nKieran’s selection is denoted by Y. NI\nETIRW\n(a) Find the probability generating functions, G (t) G (t) of X and of Y. [4]\nX Y\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n..........................................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2025-mj-43-q06-c2490e807e.pdf"
    },
    {
      "id": "9231-2025-mj-44-q01",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "44",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_44.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_44.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-mj-44-q01-p01-a08133c96c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-44-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 A random sample of 12 observations of a normal random variable is taken. The results give unbiased\nSIHT\nestimates for the population mean and variance as 10.24 and 0.52 respectively.\nNI\nETIRW Test, at the 10% significance level, the null hypothesis that the population mean is 10.6 against the\nalternative hypothesis that the population mean is less than 10.6. [4]\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n.................................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-mj-44-q01-53436c9e6a.pdf"
    },
    {
      "id": "9231-2025-mj-44-q02",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "44",
      "question_number": 2,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_44.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_44.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2025-mj-44-q02-p01-5fad708269.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-44-q02",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n2 The level of sound produced by a particular type of machine was measured for a random sample of 1 1\nSIHT\nsuch machines. The results, in suitable units, are shown below.\nNI\nETIRW\nMachine A B C D E F G H I J K\nTON Sound level 7.66 8.48 8.21 7.98 8.01 7.77 8.25 8.11 8.03 8.16 7.92\nOD\n(a) Use a Wilcoxon signed-rank test to test whether the average sound level produced by this type of\nmachine is more than 8.00. Use a 5% significance level. [6]\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n..............",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2025-mj-44-q02-6ea4232a93.pdf"
    },
    {
      "id": "9231-2025-mj-44-q03",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "44",
      "question_number": 3,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_44.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_44.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2025-mj-44-q03-p01-25c1129cfd.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2025-mj-44-q03-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-44-q03",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 A shop selling electrical goods has a team of three salespeople: Avril, Ben and Charlie. The manager\nSIHT\nwishes to investigate whether the salespeople are equally successful at selling particular types of items.\nThe following table gives a record of a random sample of 250 sales of laptops, cameras and televisions, NI\nETIRW with the number sold by each of the three salespeople.\nTON\nType of item\nOD\nLaptop Camera Television Total\nAvril 31 40 24 95\nBen 23 45 29 97\nNIGRAM\nCharlie 21 25 12 58\nSIHT\nTotal 75 110 65 250\nNI\nETIRW Test, at the 10% significance level, whether there is independence between the type of item sold and the\nsalesperson. [7]\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n..........................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2025-mj-44-q03-80d109ea97.pdf"
    },
    {
      "id": "9231-2025-mj-44-q04",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "44",
      "question_number": 4,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_44.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_44.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2025-mj-44-q04-p01-b91e7f51dd.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2025-mj-44-q04-p02-87c085259d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-44-q04",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 The continuous random variable X has probability density function f given by\nSIHT\nNI kx 0 G x 1 1,\nETIRW * kx2 f(x) x 2, = 1 G G\notherwise. 0\nTON\n6 k = (a) Show that . [2] OD\n17\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n(b) Find the cumulative distribution function of X. [3]\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2025-mj-44-q04-a09bc40d67.pdf"
    },
    {
      "id": "9231-2025-mj-44-q05",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "44",
      "question_number": 5,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_44.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_44.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2025-mj-44-q05-p01-c6e1511199.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2025-mj-44-q05-p02-8fd3878776.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-44-q05",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 Eric has three identical coins, each of which is biased so that the probability of obtaining a head when it\nSIHT\n1. is thrown is The random variable X is the number of heads obtained when Eric throws the three coins\n3 NI\nat the same time.\nETIRW\n(a) Find the probability generating function G (t) of X. [2]\nTON X\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\nEric also has two fair 6-sided dice with faces numbered 1 to 6. The random variable Y is the number of\nNIGRAM\nsixes obtained when Eric throws the two dice at the same time. It is given that the probability generating\n25 10 1 t2. + t+ function of Y is\n36 36 36\nSIHT\nEric throws the three coins and the two dice. The random variable Z is the sum of the number of heads\nNI obtained and the number of sixes obtained.\nETIRW\n(b) Find the probability generating function G (t) of Z, expressing your answer as a polynomial in t.\nZ TON\n[3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n...........................................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2025-mj-44-q05-180f276b0f.pdf"
    },
    {
      "id": "9231-2025-mj-44-q06",
      "subject": "9231",
      "year": 2025,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "44",
      "question_number": 6,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-may-june/9231_s25_qp_44.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-May-June/qp/9231_s25_qp_44.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-mj-44-q06-p01-bc0ca6ff5f.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-mj-44-q06-p02-111aa1e80e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-mj-44-q06",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6 Lina and Mona are two statisticians who also write songs.T he ‘time’ of a song is the number of minutes\nSIHT\nfor which it lasts. For a random sample of 10 of her songs, Lina calculates a 95% confidence interval\nfor the population mean time, minutes. This confidence interval is 3.13. The times, 2.95 G G NI n n\nETIRW x minutes, of Lina’s songs are normally distributed.\n/x2 /x TON (a) Find the values of and for the 10 songs in Lina’s sample. [5]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n...............",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-mj-44-q06-93bde727db.pdf"
    },
    {
      "id": "9231-2025-on-11-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_11.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_11.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2025-on-11-q01-p01-058239e2df.png",
        "9231-topic-13-summation-of-series/assets/9231-2025-on-11-q01-p02-8677632ea7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-11-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n/n\nSIHT\n`8r3 +12r2 +4r+3j 1 (a) Use standard results from the list of formulae (MF19) to find in terms of\nn, simplifying your answer. [3] NI r=1\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\n................................................................................................................................................... ETIRW\n...................................................................................................................................................\nTON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n...................................................................................................................................................\nSIHT\nNI ....................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2025-on-11-q01-ee02e24745.pdf"
    },
    {
      "id": "9231-2025-on-11-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 2,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_11.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_11.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-on-11-q02-p01-748141343d.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-on-11-q02-p02-1c56097d17.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-11-q02",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 The matrices A and B are given by\nSIHT\n1 0 3 1 NI\nf p f p. and B = A = 2 3 ETIRW 1 0 1\n2\n(a) Give full details of the geometrical transformation in the x-y plane represented by A. [2] TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\n(b) Give full details of the geometrical transformation in the x-y plane represented by B. [2] OD\n...................................................................................................................................................\n................................................................................................................................................... NIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\nThe triangle DEF in the x-y plane is transformed by AB onto triangle PQR.\n(c) Show that the triangles DEF and PQR have the same area. [2]\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON ...................................................................................................................................................\nOD\n.......",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2025-on-11-q02-d5d6b5f71b.pdf"
    },
    {
      "id": "9231-2025-on-11-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 3,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_11.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_11.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2025-on-11-q03-p01-96dc918388.png",
        "9231-topic-17-proof-by-induction/assets/9231-2025-on-11-q03-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-11-q03",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 Prove by mathematical induction that, for every positive integer n,\nSIHT\nd2n-1\n(-1)n(xsinx- (xcosx) (2n-1)cosx). = [7] NI\ndx2n-1 ETIRW\nTON ....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n......................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2025-on-11-q03-3fe5cdfa76.pdf"
    },
    {
      "id": "9231-2025-on-11-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 4,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_11.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_11.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-on-11-q04-p01-abf90eb78a.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-on-11-q04-p02-dd56edf82a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-11-q04",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\nx4 +x32+x +x+1 = 0 4 The quartic equation has roots a, b, c, d. SIHT\nNI\n2a+1, 2b+1, 2c+1, 2d+1 is (a) Show that a quartic equation with roots\nETIRW\ny4 -2y32+4y2 +1y+ 1 = 0. [4]\nTON\n................................................................................................................................................... OD\n...................................................................................................................................................\n................................................................................................................................................... NIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON ...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT ...................................................................................................................................................\nNI\n..................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2025-on-11-q04-2d7872f15c.pdf"
    },
    {
      "id": "9231-2025-on-11-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 5,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_11.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_11.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2025-on-11-q05-p01-20b91f85d3.png",
        "9231-topic-16-vectors/assets/9231-2025-on-11-q05-p02-b3f4458456.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-11-q05",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\nP 5 The plane has equation r =-3i-j-k+m(j+2k) +n(i+3j+k).\nSIHT\n1\nNI P ax+by+cz = d. in the form [4] (a) Find an equation for\n1 ETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\nP -i-2k to . [3] (b) Find the perpendicular distance from the point with position vector\n1\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI\n...................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2025-on-11-q05-981dc7b17b.pdf"
    },
    {
      "id": "9231-2025-on-11-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_11.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_11.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2025-on-11-q06-p01-8733ab1bcb.png",
        "9231-topic-15-polar-coordinates/assets/9231-2025-on-11-q06-p02-189783568b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-11-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n1 r = sin3i, 6 The curve C has polar equation for r. 0 G G i SIHT\n3\nNI\n(a) Sketch C and state the equation of the line of symmetry. [3]\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\n...................................................................................................................................................\nTON (b) Find the exact value of the area of the region enclosed by C. [4]\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT ...................................................................................................................................................\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\nOD ...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM ...................................................................................................................................................\n...................................................................................................................................................\nSIHT\n................................................................................................................................................... NI\nETIRW\n...................................................................................................................................................\nTON\nNIGRAM\n3sini-4sin3 sin3i / In parts (c) and (d) you may use the identity i. SIHT\nNI\n(c) Find the maximum distance of a point on C from the initial line. [5]\nETIRW\n...................................................................................................................................................\nTON\n................................................................................................................................................... OD\n...................................................................................................................................................\n...................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2025-on-11-q06-424446ed3a.pdf"
    },
    {
      "id": "9231-2025-on-11-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "11",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_11.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_11.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-on-11-q07-p01-c07f313af1.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-on-11-q07-p02-59f5c9b076.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-on-11-q07-p03-8b44550d46.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-11-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\nx+2\nSIHT 7 The curve C has equation y = .\nx2 +3x+1\nNI\n(a) Find the equations of the asymptotes of C. [2]\nETIRW\n...................................................................................................................................................\nTON\nOD ...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM ...................................................................................................................................................\n...................................................................................................................................................\nSIHT\n................................................................................................................................................... NI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n(b) Show that C has no stationary points. [3]\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\n......................................................................................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2025-on-11-q07-06e35b8ecd.pdf"
    },
    {
      "id": "9231-2025-on-12-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_12.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_12.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2025-on-12-q01-p01-020f43c8ee.png",
        "9231-topic-13-summation-of-series/assets/9231-2025-on-12-q01-p02-a66ffcac63.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-12-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n/n\nSIHT\n`r3 -rj 1 (a) Use standard results from the list of formulae (MF19) to find in terms of n, fully\nNI r=1\nfactorising your answer. [3] ETIRW\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n...................................................................................................................................................\nSIHT\nNI ...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ..................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2025-on-12-q01-e7ba1e4ea4.pdf"
    },
    {
      "id": "9231-2025-on-12-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 2,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_12.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_12.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-on-12-q02-p01-4ef7afde49.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-on-12-q02-p02-53428c454a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-12-q02",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\nx3 +bx2 +cx+d = 0, 2 The cubic equation where b, c and d are constants, has roots a, and c. It is b SIHT\ngiven that\nNI\nETIRW\na+bc+ = 2,\n2 2 2 +b +c = 3, a\nTON\n4 4 4 +b +c = 5. a\nOD\n(a) Find the values of b and c. [3]\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON ...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT ...................................................................................................................................................\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\nOD ...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM ...................................................................................................................................................\n...................................................................................................................................................\nSIHT\n................................................................................................................................................... NI\nETIRW\n...................................................................................................................................................\nTON\n.......................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2025-on-12-q02-a6693e1bf7.pdf"
    },
    {
      "id": "9231-2025-on-12-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 3,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_12.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_12.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2025-on-12-q03-p01-df7b347fec.png",
        "9231-topic-17-proof-by-induction/assets/9231-2025-on-12-q03-p02-0ce4929f9d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-12-q03",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\nu , u ,,u u 1 5 3 The sequence of positive numbers … is such that and, for n H 1,\nSIHT 1 23 1\nNI\n6u +5\nETIRW n u = .\nu +2 n+1\nn\nTON\n(a) By considering 5-u , u 1 5 prove by mathematical induction that for all positive integers n.\nOD\nn+1 n\n[5]\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT ...................................................................................................................................................\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\nOD ...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM ...................................................................................................................................................\n...................................................................................................................................................\nSIHT\n................................................................................................................................................... NI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n..............................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2025-on-12-q03-45378d9cfa.pdf"
    },
    {
      "id": "9231-2025-on-12-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_12.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_12.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-on-12-q04-p01-fd18079b75.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-on-12-q04-p02-4ec106f360.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-12-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 Let k and m be non-zero constants. The matrices A, B and C are given by\nSIHT\n0 1 NI\n2 -1 1 0 k\nf p ETIRW e o e o. B = C = A = -1 1 , and\n1 1 2 0 m\n1 1\nTON\n(a) Give full details of the geometrical transformation in the x-y plane represented by the matrix B in\nOD\neach of the following cases.\n(i) m = 1 [2]\nNIGRAM\n....................................................................................................................................................\n.................................................................................................................................................... SIHT\nNI\n(ii) m = k [2]\nETIRW\n....................................................................................................................................................\nTON\n.................................................................................................................................................... OD\n(b) Show that the matrix ABC is singular. [6]\nNIGRAM ...................................................................................................................................................\n...................................................................................................................................................\nSIHT\n................................................................................................................................................... NI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n.............................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2025-on-12-q04-0cc322ca63.pdf"
    },
    {
      "id": "9231-2025-on-12-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_12.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_12.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2025-on-12-q05-p01-c91fb130b6.png",
        "9231-topic-15-polar-coordinates/assets/9231-2025-on-12-q05-p02-13edc1cd15.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-12-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n1 r2 = tan2i, 5 The curve C has polar equation where r. 0 G G i SIHT\n8\nNI\n(a) Sketch C and state the greatest distance of a point on C from the pole. [2]\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\n...................................................................................................................................................\nNI\nETIRW 1 (b) Find the exact value of the area of the region bounded by C and the half-line r. [4] = i\n8\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nNIGRAM\ncosb1 SIHT x4 -2xy-y4 rl = 0 given that (c) Show that C has Cartesian equation and 0 G x G\n8\nsinb1 NI rl. 0 G y G\n8 ETIRW\n[4]\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n....",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2025-on-12-q05-f5f52481b5.pdf"
    },
    {
      "id": "9231-2025-on-12-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 6,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_12.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_12.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2025-on-12-q06-p01-70f6288d51.png",
        "9231-topic-16-vectors/assets/9231-2025-on-12-q06-p02-51544458d9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-12-q06",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 The plane has equation x+3yz+2 = 1. P\nSIHT\nNI P. [2] (a) Find the perpendicular distance from the origin O to the plane\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\nRelative to O, the points A, B, C have position vectors\nSIHT\n-j+2k, 2i-k, 2i-jk- ,\nNI\nETIRW\nrespectively.\nTON\nP. [4] (b) Find the acute angle between the planes OAB and\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\n................................................................................................................................................... OD",
      "pdf_path": "9231-topic-16-vectors/print/9231-2025-on-12-q06-5558b1f37d.pdf"
    },
    {
      "id": "9231-2025-on-12-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "12",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_12.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_12.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-on-12-q07-p01-3570256d03.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-on-12-q07-p02-4eeff32590.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-on-12-q07-p03-e12838f4a9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-12-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\nx2 +x+1 SIHT\n= y 7 The curve C has equation .\nx+1\nNI\n(a) Find the equations of the asymptotes of C. [3] ETIRW\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n...................................................................................................................................................\nSIHT\nNI ...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n(b) Find the coordinates of any stationary points on C. [3]\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n...........................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2025-on-12-q07-f1f99a435b.pdf"
    },
    {
      "id": "9231-2025-on-13-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_13.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_13.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2025-on-13-q01-p01-058239e2df.png",
        "9231-topic-13-summation-of-series/assets/9231-2025-on-13-q01-p02-8677632ea7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-13-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n/n\nSIHT\n`8r3 +12r2 +4r+3j 1 (a) Use standard results from the list of formulae (MF19) to find in terms of\nn, simplifying your answer. [3] NI r=1\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\n................................................................................................................................................... ETIRW\n...................................................................................................................................................\nTON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n...................................................................................................................................................\nSIHT\nNI ....................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2025-on-13-q01-aab1c7ed9d.pdf"
    },
    {
      "id": "9231-2025-on-13-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 2,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_13.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_13.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-on-13-q02-p01-748141343d.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-on-13-q02-p02-1c56097d17.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-13-q02",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 The matrices A and B are given by\nSIHT\n1 0 3 1 NI\nf p f p. and B = A = 2 3 ETIRW 1 0 1\n2\n(a) Give full details of the geometrical transformation in the x-y plane represented by A. [2] TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\n(b) Give full details of the geometrical transformation in the x-y plane represented by B. [2] OD\n...................................................................................................................................................\n................................................................................................................................................... NIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\nThe triangle DEF in the x-y plane is transformed by AB onto triangle PQR.\n(c) Show that the triangles DEF and PQR have the same area. [2]\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON ...................................................................................................................................................\nOD\n.......",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2025-on-13-q02-3d2935b5f6.pdf"
    },
    {
      "id": "9231-2025-on-13-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 3,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_13.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_13.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2025-on-13-q03-p01-96dc918388.png",
        "9231-topic-17-proof-by-induction/assets/9231-2025-on-13-q03-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-13-q03",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 Prove by mathematical induction that, for every positive integer n,\nSIHT\nd2n-1\n(-1)n(xsinx- (xcosx) (2n-1)cosx). = [7] NI\ndx2n-1 ETIRW\nTON ....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n......................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2025-on-13-q03-b8297576cb.pdf"
    },
    {
      "id": "9231-2025-on-13-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 4,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_13.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_13.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-on-13-q04-p01-abf90eb78a.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-on-13-q04-p02-dd56edf82a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-13-q04",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\nx4 +x32+x +x+1 = 0 4 The quartic equation has roots a, b, c, d. SIHT\nNI\n2a+1, 2b+1, 2c+1, 2d+1 is (a) Show that a quartic equation with roots\nETIRW\ny4 -2y32+4y2 +1y+ 1 = 0. [4]\nTON\n................................................................................................................................................... OD\n...................................................................................................................................................\n................................................................................................................................................... NIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON ...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT ...................................................................................................................................................\nNI\n..................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2025-on-13-q04-68789f6c42.pdf"
    },
    {
      "id": "9231-2025-on-13-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 5,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_13.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_13.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2025-on-13-q05-p01-20b91f85d3.png",
        "9231-topic-16-vectors/assets/9231-2025-on-13-q05-p02-b3f4458456.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-13-q05",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\nP 5 The plane has equation r =-3i-j-k+m(j+2k) +n(i+3j+k).\nSIHT\n1\nNI P ax+by+cz = d. in the form [4] (a) Find an equation for\n1 ETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\nP -i-2k to . [3] (b) Find the perpendicular distance from the point with position vector\n1\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI\n...................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2025-on-13-q05-382248e6b5.pdf"
    },
    {
      "id": "9231-2025-on-13-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_13.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_13.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2025-on-13-q06-p01-8733ab1bcb.png",
        "9231-topic-15-polar-coordinates/assets/9231-2025-on-13-q06-p02-189783568b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-13-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n1 r = sin3i, 6 The curve C has polar equation for r. 0 G G i SIHT\n3\nNI\n(a) Sketch C and state the equation of the line of symmetry. [3]\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\n...................................................................................................................................................\nTON (b) Find the exact value of the area of the region enclosed by C. [4]\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT ...................................................................................................................................................\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\nOD ...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM ...................................................................................................................................................\n...................................................................................................................................................\nSIHT\n................................................................................................................................................... NI\nETIRW\n...................................................................................................................................................\nTON\nNIGRAM\n3sini-4sin3 sin3i / In parts (c) and (d) you may use the identity i. SIHT\nNI\n(c) Find the maximum distance of a point on C from the initial line. [5]\nETIRW\n...................................................................................................................................................\nTON\n................................................................................................................................................... OD\n...................................................................................................................................................\n...................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2025-on-13-q06-9e58f9918b.pdf"
