Practical skills

1 Defining the problem Mass of cylinder m is the independent variable and period T is the dependent variable, or vary mass of cylinder m and 1 measure period T. Keep radius of cylinder constant. 1 Methods of data collection Labelled diagram of workable experiment including: 1 • beaker with (cooking) oil on a bench or container supported by stand where stand is on a bench • cylinder partially submerged in (cooking) oil • cylinder and (cooking) oil labelled. Method to determine mass m of cylinder, e.g. use a (top pan) balance. 1 Method to determine period or T, e.g. use a stopwatch / timer to time oscillations. 1 Method to determine diameter of cylinder, e.g. micrometer or calliper 1 Method of Analysis Plots a graph of T2 against m. 1 (Allow other valid graphs, e.g. lg T against lg m) Relationship valid if a straight line passing through the origin is produced. 1 (Allow gradient = 0.5 for log T against log m). 4π 1 K = gradient×σr2 4π (K = for lg T against lg m). 102×y-intercept ×σr2 © UCLES 2021 Page 5 of 9 1 Additional detail including safety considerations 6 Max 6 Use gloves to prevent oil contacting skin / slippery hands OR D1 Perform experiment in a tray to prevent oil spillages. Keep density / temperature of the (cooking) oil constant or keep σ constant. D2 Mass of oil = mass of beaker and oil – mass of beaker and D3 use a measuring cylinder to determine the volume of the oil. Do not accept (calibrated) beaker. Methods to measure volume of oil and determine mass of oil and use equation density σ = mass / volume for D4 measurements. Time n oscillations and divide nT by n D5 where n ⩾ 5. Description of method of counting oscillations with position of fiducial mark / mark on cylinder / beaker / fixed point shown in D6 diagram. Repeat experiment for each value of m and average T. D7 r = diameter / 2 provided diameter measured. D8 Repeat measurements of diameter in different directions and average. D9 Wait for oscillations to become even / steady. D10 © UCLES 2021 Page 6 of 9

2(a) 1 1 Gradient = 2uA 1 . y-intercept = 2u 2(b) 0.046 0.052 0.062 0.072 0.080 0.088 1 1 First mark for values of / s cm–1; allow 3sf. v Second mark for absolute uncertainties from 1 ± 0.003 to ± 0.004. 2(c)(i) Six points plotted correctly. 1 Must be accurate to the nearest half small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. v All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2021 Page 7 of 9 2(c)(ii) Line of best fit drawn. 1 Points must be balanced. Do not allow line from top plot to bottom plot. Line must pass between (320, 0.050) and (345, 0.050) and between (795, 0.085) and (815, 0.085). Worst acceptable line drawn. 1 Steepest or shallowest possible line. Mark scored only if all error bars are plotted. 2(c)(iii) Gradient determined with clear substitution of data points into Δy / Δx; distance between data points must be at least half 1 the length of the drawn line. Gradient of WAL determined and 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point into y = mx + c 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line, or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ecf from false origin method. © UCLES 2021 Page 8 of 9 2(d)(i) u determined using y-intercept and 1 u and A given to 2 or 3 sf. 1 u = 2×y −intercept A determined using gradient with correct substitution and 1 Units with correct power of ten for u and A. y −intercept 1 A= or A= gradient 2×u× gradient 2(d)(ii) Percentage uncertainty in A. 1 Δgradient Δy-intercept %uncert.=  +  ×100  gradient y-intercept  OR Δu clearly determined and Δgradient Δu %uncert.=  +  ×100  gradient u  OR Correct substitution for max/min methods. 2(e) Value of m determined from (d)(i) OR (c)(iii) and (c)(iv) with correct number substitution into relevant equation and correct 1 power of ten. 2uAt 2uA e.g. m= −A= −A, or L 10  t 1  m= − ×2uA or   L 2u t −y-intercept m= L . gradient © UCLES 2021 Page 9 of 9

1 Defining the problem R is the independent variable and t is the dependent variable or vary R and measure t 1 keep the number of turns on the coil/N constant 1 Methods of data collection labelled diagram or correct symbols including: 1 • labelled (d.c.) power supply • switch in series with power supply, resistor and coil • complete workable circuit circuit diagram to measure R, e.g. ammeter and voltmeter correctly positioned or R connected to ohmmeter with no other 1 connections (not ohmmeter in main circuit) method to determine t (of a few milliseconds) e.g. use (storage) oscilloscope or current/voltage sensor connected to 1 datalogger/computer method to determine A, e.g. micrometer/calipers to determine diameter of coil and A = πd2/4 1 Method of analysis plot a graph of t against 1 / R 1 (allow log t against log R) relationship valid if a straight line passing through the origin is produced 1 (allow gradient = –1 for graph of log t against log R) gradient×L 1 K = . AN2 © UCLES 2021 Page 6 of 10 1 Additional detail including safety considerations 6 D1 open switch/switch off (high voltage) circuit before changing the resistor/touching components or ensure no bare wires/use shrouded connectors D2 wear (insulating) gloves to prevent electric shock/electrocution D3 keep A and L constant D4 use ruler/calipers to measure L D5 repeat measurements of diameter in different directions/at points along the coil and average D6 method to determine R e.g. R = V / I linked to correct circuit diagram for ammeter/voltmeter method or measure resistance using ohmmeter D7 repeat experiment for each value of R and average t D8 method to determine t: use of time-base from oscilloscope explained or use of time axis of output from data logger/computer explained D9 use smaller values of R to increase I D10 reduce L or increase N or increase A to increase t © UCLES 2021 Page 7 of 10