    },
    {
      "id": "9231-2025-on-13-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "13",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_13.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_13.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-on-13-q07-p01-c07f313af1.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-on-13-q07-p02-59f5c9b076.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-on-13-q07-p03-8b44550d46.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-13-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\nx+2\nSIHT 7 The curve C has equation y = .\nx2 +3x+1\nNI\n(a) Find the equations of the asymptotes of C. [2]\nETIRW\n...................................................................................................................................................\nTON\nOD ...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM ...................................................................................................................................................\n...................................................................................................................................................\nSIHT\n................................................................................................................................................... NI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n(b) Show that C has no stationary points. [3]\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\n......................................................................................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2025-on-13-q07-bb7413eedc.pdf"
    },
    {
      "id": "9231-2025-on-14-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "14",
      "question_number": 1,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_14.pdf?download=true",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_14.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2025-on-14-q01-p01-dd60aff18b.png",
        "9231-topic-17-proof-by-induction/assets/9231-2025-on-14-q01-p02-6498a91e69.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-14-q01",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\nn\nb6l 1n SIHT 1 Prove by mathematical induction that for all positive integers n. [5] 1+ H\n5 5\nNI\n.......................................................................................................................................................... ETIRW\n..........................................................................................................................................................\nTON\nOD ..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM ..........................................................................................................................................................\n..........................................................................................................................................................\nSIHT\n.......................................................................................................................................................... NI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n........................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2025-on-14-q01-a99c8c9097.pdf"
    },
    {
      "id": "9231-2025-on-14-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "14",
      "question_number": 2,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_14.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_14.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-on-14-q02-p01-3d91d66c60.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2025-on-14-q02-p02-cbbbaa1781.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-14-q02",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n1\nx3 +bx2 SIHT +cx+d = 0, d ! 0, where has roots a, b, such that = 2 The cubic equation . c c\nb\nNI 1\nd-b. [3] b+ = (a) Show that\nETIRW\nb\n...................................................................................................................................................\nTON\nOD ...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM ...................................................................................................................................................\n...................................................................................................................................................\nSIHT\n................................................................................................................................................... NI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW 1 1-c\nb+ = (b) Show also that . [2]\nd b\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\n..................................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2025-on-14-q02-ad39deded9.pdf"
    },
    {
      "id": "9231-2025-on-14-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "14",
      "question_number": 3,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_14.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_14.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2025-on-14-q03-p01-0e56966109.png",
        "9231-topic-13-summation-of-series/assets/9231-2025-on-14-q03-p02-82dd8d0d7f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-14-q03",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 (a) Use standard results from the list of formulae (MF19) to show that\nSIHT\nNI /n\nETIRW 1n`n3 +bn2 +cn+dj, (r+1)(r+23)(r+ ) =\n4\nr=1\nTON\nwhere b, c and d are integers to be determined. [3]\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n...................................................................................................................................................\nSIHT\nNI ...................................................................................................................................................\nETIRW\n...................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2025-on-14-q03-bfdb8426b9.pdf"
    },
    {
      "id": "9231-2025-on-14-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "14",
      "question_number": 4,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_14.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_14.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2025-on-14-q04-p01-d6b335658c.png",
        "9231-topic-16-vectors/assets/9231-2025-on-14-q04-p02-c8280c59cb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-14-q04",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 The points A, B, C have position vectors\nSIHT\n3i+5jk+5 , 2i+2jk+2 , 2i+jk- , NI\nETIRW\nrespectively, relative to the origin O.\nTON\nax+by+cz = d. [5] (a) Find the equation of the plane ABC, giving your answer in the form\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n...................................................................................................................................................\nSIHT\nNI ...................................................................................................................................................\nETIRW\n....................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2025-on-14-q04-14c8cf5fd7.pdf"
    },
    {
      "id": "9231-2025-on-14-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "14",
      "question_number": 5,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_14.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_14.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-on-14-q05-p01-2484d45f4b.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2025-on-14-q05-p02-8e9d4d0e49.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-14-q05",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n1 cosb1 1 1 1` SIHT cosb rl rl, cosb rl 2j. 5 (a) (i) By writing as show that [1] r- = 6 +\n12 4 6 12 4\nNI\n.................................................................................................................................................... ETIRW\n....................................................................................................................................................\nTON\nOD ....................................................................................................................................................\n....................................................................................................................................................\nNIGRAM 1 1` sinb rl 2j. [1] = 6 - (ii) Show also that\n12 4\n....................................................................................................................................................\nSIHT\n.................................................................................................................................................... NI\nETIRW\n....................................................................................................................................................\nTON\n....................................................................................................................................................\nOD\n6 2 2 6 + - 1 0\n1f pe o. M = The matrix M is such that\n4 0 -1 6 2 6 2 - +\nNIGRAM\n(b) The matrix M represents a sequence of two geometrical transformations in the x-y plane.\nSIHT\nGive full details of each transformation, and make clear the order in which they are applied. [4]\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\n................................................................................................................................................... ETIRW\n................................................................................................................................................... TON\nOD\n..................................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2025-on-14-q05-fb236d3c46.pdf"
    },
    {
      "id": "9231-2025-on-14-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "14",
      "question_number": 6,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_14.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_14.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2025-on-14-q06-p01-8991800807.png",
        "9231-topic-15-polar-coordinates/assets/9231-2025-on-14-q06-p02-ffed34555b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-14-q06",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n1 6 The curve C has polar equation i, for 0 G G r. r = cos i SIHT\n2\nNI\n(a) Sketch C. [2]\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\n1 1 1 2cos2 and -1. cosi / sini / 2sin icos In parts (b) and (c) you may use the identities i i\n2 2 2 OD\n(b) Find the exact value of the area of the region enclosed by C and the initial line. [4]\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON ...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT ...................................................................................................................................................\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\nOD ...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM ...................................................................................................................................................\n...................................................................................................................................................\nSIHT\n................................................................................................................................................... NI\nETIRW\n.....................................................................................................................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2025-on-14-q06-00d8002301.pdf"
    },
    {
      "id": "9231-2025-on-14-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 1,
      "variant": "14",
      "question_number": 7,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_14.pdf?download=true",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_14.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-on-14-q07-p01-ca79c5d5ce.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-on-14-q07-p02-c1cc4b3ba0.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2025-on-14-q07-p03-2b48af5192.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-14-q07",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\n10x2 -11x-18 SIHT\ny = 7 The curve C has equation .\n10x-18\nNI\n(a) Find the equations of the asymptotes of C. [3] ETIRW\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n...................................................................................................................................................\nSIHT\nNI ...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n(b) Show that C has no stationary points. [4]\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n.................................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2025-on-14-q07-8e3c8b4a62.pdf"
    },
    {
      "id": "9231-2025-on-21-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 1,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 3,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_21.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-on-21-q01-p01-ac5bee8601.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-21-q01",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 Find the values of k for which the system of equations\nSIHT\nx- y+ z = 0,\nNI\nETIRW x+ky+3z = 0,\nx+2y+ kz = 0,\nTON\ndoes not have a unique solution. [3]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ...................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2025-on-21-q01-29c5bb0f8c.pdf"
    },
    {
      "id": "9231-2025-on-21-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 2,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_21.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_21.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2025-on-21-q02-p01-bab47c1182.png",
        "9231-topic-23-differentiation/assets/9231-2025-on-21-q02-p02-3ae7e77a97.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-21-q02",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 The curve C has parametric equations\nSIHT\nx = sinht y = t+cosht. and NI\nETIRW\ndy\nin terms of t. [2] (a) Find\ndx\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\nd2y\n1-sinht\n= . [4] (b) Show that NIGRAM\ndx2 cosh3t\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n..................................................................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2025-on-21-q02-1fb961dffa.pdf"
    },
    {
      "id": "9231-2025-on-21-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 3,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_21.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-on-21-q03-p01-c6d76a213b.png",
        "9231-topic-24-integration/assets/9231-2025-on-21-q03-p02-688f5e2544.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-21-q03",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\nn y 1\nSIHT `1+x2j I 3 The integral is defined by dx. I =\nn n\n0\nNI\nETIRW\nd\n`x(1+x2)nj, (a) By considering or otherwise, show that\ndx\nTON\n2n . [5] (2n+1)I = +2nI\nn n-1 OD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n..............................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-on-21-q03-c8c3669459.pdf"
    },
    {
      "id": "9231-2025-on-21-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 4,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_21.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-on-21-q04-p01-f6ea492625.png",
        "9231-topic-26-differential-equations/assets/9231-2025-on-21-q04-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-21-q04",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 Find the particular solution of the differential equation\nSIHT\nNI\nd2y dy\nETIRW x2 5 +2 +y = +5x+3,\ndx dx2\nTON\ndy\n= = 0. given that, when y x = 0, [10]\nOD dx\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ...............",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-on-21-q04-151844200a.pdf"
    },
    {
      "id": "9231-2025-on-21-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 5,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_21.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-on-21-q05-p01-c72238ebf0.png",
        "9231-topic-24-integration/assets/9231-2025-on-21-q05-p02-3cb995e920.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-21-q05",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5\nSIHT\ny\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nx\nETIRW 2 n-1 1\n0 1\nn n n\nTON\n1\nx3 The diagram shows the curve with equation y = +x for 0 G x G 1, together with a set of n\nOD 3\n1\nrectangles of width .\nn\ny 1 1\nx3 e +xodx 1 U , where (a) By considering the sum of the areas of these rectangles, show that\n3 n\nNIGRAM 0\n1 1 1\nb1+ lb7+ l. U = [5]\nn n 12 SIHT n\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n......................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-on-21-q05-87e3875042.pdf"
    },
    {
      "id": "9231-2025-on-21-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 6,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_21.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-on-21-q06-p01-2b8c2e619b.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-on-21-q06-p02-f1264ffbd2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-21-q06",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 The matrix P has non-zero eigenvalues and is given by\nSIHT\na 1 1 NI\nf p ETIRW P = -1 . 0 2a\n-3a 0 0\nTON\n(a) State, in terms of a, the eigenvalues of P. [1]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\nP2 in terms of a. [1] (b) (i) Find\nSIHT\n....................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................\nTON\n....................................................................................................................................................\nOD\n....................................................................................................................................................\n....................................................................................................................................................\nNIGRAM\n....................................................................................................................................................\nSIHT\n....................................................................................................................................................\nNI\nP-1 in terms of a. [3] (ii) Use the characteristic equation of P to find ETIRW\n.................................................................................................................................................... TON\nOD\n....................................................................................................................................................\n....................................................................................................................................................\nNIGRAM\n....................................................................................................................................................\n....................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................\nETIRW\n....................................................................................................................................................\nTON\n.......................................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2025-on-21-q06-c3cade5c00.pdf"
    },
    {
      "id": "9231-2025-on-21-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_21.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-on-21-q07-p01-3baca62153.png",
        "9231-topic-26-differential-equations/assets/9231-2025-on-21-q07-p02-c070135150.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-21-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\nd 1\n`tanh-1xj SIHT = 7 (a) Show that . [3]\ndx 1-x2\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n(b) Find the solution of the differential equation\nETIRW\ndy\nTON\nx2tanh-1x, x -y =\ndx\nOD\n1. for 0 1 x 1 1, given that y = 0 when Give your answer in an exact form. [9] x =\n2\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n.............",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-on-21-q07-a5fbcd176b.pdf"
    },
    {
      "id": "9231-2025-on-21-q08",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "21",
      "question_number": 8,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_21.pdf?download=true",
      "source_pages": [
        16,
        17,
        18
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_21.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2025-on-21-q08-p01-7925da872b.png",
        "9231-topic-25-complex-numbers/assets/9231-2025-on-21-q08-p02-e2effa0da7.png",
        "9231-topic-25-complex-numbers/assets/9231-2025-on-21-q08-p03-a0a05d4f18.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-21-q08",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:7)(cid:44)\n2 n 8 (a) State the sum of the series ᴢ + ᴢ + ... + ᴢ , for ᴢ ! 1. [1]\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n1(cosi+isini) use de Moivre’s theorem to deduce that = (b) By letting ᴢ\nOD\n2\nn+2 n+1\nb1l sinni-b1l 1 sin(n+1)i+ sini /n\nm 2 2 2\nb1l . [6] = sinmi\nNIGRAM\n5 2 -cosi\n4 m=1\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n.............................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2025-on-21-q08-91c87cf4db.pdf"
    },
    {
      "id": "9231-2025-on-22-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 1,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_22.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-on-22-q01-p01-14a83955f3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-22-q01",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n4 i SIHT r = e 1 The curve C has polar equation for 0 G G r. i 3\nNI\nFind the length of C. Give your answer in an exact form. [4]\nETIRW\n..........................................................................................................................................................\nTON\nOD ..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM ..........................................................................................................................................................\n..........................................................................................................................................................\nSIHT\n.......................................................................................................................................................... NI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT ........................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-on-22-q01-0590fa0c9d.pdf"
    },
    {
      "id": "9231-2025-on-22-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 2,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_22.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_22.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2025-on-22-q02-p01-53877ad702.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-22-q02",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n4 reii, r 2 0 2 Find the roots of the equation ᴢ 3, giving your answers in the form where and = 8-8i SIHT\n-r G 1 r. [5] i\nNI\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\nSIHT\n..........................................................................................................................................................\nNI\n.......................................................................................................................................................... ETIRW\n.......................................................................................................................................................... TON\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n..........................................................................................................................................................\n..........................................................................................................................................................\nSIHT\nNI ..........................................................................................................................................................\nETIRW\n..........................................................................................................................................................\nTON\n..........................................................................................................................................................\nOD\n..........................................................................................................................................................\n..........................................................................................................................................................\nNIGRAM\n.................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2025-on-22-q02-4d31eabadc.pdf"
    },
    {
      "id": "9231-2025-on-22-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 3,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_22.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_22.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2025-on-22-q03-p01-cdb881a924.png",
        "9231-topic-23-differentiation/assets/9231-2025-on-22-q03-p02-8615c1d5e8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-22-q03",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 The variables x and y are such that y = 2 when x =-1 and\nSIHT\nx2y+ (x+y)3 NI = 3.\nETIRW\ndy 1\n= when x =-1. [3] (a) Show that\n4 dx\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI\n...................................................................................................................................................\nETIRW\n.....",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2025-on-22-q03-ebe47942b9.pdf"
    },
    {
      "id": "9231-2025-on-22-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 4,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_22.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_22.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2025-on-22-q04-p01-771ac80243.png",
        "9231-topic-25-complex-numbers/assets/9231-2025-on-22-q04-p02-162f8e9614.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-22-q04",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n6 1 SIHT bz+ l z = cosi+isini, 4 (a) By considering the binomial expansion of , where use de Moivre’ s\nz\nNI theorem to show that\nETIRW\ncos6 = acos6i+bcos4i+ccos2i+d, i\nTON\nwhere a, b, c and d are constants to be determined. [5]\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW ...................................................................................................................................................\n................................................................................................................................................... TON\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\n.................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2025-on-22-q04-90cacf42fd.pdf"
    },
    {
      "id": "9231-2025-on-22-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_22.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-on-22-q05-p01-87db6c4658.png",
        "9231-topic-26-differential-equations/assets/9231-2025-on-22-q05-p02-32335c83ec.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-22-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 (a) Find the general solution of the differential equation\nSIHT\nNI\nd2y dy\nETIRW +3y 5cosx. +4 = [7]\ndx dx2\nTON ...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI\n...................................................................................................................................................\nETIRW\n...................................................................................................................................................\nTON\nOD ...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM ...................................................................................................................................................\n...................................................................................................................................................\nSIHT\n................................................................................................................................................... NI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n............................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-on-22-q05-04961f9d7e.pdf"
    },
    {
      "id": "9231-2025-on-22-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 6,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_22.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-on-22-q06-p01-e5a09c9503.png",
        "9231-topic-24-integration/assets/9231-2025-on-22-q06-p02-50ead50184.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-22-q06",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\ny 1\nSIHT 1 dx. 6 (a) Use the substitution to find [3] x = 2sinhu\n2 2x2 +1\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\ny\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nx\nn-1 0 1 2 1\nn n n\nSIHT\n1\nThe diagram shows the curve with equation y = for 0 G x G 1, together with a set of NI\n2x2 +1 ETIRW 1\nn rectangles of width .\nn\nTON\n(b) By considering the sum of the areas of these rectangles, show that\nOD\n/n\n1\n1 2ln` 3j. 2 1 + [5]\n2 2r2 +n2\nr=1\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\n................................................................................................................................................... ETIRW\n................................................................................................................................................... TON\nNIGRAM\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON ...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\n................................................................................................................................................... SIHT\nNI\n.................................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-on-22-q06-02a3081fcf.pdf"
    },
    {
      "id": "9231-2025-on-22-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_22.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-on-22-q07-p01-11356f0e0e.png",
        "9231-topic-26-differential-equations/assets/9231-2025-on-22-q07-p02-f5b798a703.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-22-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\nd SIHT 1x +2sin-1b1xlo 4-x2 4-x2. = 7 (a) Show that [3] e\ndx 2 2\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT ...................................................................................................................................................\nNI\n(b) Find the solution of the differential equation\nETIRW\ndy y TON\n2 + 2 2-x =\n2+x dx\nOD\n1 for which when x = 1. Give your answer in an exact form. [8] y =\n2\nNIGRAM ...................................................................................................................................................\n...................................................................................................................................................\nSIHT\n................................................................................................................................................... NI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n....................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-on-22-q07-b1b4344fc1.pdf"
    },
    {
      "id": "9231-2025-on-22-q08",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "22",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_22.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-on-22-q08-p01-e73c1fa2ee.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-on-22-q08-p02-01a82882d9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-22-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n8 (a) Find the values of k for which the system of equations\nSIHT\nx-y+2kz = 1,\nNI\nETIRW kx+y+ 2z = 2,\n2x-y+ z = 3,\nTON\ndoes not have a unique solution. [3]\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\n...................................................................................................................................................\nTON\n...................................................................................................................................................\nOD\n...................................................................................................................................................\n...................................................................................................................................................\nNIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\n=-1, show that the system of equations in part (a) is inconsistent. Interpret this (b) Given that k ETIRW\n2\nsituation geometrically. [3]\nTON\n................................................................................................................................................... OD\n...................................................................................................................................................\n................................................................................................................................................... NIGRAM\n...................................................................................................................................................\nSIHT\n...................................................................................................................................................\nNI\nETIRW\nk =-1, show that the system of equations in part (a) is also inconsistent. (c) Given instead that\nInterpret this situation geometrically. [4]\nTON\nOD ...................................................................................................................................................\n.",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2025-on-22-q08-57d2873063.pdf"
    },
    {
      "id": "9231-2025-on-23-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 1,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 3,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_23.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-on-23-q01-p01-ac5bee8601.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-23-q01",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 Find the values of k for which the system of equations\nSIHT\nx- y+ z = 0,\nNI\nETIRW x+ky+3z = 0,\nx+2y+ kz = 0,\nTON\ndoes not have a unique solution. [3]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ...................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2025-on-23-q01-3c0028320b.pdf"
    },
    {
      "id": "9231-2025-on-23-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 2,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_23.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_23.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2025-on-23-q02-p01-bab47c1182.png",
        "9231-topic-23-differentiation/assets/9231-2025-on-23-q02-p02-3ae7e77a97.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-23-q02",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 The curve C has parametric equations\nSIHT\nx = sinht y = t+cosht. and NI\nETIRW\ndy\nin terms of t. [2] (a) Find\ndx\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\nd2y\n1-sinht\n= . [4] (b) Show that NIGRAM\ndx2 cosh3t\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n..................................................................................................................................",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2025-on-23-q02-0f01119030.pdf"
    },
    {
      "id": "9231-2025-on-23-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 3,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_23.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-on-23-q03-p01-c6d76a213b.png",
        "9231-topic-24-integration/assets/9231-2025-on-23-q03-p02-688f5e2544.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-23-q03",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\nn y 1\nSIHT `1+x2j I 3 The integral is defined by dx. I =\nn n\n0\nNI\nETIRW\nd\n`x(1+x2)nj, (a) By considering or otherwise, show that\ndx\nTON\n2n . [5] (2n+1)I = +2nI\nn n-1 OD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n..............................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-on-23-q03-88999c0c03.pdf"
    },
    {
      "id": "9231-2025-on-23-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 4,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_23.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-on-23-q04-p01-f6ea492625.png",
        "9231-topic-26-differential-equations/assets/9231-2025-on-23-q04-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-23-q04",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 Find the particular solution of the differential equation\nSIHT\nNI\nd2y dy\nETIRW x2 5 +2 +y = +5x+3,\ndx dx2\nTON\ndy\n= = 0. given that, when y x = 0, [10]\nOD dx\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ...............",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-on-23-q04-8583f0e08f.pdf"
    },
    {
      "id": "9231-2025-on-23-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 5,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_23.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-on-23-q05-p01-c72238ebf0.png",
        "9231-topic-24-integration/assets/9231-2025-on-23-q05-p02-3cb995e920.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-23-q05",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5\nSIHT\ny\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nx\nETIRW 2 n-1 1\n0 1\nn n n\nTON\n1\nx3 The diagram shows the curve with equation y = +x for 0 G x G 1, together with a set of n\nOD 3\n1\nrectangles of width .\nn\ny 1 1\nx3 e +xodx 1 U , where (a) By considering the sum of the areas of these rectangles, show that\n3 n\nNIGRAM 0\n1 1 1\nb1+ lb7+ l. U = [5]\nn n 12 SIHT n\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n......................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-on-23-q05-fdd5632550.pdf"
    },
    {
      "id": "9231-2025-on-23-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 6,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_23.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-on-23-q06-p01-2b8c2e619b.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-on-23-q06-p02-f1264ffbd2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-23-q06",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 The matrix P has non-zero eigenvalues and is given by\nSIHT\na 1 1 NI\nf p ETIRW P = -1 . 0 2a\n-3a 0 0\nTON\n(a) State, in terms of a, the eigenvalues of P. [1]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\nP2 in terms of a. [1] (b) (i) Find\nSIHT\n....................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................\nTON\n....................................................................................................................................................\nOD\n....................................................................................................................................................\n....................................................................................................................................................\nNIGRAM\n....................................................................................................................................................\nSIHT\n....................................................................................................................................................\nNI\nP-1 in terms of a. [3] (ii) Use the characteristic equation of P to find ETIRW\n.................................................................................................................................................... TON\nOD\n....................................................................................................................................................\n....................................................................................................................................................\nNIGRAM\n....................................................................................................................................................\n....................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................\nETIRW\n....................................................................................................................................................\nTON\n.......................................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2025-on-23-q06-2f4d37d789.pdf"
    },
    {
      "id": "9231-2025-on-23-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 7,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_23.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-on-23-q07-p01-3baca62153.png",
        "9231-topic-26-differential-equations/assets/9231-2025-on-23-q07-p02-c070135150.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-23-q07",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\nd 1\n`tanh-1xj SIHT = 7 (a) Show that . [3]\ndx 1-x2\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n(b) Find the solution of the differential equation\nETIRW\ndy\nTON\nx2tanh-1x, x -y =\ndx\nOD\n1. for 0 1 x 1 1, given that y = 0 when Give your answer in an exact form. [9] x =\n2\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n.............",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-on-23-q07-9d0036a57a.pdf"
    },
    {
      "id": "9231-2025-on-23-q08",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "23",
      "question_number": 8,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_23.pdf?download=true",
      "source_pages": [
        16,
        17,
        18
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_23.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2025-on-23-q08-p01-7925da872b.png",
        "9231-topic-25-complex-numbers/assets/9231-2025-on-23-q08-p02-e2effa0da7.png",
        "9231-topic-25-complex-numbers/assets/9231-2025-on-23-q08-p03-a0a05d4f18.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-23-q08",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:7)(cid:44)\n2 n 8 (a) State the sum of the series ᴢ + ᴢ + ... + ᴢ , for ᴢ ! 1. [1]\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n1(cosi+isini) use de Moivre’s theorem to deduce that = (b) By letting ᴢ\nOD\n2\nn+2 n+1\nb1l sinni-b1l 1 sin(n+1)i+ sini /n\nm 2 2 2\nb1l . [6] = sinmi\nNIGRAM\n5 2 -cosi\n4 m=1\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n.............................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2025-on-23-q08-09639d4305.pdf"
    },
    {
      "id": "9231-2025-on-24-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "24",
      "question_number": 1,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_24.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_24.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-on-24-q01-p01-30602a19b2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-24-q01",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\ny 1\nSIHT xnsinhx = I dx, 1 It is given that where n is a non-negative integer.\nn\n0\nNI\nETIRW\n(a) Show that, for n H 2,\nTON\nI = cosh1-nsinh1+n(n-1)I . [3] OD\nn n-2\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\nI . [2] (b) Find the exact value of\n2\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n........................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-on-24-q01-87e49a3d78.pdf"
    },
    {
      "id": "9231-2025-on-24-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "24",
      "question_number": 2,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_24.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_24.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2025-on-24-q02-p01-a4e43ff051.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-24-q02",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n6sinh2x cosh2x = 2 (a) Find the exact values of x for which , giving your answers in logarithmic form. SIHT\n[4]\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW 6sinh2x C |y = cosh2x C |y = and on the same diagram. [2] (b) Sketch the curves\n1 2\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2025-on-24-q02-74fd8e956e.pdf"
    },
    {
      "id": "9231-2025-on-24-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "24",
      "question_number": 3,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_24.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_24.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2025-on-24-q03-p01-e0ef78af26.png",
        "9231-topic-25-complex-numbers/assets/9231-2025-on-24-q03-p02-9eb3ea43fe.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-24-q03",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 (a) Use de Moivre’s theorem to show that\nSIHT\n16sin5 i-20sin3 sin5i = i+5sini. [4] NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n....................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2025-on-24-q03-f2d036e45c.pdf"
    },
    {
      "id": "9231-2025-on-24-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "24",
      "question_number": 4,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_24.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_24.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2025-on-24-q04-p01-406931f901.png",
        "9231-topic-24-integration/assets/9231-2025-on-24-q04-p02-38a3b9dd37.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-24-q04",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4\nSIHT\ny\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI x 0 1 2 N-1 1\nETIRW\nN N N\nTON\nOD\n1`3xj = y The diagram shows the curve with equation for 0 G x G 1, together with a set of N rectangles\n2\n1\neach of width .\nN\ny 1\n1`3xjdx NIGRAM 1 U , where (a) By considering the sum of the areas of these rectangles, show that\n2 N\n0\nSIHT\n1\n3\nN . [4] U = NI\nN 1 ` j N 3 -1 ETIRW N\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\ny 1\n1`3xjdx. for [4] (b) Use a similar method to find, in terms of N, a lower bound L\nN NI 2\n0\nETIRW\n............................................................................................................................................................\nTON\n.................................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2025-on-24-q04-6c7472cd3d.pdf"
    },
    {
      "id": "9231-2025-on-24-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "24",
      "question_number": 5,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_24.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_24.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-on-24-q05-p01-4d07538fd3.png",
        "9231-topic-26-differential-equations/assets/9231-2025-on-24-q05-p02-8de723fbae.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-24-q05",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 (a) Find the general solution of the differential equation\nSIHT\nNI d2y dy\nETIRW sin2x+cos2x. +y + = [7]\ndx dx2\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-on-24-q05-8d82ecad16.pdf"
    },
    {
      "id": "9231-2025-on-24-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "24",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_24.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_24.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2025-on-24-q06-p01-b4cbcfba68.png",
        "9231-topic-26-differential-equations/assets/9231-2025-on-24-q06-p02-26bb69daf8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-24-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6 Find the particular solution of the differential equation\nSIHT\ndy 1+x NI\n= 1 + y\nETIRW dx 1-2x-x2\ny = when x = 0. Give your answer in an exact form. [10] given that TON r\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n.",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2025-on-24-q06-a076ccc044.pdf"
    },
    {
      "id": "9231-2025-on-24-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "24",
      "question_number": 7,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_24.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_24.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2025-on-24-q07-p01-9fe3340165.png",
        "9231-topic-23-differentiation/assets/9231-2025-on-24-q07-p02-8427e0cc97.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-24-q07",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n7 The curve C has parametric equations\nSIHT\nNI\n1 1 1t) 1t) -3cott-3t, +4cott+4t, for r. G t G x = 8ln(tan y = 6ln(tan r\nETIRW 6 3 2 2\nTON\ndx\n8cosect+3cot2t. = [1] (a) (i) Show that\ndt OD\n....................................................................................................................................................\n....................................................................................................................................................\nNIGRAM\n....................................................................................................................................................\nSIHT\ndy\nNI\n. [1] (ii) Find\nETIRW dt\n....................................................................................................................................................\nTON\nOD ....................................................................................................................................................\n....................................................................................................................................................\nNIGRAM\n(iii) Find the exact value of the length of C. [6]\nSIHT\n....................................................................................................................................................\nNI\n.................................................................................................................................................... ETIRW\n.................................................................................................................................................... TON\nOD\n....................................................................................................................................................\n....................................................................................................................................................\nNIGRAM\n....................................................................................................................................................\n....................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................\nETIRW\n....................................................................................................................................................\nTON\n....................................................................................................................................................\nOD\n....................................................................................................................................................\n........",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2025-on-24-q07-699c61114c.pdf"
    },
    {
      "id": "9231-2025-on-24-q08",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 2,
      "variant": "24",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_24.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_24.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-on-24-q08-p01-5aab5eb682.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2025-on-24-q08-p02-d1dadcbc33.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-24-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n8 (a) Find the set of values of a for which the system of equations\nSIHT\n5x+ay = 3,\nNI\nETIRW 20x-5y = 2,\n2x-3y+z = 1,\nTON\nhas a unique solution and interpret this situation geometrically. [3] OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n5\na =- , show that the system of equations in part (a) is inconsistent and interpret this (b) Given that\nNI\n4\nETIRW\nsituation geometrically. [3]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n..........................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2025-on-24-q08-aec655a272.pdf"
    },
    {
      "id": "9231-2025-on-31-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 1,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_31.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_31.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2025-on-31-q01-p01-bca2dea1a6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-31-q01",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 Two uniform smooth spheres A and B of equal radii have masses 4m and m respectively . Sphere B is\nSIHT\nat rest on a smooth horizontal surface. Sphere A is moving on the surface with speed u and collides\ndirectly with sphere B. After the collision, the momentum of A is three times the momentum of B. NI\nETIRW\nFind the value of the coefficient of restitution e. [4]\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n.................................................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2025-on-31-q01-ae1acf9101.pdf"
    },
    {
      "id": "9231-2025-on-31-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 2,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_31.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-on-31-q02-p01-4c939df9da.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-31-q02",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n2 A particle P of mass m is moving in a horizontal circle with angular speed on the smooth inner ~\nSIHT 1\nsurface of a hemispherical shell of radius r. The angle between the upward vertical and the normal\nNI 3 reaction of the surface on P is tani = , where . i\n4 1 1 ETIRW\n, the angle between the upward vertical and the normal When the angular speed is increased to ~\n2 TON 4 reaction of the surface on P becomes , where . tani = i\n3 2 2\nOD\n~\n1 . [4] Find the ratio\n~\n2\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n.........................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-on-31-q02-1e2d54ca38.pdf"
    },
    {
      "id": "9231-2025-on-31-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 3,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_31.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_31.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2025-on-31-q03-p01-af5b142dfe.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2025-on-31-q03-p02-3a094381ea.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-31-q03",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 A particle P is moving in a straight horizontal line. At time t s, the displacement of P from a fixed point\nSIHT\n1(v2 vms-1. ms-2 O on the line is x m and the velocity of P is The acceleration of P is in the +4)\n2\n2ms-1. NI direction PO. Initially P is at O and is moving with velocity\nETIRW\n(a) Find an expression for x in terms of t. [5]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n......................................................................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2025-on-31-q03-d7b77f0852.pdf"
    },
    {
      "id": "9231-2025-on-31-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 4,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_31.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_31.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2025-on-31-q04-p01-021d8811f5.png",
        "9231-topic-34-hooke-s-law/assets/9231-2025-on-31-q04-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-31-q04",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 One end of a light elastic string of natural length a and modulus of elasticity 5mg is attached to a fixed\nSIHT\npoint O. Two particles, P and Q, of masses m and 4m respectively are attached to the other end of\nthe string and they hang vertically in equilibrium. Particle Q is then detached from the string, hence NI\nETIRW releasing particle P from rest.\n7 TON Find, in terms of a, the length of the string when the speed of particle P is first equal to ag. [6]\n5\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n...................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2025-on-31-q04-9610da6059.pdf"
    },
    {
      "id": "9231-2025-on-31-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 5,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_31.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_31.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-on-31-q05-p01-07743b0b4e.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-on-31-q05-p02-636d60ef7f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-31-q05",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5\nSIHT\ny\nNI\nD C ETIRW\nTON\nOD\nB\nh\n2a\nNIGRAM\nSIHT\nNI\nETIRW 45°\nO TON x ka A\nOD\nA uniform lamina OABCD consists of a rectangle OACD and a triangle ABC. The length of OA is ka,\nthe length of OD is 2a, the height of triangle ABC is h and angle CAB is 45° (see diagram). Relative to\naxes through O, parallel and perpendicular to OA as shown, the centre of mass of triangle ABC is( x, y).\nNIGRAM\n1(3ka+h), is and find an expression for y. [3] x (a) Show that\n3\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n......",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2025-on-31-q05-4701d3ad67.pdf"
    },
    {
      "id": "9231-2025-on-31-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 6,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_31.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-on-31-q06-p01-3b0363407f.png",
        "9231-topic-33-circular-motion/assets/9231-2025-on-31-q06-p02-acf62e8bd3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-31-q06",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6\nSIHT\nNI 17ag\nP\nETIRW\n60° a\nTON\nO\nOD\nNIGRAM\nA particle P of mass m is attached to one end of a light inextensible string of length a. The other end\nof the string is attached to a fixed point O. Initially P is held with the string taut and making an angle SIHT\nof 60° with the upward vertical through O. The particle P is projected perpendicular to the string in a\nNI\ndownwards direction with speed 17ag. It then starts to move along a circular path in a vertical plane\nETIRW\nwith centre O (see diagram). At the lowest point of its path, vertically below O, the particle P collides\nwith a stationary particle Q.\nTON\n(a) Find, in terms of a and g, an expression for the speed of P immediately before the collision with Q. OD\n[2]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-on-31-q06-bb2d0b3c35.pdf"
    },
    {
      "id": "9231-2025-on-31-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "31",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_31.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_31.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-on-31-q07-p01-21cad85efe.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-on-31-q07-p02-67d6c76b32.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-31-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n7 A particle P is projected from a point O on a horizontal plane and moves freely under gravity. The\nSIHT\n4. 25ms-1 initial velocity of P is at an angle above the horizontal, where tan At point A, the = i i\n3\nNI\ndirection of motion of P makes an angle of 45° with the downward vertical through A.\nETIRW\n(a) By differentiating the equation of the trajectory or otherwise, find the coordinates of A. [5]\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n..........................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2025-on-31-q07-f38ea7a690.pdf"
    },
    {
      "id": "9231-2025-on-32-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 1,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_32.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-on-32-q01-p01-538d1051df.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-32-q01",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 A particle P of mass m is attached to two light inextensible strings each of length l. The end of one\nSIHT\nstring is attached to a fixed point A and the end of the other string is attached to a fixed point B, with A\nNI vertically above B. Angle APB is a right angle. The particle P rotates in a horizontal circle at a constant\nETIRW angular speed with both strings taut (see diagram). ~\nTON\nA\nOD\nl\nNIGRAM\n~\nP\nSIHT\nNI\nl ETIRW\nTON B\nOD\nFind the tension in string AP in terms of m, g, l and ~. [4]\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW ....................................................................................................................................................................\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n...............",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-on-32-q01-d07660adff.pdf"
    },
    {
      "id": "9231-2025-on-32-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 2,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_32.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_32.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2025-on-32-q02-p01-c8f87d29c4.png",
        "9231-topic-34-hooke-s-law/assets/9231-2025-on-32-q02-p02-66583d64a9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-32-q02",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 One end of a light elastic string of natural length a and modulus of elasticity 2mg is attached to a fixed\nSIHT\npoint A on a rough horizontal surface. The other end of the string is attached to a particle P of mass m.\nThe particle and string rest on the surface.T he coefficient of friction between P and the surface is n. The NI\nETIRW\n4a particle P is initially held in equilibrium at a distance from A. The particle is then released from rest.\n3\nTON\nn. [3] (a) Given that the string never becomes slack, find the minimum value of\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n..............................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2025-on-32-q02-d1c6207e76.pdf"
    },
    {
      "id": "9231-2025-on-32-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 3,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_32.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_32.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-on-32-q03-p01-b87fbb1aed.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-on-32-q03-p02-1327155adb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-32-q03",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 A uniform lamina OABCD is in the form of a rectangle, OBCD, joined along the edge OB to a quarter\nSIHT\ncircle OAB. The length of DO is ka and the length of OB is a. The lamina rests in a vertical plane with\nits edge CB on a horizontal surface (see diagram). NI\nETIRW\nka\nD O A\nTON\nOD\na\nNIGRAM\nC B\nSIHT\n(a) Find, in terms of k, a and r, an expression for the distance of the centre of mass above the NI\nETIRW horizontal surface. [4]\n[Y ou may use without proof the result for the centre of mass of a circular sector in the list of TON\nformulae (MF19).]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n...........................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2025-on-32-q03-76a78dd86b.pdf"
    },
    {
      "id": "9231-2025-on-32-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 4,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_32.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_32.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-on-32-q04-p01-f1e5f4d505.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-on-32-q04-p02-0a2f08c5ff.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-32-q04",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 A particle Q is initially positioned at a distance d vertically above a particle P. Particle P is projected\nSIHT\nwith speed U at an angle above the horizontal. At the same time, Q is projected at an angle a b\nbelow the horizontal. Both particles move freely under gravity. The particles collide at time T after the NI\nETIRW projections.\nTON (a) Show that d = UT(sina+cosatanb). [4]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2025-on-32-q04-cd856dfba8.pdf"
    },
    {
      "id": "9231-2025-on-32-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 5,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_32.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_32.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2025-on-32-q05-p01-49e405e390.png",
        "9231-topic-36-momentum/assets/9231-2025-on-32-q05-p02-29279b7b86.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-32-q05",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 Two uniform smooth spheres, A and B, of equal radii are on a horizontal surface. They have masses of\nSIHT\nm and km respectively. Sphere A is moving with speed u at an angle with the line of centres when it a\ncollides with sphere B which is stationary. Immediately after the collision, sphere A moves with speed v NI\nETIRW at an angle 2a with the line of centres (see diagram).\nTON\nOD\nA v B\n2α\nα\nNIGRAM u\nm km\nSIHT\n3 tana = It is given that .\nNI\n4\nETIRW\n(a) Find v in terms of u. [2]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON (b) Find the coefficient of restitution between the spheres in terms of k. [4]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM .....................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2025-on-32-q05-677ea977a6.pdf"
    },
    {
      "id": "9231-2025-on-32-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 6,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_32.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-on-32-q06-p01-8f26f30b93.png",
        "9231-topic-33-circular-motion/assets/9231-2025-on-32-q06-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-32-q06",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6\nSIHT\nu\nNI\nA\nETIRW\nB\na\nTON θ\nOD\nO\n3u\nNIGRAM\nβ\nSIHT\nA fixed smooth sphere with radius a and centre O rests on horizontal ground. A particle is projected\nhorizontally from the highest point, A, of the sphere with speed u. The particle begins to move in a\nNI\nvertical circle along the surface of the sphere. The particle loses contact with the sphere at the point B, ETIRW\nwhere the angle AOB is i.\nTON\nAfter leaving the surface of the sphere, the particle moves freely under gravity before striking the\nOD horizontal ground with speed 3u at an angle to the horizontal (see diagram). b\nb. [8] Find the value of\nNIGRAM ....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\n.................................................................................................................................................................... NI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-on-32-q06-bd4c362968.pdf"
    },
    {
      "id": "9231-2025-on-32-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "32",
      "question_number": 7,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_32.pdf?download=true",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_32.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2025-on-32-q07-p01-a98ae8d1c9.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2025-on-32-q07-p02-da9d1e4bcf.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-32-q07",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n7 A particle P of mass m kg moving along a rough horizontal table has displacement x m from a fixed\nSIHT\npoint O on the table and velocity v m s-1 at time t s. The particle P is subject to a resistive force of\nmagnitude mgkv N, where k is a positive constant, and a frictional force of magnitude nmg. The particle NI\nETIRW P is initially at O with speed U m s-1.\nTON\nkU+n 1\nlne o. (a) Show that t = [4]\nOD gk kv+n\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n.....................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2025-on-32-q07-4fee833790.pdf"
    },
    {
      "id": "9231-2025-on-33-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 1,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_33.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_33.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2025-on-33-q01-p01-bca2dea1a6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-33-q01",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 Two uniform smooth spheres A and B of equal radii have masses 4m and m respectively . Sphere B is\nSIHT\nat rest on a smooth horizontal surface. Sphere A is moving on the surface with speed u and collides\ndirectly with sphere B. After the collision, the momentum of A is three times the momentum of B. NI\nETIRW\nFind the value of the coefficient of restitution e. [4]\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n.................................................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2025-on-33-q01-4fdc42122c.pdf"
    },
    {
      "id": "9231-2025-on-33-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 2,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_33.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-on-33-q02-p01-4c939df9da.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-33-q02",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n2 A particle P of mass m is moving in a horizontal circle with angular speed on the smooth inner ~\nSIHT 1\nsurface of a hemispherical shell of radius r. The angle between the upward vertical and the normal\nNI 3 reaction of the surface on P is tani = , where . i\n4 1 1 ETIRW\n, the angle between the upward vertical and the normal When the angular speed is increased to ~\n2 TON 4 reaction of the surface on P becomes , where . tani = i\n3 2 2\nOD\n~\n1 . [4] Find the ratio\n~\n2\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n.........................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-on-33-q02-7d39e19fc7.pdf"
    },
    {
      "id": "9231-2025-on-33-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 3,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_33.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_33.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2025-on-33-q03-p01-af5b142dfe.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2025-on-33-q03-p02-3a094381ea.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-33-q03",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 A particle P is moving in a straight horizontal line. At time t s, the displacement of P from a fixed point\nSIHT\n1(v2 vms-1. ms-2 O on the line is x m and the velocity of P is The acceleration of P is in the +4)\n2\n2ms-1. NI direction PO. Initially P is at O and is moving with velocity\nETIRW\n(a) Find an expression for x in terms of t. [5]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n......................................................................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2025-on-33-q03-ea7ed38b4b.pdf"
    },
    {
      "id": "9231-2025-on-33-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 4,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_33.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_33.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2025-on-33-q04-p01-021d8811f5.png",
        "9231-topic-34-hooke-s-law/assets/9231-2025-on-33-q04-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-33-q04",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 One end of a light elastic string of natural length a and modulus of elasticity 5mg is attached to a fixed\nSIHT\npoint O. Two particles, P and Q, of masses m and 4m respectively are attached to the other end of\nthe string and they hang vertically in equilibrium. Particle Q is then detached from the string, hence NI\nETIRW releasing particle P from rest.\n7 TON Find, in terms of a, the length of the string when the speed of particle P is first equal to ag. [6]\n5\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n...................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2025-on-33-q04-a1e54d01bd.pdf"
    },
    {
      "id": "9231-2025-on-33-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 5,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_33.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_33.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-on-33-q05-p01-07743b0b4e.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-on-33-q05-p02-636d60ef7f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-33-q05",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5\nSIHT\ny\nNI\nD C ETIRW\nTON\nOD\nB\nh\n2a\nNIGRAM\nSIHT\nNI\nETIRW 45°\nO TON x ka A\nOD\nA uniform lamina OABCD consists of a rectangle OACD and a triangle ABC. The length of OA is ka,\nthe length of OD is 2a, the height of triangle ABC is h and angle CAB is 45° (see diagram). Relative to\naxes through O, parallel and perpendicular to OA as shown, the centre of mass of triangle ABC is( x, y).\nNIGRAM\n1(3ka+h), is and find an expression for y. [3] x (a) Show that\n3\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n......",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2025-on-33-q05-81ba1401da.pdf"
    },
    {
      "id": "9231-2025-on-33-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 6,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_33.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-on-33-q06-p01-3b0363407f.png",
        "9231-topic-33-circular-motion/assets/9231-2025-on-33-q06-p02-acf62e8bd3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-33-q06",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6\nSIHT\nNI 17ag\nP\nETIRW\n60° a\nTON\nO\nOD\nNIGRAM\nA particle P of mass m is attached to one end of a light inextensible string of length a. The other end\nof the string is attached to a fixed point O. Initially P is held with the string taut and making an angle SIHT\nof 60° with the upward vertical through O. The particle P is projected perpendicular to the string in a\nNI\ndownwards direction with speed 17ag. It then starts to move along a circular path in a vertical plane\nETIRW\nwith centre O (see diagram). At the lowest point of its path, vertically below O, the particle P collides\nwith a stationary particle Q.\nTON\n(a) Find, in terms of a and g, an expression for the speed of P immediately before the collision with Q. OD\n[2]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-on-33-q06-ff74ce6332.pdf"
    },
    {
      "id": "9231-2025-on-33-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "33",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_33.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_33.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-on-33-q07-p01-21cad85efe.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-on-33-q07-p02-67d6c76b32.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-33-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n7 A particle P is projected from a point O on a horizontal plane and moves freely under gravity. The\nSIHT\n4. 25ms-1 initial velocity of P is at an angle above the horizontal, where tan At point A, the = i i\n3\nNI\ndirection of motion of P makes an angle of 45° with the downward vertical through A.\nETIRW\n(a) By differentiating the equation of the trajectory or otherwise, find the coordinates of A. [5]\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n..........................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2025-on-33-q07-83a6eb92c9.pdf"
    },
    {
      "id": "9231-2025-on-34-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "34",
      "question_number": 1,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_34.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_34.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-on-34-q01-p01-7613823e97.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-34-q01",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1\nSIHT\nR\nNI\nETIRW\ni\nTON\nA B OD\nA light inextensible string of length 12a is threaded through a fixed smooth ring R. One end of the\nstring is attached to a particle A of mass m. The other end of the string is attached to a particle B of\nNIGRAM\nmass 0.5m. Particle A hangs in equilibrium vertically below the ring. Particle B moves with constant\nangular speed in a horizontal circle with particle A at its centre. The angle between AR and BR is ~ i\n(see diagram).\nSIHT\nNI Express in terms of g and a. [5] ~\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW ....................................................................................................................................................................\nTON ....................................................................................................................................................................\nOD\n.",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-on-34-q01-9c03a3d924.pdf"
    },
    {
      "id": "9231-2025-on-34-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "34",
      "question_number": 2,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_34.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_34.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2025-on-34-q02-p01-aa42fb4b25.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-34-q02",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\nums-1 2 A particle P is projected with speed at an angle i, where tani = 2, above the horizontal from a SIHT\npoint O on a horizontal plane and moves freely under gravity. The horizontal and vertical displacements\nNI\nof P from O at a time t s are denoted by x m and y m respectively.\nETIRW\n(a) Use the equation of the trajectory given in the list of formulae (MF 19) to show that\nTON\n25x2\n= 2x- y . OD\nu2\n[1]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n(b) In the subsequent motion, P passes through the point with coordinates (8,12). The particle then hits\na fixed vertical barrier 7 m high that is at a horizontal distance of D m from the point of projection.\nFind the set of possible values of D. [5]\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n..................................................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2025-on-34-q02-b5851dc094.pdf"
    },
    {
      "id": "9231-2025-on-34-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "34",
      "question_number": 3,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_34.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_34.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-on-34-q03-p01-b7efde3947.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2025-on-34-q03-p02-08075dbca9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-34-q03",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3\nSIHT\nD C F h\nNI\nETIRW\nTON 2a\nOD\nA a E B\nNIGRAM\nThe lamina BFDE is obtained by removing triangles AED and BCF from a uniform square lamina\nABCD of side 2a. The length of side AE is a and the length of side FC is h (see diagram). The centre of\nSIHT mass of BFDE is at a distance from AD, and at a distance from AB. x y\nh2 -6ah+11a2 NI\nx = and find a corresponding expression for . [5] y (a) Show that ETIRW\n3(3a-h)\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n....................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2025-on-34-q03-0167c3077e.pdf"
    },
    {
      "id": "9231-2025-on-34-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "34",
      "question_number": 4,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_34.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_34.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2025-on-34-q04-p01-c7ab2724f0.png",
        "9231-topic-33-circular-motion/assets/9231-2025-on-34-q04-p02-c4a04128f7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-34-q04",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4\nSIHT\nNI\nETIRW\ni A v\na\nTON O\nOD\nNIGRAM A fixed smooth spherical shell has centre O and radius a. A particle of mass m moves in complete\nvertical circles on the smooth inner surface of the shell, where the plane of the circular motion is\nvertical and passes through O. The particle has speed v when it is at point A, where OA makes an angle\nSIHT 1 with the upward vertical through O, and (see diagram). cosi = i\n18\nNI\n1 ETIRW 26ag. [5] v H (a) Show that\n3\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n...........",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2025-on-34-q04-e10fcae1b9.pdf"
    },
    {
      "id": "9231-2025-on-34-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "34",
      "question_number": 5,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_34.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_34.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2025-on-34-q05-p01-1a516ae177.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2025-on-34-q05-p02-625abad3f8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-34-q05",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n20ms-1 5 A particle P of mass m kg is projected vertically upwards from a point O with an initial speed of SIHT\n0.025mv2 vms-1 and moves under gravity. There is a resistive force of magnitude N, where is the\nNI\nspeed of P at time t s after projection. The displacement of P from O is x m at time t s after projection.\nETIRW\n(a) Find an expression for v in terms of x, while P is moving upwards. [5]\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n.............................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2025-on-34-q05-2ffdf8b401.pdf"
    },
    {
      "id": "9231-2025-on-34-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "34",
      "question_number": 6,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_34.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_34.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2025-on-34-q06-p01-37ee6f80b7.png",
        "9231-topic-34-hooke-s-law/assets/9231-2025-on-34-q06-p02-786b9bea5c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-34-q06",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6 A and B are two fixed points at a distance 22a apart, with B vertically below A. A light elastic string of\nSIHT\nnatural length 4a and modulus of elasticity 5mg has one end attached to A and the other end attached to\na particle P of mass km. Another light elastic string of natural length 8a and modulus of elasticity 6mg NI\nETIRW has one end attached to B and the other end attached to P. Particle P is vertically above B.\n57a-2ak\nTON (a) Show that, when the system is in equilibrium, BP = . [5]\n4\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n................................................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2025-on-34-q06-390c04e8ee.pdf"
    },
    {
      "id": "9231-2025-on-34-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 3,
      "variant": "34",
      "question_number": 7,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_34.pdf?download=true",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_34.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2025-on-34-q07-p01-7d8154df0d.png",
        "9231-topic-36-momentum/assets/9231-2025-on-34-q07-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-34-q07",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n7\nSIHT\nP\nWall Y\n2\nNI\ni a ETIRW\nu\nTON\nd\nOD\nNIGRAM P P Wall X\n1 3\nSIHT X and Y are two fixed smooth vertical walls on a smooth horizontal surface. The walls are parallel and\nat a distance d apart. The points P P P all lie on the surface. and ,\nNI 2 1 3\nETIRW\nP on Wall X with speed u, and moves along the A particle Q is projected horizontally from the point\n1\nsurface. The particle Q strikes Wall Y at the point P . Immediately before the collision, the direction\nTON 2\n4. of motion of Q makes an angle with Wall Y, where Immediately after the collision, the sina = a\n5\nOD direction of motion of Q makes an angle with Wall Y. The particle Q then strikes Wall X at the point i\nP (see diagram).\n3\nP P is T. The time that it takes Q to travel the distance The time that it takes Q to travel the distance\n1 2\nNIGRAM P P is kT.\n2 3\nFind, in terms of k, the coefficient of restitution between Q and Wall Y. [6]\nSIHT\n.................................................................................................................................................................... NI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n..............................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2025-on-34-q07-b365fafed7.pdf"
    },
    {
      "id": "9231-2025-on-41-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_41.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-41-q01-p01-30ba73e55a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-41-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 A group of 10 school children are asked to estimate the size of an angle i° in a given acute angled\nSIHT\ntriangle. These estimates, in degrees, are as follows.\nNI\nETIRW\n84 85 77 85 84 87 86 88 83 85\nTON (a) Stating any assumptions you make, calculate a 95% confidence interval for i. [5]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-on-41-q01-28ce636a2e.pdf"
    },
    {
      "id": "9231-2025-on-41-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_41.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_41.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2025-on-41-q02-p01-ad789f1ed5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-41-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n2 The manager of a car park claims that the number of cars entering the car park follows a Poisson\nSIHT\ndistribution with mean 2.8. The numbers of cars entering the car park are recorded on a working day\nNI during successive 5-minute periods. The following table contains the observed frequencies, together\nETIRW |2-test with most of the expected frequencies and their contributions to the statistic.\nTON\nNumber of cars 0 1 2 3 4 5 H 6\nOD\nObserved frequency 2 15 31 29 13 3 7\nExpected frequency 6.081 17.03 23.84 p 15.57 8.721 6.511\n|2-test statistic 2.739 0.241 2.152 q 0.425 3.753 0.037 NIGRAM\nSIHT\n(a) Find the value of p and the value of q. [2]\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n(b) Carry out a goodness of fit test at the 5% significance level to investigate the manager ’s claim. SIHT\n[4]\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD ................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2025-on-41-q02-14b8ad52bb.pdf"
    },
    {
      "id": "9231-2025-on-41-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 3,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_41.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-41-q03-p01-c879c3364f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-41-q03",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 A random sample of 10 newborn baby boys is taken and their masses in kg are recorded. From this\nSIHT\nsample, the population standard deviation of all newborn baby boys is estimated as 0.6 kg. A random\nsample of 5 newborn baby girls is taken and their masses in kg are recorded as follows. NI\nETIRW\n3.9 3.1 2.9 3.1 3.6\nTON\nIt is assumed that the masses of newborn baby boys and girls have the same population standard\nOD\ndeviation, kg. v\nv. [4] By pooling the two samples, calculate an estimate of\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n........",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-on-41-q03-48e786eb0d.pdf"
    },
    {
      "id": "9231-2025-on-41-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 4,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_41.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_41.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2025-on-41-q04-p01-92031badd0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-41-q04",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:6)(cid:44)\n4 A researcher believes that the median m of a population has changed from its known previous value m .\nSIHT 0\nThe researcher collects a random sample of size 28. She ranks the data and calculates a test statistic T\nNI using the Wilcoxon signed-rank test. The conclusion of the test carried out at a 1% significance level is\nETIRW\nthat there is not sufficient evidence to support her belief.\nTON Using a normal approximation, find the least possible value of T. [5]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT ....................................................................................................................................................................\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\nOD ....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM ....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\n.................................................................................................................................................................... NI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n..................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2025-on-41-q04-42e13a1246.pdf"
    },
    {
      "id": "9231-2025-on-41-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 5,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_41.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_41.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2025-on-41-q05-p01-71157e2ae8.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2025-on-41-q05-p02-771eac4acb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-41-q05",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n5 A continuous random variable X has probability density function f given by\nSIHT\nZ\nNI 1 ]\nx x 4, G 1 0\nETIRW ]16\n]\n1 f(x) = [ 4 G x G 9,\nTON ]k x\n]\n0 otherwise,\nOD\n\\\nwhere k is a constant.\nk = 3. [2] (a) Show that\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW (b) Find the median value of X. [3]\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n......................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2025-on-41-q05-847f63aa59.pdf"
    },
    {
      "id": "9231-2025-on-41-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 6,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_41.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-41-q06-p01-b57c0e67c5.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-41-q06-p02-a102e17cbe.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-41-q06",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n6 Nine athletes in a club have a new coach. The coach adopts a new training programme which he\nSIHT\nbelieves will reduce the race times of these athletes. Each athlete completes a 1500 m time trial before\nand after completing the new training programme. Their times, in seconds (s), are recorded. NI\nETIRW\nAthlete A B C D E F G H I\nTON\nTime before training (s) 250 251 252 267 276 291 310 320 335\nOD\nTime after training (s) 245 251 253 261 275 293 302 313 320\n(a) Carry out a paired t-test at the 5% significance level to test the coach’s belief. [7]\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n....................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-on-41-q06-3037c2be5f.pdf"
    },
    {
      "id": "9231-2025-on-41-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "41",
      "question_number": 7,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_41.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_41.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2025-on-41-q07-p01-33f5968183.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2025-on-41-q07-p02-44076b74ac.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-41-q07",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\nP(X = r) 7 A discrete random variable X takes values r = 0,1, 2 with probabilities as given in the\nSIHT\nfollowing table.\nNI\nETIRW\nr 0 1 2\nTON a 2a b P(X = r)\nOD\n(a) Write down the probability generating function of X, and use it to find an expression for E(X ) in\nterms of a and b. [2]\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\nVar(X) = 2b+2(a+b)(1-2a-2b). [3] (b) Show that\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n........................................................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2025-on-41-q07-e340f0c17e.pdf"
    },
    {
      "id": "9231-2025-on-42-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 1,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_42.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_42.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2025-on-42-q01-p01-a78014ea76.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-42-q01",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 A large company claims that the median salary of its employees is $32 500. The salaries ($) of 15\nSIHT\nrandomly selected employees are listed below.\nNI\nETIRW 18 750 30 500 125 000 42 500 25 000\n26 000 52 500 23 000 27 500 19 500\nTON 25 500 33 000 30 000 21 500 29 000\nOD\n(a) Explain why a Wilcoxon signed-rank test may not be appropriate to test the company’s claim in\nthis case. [1]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n(b) Carry out a sign test at the 10% significance level to investigate the company’s claim. [5]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ......................................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2025-on-42-q01-f6779da31a.pdf"
    },
    {
      "id": "9231-2025-on-42-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 2,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_42.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-42-q02-p01-1f2dc9e06b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-42-q02",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 A factory produces packets of biscuits. The total mass of biscuits in a packet has a normal distribution\nSIHT\nwith mean n. A random sample of 12 packets is taken and the mass of the contents of each packet, x g,\nis recorded. The results are summarised as follows. NI\nETIRW\n/x2 /x = 2390 = 476117\nTON\n(a) Find a 99% confidence interval for n. [4]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n= k is carried out on this sample using a 5% significance level. The test A test of the null hypothesis n TON\ndoes not support the alternative hypothesis 1 k. n\nOD\n(b) Find the greatest possible value of k. [3]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n.......................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-on-42-q02-752c72d595.pdf"
    },
    {
      "id": "9231-2025-on-42-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 3,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_42.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_42.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2025-on-42-q03-p01-3808f96f07.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-42-q03",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:6)(cid:44)\n3 A traffic expert claims that the number of breakdowns occurring each day on a busy section of a\nSIHT\nmotorway follows a Poisson distribution with mean 0.7. The number of breakdowns each day over\nNI a 200-day period was recorded. The following table contains the observed frequencies together with\nETIRW some of the expected frequencies using the expert’s distribution.\nTON\nNumber of breakdowns per day 0 1 2 3 4 H 5\nOD\nObserved frequency 88 73 26 7 3 3\nExpected frequency 99.317 m 24.333 5.678 0.994 n\nNIGRAM\n(a) Find the value of m and the value of n, correct to 3 decimal places. [2]\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n(b) Carry out a goodness of fit test at the 5% significance level to investigate the expert’s claim. [6] OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n.................................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2025-on-42-q03-d6da047644.pdf"
    },
    {
      "id": "9231-2025-on-42-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 4,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_42.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_42.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2025-on-42-q04-p01-7dd10231c1.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2025-on-42-q04-p02-82bf543da0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-42-q04",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 A continuous random variable X has cumulative distribution function F given by\nSIHT\nZ\nNI 1 1, x ]0\nETIRW\n]]1x+a\n1 4, G 1 x\n[5 F(x) =\n1 x2 4 6, G G +b x\n] TON\n50\n]\n6, 2 1 x\n\\ OD\nwhere a and b are constants.\n(a) Find the value of a and the value of b. [2]\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n(b) Find the probability density function of X. [2] ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n.............................................................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2025-on-42-q04-9baf570d73.pdf"
    },
    {
      "id": "9231-2025-on-42-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 5,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_42.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-42-q05-p01-419c084067.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-42-q05-p02-473dcb251b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-42-q05",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 An engineer is comparing the tensile strengths of steel rods made from two machines, A and B. The\nSIHT\nengineer randomly selects 8 rods from machine A and 6 rods from machine B. The tensile strengths, in\nappropriate units, are given in the following table. NI\nETIRW\nMachine A 402 403 415 412 409 407 406 410\nTON\nMachine B 401 398 395 397 410 405\nOD\nYou should assume that the two distributions are normal and have the same population variance.\nUse a t-test at the 5% significance level to test whether there is any dif ference in the mean tensile\nNIGRAM\nstrengths of steel rods from the two machines. [9]\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n........................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-on-42-q05-3729b51540.pdf"
    },
    {
      "id": "9231-2025-on-42-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "42",
      "question_number": 6,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_42.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_42.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2025-on-42-q06-p01-9dbe9f2c79.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2025-on-42-q06-p02-23ed5f5b92.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-42-q06",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\nG (t) 6 The discrete random variable X has probability generating function given by\nSIHT\nX\nt\nNI G (t) = .\n(3-2t)2 X ETIRW\nTON (a) Find E(X) and Var (X ). [5]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n................................................................................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2025-on-42-q06-ab3ee10187.pdf"
    },
    {
      "id": "9231-2025-on-43-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_43.pdf?download=true",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-43-q01-p01-30ba73e55a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-43-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 A group of 10 school children are asked to estimate the size of an angle i° in a given acute angled\nSIHT\ntriangle. These estimates, in degrees, are as follows.\nNI\nETIRW\n84 85 77 85 84 87 86 88 83 85\nTON (a) Stating any assumptions you make, calculate a 95% confidence interval for i. [5]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-on-43-q01-4be010f9cd.pdf"
    },
    {
      "id": "9231-2025-on-43-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_43.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_43.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2025-on-43-q02-p01-ad789f1ed5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-43-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n2 The manager of a car park claims that the number of cars entering the car park follows a Poisson\nSIHT\ndistribution with mean 2.8. The numbers of cars entering the car park are recorded on a working day\nNI during successive 5-minute periods. The following table contains the observed frequencies, together\nETIRW |2-test with most of the expected frequencies and their contributions to the statistic.\nTON\nNumber of cars 0 1 2 3 4 5 H 6\nOD\nObserved frequency 2 15 31 29 13 3 7\nExpected frequency 6.081 17.03 23.84 p 15.57 8.721 6.511\n|2-test statistic 2.739 0.241 2.152 q 0.425 3.753 0.037 NIGRAM\nSIHT\n(a) Find the value of p and the value of q. [2]\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n(b) Carry out a goodness of fit test at the 5% significance level to investigate the manager ’s claim. SIHT\n[4]\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD ................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2025-on-43-q02-7f757794b0.pdf"
    },
    {
      "id": "9231-2025-on-43-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 3,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_43.pdf?download=true",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-43-q03-p01-c879c3364f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-43-q03",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 A random sample of 10 newborn baby boys is taken and their masses in kg are recorded. From this\nSIHT\nsample, the population standard deviation of all newborn baby boys is estimated as 0.6 kg. A random\nsample of 5 newborn baby girls is taken and their masses in kg are recorded as follows. NI\nETIRW\n3.9 3.1 2.9 3.1 3.6\nTON\nIt is assumed that the masses of newborn baby boys and girls have the same population standard\nOD\ndeviation, kg. v\nv. [4] By pooling the two samples, calculate an estimate of\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n........",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-on-43-q03-f2e799bf47.pdf"
    },
    {
      "id": "9231-2025-on-43-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 4,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_43.pdf?download=true",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_43.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2025-on-43-q04-p01-92031badd0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-43-q04",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:6)(cid:44)\n4 A researcher believes that the median m of a population has changed from its known previous value m .\nSIHT 0\nThe researcher collects a random sample of size 28. She ranks the data and calculates a test statistic T\nNI using the Wilcoxon signed-rank test. The conclusion of the test carried out at a 1% significance level is\nETIRW\nthat there is not sufficient evidence to support her belief.\nTON Using a normal approximation, find the least possible value of T. [5]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT ....................................................................................................................................................................\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\nOD ....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM ....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\n.................................................................................................................................................................... NI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n..................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2025-on-43-q04-6fd6cca5c0.pdf"
    },
    {
      "id": "9231-2025-on-43-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 5,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_43.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_43.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2025-on-43-q05-p01-71157e2ae8.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2025-on-43-q05-p02-771eac4acb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-43-q05",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n5 A continuous random variable X has probability density function f given by\nSIHT\nZ\nNI 1 ]\nx x 4, G 1 0\nETIRW ]16\n]\n1 f(x) = [ 4 G x G 9,\nTON ]k x\n]\n0 otherwise,\nOD\n\\\nwhere k is a constant.\nk = 3. [2] (a) Show that\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW (b) Find the median value of X. [3]\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n......................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2025-on-43-q05-d373ad2a27.pdf"
    },
    {
      "id": "9231-2025-on-43-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 6,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_43.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-43-q06-p01-b57c0e67c5.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-43-q06-p02-a102e17cbe.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-43-q06",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n6 Nine athletes in a club have a new coach. The coach adopts a new training programme which he\nSIHT\nbelieves will reduce the race times of these athletes. Each athlete completes a 1500 m time trial before\nand after completing the new training programme. Their times, in seconds (s), are recorded. NI\nETIRW\nAthlete A B C D E F G H I\nTON\nTime before training (s) 250 251 252 267 276 291 310 320 335\nOD\nTime after training (s) 245 251 253 261 275 293 302 313 320\n(a) Carry out a paired t-test at the 5% significance level to test the coach’s belief. [7]\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n....................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-on-43-q06-049e5b55f9.pdf"
    },
    {
      "id": "9231-2025-on-43-q07",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "43",
      "question_number": 7,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_43.pdf?download=true",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_43.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2025-on-43-q07-p01-33f5968183.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2025-on-43-q07-p02-44076b74ac.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-43-q07",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\nP(X = r) 7 A discrete random variable X takes values r = 0,1, 2 with probabilities as given in the\nSIHT\nfollowing table.\nNI\nETIRW\nr 0 1 2\nTON a 2a b P(X = r)\nOD\n(a) Write down the probability generating function of X, and use it to find an expression for E(X ) in\nterms of a and b. [2]\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\nVar(X) = 2b+2(a+b)(1-2a-2b). [3] (b) Show that\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n........................................................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2025-on-43-q07-4ea0283188.pdf"
    },
    {
      "id": "9231-2025-on-44-q01",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "44",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_44.pdf?download=true",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_44.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-44-q01-p01-e354a4967f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-44-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 Local residents are concerned about the speed of cars travelling through their village. They record the\nSIHT\nkmh-1, speed, in of a random sample of 13 cars travelling through their village. The recorded speeds\nNI are as follows.\nETIRW\n40 53 59 42 43 48 62 67 46 82 66 45 70\nTON\n(a) Construct a 90% confidence interval for the population mean speed of cars passing through the\nOD\nvillage. [5]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n....................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-on-44-q01-faff4ff19e.pdf"
    },
    {
      "id": "9231-2025-on-44-q02",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "44",
      "question_number": 2,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_44.pdf?download=true",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_44.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2025-on-44-q02-p01-7d76b29512.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2025-on-44-q02-p02-77751a5a6a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-44-q02",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 The continuous random variable X has probability density function f given by\nSIHT\nNI 4(x+1) x 1, 0 G G\nETIRW 9\n*\nf(x) =\n(x-2)2 x 2, 1 1 G\nTON\notherwise. 0\nOD\n(a) Find the cumulative distribution function of X. [3]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2025-on-44-q02-60d8ccd70a.pdf"
    },
    {
      "id": "9231-2025-on-44-q03",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "44",
      "question_number": 3,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_44.pdf?download=true",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_44.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-44-q03-p01-f1a0c87d83.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2025-on-44-q03-p02-65134eae05.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-44-q03",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 Two different types of juice extractor, machine X and machine Y, are being compared. The manufacturer\nSIHT\nclaims that machine Y extracts more juice per orange on average than machine X.\nNI\nETIRW A random sample of 20 oranges is selected and each carefully measured. Oranges of a similar size and\nmass are then paired and numbered from 1 to 10. One orange from each pair is randomly allocated\nTON to machine X with the other allocated to machine Y. The amount of juice, in ml, extracted from each\norange is recorded in the following table.\nOD\npair\n1 2 3 4 5 6 7 8 9 10\nNIGRAM\nmachine X 65 73 58 61 72 79 64 65 69 71\nSIHT machine Y 68 72 64 63 75 82 63 63 72 74\nNI\nETIRW\n(a) Use a t-test to test the manufacturer’s claim at the 1% significance level. [7]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD .........................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2025-on-44-q03-701bca586f.pdf"
    },
    {
      "id": "9231-2025-on-44-q04",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "44",
      "question_number": 4,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_44.pdf?download=true",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_44.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2025-on-44-q04-p01-a22d2aa8e9.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2025-on-44-q04-p02-6dcfdd1bc8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-44-q04",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n3 2 4 The random variable X takes values 1 and 2 with probabilities and respectively. SIHT\n5 5\nNI\nG (t) of X. [1] (a) Write down the probability generating function\nETIRW X\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\nThe random variable Y is the sum of four independent observations of X. NIGRAM\nG (t) of Y. Give your answer in the form (b) Find the probability generating function\nY SIHT atm(b+ct)n, where a, b, c, m and n are constants to be determined. [2] G (t) =\nY\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\nP(Y = 6). G (t) to find [2] (c) Use\nY\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2025-on-44-q04-a1793d7f77.pdf"
    },
    {
      "id": "9231-2025-on-44-q05",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "44",
      "question_number": 5,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_44.pdf?download=true",
      "source_pages": [
        10
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_44.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2025-on-44-q05-p01-d596fd26b8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-44-q05",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 A driving instructor believes that the performance (pass or fail) of students when taking a driving test\nSIHT\nis associated with their age. The following table summarises the number of students who pass and who\nfail, and the ages in years of the students taking the test, over a period of three years. NI\nETIRW\nage of student\nTON\nunder 20 20 – 40 over 40 total\nOD\npass 34 41 6 81\nfail 16 39 9 64\nNIGRAM\ntotal 50 80 15 145\nSIHT\nTest, at the 10% significance level, whether performance is independent of age of student. [7]\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n.......................................................................................................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2025-on-44-q05-f548f3435f.pdf"
    },
    {
      "id": "9231-2025-on-44-q06",
      "subject": "9231",
      "year": 2025,
      "session": "Oct/Nov",
      "session_code": "on",
      "paper": 4,
      "variant": "44",
      "question_number": 6,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2025-oct-nov/9231_w25_qp_44.pdf?download=true",
      "source_pages": [
        11
      ],
      "local_pdf": "_source-pdfs/2025-Oct-Nov/qp/9231_w25_qp_44.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2025-on-44-q06-p01-1b36f60a92.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2025-on-44-q06",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:2)(cid:44)\n6 Students of the same age from two schools, school A and school B, take a large number of quizzes\nSIHT\nthroughout the year and are each awarded a mark out of 1000. The marks of 123 students in school A\nNI and 147 students in school B are ranked from lowest (rank 1) to highest (rank 270). The sum of the\nETIRW ranks of the students from school A is 15 355.\nTON (a) Carry out a Wilcoxon rank-sum test at the 5% significance level to investigate whether there is a\ndifference in average marks between the students in school A and school B. [6]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n..............................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2025-on-44-q06-a8f731929f.pdf"
    },
    {
      "id": "9231-2026-mj-11-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-11.pdf?download=1",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_11.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2026-mj-11-q01-p01-f5c6c2f2c5.png",
        "9231-topic-13-summation-of-series/assets/9231-2026-mj-11-q01-p02-08a7b9e8ac.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-11-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 (a) Use standard results from the list of formulae (MF19) to show that\nSIHT\n/n NI\nan3 +bn2 (4r-3)(12r+3) +cn , = ETIRW\nr=1\nTON\nwhere a, b and c are integers to be determined. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n.........................................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2026-mj-11-q01-3c48aca519.pdf"
    },
    {
      "id": "9231-2026-mj-11-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-11.pdf?download=1",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_11.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2026-mj-11-q02-p01-e405d0ee6d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-11-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n36n +92n +89 2 Prove by mathematical induction that is divisible by 91 for all positive integers n. [6] SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON ....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2026-mj-11-q02-64d88ed733.pdf"
    },
    {
      "id": "9231-2026-mj-11-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 3,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-11.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_11.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2026-mj-11-q03-p01-bc8450f107.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2026-mj-11-q03-p02-f64b0f2c0e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-11-q03",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 The matrix M represents the sequence of two transformations in the x-y plane given by a stretch\nSIHT\nparallel to the x-axis, scale factor k (k ! 0,1), followed by a shear, x-axis fixed, with (0,1) mapped\nto (2,1). NI\nETIRW\n2 k\ne o. (a) Show that M = [4]\n0 1\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n......................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2026-mj-11-q03-b088cc6cee.pdf"
    },
    {
      "id": "9231-2026-mj-11-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 4,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-11.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_11.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2026-mj-11-q04-p01-9954e7c629.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2026-mj-11-q04-p02-e82f49f2ed.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-11-q04",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5x3 -2x+1 = 0 has roots a, b, c. 4 The cubic equation SIHT\n1 1 1 NI\n, , . [3] (a) Find a cubic equation whose roots are\nETIRW a2 b2 c2\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n1 1 1\n+ + [3] . (b) Find the value of\na4 b4 c4\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n...........................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2026-mj-11-q04-bfe26732d4.pdf"
    },
    {
      "id": "9231-2026-mj-11-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 5,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-11.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_11.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2026-mj-11-q05-p01-d388a96eb3.png",
        "9231-topic-16-vectors/assets/9231-2026-mj-11-q05-p02-086cf6d12e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-11-q05",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\nl l have equations and 5 The lines r =-2i+4j+4k+m(3i-4jk+2 ) and\nSIHT\n2 1\nr =-3i-2j+4k+n(2i+4jk+3 ) respectively.\nNI\nETIRW\nl l . [5] and (a) Find the shortest distance between\n2 1\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ......................................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2026-mj-11-q05-ca2cbf3b08.pdf"
    },
    {
      "id": "9231-2026-mj-11-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-11.pdf?download=1",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_11.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2026-mj-11-q06-p01-f00bc3607d.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2026-mj-11-q06-p02-4ec10a526e.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2026-mj-11-q06-p03-cc4ad89ab0.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-11-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\nx2 -x+2 SIHT\ny = 6 The curve C has equation .\n4x+4\nNI\n(a) Find the equations of the asymptotes of C. [3] ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n(b) Find the coordinates of any stationary points on C. [3]\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n.....................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2026-mj-11-q06-ad949b3c47.pdf"
    },
    {
      "id": "9231-2026-mj-11-q07",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "11",
      "question_number": 7,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 17,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-11.pdf?download=1",
      "source_pages": [
        15,
        16,
        17
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_11.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2026-mj-11-q07-p01-2b5ad03792.png",
        "9231-topic-15-polar-coordinates/assets/9231-2026-mj-11-q07-p02-db37325b3e.png",
        "9231-topic-15-polar-coordinates/assets/9231-2026-mj-11-q07-p03-c7330705b8.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-11-q07",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:6)(cid:44)\na\n1 r2 = 7 The curve C has polar equation for r, where a is a positive constant. G 1 0 SIHT i\n1+tan2i 4\nNI\n(a) Show that, at the point on C furthest from the initial line,\nETIRW\nsini\n0 cosi- = TON\ncos2i(sin2i+cos2i)\nOD\nand verify that this equation has a root between 0.5 and 0.6. [6]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n...........",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2026-mj-11-q07-6803c7e5e0.pdf"
    },
    {
      "id": "9231-2026-mj-12-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-12.pdf?download=1",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_12.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2026-mj-12-q01-p01-425e938199.png",
        "9231-topic-13-summation-of-series/assets/9231-2026-mj-12-q01-p02-f5818fd473.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-12-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 (a) Use standard results from the list of formulae (MF19) to show that\nSIHT\n/n NI\nan3 +bn2 (4r-3)(12r+3) +cn , = ETIRW\nr=1\nTON\nwhere a, b and c are integers to be determined. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n.........................................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2026-mj-12-q01-7d311aefaa.pdf"
    },
    {
      "id": "9231-2026-mj-12-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 2,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-12.pdf?download=1",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_12.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2026-mj-12-q02-p01-9977571537.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-12-q02",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n36n +92n +89 2 Prove by mathematical induction that is divisible by 91 for all positive integers n. [6] SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON ....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2026-mj-12-q02-62a0277d5e.pdf"
    },
    {
      "id": "9231-2026-mj-12-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 3,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-12.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_12.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2026-mj-12-q03-p01-4fc4ba07cf.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2026-mj-12-q03-p02-6f44e0e2dc.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-12-q03",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 The matrix M represents the sequence of two transformations in the x-y plane given by a stretch\nSIHT\nparallel to the x-axis, scale factor k (k ! 0,1), followed by a shear, x-axis fixed, with (0,1) mapped\nto (2,1). NI\nETIRW\n2 k\ne o. (a) Show that M = [4]\n0 1\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n......................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2026-mj-12-q03-2c4a4503d7.pdf"
    },
    {
      "id": "9231-2026-mj-12-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 4,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-12.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_12.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2026-mj-12-q04-p01-6efb4afd9e.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2026-mj-12-q04-p02-58a0edb016.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-12-q04",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5x3 -2x+1 = 0 has roots a, b, c. 4 The cubic equation SIHT\n1 1 1 NI\n, , . [3] (a) Find a cubic equation whose roots are\nETIRW a2 b2 c2\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n1 1 1\n+ + [3] . (b) Find the value of\na4 b4 c4\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n...........................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2026-mj-12-q04-1357297992.pdf"
    },
    {
      "id": "9231-2026-mj-12-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 5,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-12.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_12.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2026-mj-12-q05-p01-e75a659445.png",
        "9231-topic-16-vectors/assets/9231-2026-mj-12-q05-p02-e25698aa00.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-12-q05",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\nl l have equations and 5 The lines r =-2i+4j+4k+m(3i-4jk+2 ) and\nSIHT\n2 1\nr =-3i-2j+4k+n(2i+4jk+3 ) respectively.\nNI\nETIRW\nl l . [5] and (a) Find the shortest distance between\n2 1\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ......................................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2026-mj-12-q05-ce4132609f.pdf"
    },
    {
      "id": "9231-2026-mj-12-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-12.pdf?download=1",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_12.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2026-mj-12-q06-p01-d1b1ac7dcf.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2026-mj-12-q06-p02-7905aa9e39.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2026-mj-12-q06-p03-8d6713a9b3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-12-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\nx2 -x+2 SIHT\ny = 6 The curve C has equation .\n4x+4\nNI\n(a) Find the equations of the asymptotes of C. [3] ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n(b) Find the coordinates of any stationary points on C. [3]\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n.....................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2026-mj-12-q06-95712549a4.pdf"
    },
    {
      "id": "9231-2026-mj-12-q07",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "12",
      "question_number": 7,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 17,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-12.pdf?download=1",
      "source_pages": [
        15,
        16,
        17
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_12.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2026-mj-12-q07-p01-f39339c036.png",
        "9231-topic-15-polar-coordinates/assets/9231-2026-mj-12-q07-p02-ce112a78e4.png",
        "9231-topic-15-polar-coordinates/assets/9231-2026-mj-12-q07-p03-bb46d261b3.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-12-q07",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:6)(cid:44)\na\n1 r2 = 7 The curve C has polar equation for r, where a is a positive constant. G 1 0 SIHT i\n1+tan2i 4\nNI\n(a) Show that, at the point on C furthest from the initial line,\nETIRW\nsini\n0 cosi- = TON\ncos2i(sin2i+cos2i)\nOD\nand verify that this equation has a root between 0.5 and 0.6. [6]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n...........",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2026-mj-12-q07-70322970d6.pdf"
    },
    {
      "id": "9231-2026-mj-13-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 1,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-13.pdf?download=1",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_13.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2026-mj-13-q01-p01-84a23bd3ed.png",
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2026-mj-13-q01-p02-c0b145d262.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-13-q01",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\nx3 +bx2 +cx+d = 0 , where b, c and d are constants, has roots a, and . It is 1 The cubic equation b c SIHT\ngiven that\nNI\nETIRW\n5, +b +c = a\n2 2 2 =-19, +b +c a\nTON\n-1 -1 -1 11. +bc + = a\nOD\n(a) Find the values of b, c and d. [5]\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n...................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2026-mj-13-q01-a44cb67e1c.pdf"
    },
    {
      "id": "9231-2026-mj-13-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 2,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-13.pdf?download=1",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_13.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2026-mj-13-q02-p01-6bcc09d5bf.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2026-mj-13-q02-p02-7c40810f41.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-13-q02",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "1 -1 SIHT 2 2 1 2\nf2 p 2 e o. M = 2 The matrix M is given by\n1 1 0 1 2 2 NI\n2 2\nETIRW\n(a) The matrix M represents a sequence of two geometrical transformations in the x-y plane.\nTON\nGive full details of each transformation, and make clear the order in which they are applied. [4] OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n......................................................................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2026-mj-13-q02-6873f74aec.pdf"
    },
    {
      "id": "9231-2026-mj-13-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 3,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-13.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_13.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2026-mj-13-q03-p01-5dfa9db21c.png",
        "9231-topic-17-proof-by-induction/assets/9231-2026-mj-13-q03-p02-8cda41a4e7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-13-q03",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 The matrices A, P and D are given by\nSIHT\n-5 8 5 3 1 0 NI\ne o, e o, e o A P D = = = .\nETIRW\n-4 7 7 3 0 3\nTON\nAn PDnP-1. Prove by mathematical induction that, for all positive integers n, = [7]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW ....................................................................................................................................................................\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n.....................................................................................................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2026-mj-13-q03-d7b21bd6a0.pdf"
    },
    {
      "id": "9231-2026-mj-13-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 4,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-13.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_13.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2026-mj-13-q04-p01-e546b4fa99.png",
        "9231-topic-13-summation-of-series/assets/9231-2026-mj-13-q04-p02-256f537dac.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-13-q04",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n(2r+1)4 (2r-1)4, - 4 (a) By considering use the method of differences to prove that SIHT\nNI /n\n1n2(n+1)2. ETIRW r3 = [6]\n4\nr=1\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ...........",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2026-mj-13-q04-9be7aa63e2.pdf"
    },
    {
      "id": "9231-2026-mj-13-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 5,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-13.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_13.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2026-mj-13-q05-p01-d6e40b250a.png",
        "9231-topic-15-polar-coordinates/assets/9231-2026-mj-13-q05-p02-2f9703bb36.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-13-q05",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n1\nr2 isec2 SIHT = 5 The curve C has polar equation . , where 0 G G i r i\n3\nNI\n(a) Sketch C and state the greatest distance of a point on C from the pole. [2]\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\n............................................................................................................................................................\nTON\nOD\n1\n(b) Find the exact value of the area of the region bounded by C and the half-line r. [5] = i\n3\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n.........",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2026-mj-13-q05-30cb3d7064.pdf"
    },
    {
      "id": "9231-2026-mj-13-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-13.pdf?download=1",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_13.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2026-mj-13-q06-p01-1f1b1b7013.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2026-mj-13-q06-p02-5cc4873ef2.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2026-mj-13-q06-p03-4467ba43e6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-13-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\nx+1\nSIHT = 6 The curve C has equation y .\nx2 +x+1\nNI\n(a) Show that C has no vertical asymptotes and state the equation of the horizontal asymptote. [2]\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\nOD\n(b) Find the coordinates of any stationary points on C. [3]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n................................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2026-mj-13-q06-592e87afcf.pdf"
    },
    {
      "id": "9231-2026-mj-13-q07",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "13",
      "question_number": 7,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-13.pdf?download=1",
      "source_pages": [
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_13.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2026-mj-13-q07-p01-f869b1540e.png",
        "9231-topic-16-vectors/assets/9231-2026-mj-13-q07-p02-31001b053f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-13-q07",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:6)(cid:44)\nr = j+2k+s(i-j) +t(-3i+j-k). 7 The plane has equation P\nSIHT\nNI ax+by+cz = d. (a) Obtain an equation of in the form [4] P\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\nNIGRAM (6+6i)i+ (1+i)j+ (2+5i)k. (b) The point P has position vector\n6. [4] for which the perpendicular distance of P from is less than Find the set of values of P i\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n..",
      "pdf_path": "9231-topic-16-vectors/print/9231-2026-mj-13-q07-85c2601c69.pdf"
    },
    {
      "id": "9231-2026-mj-14-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 1,
      "topic_number": 13,
      "topic": "Summation of series",
      "topic_slug": "9231-topic-13-summation-of-series",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-14.pdf?download=1",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_14.pdf",
      "image_paths": [
        "9231-topic-13-summation-of-series/assets/9231-2026-mj-14-q01-p01-a9ede480c8.png",
        "9231-topic-13-summation-of-series/assets/9231-2026-mj-14-q01-p02-f353429393.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-14-q01",
        "html": "9231-topic-13-summation-of-series/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 (a) Use standard results from the list of formulae (MF19) to show that\nSIHT\nNI /n\nETIRW (r-1)2 an3 +bn2 = +cn,\nr=1\nTON\nwhere a, b and c are constants to be determined. [2]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI .............................................................................................................................",
      "pdf_path": "9231-topic-13-summation-of-series/print/9231-2026-mj-14-q01-8f8ddafc68.pdf"
    },
    {
      "id": "9231-2026-mj-14-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 2,
      "topic_number": 11,
      "topic": "Roots of polynomial equations",
      "topic_slug": "9231-topic-11-roots-of-polynomial-equations",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-14.pdf?download=1",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_14.pdf",
      "image_paths": [
        "9231-topic-11-roots-of-polynomial-equations/assets/9231-2026-mj-14-q02-p01-02cdcc470f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-14-q02",
        "html": "9231-topic-11-roots-of-polynomial-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2x4 +3x3 a, b, c, +x+1 = 0 has roots d. 2 The equation SIHT\n1 1 1 1 NI\n, , , and state the value of (a) Find a quartic equation whose roots are\nETIRW 3 3 3 3 a b c d\n1 1 1 1\n+ + + . [5]\n3 3 3 3 TON a b c d\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n.....................................................................................................",
      "pdf_path": "9231-topic-11-roots-of-polynomial-equations/print/9231-2026-mj-14-q02-c76eee759e.pdf"
    },
    {
      "id": "9231-2026-mj-14-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 3,
      "topic_number": 17,
      "topic": "Proof by induction",
      "topic_slug": "9231-topic-17-proof-by-induction",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-14.pdf?download=1",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_14.pdf",
      "image_paths": [
        "9231-topic-17-proof-by-induction/assets/9231-2026-mj-14-q03-p01-cdb2bd1690.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-14-q03",
        "html": "9231-topic-17-proof-by-induction/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:6)(cid:44)\nu 2 1 u , u ,,u. .. is such that and, for n H 1, 3 The sequence of positive numbers\nSIHT 1 1 23\nNI u2 +2u +2\nETIRW n n u = .\n+4 u n+1\nn\nTON\n(a) By considering u 2 1 u -1, or otherwise, prove by mathematical induction that for all\nOD\nn n+1\npositive integers n. [5]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\nNI u 1 u for n H 1. [3] (b) Show that\nETIRW n+1 n\n....................................................................................................................................................",
      "pdf_path": "9231-topic-17-proof-by-induction/print/9231-2026-mj-14-q03-71ed70cb60.pdf"
    },
    {
      "id": "9231-2026-mj-14-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 4,
      "topic_number": 14,
      "topic": "Matrices (Further Pure 1)",
      "topic_slug": "9231-topic-14-matrices-further-pure-1",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-14.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_14.pdf",
      "image_paths": [
        "9231-topic-14-matrices-further-pure-1/assets/9231-2026-mj-14-q04-p01-f6a7772a95.png",
        "9231-topic-14-matrices-further-pure-1/assets/9231-2026-mj-14-q04-p02-655dbcab3b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-14-q04",
        "html": "9231-topic-14-matrices-further-pure-1/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 The matrix A is given by\nSIHT\n1 k -1 NI\nf p ETIRW A = 3 1 k ,\n-1 3 1\nTON\nwhere k is a real constant.\nOD\n(a) Find the values of k for which A is singular. [3]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\nThe matrices B and C are given by\nNI\nETIRW\n1 0\n1 0 0\nf p\ne o. B 0 = -1 C = and TON\n1 0 1\n1 0\nOD\nk+3 -1\ne o. = CAB [3] (b) Show that\n-k-1 2\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................",
      "pdf_path": "9231-topic-14-matrices-further-pure-1/print/9231-2026-mj-14-q04-afb610dbd4.pdf"
    },
    {
      "id": "9231-2026-mj-14-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 5,
      "topic_number": 16,
      "topic": "Vectors",
      "topic_slug": "9231-topic-16-vectors",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-14.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_14.pdf",
      "image_paths": [
        "9231-topic-16-vectors/assets/9231-2026-mj-14-q05-p01-95d2f190c4.png",
        "9231-topic-16-vectors/assets/9231-2026-mj-14-q05-p02-1caf26c581.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-14-q05",
        "html": "9231-topic-16-vectors/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 The points A, B, C have position vectors\nSIHT\n4i+3jk+,3 2i+2j+k, 5i+2jk+,2 NI\nETIRW\nrespectively, relative to the origin O.\nTON\nax+by+cz = d. [5] (a) Find the equation of the plane ABC, giving your answer in the form\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n2i+j+3k. The point D has position vector\nTON\n(b) Find the acute angle between the line OD and the plane ABC. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ......................................................................................................................",
      "pdf_path": "9231-topic-16-vectors/print/9231-2026-mj-14-q05-3b97e76231.pdf"
    },
    {
      "id": "9231-2026-mj-14-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 6,
      "topic_number": 12,
      "topic": "Rational functions and graphs",
      "topic_slug": "9231-topic-12-rational-functions-and-graphs",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-14.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_14.pdf",
      "image_paths": [
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2026-mj-14-q06-p01-f8afae9e4d.png",
        "9231-topic-12-rational-functions-and-graphs/assets/9231-2026-mj-14-q06-p02-0272f84bd1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-14-q06",
        "html": "9231-topic-12-rational-functions-and-graphs/answers.html"
      },
      "text_excerpt": "NIGRAM\n2x2 -x+1 SIHT\n= f(x) y = f(x), 6 The curve C has equation . where\n2x2 -2x\nNI\n(a) Find the equations of the asymptotes of C. [2] ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD\n(b) Find the coordinates of any stationary points on C, giving your answers correct to\n1 decimal place. [4]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ...........................................................................................................................................................",
      "pdf_path": "9231-topic-12-rational-functions-and-graphs/print/9231-2026-mj-14-q06-b725f82e29.pdf"
    },
    {
      "id": "9231-2026-mj-14-q07",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 1,
      "variant": "14",
      "question_number": 7,
      "topic_number": 15,
      "topic": "Polar coordinates",
      "topic_slug": "9231-topic-15-polar-coordinates",
      "marks": 16,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-14.pdf?download=1",
      "source_pages": [
        12,
        13,
        14
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_14.pdf",
      "image_paths": [
        "9231-topic-15-polar-coordinates/assets/9231-2026-mj-14-q07-p01-4b7a063f26.png",
        "9231-topic-15-polar-coordinates/assets/9231-2026-mj-14-q07-p02-6ad7d594ae.png",
        "9231-topic-15-polar-coordinates/assets/9231-2026-mj-14-q07-p03-f647e0e5b2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-14-q07",
        "html": "9231-topic-15-polar-coordinates/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n7 (a) Show that the curve with Cartesian equation\nSIHT\n3\n`x2 +y2j2 xy = NI\nETIRW\n1 has polar equation sin2i. [3] r =\n2\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\nC and C have polar equations The curves OD\n1 2\n1\n1 r2 and = r = sin2i\n2 2\n1 NIGRAM bi- rl +5\n4\n1 respectively, where r. 0 G G i\n2 SIHT\nC and C intersect where (b) Show that\nNI\n1 2\nETIRW\n2\n1 +5osin22i ebi- rl = 4 TON\n4\nOD\nand verify by calculation that this equation has a root between = 0.5 and = 0.6. [3] a i i\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n........................",
      "pdf_path": "9231-topic-15-polar-coordinates/print/9231-2026-mj-14-q07-f5b48920d3.pdf"
    },
    {
      "id": "9231-2026-mj-21-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 1,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-21.pdf?download=1",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_21.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2026-mj-21-q01-p01-c9043269cd.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-21-q01",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\nlna+bx+cx2, sinh-1(x+1) 1 Find the first three terms in the Maclaurin’s series for in the form giving SIHT\nthe exact values of the constants a, b and c. [5]\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2026-mj-21-q01-a2b03b160d.pdf"
    },
    {
      "id": "9231-2026-mj-21-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 2,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-21.pdf?download=1",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2026-mj-21-q02-p01-df60a2440e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-21-q02",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\ny 1 n SIHT\n`1+tj 2 Let sinrtdt, where n is a non-negative integer. = I\nn\n0\nNI\n2I `2n +1j-n(n-1)I = . [4] n H 2, (a) Show that, for r r ETIRW\nn n-2\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\nI . [3] (b) Find the exact value of\n2\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n.......................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2026-mj-21-q02-83d2ec46fb.pdf"
    },
    {
      "id": "9231-2026-mj-21-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 3,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-21.pdf?download=1",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2026-mj-21-q03-p01-6c55a3ffd2.png",
        "9231-topic-26-differential-equations/assets/9231-2026-mj-21-q03-p02-4ec0cd0771.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-21-q03",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\ny\nSIHT sinh3xdx. u = coshx, 3 (a) Using the substitution find [4]\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n.................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2026-mj-21-q03-46ead49107.pdf"
    },
    {
      "id": "9231-2026-mj-21-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 4,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-21.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_21.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2026-mj-21-q04-p01-bde10d1664.png",
        "9231-topic-23-differentiation/assets/9231-2026-mj-21-q04-p02-a72d21f086.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-21-q04",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 The curve C has equation\nSIHT\n(x-y)5 +ln(xy) = 1+ln2. NI\nETIRW\ndy 8\n. [3] = (a) Show that, at the point (-1, -2) on C,\n11 dx\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ...........",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2026-mj-21-q04-dc65f40ef4.pdf"
    },
    {
      "id": "9231-2026-mj-21-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 5,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-21.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_21.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2026-mj-21-q05-p01-c0e53f5c07.png",
        "9231-topic-24-integration/assets/9231-2026-mj-21-q05-p02-777b838a6b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-21-q05",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5\nSIHT\ny\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nx\n0 1 1 2 n-1\nn n n\nSIHT\n1 11 x3 = + y The diagram shows the curve with equation for 0 G x G 1, together with a set of n NI\n10 10\nETIRW 1\nrectangles of width .\nn\ny 1 TON b11 1 x3 ldx , where + 1 U (a) By considering the sum of the areas of these rectangles, show that\n10 10 n\n0 OD\n2 11 1 1\nb1+ l . [4] U = +\nn 40 10 n\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\nTON\nNIGRAM\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\ny 1\nb11 1 x3 ldx. L for [4] + (b) Use a similar method to find, in terms of n, a lower bound\nNIGRAM 10 10 n\n0\n......................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2026-mj-21-q05-424ab91e0f.pdf"
    },
    {
      "id": "9231-2026-mj-21-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-21.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_21.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2026-mj-21-q06-p01-b909bd1dc4.png",
        "9231-topic-26-differential-equations/assets/9231-2026-mj-21-q06-p02-55f247868e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-21-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6 Find the particular solution of the differential equation\nSIHT\nNI d2y dy\nETIRW x2 +7x+1, 5 +4 +y =\ndx dx2\nTON\ndy\ngiven that, when x = 0, y = 0 and [10] 0. =\nOD dx\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI .........",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2026-mj-21-q06-7bf2a9023b.pdf"
    },
    {
      "id": "9231-2026-mj-21-q07",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 7,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-21.pdf?download=1",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_21.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2026-mj-21-q07-p01-3afe742c59.png",
        "9231-topic-25-complex-numbers/assets/9231-2026-mj-21-q07-p02-79f65573bd.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-21-q07",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\nz+z2 +z3 +...+zn, 7 (a) State the sum of the series for z ! 1. [1] SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n1+z+z2 +...+z6 eiqr, = 0 in the form where q is rational. [2] (b) Find all roots of the equation\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\ne-1+ii, z = use de Moivre’s theorem to show that (c) Given instead that\nNI\n3 ecosi-1 ETIRW / -mcosmi . [7] = e\ne2 -2ecosi+1\nm=1\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n...........................................................................................................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2026-mj-21-q07-74364652fa.pdf"
    },
    {
      "id": "9231-2026-mj-21-q08",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "21",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-21.pdf?download=1",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_21.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2026-mj-21-q08-p01-afbdbdae15.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2026-mj-21-q08-p02-4f4b06f849.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-21-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n8 (a) The constants a and b are such that the system of equations\nSIHT\n-x-2y +z = 1,\nNI\nETIRW ax+ y -z = 2,\nby+2z = 3\nTON\ndoes not have a unique solution. Find a in terms of b. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\nThe matrix A is given by\n-1 -2 1 NIGRAM\nf p\nA = 0 1 -1 .\n0 0 2\nSIHT\nA4 pA2 (b) Use the characteristic equation of A to show that = +qI, where p and q are integers to\nNI\nbe determined. [4] ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n.........................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2026-mj-21-q08-eab7af895a.pdf"
    },
    {
      "id": "9231-2026-mj-22-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 1,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-22.pdf?download=1",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_22.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2026-mj-22-q01-p01-725aee413e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-22-q01",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\nlna+bx+cx2, sinh-1(x+1) 1 Find the first three terms in the Maclaurin’s series for in the form giving SIHT\nthe exact values of the constants a, b and c. [5]\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2026-mj-22-q01-53e366bfc3.pdf"
    },
    {
      "id": "9231-2026-mj-22-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 2,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-22.pdf?download=1",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2026-mj-22-q02-p01-0cf3f16b48.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-22-q02",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\ny 1 n SIHT\n`1+tj 2 Let sinrtdt, where n is a non-negative integer. = I\nn\n0\nNI\n2I `2n +1j-n(n-1)I = . [4] n H 2, (a) Show that, for r r ETIRW\nn n-2\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\nI . [3] (b) Find the exact value of\n2\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n.......................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2026-mj-22-q02-5c78024c13.pdf"
    },
    {
      "id": "9231-2026-mj-22-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 3,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-22.pdf?download=1",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2026-mj-22-q03-p01-b205673dc0.png",
        "9231-topic-26-differential-equations/assets/9231-2026-mj-22-q03-p02-43974d127f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-22-q03",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\ny\nSIHT sinh3xdx. u = coshx, 3 (a) Using the substitution find [4]\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n.................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2026-mj-22-q03-b9715220ce.pdf"
    },
    {
      "id": "9231-2026-mj-22-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 4,
      "topic_number": 23,
      "topic": "Differentiation",
      "topic_slug": "9231-topic-23-differentiation",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-22.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_22.pdf",
      "image_paths": [
        "9231-topic-23-differentiation/assets/9231-2026-mj-22-q04-p01-c4fd4ebacd.png",
        "9231-topic-23-differentiation/assets/9231-2026-mj-22-q04-p02-2226092204.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-22-q04",
        "html": "9231-topic-23-differentiation/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 The curve C has equation\nSIHT\n(x-y)5 +ln(xy) = 1+ln2. NI\nETIRW\ndy 8\n. [3] = (a) Show that, at the point (-1, -2) on C,\n11 dx\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ...........",
      "pdf_path": "9231-topic-23-differentiation/print/9231-2026-mj-22-q04-1e8367b91d.pdf"
    },
    {
      "id": "9231-2026-mj-22-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 5,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-22.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_22.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2026-mj-22-q05-p01-d5415e5673.png",
        "9231-topic-24-integration/assets/9231-2026-mj-22-q05-p02-2b9f5b8836.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-22-q05",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5\nSIHT\ny\nNI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nNIGRAM\nx\n0 1 1 2 n-1\nn n n\nSIHT\n1 11 x3 = + y The diagram shows the curve with equation for 0 G x G 1, together with a set of n NI\n10 10\nETIRW 1\nrectangles of width .\nn\ny 1 TON b11 1 x3 ldx , where + 1 U (a) By considering the sum of the areas of these rectangles, show that\n10 10 n\n0 OD\n2 11 1 1\nb1+ l . [4] U = +\nn 40 10 n\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\nTON\nNIGRAM\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\ny 1\nb11 1 x3 ldx. L for [4] + (b) Use a similar method to find, in terms of n, a lower bound\nNIGRAM 10 10 n\n0\n......................................................................................................................",
      "pdf_path": "9231-topic-24-integration/print/9231-2026-mj-22-q05-dac6736ade.pdf"
    },
    {
      "id": "9231-2026-mj-22-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-22.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_22.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2026-mj-22-q06-p01-a26a2a4e96.png",
        "9231-topic-26-differential-equations/assets/9231-2026-mj-22-q06-p02-626c2111fe.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-22-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6 Find the particular solution of the differential equation\nSIHT\nNI d2y dy\nETIRW x2 +7x+1, 5 +4 +y =\ndx dx2\nTON\ndy\ngiven that, when x = 0, y = 0 and [10] 0. =\nOD dx\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI .........",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2026-mj-22-q06-ec7fb25d64.pdf"
    },
    {
      "id": "9231-2026-mj-22-q07",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 7,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-22.pdf?download=1",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_22.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2026-mj-22-q07-p01-3278078d59.png",
        "9231-topic-25-complex-numbers/assets/9231-2026-mj-22-q07-p02-4c481372a2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-22-q07",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\nz+z2 +z3 +...+zn, 7 (a) State the sum of the series for z ! 1. [1] SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n1+z+z2 +...+z6 eiqr, = 0 in the form where q is rational. [2] (b) Find all roots of the equation\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\ne-1+ii, z = use de Moivre’s theorem to show that (c) Given instead that\nNI\n3 ecosi-1 ETIRW / -mcosmi . [7] = e\ne2 -2ecosi+1\nm=1\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n...........................................................................................................................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2026-mj-22-q07-c46be98500.pdf"
    },
    {
      "id": "9231-2026-mj-22-q08",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "22",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 14,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-22.pdf?download=1",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_22.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2026-mj-22-q08-p01-06a65bb39e.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2026-mj-22-q08-p02-6234cce243.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-22-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n8 (a) The constants a and b are such that the system of equations\nSIHT\n-x-2y +z = 1,\nNI\nETIRW ax+ y -z = 2,\nby+2z = 3\nTON\ndoes not have a unique solution. Find a in terms of b. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\nThe matrix A is given by\n-1 -2 1 NIGRAM\nf p\nA = 0 1 -1 .\n0 0 2\nSIHT\nA4 pA2 (b) Use the characteristic equation of A to show that = +qI, where p and q are integers to\nNI\nbe determined. [4] ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n.........................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2026-mj-22-q08-c7afb63107.pdf"
    },
    {
      "id": "9231-2026-mj-23-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 1,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-23.pdf?download=1",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_23.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2026-mj-23-q01-p01-5977b29e11.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-23-q01",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\nw5 reii, +32 = 0, r 2 0 giving your answers in the form where and 1 Find the roots of the equation SIHT\n–r G i 1 r. [5]\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n.....................................................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2026-mj-23-q01-d7aca5ad37.pdf"
    },
    {
      "id": "9231-2026-mj-23-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 2,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-23.pdf?download=1",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2026-mj-23-q02-p01-f5ba7c3c22.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-23-q02",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n2 A curve has equation x, for 0 G x G 2. Find, in terms of r, the area of the surface generated y = SIHT\nwhen the curve is rotated through 2r radians about the x-axis. [5]\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT ....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD ....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM ....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\n.................................................................................................................................................................... NI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n.",
      "pdf_path": "9231-topic-24-integration/print/9231-2026-mj-23-q02-42cc9c752f.pdf"
    },
    {
      "id": "9231-2026-mj-23-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 3,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-23.pdf?download=1",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2026-mj-23-q03-p01-6b15f6c688.png",
        "9231-topic-24-integration/assets/9231-2026-mj-23-q03-p02-9485aa05f5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-23-q03",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\na y n SIHT\na 2 0. 3 Let dx, where n is a non-negative integer and `ln(x+1)j I =\nn\n0 NI\nn\n(x+1)`ln(x+1)j , or otherwise, show that for n H 1 (a) By considering the derivative of ETIRW\nn\n(a+1)`ln(a+1)j . I = -nI TON\nn n–1\n[3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n....................",
      "pdf_path": "9231-topic-24-integration/print/9231-2026-mj-23-q03-a89e3915d3.pdf"
    },
    {
      "id": "9231-2026-mj-23-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 4,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-23.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2026-mj-23-q04-p01-6535a57430.png",
        "9231-topic-26-differential-equations/assets/9231-2026-mj-23-q04-p02-fe46e80ebb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-23-q04",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 Find the particular solution of the differential equation\nSIHT\nd2x NI dx 1t\n196e 49 +14 +x = , ETIRW 7\ndt dt2\nTON\ndx\n0. = x = 7 [10] given that, when t = 0, and\nOD dt\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\nNI .....",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2026-mj-23-q04-eea51c4035.pdf"
    },
    {
      "id": "9231-2026-mj-23-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 5,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-23.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_23.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2026-mj-23-q05-p01-3e365532ec.png",
        "9231-topic-21-hyperbolic-functions/assets/9231-2026-mj-23-q05-p02-b0825eb487.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-23-q05",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 (a) Starting from the definitions of sech and tanh in terms of exponentials, prove that\nSIHT\n1-sech2t tanh2t. NI =\nETIRW\n[3]\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\nSIHT\nIt is given that\nNI\nETIRW x = tsinht y = tcosht. and\nTON\ndy ttanht+1\n(b) Show that = . [3]\nt+tanht dx OD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n..................................................................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2026-mj-23-q05-c20bfb3cc8.pdf"
    },
    {
      "id": "9231-2026-mj-23-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 6,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-23.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_23.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2026-mj-23-q06-p01-97cc07733a.png",
        "9231-topic-26-differential-equations/assets/9231-2026-mj-23-q06-p02-bc0ace2950.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-23-q06",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6 (a) Show that an appropriate integrating factor for\nSIHT\nNI\ndy\nETIRW `1-x2j `1-x2j +y = 1-x\ndx\nTON\n1+x\nOD is . [4]\n1-x\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n..............",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2026-mj-23-q06-4e8eab7c96.pdf"
    },
    {
      "id": "9231-2026-mj-23-q07",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 7,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 12,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-23.pdf?download=1",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_23.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2026-mj-23-q07-p01-04ba8cbd5a.png",
        "9231-topic-24-integration/assets/9231-2026-mj-23-q07-p02-256b0418c1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-23-q07",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\ny x\nSIHT 1-x2 dx. u = 7 (a) Use the substitution [3] to find\n1-x2\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\ny\nTON\nOD\nNIGRAM\nSIHT\nNI\nETIRW\nTON\nOD\nx\n0 1 2 n – 1 1\nn n n\nNIGRAM\nsin–1x y = The diagram shows the curve with equation for for 0 G x G 1, together with a set of\n1\nn rectangles of width . SIHT\nn\nNI (b) By considering the sum of the areas of these rectangles, show that\nETIRW\n/n\n1 r TON 1 sin–1b l r-1. H\nn n 2\n[5] OD r=1\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n.....",
      "pdf_path": "9231-topic-24-integration/print/9231-2026-mj-23-q07-39582fb95d.pdf"
    },
    {
      "id": "9231-2026-mj-23-q08",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "23",
      "question_number": 8,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 15,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-23.pdf?download=1",
      "source_pages": [
        14,
        15,
        16
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_23.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2026-mj-23-q08-p01-9f9b637f80.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2026-mj-23-q08-p02-125c6029cc.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2026-mj-23-q08-p03-10d7125623.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-23-q08",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n8 (a) The constants a and b are such that the system of equations\nSIHT\n6x-12y-12z = 1, NI\nETIRW\nax+ 4y+ z = 2,\n2x+ by- 8z = 3,\nTON\ndoes not have a unique solution. Find a in terms of b. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON\n–1 b = –32, and show that the system of equations in part (a) is inconsistent. a = (b) Given that\nOD 4\nInterpret this situation geometrically. [3]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n....................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2026-mj-23-q08-40989b3db2.pdf"
    },
    {
      "id": "9231-2026-mj-24-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 1,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-24.pdf?download=1",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_24.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2026-mj-24-q01-p01-e1dd93e5f4.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-24-q01",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 (a) Find the set of values of k for which the system of equations\nSIHT\n3x+ky+7z 1, = NI\nETIRW 8x+ y+2z = 2,\n8x+2y+4z 3, =\nTON\nhas a unique solution and interpret this situation geometrically. [4]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n7\nk = , show that the system of equations in part 1(a) is inconsistent and interpret this (b) Given that\n2 NIGRAM\nsituation geometrically. [2]\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n....................................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2026-mj-24-q01-6c5930733d.pdf"
    },
    {
      "id": "9231-2026-mj-24-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 2,
      "topic_number": 22,
      "topic": "Matrices (Further Pure 2)",
      "topic_slug": "9231-topic-22-matrices-further-pure-2",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-24.pdf?download=1",
      "source_pages": [
        3,
        4
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_24.pdf",
      "image_paths": [
        "9231-topic-22-matrices-further-pure-2/assets/9231-2026-mj-24-q02-p01-08356eacac.png",
        "9231-topic-22-matrices-further-pure-2/assets/9231-2026-mj-24-q02-p02-63bf27ba1d.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-24-q02",
        "html": "9231-topic-22-matrices-further-pure-2/answers.html"
      },
      "text_excerpt": "NIGRAM\n-1 a SIHT\ne o, A = 2 Let where a is a constant. It is given that 2 is an eigenvalue of A.\n5 1\nNI\nETIRW (a) Find a and the other eigenvalue of A. [4]\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n....................................................................................................",
      "pdf_path": "9231-topic-22-matrices-further-pure-2/print/9231-2026-mj-24-q02-46fbbb2e79.pdf"
    },
    {
      "id": "9231-2026-mj-24-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 3,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-24.pdf?download=1",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_24.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2026-mj-24-q03-p01-748f96a80a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-24-q03",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:6)(cid:44)\n3 Find the particular solution of the differential equation\nSIHT\ndy\n1x,\nNI +y e =\n2\ndx\nETIRW\ny = f(x). [7] given that y = 3 when x = 0. Give your answer in the form\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n..........",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2026-mj-24-q03-b08c89571e.pdf"
    },
    {
      "id": "9231-2026-mj-24-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 4,
      "topic_number": 21,
      "topic": "Hyperbolic functions",
      "topic_slug": "9231-topic-21-hyperbolic-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-24.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_24.pdf",
      "image_paths": [
        "9231-topic-21-hyperbolic-functions/assets/9231-2026-mj-24-q04-p01-70409eb5b3.png",
        "9231-topic-21-hyperbolic-functions/assets/9231-2026-mj-24-q04-p02-9bb5ed643e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-24-q04",
        "html": "9231-topic-21-hyperbolic-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 The curve C has parametric equations\nSIHT\n3\n= cosh2t-4t, x = 2cosh2t+3t, y for 0 G t G 1. NI\n2\nETIRW\n(a) Find, in terms of e, the length of C. [5]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n..........................................................................................................................................................",
      "pdf_path": "9231-topic-21-hyperbolic-functions/print/9231-2026-mj-24-q04-5a92bcc090.pdf"
    },
    {
      "id": "9231-2026-mj-24-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 5,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-24.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_24.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2026-mj-24-q05-p01-76af21eb1c.png",
        "9231-topic-24-integration/assets/9231-2026-mj-24-q05-p02-976c24f38f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-24-q05",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5\nSIHT\ny NI\nETIRW\nTON\nOD\nNIGRAM\nSIHT\nx 0 1 2 n – 1 n\nNI\nETIRW\n1\nThe diagram shows the curve y for x H 0, together with a set of n rectangles of unit =\n4x2 +4x+3\nTON\nwidth. By considering the sum of the areas of these rectangles, show that\nOD\n/n 4n2 1+2n+ +4n+3 1\n1 lnf p. 1 [9]\n2 1+ 3 4r2 +4r+3\nr=1\nNIGRAM\n.................................................................................................................................................................... SIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW ....................................................................................................................................................................\nTON\nNIGRAM\nSIHT\n..................",
      "pdf_path": "9231-topic-24-integration/print/9231-2026-mj-24-q05-a87fb4822c.pdf"
    },
    {
      "id": "9231-2026-mj-24-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 6,
      "topic_number": 24,
      "topic": "Integration",
      "topic_slug": "9231-topic-24-integration",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-24.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_24.pdf",
      "image_paths": [
        "9231-topic-24-integration/assets/9231-2026-mj-24-q06-p01-2b08341882.png",
        "9231-topic-24-integration/assets/9231-2026-mj-24-q06-p02-b5d4e82003.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-24-q06",
        "html": "9231-topic-24-integration/answers.html"
      },
      "text_excerpt": "NIGRAM\n1r y SIHT\n2 excosnxdx, 6 Let where n is a non-negative integer. = I\nn\n-1r\nNI\n2\nETIRW\n(a) Show that, for n H 2,\nTON\n1r y OD\n2 excosn-2xsin2xdx-nI , = n(n-1) I\nn n\n-1r\n2\nand deduce the reduction formula\nNIGRAM\n`n2 +1jI = n(n-1)I . [6]\nn n-2\nSIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n.",
      "pdf_path": "9231-topic-24-integration/print/9231-2026-mj-24-q06-2ee1e77e0d.pdf"
    },
    {
      "id": "9231-2026-mj-24-q07",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 7,
      "topic_number": 25,
      "topic": "Complex numbers",
      "topic_slug": "9231-topic-25-complex-numbers",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-24.pdf?download=1",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_24.pdf",
      "image_paths": [
        "9231-topic-25-complex-numbers/assets/9231-2026-mj-24-q07-p01-8867facbca.png",
        "9231-topic-25-complex-numbers/assets/9231-2026-mj-24-q07-p02-54c0556f50.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-24-q07",
        "html": "9231-topic-25-complex-numbers/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n7 (a) Use de Moivre’s theorem to show that\nSIHT\nNI\nsec7 i ETIRW sec7i . [6] =\n64-112sec2 i+56sec4 i-7sec6 i\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ..............................",
      "pdf_path": "9231-topic-25-complex-numbers/print/9231-2026-mj-24-q07-c48b6e5ee0.pdf"
    },
    {
      "id": "9231-2026-mj-24-q08",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 2,
      "variant": "24",
      "question_number": 8,
      "topic_number": 26,
      "topic": "Differential equations",
      "topic_slug": "9231-topic-26-differential-equations",
      "marks": 13,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-24.pdf?download=1",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_24.pdf",
      "image_paths": [
        "9231-topic-26-differential-equations/assets/9231-2026-mj-24-q08-p01-a49006ce2a.png",
        "9231-topic-26-differential-equations/assets/9231-2026-mj-24-q08-p02-52e8b8f6f5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-24-q08",
        "html": "9231-topic-26-differential-equations/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\nxy3 v = 8 It is given that and SIHT\nNI\n2 d2y dy dy ETIRW\n(3x+2)y2 (x+1)y3 xy2 +2xye o x+5. + + =\ndx dx dx2\nTON\n(a) Show that OD\nd2v 1 dv\n+ +v = x+5. [4]\n3 dx dx2\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD\n(b) Find the general solution for y in terms of x. [7]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n........................................................................",
      "pdf_path": "9231-topic-26-differential-equations/print/9231-2026-mj-24-q08-484cb91164.pdf"
    },
    {
      "id": "9231-2026-mj-31-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 1,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-31.pdf?download=1",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_31.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2026-mj-31-q01-p01-ef741cf61b.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2026-mj-31-q01-p02-03c6176822.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-31-q01",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1\nSIHT\nNI\nETIRW\nTON\nOD\n3r\nx\nNIGRAM\nr\nSIHT\nNI\nAn object consists of a uniform solid hemisphere of radius 3r attached to a uniform solid cylinder of ETIRW\nradius r and height x. The hemisphere and the cylinder are made of the same material. The axes of\nsymmetry of the hemisphere and the cylinder coincide (see diagram).\nTON\nOD (a) Find, in terms of r and x, the distance of the centre of mass of the object from the base of\nthe cylinder. [4]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n.....................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2026-mj-31-q01-ec54c90734.pdf"
    },
    {
      "id": "9231-2026-mj-31-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 2,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-31.pdf?download=1",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2026-mj-31-q02-p01-1dc4f21867.png",
        "9231-topic-33-circular-motion/assets/9231-2026-mj-31-q02-p02-d89d0feb94.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-31-q02",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2\nSIHT\nA\nNI\nETIRW\n15 m\n9 m\nTON\nOD\nP\nNIGRAM\nB SIHT\nNI\nThe point B is vertically below the point A. One end of a light inextensible string is attached to A, and ETIRW\nthe other end is attached to B. A particle P of mass 7 kg is attached to the string so that the length AP\nis 15 m. The particle P moves at a constant angular speed of 4 radians per second in a horizontal circle TON\nwith centre between A and B at a distance 9 m vertically below A (see diagram). The string has length\nOD\n28 m, and both sections of the string are taut.\nThe tensions in the sections of the string AP and BP are T T respectively. and\nB A\nNIGRAM\nT T . [5] and the value of Find the value of\nB A\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n.......",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2026-mj-31-q02-ad3d8f95d4.pdf"
    },
    {
      "id": "9231-2026-mj-31-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 3,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-31.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_31.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2026-mj-31-q03-p01-17c5ce7615.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2026-mj-31-q03-p02-1b068a1795.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-31-q03",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\nms–1 v 3 A particle P of mass 6.4 kg moves along a straight horizontal line. At time t s, the speed of P is SIHT\nand the distance of P from its starting position is x m. The forces acting on P are a constant driving\nNI 0.5v2 force of magnitude 8 N and a resistive force of magnitude N. Initially, P is at rest.\nETIRW\n(a) Find the speed of P when it has travelled 0.8 m. [5]\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n......................................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2026-mj-31-q03-73273eff7b.pdf"
    },
    {
      "id": "9231-2026-mj-31-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 4,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-31.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_31.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2026-mj-31-q04-p01-8bcc891242.png",
        "9231-topic-36-momentum/assets/9231-2026-mj-31-q04-p02-ece0052f00.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-31-q04",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 A particle P is moving on a fixed smooth horizontal surface with speed u. The particle collides with a\nSIHT\nfixed smooth vertical barrier. Immediately before the collision, the direction of motion of P makes an\nNI 12. angle with the barrier, where tana = a\n5 ETIRW\n4 times its kinetic energy before the collision. The kinetic energy of P after the collision is\n13 TON\n(a) Find the coefficient of restitution between P and the barrier. [4] OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n.......................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2026-mj-31-q04-c4db95f309.pdf"
    },
    {
      "id": "9231-2026-mj-31-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-31.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_31.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2026-mj-31-q05-p01-2446d5869a.png",
        "9231-topic-33-circular-motion/assets/9231-2026-mj-31-q05-p02-8a6ab2e942.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-31-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 A particle of mass m is attached to a fixed point O by a light inextensible string of length a. The\nSIHT\nparticle initially hangs directly below O at the point A. The particle is then projected horizontally with\nspeed ag, and begins to move in a vertical circle. 2 NI\nETIRW\ni. Give your (a) Find an expression for the tension in the string when it has turned through an angle\nTON answer in terms of m, g and i. [4]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n...............................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2026-mj-31-q05-083b4cb77f.pdf"
    },
    {
      "id": "9231-2026-mj-31-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 6,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-31.pdf?download=1",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_31.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2026-mj-31-q06-p01-996864419d.png",
        "9231-topic-34-hooke-s-law/assets/9231-2026-mj-31-q06-p02-1297cc6bf1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-31-q06",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 A light elastic string has natural length 2.5 m and modulus of elasticity 15 N. One end of the string\nSIHT\nis attached to the fixed point O, and the other end is attached to a particle P of mass 1.5 kg. The\nparticle P is held at O and then released from rest so that it begins to fall vertically. In its subsequent NI\nETIRW ms–1 motion, P moves with velocity v when the extension of the string is x m. The maximum\nextension of the string during this motion is E m.\nTON\n(a) Show that, for 0 1 x 1 E,\nOD\nv2 –4(x2 = -5x-12.5). [3]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n..............................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2026-mj-31-q06-14ad37523f.pdf"
    },
    {
      "id": "9231-2026-mj-31-q07",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "31",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-31.pdf?download=1",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_31.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2026-mj-31-q07-p01-4f1a95d159.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2026-mj-31-q07-p02-fe819fe045.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-31-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n7 The points O and A lie on a horizontal plane 10 m apart. The point A lies at the foot of another plane\nSIHT\n4 which is inclined at an angle of above the horizontal such that . The point B is on the tana = a\n3\ninclined plane at a distance of 5 m from A measured along a line of greatest slope. The points O, A and NI\nETIRW B are in the same vertical plane and angle OAB is obtuse.\nTON ms–1 A particle is projected from O at an angle u to the horizontal with initial speed towards the i\ninclined plane. The particle moves freely under gravity. The particle strikes the inclined plane at the\nOD\npoint B when its direction of motion is perpendicular to the inclined plane.\nFind, in either order, the value of u and the value of i. [6]\nNIGRAM\n....................................................................................................................................................................\nSIHT ....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD ....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM ....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\n.................................................................................................................................................................... NI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2026-mj-31-q07-897c50ffae.pdf"
    },
    {
      "id": "9231-2026-mj-32-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 1,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-32.pdf?download=1",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_32.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2026-mj-32-q01-p01-7772d31b84.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2026-mj-32-q01-p02-67144e9d03.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-32-q01",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1\nSIHT\nNI\nETIRW\nTON\nOD\n3r\nx\nNIGRAM\nr\nSIHT\nNI\nAn object consists of a uniform solid hemisphere of radius 3r attached to a uniform solid cylinder of ETIRW\nradius r and height x. The hemisphere and the cylinder are made of the same material. The axes of\nsymmetry of the hemisphere and the cylinder coincide (see diagram).\nTON\nOD (a) Find, in terms of r and x, the distance of the centre of mass of the object from the base of\nthe cylinder. [4]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n.....................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2026-mj-32-q01-f86b9561d4.pdf"
    },
    {
      "id": "9231-2026-mj-32-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 2,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-32.pdf?download=1",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2026-mj-32-q02-p01-05d4b6a2e6.png",
        "9231-topic-33-circular-motion/assets/9231-2026-mj-32-q02-p02-bfd40f396f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-32-q02",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2\nSIHT\nA\nNI\nETIRW\n15 m\n9 m\nTON\nOD\nP\nNIGRAM\nB SIHT\nNI\nThe point B is vertically below the point A. One end of a light inextensible string is attached to A, and ETIRW\nthe other end is attached to B. A particle P of mass 7 kg is attached to the string so that the length AP\nis 15 m. The particle P moves at a constant angular speed of 4 radians per second in a horizontal circle TON\nwith centre between A and B at a distance 9 m vertically below A (see diagram). The string has length\nOD\n28 m, and both sections of the string are taut.\nThe tensions in the sections of the string AP and BP are T T respectively. and\nB A\nNIGRAM\nT T . [5] and the value of Find the value of\nB A\n....................................................................................................................................................................\nSIHT\nNI ....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n.......",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2026-mj-32-q02-ca4c02e31e.pdf"
    },
    {
      "id": "9231-2026-mj-32-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 3,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-32.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_32.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2026-mj-32-q03-p01-e44488a5cb.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2026-mj-32-q03-p02-aaf07d4c47.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-32-q03",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\nms–1 v 3 A particle P of mass 6.4 kg moves along a straight horizontal line. At time t s, the speed of P is SIHT\nand the distance of P from its starting position is x m. The forces acting on P are a constant driving\nNI 0.5v2 force of magnitude 8 N and a resistive force of magnitude N. Initially, P is at rest.\nETIRW\n(a) Find the speed of P when it has travelled 0.8 m. [5]\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n......................................................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2026-mj-32-q03-b0ffbb6e74.pdf"
    },
    {
      "id": "9231-2026-mj-32-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 4,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-32.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_32.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2026-mj-32-q04-p01-de13398d7d.png",
        "9231-topic-36-momentum/assets/9231-2026-mj-32-q04-p02-4020d27bee.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-32-q04",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 A particle P is moving on a fixed smooth horizontal surface with speed u. The particle collides with a\nSIHT\nfixed smooth vertical barrier. Immediately before the collision, the direction of motion of P makes an\nNI 12. angle with the barrier, where tana = a\n5 ETIRW\n4 times its kinetic energy before the collision. The kinetic energy of P after the collision is\n13 TON\n(a) Find the coefficient of restitution between P and the barrier. [4] OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n.......................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2026-mj-32-q04-2d8e48e4ab.pdf"
    },
    {
      "id": "9231-2026-mj-32-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 5,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-32.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_32.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2026-mj-32-q05-p01-a1361f289b.png",
        "9231-topic-33-circular-motion/assets/9231-2026-mj-32-q05-p02-4178310838.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-32-q05",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 A particle of mass m is attached to a fixed point O by a light inextensible string of length a. The\nSIHT\nparticle initially hangs directly below O at the point A. The particle is then projected horizontally with\nspeed ag, and begins to move in a vertical circle. 2 NI\nETIRW\ni. Give your (a) Find an expression for the tension in the string when it has turned through an angle\nTON answer in terms of m, g and i. [4]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n...............................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2026-mj-32-q05-ac1e4ef174.pdf"
    },
    {
      "id": "9231-2026-mj-32-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 6,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-32.pdf?download=1",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_32.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2026-mj-32-q06-p01-b2562de879.png",
        "9231-topic-34-hooke-s-law/assets/9231-2026-mj-32-q06-p02-e48ca4e83f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-32-q06",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6 A light elastic string has natural length 2.5 m and modulus of elasticity 15 N. One end of the string\nSIHT\nis attached to the fixed point O, and the other end is attached to a particle P of mass 1.5 kg. The\nparticle P is held at O and then released from rest so that it begins to fall vertically. In its subsequent NI\nETIRW ms–1 motion, P moves with velocity v when the extension of the string is x m. The maximum\nextension of the string during this motion is E m.\nTON\n(a) Show that, for 0 1 x 1 E,\nOD\nv2 –4(x2 = -5x-12.5). [3]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n..............................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2026-mj-32-q06-12e87931b4.pdf"
    },
    {
      "id": "9231-2026-mj-32-q07",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "32",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-32.pdf?download=1",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_32.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2026-mj-32-q07-p01-c96d652312.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2026-mj-32-q07-p02-6d6250129e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-32-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n7 The points O and A lie on a horizontal plane 10 m apart. The point A lies at the foot of another plane\nSIHT\n4 which is inclined at an angle of above the horizontal such that . The point B is on the tana = a\n3\ninclined plane at a distance of 5 m from A measured along a line of greatest slope. The points O, A and NI\nETIRW B are in the same vertical plane and angle OAB is obtuse.\nTON ms–1 A particle is projected from O at an angle u to the horizontal with initial speed towards the i\ninclined plane. The particle moves freely under gravity. The particle strikes the inclined plane at the\nOD\npoint B when its direction of motion is perpendicular to the inclined plane.\nFind, in either order, the value of u and the value of i. [6]\nNIGRAM\n....................................................................................................................................................................\nSIHT ....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n....................................................................................................................................................................\nTON\nOD ....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM ....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\n.................................................................................................................................................................... NI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2026-mj-32-q07-cfaacf2d05.pdf"
    },
    {
      "id": "9231-2026-mj-33-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 1,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-33.pdf?download=1",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2026-mj-33-q01-p01-5796eb8928.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-33-q01",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 A particle P of mass m is moving in a horizontal circle with constant angular speed on the smooth ~\nSIHT\ninner surface of a hemispherical shell of radius a. The plane of the circular motion is at a height\n1a above the lowest point on the shell. The angle between the vertical and the normal reaction of h = NI\n3 ETIRW the surface on P is i.\nTON in terms of a and g. [5] Find ~\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n.........................................................................................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2026-mj-33-q01-2d20fc1db3.pdf"
    },
    {
      "id": "9231-2026-mj-33-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 2,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-33.pdf?download=1",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_33.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2026-mj-33-q02-p01-eedb5b371b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-33-q02",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n2 A particle P of mass m is moving in a straight horizontal line towards a fixed point A on the line.\nSIHT\nThe particle is subject to a resistive force that is inversely proportional to the square of the distance\nNI between P and A. Initially, the distance between P and A is d and the speed of P is u. When the\nETIRW 3 1 distance between P and A is d, the particle has lost of its kinetic energy.\n2 4\nTON Find, in terms of d, the minimum distance of P from A. [6]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT ....................................................................................................................................................................\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\nOD ....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM ....................................................................................................................................................................\n....................................................................................................................................................................\nSIHT\n.................................................................................................................................................................... NI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n...................................................",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2026-mj-33-q02-d5b1b2961d.pdf"
    },
    {
      "id": "9231-2026-mj-33-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 3,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 5,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-33.pdf?download=1",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_33.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2026-mj-33-q03-p01-5dbf8563d9.png",
        "9231-topic-33-circular-motion/assets/9231-2026-mj-33-q03-p02-600ee80ed7.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-33-q03",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3\nSIHT\nu NI\nETIRW\nA\nTON\na\na\nOD\nO\nNIGRAM\nSIHT\nNI\nA fixed smooth solid sphere has centre O and radius a. A particle of mass m is projected perpendicular ETIRW\nto OA in a downward direction with speed u from the point A on the outer surface of the sphere, where\nOA makes an angle with the upward vertical through O (see diagram). a\nTON\nOD The particle moves in part of a vertical circle on the surface of the sphere. It loses contact with the\nsphere at the point B, where OB makes an angle with the upward vertical through O. The speed of b\nthe particle at the point B is v.\nNIGRAM Given that the ratio of u to v is 2 : 3 find the ratio cosa cosb. : [5]\n....................................................................................................................................................................\nSIHT\n.................................................................................................................................................................... NI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n............................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2026-mj-33-q03-1afced94cf.pdf"
    },
    {
      "id": "9231-2026-mj-33-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 4,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-33.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_33.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2026-mj-33-q04-p01-ad1d7b6acf.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2026-mj-33-q04-p02-1256948ddb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-33-q04",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4\nSIHT\nNI\nETIRW A\nTON\nOD h\nNIGRAM\nd\nSIHT\nB\nNI\nETIRW\nd kh\nTON\nAn object consists of a uniform solid cube of side length d, and two uniform solid square-based\nOD\npyramids. The cube and the pyramids are made of the same material.\nOne pyramid, of base length d and height h, is fixed to the top face of the cube so that the base of this\npyramid and the face of the cube coincide. Point A is at the top vertex of this pyramid. Point B is in the\nNIGRAM\ncentre of the bottom face of the cube, vertically below A. The second pyramid, of base length d and\nheight kh, is fixed to a vertical face of the cube so that the base of this pyramid and the vertical face of\nthe cube coincide (see diagram, which shows a cross-section, including the axes of symmetry).\nSIHT\nNI (a) Find an expression for the distance of the centre of mass of the object from AB in terms\nETIRW\nof h, k, and d. [4]\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n....................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2026-mj-33-q04-154c7a96a3.pdf"
    },
    {
      "id": "9231-2026-mj-33-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 5,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-33.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_33.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2026-mj-33-q05-p01-7d1f4de702.png",
        "9231-topic-36-momentum/assets/9231-2026-mj-33-q05-p02-4384a91d59.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-33-q05",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5\nSIHT\nNI\nu\nA ETIRW\nu A B\nB\nTON\nb a\nOD\nm m\nNIGRAM\nTwo uniform smooth spheres A and B of equal radii and equal masses m are moving with speeds\nu u respectively on a smooth horizontal surface when they collide. Immediately before the and SIHT\nB A\ncollision, the direction of motion of A makes an angle with the line of centres, and the direction of a\nNI\nmotion of B makes an angle with the line of centres (see diagram). Immediately after the collision, b\nETIRW\nthe direction of motion of A makes an angle with the line of centres, and the direction of motion i\n4 24 of B makes an angle cosa = cos{ = cosb = with the line of centres. It is given that and {\nTON 5 25\n3 and . = cosi\n5 OD\n(a) Find the percentage loss in the kinetic energy of A. [2]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ....................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2026-mj-33-q05-c5941c2e7c.pdf"
    },
    {
      "id": "9231-2026-mj-33-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 6,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-33.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_33.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2026-mj-33-q06-p01-bbd9523e2c.png",
        "9231-topic-34-hooke-s-law/assets/9231-2026-mj-33-q06-p02-9e0d25b6b1.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-33-q06",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n4 sini = 6 A smooth plane is inclined at an angle to the horizontal, where . One end of a light elastic i SIHT\n5\nspring, with natural length 10a and modulus of elasticity 1.6mg, is attached to a fixed point O on the\nNI\nplane. The other end of the spring is attached to a particle P of mass 0.2m. The point O and the particle\nETIRW\nP lie on the same line of greatest slope of the plane, with P above O. The particle P is at rest.\nTON\n(a) Show that the distance OP is 9a. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\nA particle Q of mass 0.3m is held on the plane on the line of greatest slope through O, at a distance\n10a above P. The particle Q is released from rest. The particle Q then collides and coalesces with\nSIHT\nparticle P.\nNI\n(b) Find, in terms of a and g, the speed of the combined particle immediately after the collision. [4] ETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n.................................................................................................................................................",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2026-mj-33-q06-a1542da5ee.pdf"
    },
    {
      "id": "9231-2026-mj-33-q07",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "33",
      "question_number": 7,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-33.pdf?download=1",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_33.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2026-mj-33-q07-p01-1a05e6964e.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2026-mj-33-q07-p02-bed0b5403a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-33-q07",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n7 At time t = 0 s, a particle P is projected from a point O on horizontal ground. The initial velocity of P\nSIHT\n4 s–1 is 25 sina = at an angle above the horizontal, where . At time t = T s, a second particle Q is m a\n5\nNI projected vertically upwards from the horizontal ground directly below the point where P will reach its\nETIRW\ns–1. greatest height. The initial velocity of Q is also 25 Both particles move freely under gravity. The m\ntwo particles collide when Q is travelling upwards and at the point where P reaches its greatest height.\nTON\n(a) Show that the two particles collide 2 seconds after P is projected. [1] OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n(b) Find the value of T. [3]\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2026-mj-33-q07-8ee6745bb4.pdf"
    },
    {
      "id": "9231-2026-mj-34-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 1,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 4,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-34.pdf?download=1",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_34.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2026-mj-34-q01-p01-acb66f9769.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-34-q01",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 One end of a light inextensible string of length a is attached to a fixed point O. The other end of the\nSIHT\nstring is attached to a particle of mass m. The string is taut and makes an angle with the downward i\nNI vertical through O. The particle moves in a horizontal circle with speed ag. 2\nETIRW\ncosi. [4] Find the value of\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW ....................................................................................................................................................................\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n...........",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2026-mj-34-q01-a0ce290b8b.pdf"
    },
    {
      "id": "9231-2026-mj-34-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 2,
      "topic_number": 32,
      "topic": "Equilibrium of a rigid body",
      "topic_slug": "9231-topic-32-equilibrium-of-a-rigid-body",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-34.pdf?download=1",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_34.pdf",
      "image_paths": [
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2026-mj-34-q02-p01-aa5bb2cb58.png",
        "9231-topic-32-equilibrium-of-a-rigid-body/assets/9231-2026-mj-34-q02-p02-b9f92ad413.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-34-q02",
        "html": "9231-topic-32-equilibrium-of-a-rigid-body/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2\nSIHT\n1 1 a a\nD G C 2 2\nNI\nETIRW\n1 a\n2\nTON\nOD\na F\na\nNIGRAM\nA E B\nSIHT\nA uniform lamina AEFGD is formed by removing two triangles from a square ABCD of side length a. NI\n1a. ETIRW Triangle FCG is isosceles and the two equal sides each have length The angle between EF and BF\n2\n1 is (see diagram). a, where = tana\n2\nTON\nfrom AD and at a distance from AE. y x The centre of mass of the lamina is at a distance\nOD\n5 x = a and find a corresponding expression for in terms of a. [5] y (a) Show that\n12\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n...............................................................................................................................",
      "pdf_path": "9231-topic-32-equilibrium-of-a-rigid-body/print/9231-2026-mj-34-q02-9c31607b7a.pdf"
    },
    {
      "id": "9231-2026-mj-34-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 3,
      "topic_number": 33,
      "topic": "Circular motion",
      "topic_slug": "9231-topic-33-circular-motion",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-34.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_34.pdf",
      "image_paths": [
        "9231-topic-33-circular-motion/assets/9231-2026-mj-34-q03-p01-6a5c1cb2b8.png",
        "9231-topic-33-circular-motion/assets/9231-2026-mj-34-q03-p02-08441c80e9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-34-q03",
        "html": "9231-topic-33-circular-motion/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 A particle of mass m is attached to a fixed point O by a light inextensible string of length a. The particle\nSIHT\nmoves in complete vertical circles about O. Points A and B are on the path of the particle. OA makes an\nangle of 60° with the upward vertical through O. The speed of the particle at A is u and the tension in the NI\nETIRW string at A is T . The point B is vertically below O. The speed of the particle at B is 2u and the tension in\nA\nthe string at B is T .\nB TON\nT = kT , where k is a constant to be determined. [6] (a) Show that\nOD\nB A\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n........................................................................",
      "pdf_path": "9231-topic-33-circular-motion/print/9231-2026-mj-34-q03-18cf01aae6.pdf"
    },
    {
      "id": "9231-2026-mj-34-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 4,
      "topic_number": 36,
      "topic": "Momentum",
      "topic_slug": "9231-topic-36-momentum",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-34.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_34.pdf",
      "image_paths": [
        "9231-topic-36-momentum/assets/9231-2026-mj-34-q04-p01-449dc18d40.png",
        "9231-topic-36-momentum/assets/9231-2026-mj-34-q04-p02-07f001179e.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-34-q04",
        "html": "9231-topic-36-momentum/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 Two particles A and B have masses km and m respectively. Particle A is at rest on a smooth horizontal\nSIHT\nsurface and is exactly halfway between two fixed parallel smooth vertical walls. Particle B is moving\non the surface with speed u in a direction perpendicular to the walls, and collides directly with A. The NI\nETIRW coefficient of restitution between the two particles is 0.2. As a result of the collision, the direction of\nmotion of B is reversed.\nTON\n(a) Show that k 2 5. [3]\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n......................................................................................................................................",
      "pdf_path": "9231-topic-36-momentum/print/9231-2026-mj-34-q04-362e53f452.pdf"
    },
    {
      "id": "9231-2026-mj-34-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 5,
      "topic_number": 35,
      "topic": "Linear motion under a variable force",
      "topic_slug": "9231-topic-35-linear-motion-under-a-variable-force",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-34.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_34.pdf",
      "image_paths": [
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2026-mj-34-q05-p01-bc98c609ca.png",
        "9231-topic-35-linear-motion-under-a-variable-force/assets/9231-2026-mj-34-q05-p02-5281116cf2.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-34-q05",
        "html": "9231-topic-35-linear-motion-under-a-variable-force/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\nm`v2 -ku2j, SIHT 5 A particle P of mass m is moving in a straight horizontal line under a variable force\nwhere v is the velocity of P at time t, u is the initial velocity of P, and k is a positive constant. It is\nNI\ngiven that u 2 0 k 2 1. and\nETIRW\nP have opposite directions. [1] (a) Explain why, initially, the acceleration and velocity of\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\nSIHT\n(b) Find, in terms of k, the distance travelled by P until it first comes to instantaneous rest. [4]\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n...........",
      "pdf_path": "9231-topic-35-linear-motion-under-a-variable-force/print/9231-2026-mj-34-q05-491aff2ea0.pdf"
    },
    {
      "id": "9231-2026-mj-34-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 6,
      "topic_number": 31,
      "topic": "Motion of a projectile",
      "topic_slug": "9231-topic-31-motion-of-a-projectile",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-34.pdf?download=1",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_34.pdf",
      "image_paths": [
        "9231-topic-31-motion-of-a-projectile/assets/9231-2026-mj-34-q06-p01-b6846ba318.png",
        "9231-topic-31-motion-of-a-projectile/assets/9231-2026-mj-34-q06-p02-30a0efc151.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-34-q06",
        "html": "9231-topic-31-motion-of-a-projectile/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\n6\nSIHT\nNI\nu ETIRW\nP\n2a\nTON A\nOD\nNIGRAM\nSIHT\nNI\na\nETIRW\nB\nTON\nThe points A and B lie on the line of greatest slope of a smooth inclined plane, with A above B. The\nOD plane is inclined to the horizontal at an acute angle a.\nt = 0, a particle P is projected from A with speed u at an angle 2a to the plane and moves At time\nfreely under gravity (see diagram). At the same time, a second particle Q is released from rest at A.\nNIGRAM Particle Q moves down the plane along the line of greatest slope. The two particles move in the same\nvertical plane. At time t = T , the two particles collide at point B. The length of AB is d.\nSIHT 2d\nT . [2] = (a) By considering the motion of particle Q, show that\ngsina\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\nTON\nNIGRAM\n(b) Find the value of a. [5]\nSIHT\nNI ............................",
      "pdf_path": "9231-topic-31-motion-of-a-projectile/print/9231-2026-mj-34-q06-3284f58480.pdf"
    },
    {
      "id": "9231-2026-mj-34-q07",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 3,
      "variant": "34",
      "question_number": 7,
      "topic_number": 34,
      "topic": "Hooke's law",
      "topic_slug": "9231-topic-34-hooke-s-law",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-34.pdf?download=1",
      "source_pages": [
        14,
        15
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_34.pdf",
      "image_paths": [
        "9231-topic-34-hooke-s-law/assets/9231-2026-mj-34-q07-p01-4c0b150ad3.png",
        "9231-topic-34-hooke-s-law/assets/9231-2026-mj-34-q07-p02-7ccd654ebd.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-34-q07",
        "html": "9231-topic-34-hooke-s-law/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:5)(cid:44)\n7 One end of a light elastic string of natural length 8a and modulus of elasticity 6mg is attached to a\nSIHT\n5 fixed point O on a rough plane. The plane is inclined at an angle = sini to the horizontal where . i\n13\nA particle P of mass m is attached to the other end of the string. The string lies along a line of greatest NI\nETIRW slope of the plane, with P below the level of O. The particle is held at a point such that the extension\nof the string is 3a.\nTON\n5 When P is released from rest, its acceleration is g in the direction PO.\nOD 4\n(a) Find the coefficient of friction between the particle and the plane. [4]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n.....",
      "pdf_path": "9231-topic-34-hooke-s-law/print/9231-2026-mj-34-q07-b1eca6b377.pdf"
    },
    {
      "id": "9231-2026-mj-41-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 1,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-41.pdf?download=1",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_41.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2026-mj-41-q01-p01-267549e53f.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2026-mj-41-q01-p02-9cc249ecf9.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-41-q01",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 The manager of a village store is interested in the shopping habits of his customers. From previous\nSIHT\nrecords, the median number of items bought per customer was found to be 8. The manager believes\nthe median number of items bought per customer has increased. He selects a random sample of 15 NI\nETIRW customers and records the number of items each customer buys at his store. The results are shown in\nthe stem-and-leaf diagram.\nTON\nkey\nOD\n2 3 represents 23 items\n0 6 6 7 9 9\nNIGRAM\n1 0 5 7 7\n2 3 6 SIHT\n3 4 NI\nETIRW\n4 0\nTON 5 6\nOD\n6\n7 4\nNIGRAM\n(a) Suggest a reason why using a Wilcoxon signed-rank test to examine the manager’s belief might\nnot be appropriate in this situation. [1]\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n(b) Use a sign test at the 5% significance level to investigate the manager’s belief. [6]\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n........................................................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2026-mj-41-q01-f01faef558.pdf"
    },
    {
      "id": "9231-2026-mj-41-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-41.pdf?download=1",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_41.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2026-mj-41-q02-p01-5ca4005acd.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2026-mj-41-q02-p02-0f633ff36a.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-41-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 Three different bus companies, A, B and C, record the reliability of their service. Each bus journey\nSIHT\nis recorded as being on time, late or cancelled. A reliable bus service is considered to be one that is\non time. The records from a random sample of buses for each company are taken. These records are NI\nETIRW summarised in the table.\nTON\ncompany A company B company C total\nOD\non time 7 22 20 49\nlate 14 11 12 37\ncancelled 4 6 4 14\nNIGRAM\ntotal 25 39 36 100\nSIHT\n(a) Test at the 5% significance level whether the reliability of the service is independent of the\nNI\nbus company. [8]\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n...........................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2026-mj-41-q02-25c575fdd4.pdf"
    },
    {
      "id": "9231-2026-mj-41-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 3,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-41.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2026-mj-41-q03-p01-7f7913dd25.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2026-mj-41-q03-p02-d7eec80673.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-41-q03",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 Students studying geography at a college choose either human geography or physical geography.\nSIHT\nA teacher is investigating the times taken by students to complete assignments of equal difficulty\nin human geography and physical geography. He selects a random sample of 7 human geography NI\nETIRW students and records the times taken, x minutes, to complete an assignment. He also selects a random\nsample of 9 physical geography students and records the times taken, y minutes, to complete an\nTON assignment. The results are summarised as follows.\nOD\n2 2 /x /y /x /y = 29144 = 11622 = 445 = 310\nYou should assume that the underlying distributions are normal and have equal population variances.\nNIGRAM\nFind a 95% confidence interval for the difference in population mean times taken to complete\nassignments in human geography and physical geography. [6]\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n.............................................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2026-mj-41-q03-4da7f375bf.pdf"
    },
    {
      "id": "9231-2026-mj-41-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 4,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-41.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_41.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2026-mj-41-q04-p01-ec1101f431.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2026-mj-41-q04-p02-b2b112d31b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-41-q04",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 The continuous random variable X has cumulative distribution function F given by\nSIHT\nZ\n0 x 0, NI 1 ]\nETIRW ] 1 1 x3 x2 ] G + x 3, G 0 –\n108 12 [ = F(x)\n7 1 x2 1 b, – + x+a 3 x G ] TON\n16 32\n]\nb. x 2 1\nOD \\\n(a) Verify that the median of X is 3. [1]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n17 a = – and find the value of b. [3] (b) Show that\nTON 32\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ...............................................................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2026-mj-41-q04-2d20598b9b.pdf"
    },
    {
      "id": "9231-2026-mj-41-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 5,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-41.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_41.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2026-mj-41-q05-p01-3463e1a0c5.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2026-mj-41-q05-p02-d55086eecc.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-41-q05",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 Margaret has developed a new revision course to help students improve their ability in mathematics.\nSIHT\nThe students each sit a mathematics test before attending the course and a test of similar difficulty\nafter attending the course. Margaret claims that her course will increase a student’s score by more than NI\nETIRW 5 marks on average. For a random sample of 8 students, the test scores before and after the revision\ncourse are given in the table.\nTON\nstudent A B C D E F G H OD\ntest score before 43 29 67 54 34 29 52 60\ntest score after 49 53 79 62 39 55 65 62\nNIGRAM\n(a) Use a t-test at the 10% significance level to investigate Margaret’s claim. [7]\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n...........................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2026-mj-41-q05-a426440055.pdf"
    },
    {
      "id": "9231-2026-mj-41-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "41",
      "question_number": 6,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-41.pdf?download=1",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_41.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2026-mj-41-q06-p01-363c3fa1a1.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2026-mj-41-q06-p02-b1978e6846.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-41-q06",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\nG (t) defined by 6 The discrete random variable X has probability generating function\nSIHT\nX\nk\n(at+1)2. NI G (t) =\nt X ETIRW\n1. E(X) = It is given that\n3\nTON\na = 2 and find the value of k. [5] (a) Show that\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI ............................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2026-mj-41-q06-624c22bbae.pdf"
    },
    {
      "id": "9231-2026-mj-42-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 1,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-42.pdf?download=1",
      "source_pages": [
        2,
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_42.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2026-mj-42-q01-p01-eafc9cdcdb.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2026-mj-42-q01-p02-3c9971f968.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-42-q01",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 The manager of a village store is interested in the shopping habits of his customers. From previous\nSIHT\nrecords, the median number of items bought per customer was found to be 8. The manager believes\nthe median number of items bought per customer has increased. He selects a random sample of 15 NI\nETIRW customers and records the number of items each customer buys at his store. The results are shown in\nthe stem-and-leaf diagram.\nTON\nkey\nOD\n2 3 represents 23 items\n0 6 6 7 9 9\nNIGRAM\n1 0 5 7 7\n2 3 6 SIHT\n3 4 NI\nETIRW\n4 0\nTON 5 6\nOD\n6\n7 4\nNIGRAM\n(a) Suggest a reason why using a Wilcoxon signed-rank test to examine the manager’s belief might\nnot be appropriate in this situation. [1]\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n(b) Use a sign test at the 5% significance level to investigate the manager’s belief. [6]\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n........................................................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2026-mj-42-q01-1045f49cbc.pdf"
    },
    {
      "id": "9231-2026-mj-42-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 2,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-42.pdf?download=1",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_42.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2026-mj-42-q02-p01-db9bd8879e.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2026-mj-42-q02-p02-048cda3fca.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-42-q02",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 Three different bus companies, A, B and C, record the reliability of their service. Each bus journey\nSIHT\nis recorded as being on time, late or cancelled. A reliable bus service is considered to be one that is\non time. The records from a random sample of buses for each company are taken. These records are NI\nETIRW summarised in the table.\nTON\ncompany A company B company C total\nOD\non time 7 22 20 49\nlate 14 11 12 37\ncancelled 4 6 4 14\nNIGRAM\ntotal 25 39 36 100\nSIHT\n(a) Test at the 5% significance level whether the reliability of the service is independent of the\nNI\nbus company. [8]\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n...........................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2026-mj-42-q02-f1b24000ad.pdf"
    },
    {
      "id": "9231-2026-mj-42-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 3,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-42.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_42.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2026-mj-42-q03-p01-20b916e5d2.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2026-mj-42-q03-p02-581faeab47.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-42-q03",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n3 Students studying geography at a college choose either human geography or physical geography.\nSIHT\nA teacher is investigating the times taken by students to complete assignments of equal difficulty\nin human geography and physical geography. He selects a random sample of 7 human geography NI\nETIRW students and records the times taken, x minutes, to complete an assignment. He also selects a random\nsample of 9 physical geography students and records the times taken, y minutes, to complete an\nTON assignment. The results are summarised as follows.\nOD\n2 2 /x /y /x /y = 29144 = 11622 = 445 = 310\nYou should assume that the underlying distributions are normal and have equal population variances.\nNIGRAM\nFind a 95% confidence interval for the difference in population mean times taken to complete\nassignments in human geography and physical geography. [6]\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\n.................................................................................................................................................................... ETIRW\n.................................................................................................................................................................... TON\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n.............................................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2026-mj-42-q03-e6450e9080.pdf"
    },
    {
      "id": "9231-2026-mj-42-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 4,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-42.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_42.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2026-mj-42-q04-p01-93e4473d73.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2026-mj-42-q04-p02-677247ed18.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-42-q04",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n4 The continuous random variable X has cumulative distribution function F given by\nSIHT\nZ\n0 x 0, NI 1 ]\nETIRW ] 1 1 x3 x2 ] G + x 3, G 0 –\n108 12 [ = F(x)\n7 1 x2 1 b, – + x+a 3 x G ] TON\n16 32\n]\nb. x 2 1\nOD \\\n(a) Verify that the median of X is 3. [1]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n17 a = – and find the value of b. [3] (b) Show that\nTON 32\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ...............................................................................................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2026-mj-42-q04-ac9ea2d58a.pdf"
    },
    {
      "id": "9231-2026-mj-42-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 5,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-42.pdf?download=1",
      "source_pages": [
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        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_42.pdf",
      "image_paths": [
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      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-42-q05",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n5 Margaret has developed a new revision course to help students improve their ability in mathematics.\nSIHT\nThe students each sit a mathematics test before attending the course and a test of similar difficulty\nafter attending the course. Margaret claims that her course will increase a student’s score by more than NI\nETIRW 5 marks on average. For a random sample of 8 students, the test scores before and after the revision\ncourse are given in the table.\nTON\nstudent A B C D E F G H OD\ntest score before 43 29 67 54 34 29 52 60\ntest score after 49 53 79 62 39 55 65 62\nNIGRAM\n(a) Use a t-test at the 10% significance level to investigate Margaret’s claim. [7]\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n...........................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2026-mj-42-q05-3b2b8a48b2.pdf"
    },
    {
      "id": "9231-2026-mj-42-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "42",
      "question_number": 6,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-42.pdf?download=1",
      "source_pages": [
        12,
        13
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_42.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2026-mj-42-q06-p01-3d33275665.png",
        "9231-topic-45-probability-generating-functions/assets/9231-2026-mj-42-q06-p02-4d40bd8db5.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-42-q06",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:3)(cid:44)\nG (t) defined by 6 The discrete random variable X has probability generating function\nSIHT\nX\nk\n(at+1)2. NI G (t) =\nt X ETIRW\n1. E(X) = It is given that\n3\nTON\na = 2 and find the value of k. [5] (a) Show that\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI ............................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2026-mj-42-q06-144292a519.pdf"
    },
    {
      "id": "9231-2026-mj-43-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-43.pdf?download=1",
      "source_pages": [
        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2026-mj-43-q01-p01-2834a5f9cb.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-43-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\n1 A random sample of 9 observations of a normally distributed random variable X is as follows.\nSIHT\nNI 19.4 21.3 24.0 20.7 18.9 21.2 22.5 23.4 20.8\nETIRW\nn. The population mean of X is\nTON\n= 20 against the alternative Test, at the 2.5% significance level, the null hypothesis n\nOD\nhypothesis 2 20. [6] n\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\nSIHT\n......................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2026-mj-43-q01-4077bf818f.pdf"
    },
    {
      "id": "9231-2026-mj-43-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 2,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-43.pdf?download=1",
      "source_pages": [
        4
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_43.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2026-mj-43-q02-p01-4cbc4db8e6.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-43-q02",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n2 A large school sets an English exam for all students in a particular year group. Two teachers will\nSIHT\nbe marking the scripts. To check for consistency of marking between the teachers, 10 scripts are\nselected at random and independently marked by each teacher. The marks awarded are shown in the NI\nETIRW following table.\nTON\nstudent A B C D E F G H I J\nOD\nteacher 1 42 73 53 21 61 83 63 37 46 55\nteacher 2 46 81 68 24 60 81 68 30 57 61\nNIGRAM\n(a) Give a reason why a Wilcoxon matched-pairs signed-rank test might be more appropriate than a\npaired sample t-test in this situation. [1]\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n(b) Carry out a Wilcoxon matched-pairs signed-rank test at the 5% significance level to investigate\nwhether, on average, there is a difference between the marks awarded by the two teachers. [7]\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................\nTON\nOD ............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM ............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n.....................................................................................................................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2026-mj-43-q02-9d7e7a7303.pdf"
    },
    {
      "id": "9231-2026-mj-43-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 3,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-43.pdf?download=1",
      "source_pages": [
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_43.pdf",
      "image_paths": [
        "9231-topic-45-probability-generating-functions/assets/9231-2026-mj-43-q03-p01-ee04a637ea.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-43-q03",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:6)(cid:44)\nG (t) given by 3 The discrete random variable X has probability generating function\nSIHT\nX\n0.1+0.3t+0.4t2 +0.2t3. NI (t) G =\nX ETIRW\nG (t) to find E(X ) and Var(X ). [5] (a) Use\nX\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n...........................................................................................................................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2026-mj-43-q03-85022e5087.pdf"
    },
    {
      "id": "9231-2026-mj-43-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 4,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-43.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_43.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2026-mj-43-q04-p01-4f73afdb8c.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2026-mj-43-q04-p02-0470ee71cc.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-43-q04",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 A random sample of 200 observations of the continuous random variable X was taken. The values are\nSIHT\nsummarised in the following table.\nNI\nETIRW\ninterval x H 4 1 G x 1 1.5 1.5 G x 1 2 2 G x 1 2.5 2.5 G x 1 3 3 G x 1 3.5 3.5 G x 1 4\nTON observed\n97 35 23 15 10 6 14\nfrequency\nOD\nIt is required to test the goodness of fit of the distribution with probability density function f given by\nNIGRAM 2\nx H 1,\n* x3 = f(x)\n0 otherwise. SIHT\nNI\nMost of the relevant expected frequencies, correct to 2 decimal places, are given in the\nETIRW\nfollowing table.\nTON\ninterval x H 4 1 G x 1 1.5 1.5 G x 1 2 2 G x 1 2.5 2.5 G x 1 3 3 G x 1 3.5 3.5 G x 1 4\nOD\nexpected\n111.11 a 18.00 9.78 b 3.83 12.50\nfrequency\nNIGRAM\n(a) Show that a = 38.89 and find the value of b correct to 2 decimal places. [3]\n............................................................................................................................................................\nSIHT\nNI ............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIH",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2026-mj-43-q04-fe0d9b91fd.pdf"
    },
    {
      "id": "9231-2026-mj-43-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 5,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 11,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-43.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_43.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2026-mj-43-q05-p01-787aa5cce9.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2026-mj-43-q05-p02-eadcbd4a8b.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-43-q05",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5 Rhiannon is investigating the mass of individual bananas sold in two supermarkets. Rhiannon claims\nSIHT\nthat the mean mass of bananas sold by supermarket X is less than the mean mass of bananas sold by\nsupermarket Y. NI\nETIRW\nShe takes random samples of 44 bananas from supermarket X and 52 bananas from supermarket Y.\nTON She records their masses, x kg and y kg, respectively. Her results are summarised as follows.\nOD\n/x / /x2 y2 / 6.63 = y = 8.56 1.21 1.65 = =\nYou should not assume that the population variances of the masses of bananas sold by each\nsupermarket are equal.\nNIGRAM\n(a) Test Rhiannon’s claim at the 10% significance level. [8]\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\n............................................................................................................................................................ ETIRW\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\nNI .....................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2026-mj-43-q05-8d2227ec85.pdf"
    },
    {
      "id": "9231-2026-mj-43-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "43",
      "question_number": 6,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-43.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_43.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2026-mj-43-q06-p01-2f57b8f5ae.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2026-mj-43-q06-p02-e463f2c814.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-43-q06",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6 The random variable X has cumulative distribution function F given by\nSIHT\nZ\n0 x 1-2, NI ]\nETIRW ] 1 (x2 ] +4x+4) -2 G 1 0, x\n12\n[ = F(x)\n1 (x3 +x+5) 0 G x G 2, ] TON\n15\n]\n1 x 2 2.\nOD \\\n(a) Find the exact value of the lower quartile of X. [2]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n2j. Var`X [5] (b) Find\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n.....................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2026-mj-43-q06-fe38e88c5b.pdf"
    },
    {
      "id": "9231-2026-mj-44-q01",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "44",
      "question_number": 1,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 6,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-44.pdf?download=1",
      "source_pages": [
        2
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_44.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2026-mj-44-q01-p01-52c780470c.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-44-q01",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:3)(cid:44)\n1 A random sample of 40 observations of a random variable X and a random sample of 60 observations\nSIHT\nof a random variable Y are taken. The results are summarised as follows.\nNI\nETIRW /(x-x)2 /(y-y)2 /y / = 934.3 = 756.2 = 84.1 x = 67.1\nTON Find a 95% confidence interval for the difference between the population means of X and Y. [6]\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n....................................................................................................................................................................\n.................................................................................................................................................................... SIHT\nNI\n....................................................................................................................................................................\nETIRW\n....................................................................................................................................................................\nTON\n.................................................................................................................................................................... OD\n....................................................................................................................................................................\n.................................................................................................................................................................... NIGRAM\n....................................................................................................................................................................\nSIHT\n....................................................................................................................................................................\nNI\nETIRW\n....................................................................................................................................................................\nTON\n....................................................................................................................................................................\nOD\n....................................................................................................................................................................\n....................................................................................................................................................................\nNIGRAM\n......",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2026-mj-44-q01-59028eeebf.pdf"
    },
    {
      "id": "9231-2026-mj-44-q02",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "44",
      "question_number": 2,
      "topic_number": 45,
      "topic": "Probability generating functions",
      "topic_slug": "9231-topic-45-probability-generating-functions",
      "marks": 7,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-44.pdf?download=1",
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        3
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_44.pdf",
      "image_paths": [
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      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-44-q02",
        "html": "9231-topic-45-probability-generating-functions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4)(cid:44)\nof a random variable X is given by 2 The probability generating function G (t)\nSIHT\nX\n1(t+1)(t2 NI G = +2). (t)\n6 X ETIRW\nG (t) (a) Use to find E(X ) and Var(X ). [4]\nX\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n(b) The random variable Y is the sum of three independent observations of X.\nNIGRAM\nP(Y G 2). [3] Find\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n....................",
      "pdf_path": "9231-topic-45-probability-generating-functions/print/9231-2026-mj-44-q02-0ca3192942.pdf"
    },
    {
      "id": "9231-2026-mj-44-q03",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "44",
      "question_number": 3,
      "topic_number": 43,
      "topic": "Chi-squared tests",
      "topic_slug": "9231-topic-43-chi-squared-tests",
      "marks": 9,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-44.pdf?download=1",
      "source_pages": [
        4,
        5
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_44.pdf",
      "image_paths": [
        "9231-topic-43-chi-squared-tests/assets/9231-2026-mj-44-q03-p01-41a59c436e.png",
        "9231-topic-43-chi-squared-tests/assets/9231-2026-mj-44-q03-p02-db8755854f.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-44-q03",
        "html": "9231-topic-43-chi-squared-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:5)(cid:44)\n3 A biologist believes that the masses of a particular species of frog are normally distributed. He\nSIHT\nrandomly selects 120 frogs of this species and records the mass, x, of each frog in grams. The results\nare summarised as follows. NI\nETIRW\nTON mass, x xH260 x1210 210G x1220 220G x1230 230G x1240 240G x1250 250G x1260\nOD\nobserved\n2 9 20 39 40 8 2\nfrequency\nNIGRAM The biologist uses the masses of these 120 frogs to calculate unbiased estimates of the mean and\ng2 variance of the underlying population. He calculates these as 236.5 g and 134.56 respectively.\nSIHT\nMost of the relevant expected frequencies, correct to 2 decimal places, are given in the following\ntable. The expected frequencies have been calculated using the biologist’s model. NI\nETIRW\nmass, x xH260 x1210 210G x1220 220G x1230 230G x1240 240G x1250 250G x1260 TON\nOD\nexpected\n1.34 7.94 25.22 39.73 31.11 c d\nfrequency\n(a) Find the value of c and the value of d. Give your answers correct to two decimal places. [2] NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n.................................................................................................................",
      "pdf_path": "9231-topic-43-chi-squared-tests/print/9231-2026-mj-44-q03-2ef2861e77.pdf"
    },
    {
      "id": "9231-2026-mj-44-q04",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "44",
      "question_number": 4,
      "topic_number": 44,
      "topic": "Non-parametric tests",
      "topic_slug": "9231-topic-44-non-parametric-tests",
      "marks": 8,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-44.pdf?download=1",
      "source_pages": [
        6,
        7
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_44.pdf",
      "image_paths": [
        "9231-topic-44-non-parametric-tests/assets/9231-2026-mj-44-q04-p01-9540dfee49.png",
        "9231-topic-44-non-parametric-tests/assets/9231-2026-mj-44-q04-p02-1c1e4073be.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-44-q04",
        "html": "9231-topic-44-non-parametric-tests/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:7)(cid:44)\n4 Members of a cycling club use the same circular route every week. Throughout the year, the members\nSIHT\nchange between completing the route in a clockwise direction one week and completing the route in\nan anticlockwise direction the following week. NI\nETIRW\nThe club captain believes that the anticlockwise route takes longer to complete on average than the\nTON clockwise route. To test this belief, the club captain selects a random sample of 10 members. Each\nof the 10 club members gives their personal best times, in minutes, for completing the route in both\nOD\ndirections over the previous year. These times are as follows.\nclub member A B C D E F G H I J\nNIGRAM\nclockwise 127.1 129.1 139.3 151.4 142.5 123.6 153.4 129.7 140.9 137.4\nanticlockwise 131.0 132.1 147.3 150.0 144.3 127.3 153.5 129.9 140.6 140.3 SIHT\nNI\nETIRW\n(a) Explain the advantage of using paired samples to investigate the club captain’s belief. [1]\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n(b) Carry out a Wilcoxon matched-pairs signed-rank test at the 5% significance level to test the club\nSIHT\ncaptain’s belief. [6]\nNI\nETIRW\n............................................................................................................................................................\nTON ............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD ................................................",
      "pdf_path": "9231-topic-44-non-parametric-tests/print/9231-2026-mj-44-q04-29500a74d6.pdf"
    },
    {
      "id": "9231-2026-mj-44-q05",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "44",
      "question_number": 5,
      "topic_number": 41,
      "topic": "Continuous random variables",
      "topic_slug": "9231-topic-41-continuous-random-variables",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-44.pdf?download=1",
      "source_pages": [
        8,
        9
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_44.pdf",
      "image_paths": [
        "9231-topic-41-continuous-random-variables/assets/9231-2026-mj-44-q05-p01-cc7609f929.png",
        "9231-topic-41-continuous-random-variables/assets/9231-2026-mj-44-q05-p02-5970327160.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-44-q05",
        "html": "9231-topic-41-continuous-random-variables/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:44)\n5\nSIHT\nf(x)\nNI\nETIRW\nTON\nOD\nNIGRAM\nx 0 5 7 1\nSIHT\nNI\nETIRW\nThe diagram shows the continuous random variable X with probability density function f given by\na(x-1) G G 1 5, x TON\n* b(x-7)2 = G G 5 7, f(x) x\nOD\notherwise. 0\nThe upper quartile of X is equal to 5.\nNIGRAM 3 3 (a) Show that = a b = and . [3]\n32 32\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................\nNI\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\nSIHT ............................................................................................................................................................\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\nOD\n(b) Find the cumulative distribution function of X. [3]\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................\nSIHT\n............................................................................................................................................................ NI\nETIRW\n............................................................................................................................................................\nTON\nNIGRAM\nSIHT\n...............................................................................",
      "pdf_path": "9231-topic-41-continuous-random-variables/print/9231-2026-mj-44-q05-6e768f2cc8.pdf"
    },
    {
      "id": "9231-2026-mj-44-q06",
      "subject": "9231",
      "year": 2026,
      "session": "May/June",
      "session_code": "mj",
      "paper": 4,
      "variant": "44",
      "question_number": 6,
      "topic_number": 42,
      "topic": "Inference using normal and t-distributions",
      "topic_slug": "9231-topic-42-inference-using-normal-and-t-distributions",
      "marks": 10,
      "source_pdf": "https://pastpapers.co/api/file/caie/a-level/mathematics-further-9231/2026-may-june/9231-mathematics---further-june-2026-question-paper-44.pdf?download=1",
      "source_pages": [
        10,
        11
      ],
      "local_pdf": "_source-pdfs/2026-May-June/qp/9231_s26_qp_44.pdf",
      "image_paths": [
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2026-mj-44-q06-p01-7923557b1d.png",
        "9231-topic-42-inference-using-normal-and-t-distributions/assets/9231-2026-mj-44-q06-p02-122f58e033.png"
      ],
      "answer": {
        "status": "pending",
        "id": "9231-2026-mj-44-q06",
        "html": "9231-topic-42-inference-using-normal-and-t-distributions/answers.html"
      },
      "text_excerpt": "NIGRAM\n(cid:44)(cid:1)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:2)(cid:1)(cid:44)\n6 A researcher is studying household waste recycling in two different cities, city A and city B. She\nSIHT\nrandomly selects 8 households in city A and 9 households in city B. The researcher records the\npercentage of waste that each household recycles. Her results are as follows. NI\nETIRW\nCity A 25 49 41 38 31 21 25 40\nTON\nCity B 18 35 30 22 13 23 24 36 24\nOD\nThe population variances of the percentages of waste recycled in each city are assumed to be equal.\nNIGRAM (a) Carry out a t-test at the 10% significance level to investigate whether there is a difference in the\naverage percentage of waste that is recycled in city A and city B. [9]\nSIHT\n............................................................................................................................................................\nNI\nETIRW ............................................................................................................................................................\n............................................................................................................................................................ TON\nOD\n............................................................................................................................................................\n............................................................................................................................................................\nNIGRAM\n............................................................................................................................................................\n............................................................................................................................................................ SIHT\nNI\n............................................................................................................................................................\nETIRW\n............................................................................................................................................................\nTON\n............................................................................................................................................................ OD\n............................................................................................................................................................\n............................................................................................................................................................ NIGRAM\n............................................................................................................................................................\nSIHT\n............................................................................................................................................",
      "pdf_path": "9231-topic-42-inference-using-normal-and-t-distributions/print/9231-2026-mj-44-q06-a9d3bfa463.pdf"
    }
  ]
}