2(a) 1 1 gradient = uA 1 y-intercept = u 2(b) 1 1 (M + m) / g / s cm–1 v 380 0.226 or 0.2262 480 0.255 or 0.2551 580 0.294 or 0.2941 680 0.331 or 0.3311 830 0.388 or 0.3876 930 0.429 or 0.4292 1 Values of (M + m) and as shown above. v Absolute uncertainties in (M + m) from ± (19 or 20) to ± (46.5 or 47 or 50). 1 2(c)(i) Six points plotted correctly. 1 Must be accurate to the nearest half a small square. Diameter of points must be less than half a small square. Error bars in (M + m) plotted correctly. 1 All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2021 Page 8 of 10 2(c)(ii) Line of best fit drawn covers all points. 1 Points must be balanced. Do not allow line from top point to bottom point. Line must pass between (425, 0.240) and (440, 0.240) and between (850, 0.400) and (865, 0.400). Worst acceptable line drawn (steepest or shallowest possible line that passes through all error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into Δy / Δx. 1 Distance between data points must be at least half the length of the drawn line. Gradient of worst acceptable line determined. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = (y-intercept of line of best fit – y-intercept of worst acceptable line) or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not allow ECF from false origin method. 2(d)(i) u determined using y-intercept and u and A given to two or three significant figures. 1 1 u = y-intercept A determined using gradient with correct substitution and units with correct power of ten for u and A. 1 y-intercept 1 A= or A= gradient u× gradient © UCLES 2021 Page 9 of 10 2(d)(ii) Percentage uncertainty in A determined, e.g. 1 Δgradient Δy-intercept percentage uncertainty in A=  +   gradient y-intercept  or Δu clearly determined using the value of u and Δgradient Δu percentage uncertainty in A=  +  ×100  gradient u  or correct substitution for max/min methods e.g. 1 maxA= minu×min gradient 1 minA= maxu×max gradient 2(e) Value of m determined from (d)(i) or (c)(iii) and (c)(iv), with correct number substitution and correct power of ten. 1 A×u ( ) m= − 330+A 2 or 0.5−y-intercept m = −330 gradient © UCLES 2021 Page 10 of 10

1 Defining the problem A is the independent variable and t is the dependent variable or vary A and measure t 1 keep Δθ constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • beaker of water • cylinder in water • electrical heater in water • thermometer in water • minimum of three labels from heater, thermometer, cylinder, water, beaker circuit diagram to determine power of the heater e.g. ammeter and voltmeter correctly positioned with a power supply or 1 wattmeter correctly connected to power supply and heater method to determine time for temperature of water to increase or t, e.g. use a stopwatch/timer 1 method to determine A, e.g. micrometer/calipers to determine diameter of cylinder and A = πd2/4 1 Method of analysis plot a graph of t against A (not logarithmic graphs) 1 gradient×P 1 W = hΔθ y-intercept×P 1 Z = Δθ © UCLES 2021 Page 6 of 10 1 Additional detail including safety considerations 6 D1 wear (heat proof) gloves to prevent burns from hot beaker/cylinder/heater/water D2 keep P and h constant D3 check that/ensure/keep initial temperature of the water constant or volume/mass of water constant D4 use calipers/ruler to measure h D5 repeat measurements of diameter in different directions/at different positions along cylinder and average D6 method to calculate power of heater e.g. P = VI linked to correct circuit diagram for ammeter/voltmeter method D7 repeat measurements of t for same A and average t D8 ensure heater and cylinder are (totally) submerged/immersed or stir water (using a glass rod/stirrer) D9 relationship valid if a straight line (not passing through the origin) D10 method to insulate beaker, e.g. use of a lid on the beaker or foam/insulation around outside of beaker © UCLES 2021 Page 7 of 10

2(a) 1 1 gradient = E r y-intercept = E 2(b) 1 1 (R + R ) / Ω / A–1 1 2 I 55 58.1 or 58.14 69 70.4 or 70.42 78 78.1 or 78.13 80 80.6 or 80.65 89 87.7 or 87.72 103 99.0 or 99.01 1 Values of (R + R ) and as shown above. 1 2 I Absolute uncertainties in (R + R ) from ± (2.75 or 2.8 or 3) to ± (5.15 or 5.2 or 5). 1 1 2 2(c)(i) Six points plotted correctly. 1 Must be accurate to the nearest half a small square. Diameter of points must be less than half a small square. Error bars in (R + R ) plotted correctly. 1 1 2 All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2021 Page 8 of 10 2(c)(ii) Line of best fit drawn covers all points. 1 Points must be balanced. Do not allow line from top point to bottom point. Line must pass between (61.0, 65.0) and (63.5, 65.0) and between (96.5, 95.0) and (98.5, 95.0). Worst acceptable line drawn (steepest or shallowest possible line that passes through all error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into Δy / Δx. 1 Distance between data points must be at least half the length of the drawn line. Gradient of worst acceptable line determined. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = (y-intercept of line of best fit – y-intercept of worst acceptable line) or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not allow ECF from false origin method. 2(d)(i) E determined using gradient and E and r given to two or three significant figures. 1 1 E = gradient r determined using y-intercept with correct substitution and units with correct power of ten for E and r. 1 r = y-intercept/gradient or r = E × y-intercept © UCLES 2021 Page 9 of 10 2(d)(ii) Absolute uncertainty in E determined with method shown e.g. 1 Δgradient ΔE = ×E gradient or correct substitution for max/min methods e.g. 1 ΔE = −E mingradient 1 ΔE =E − max gradient 2(e) Value of R determined from (d)(i) or (c)(iii) and (c)(iv), with correct substitution and correct power of ten. 1 2 E ( ) R = − 22+r 2 0.0075 or R = 1 −( 22+r ) 2 0.0075×gradient © UCLES 2021 Page 10 of 10

1 Defining the problem R is the independent variable and t is the dependent variable or vary R and measure t 1 keep the number of turns on the coil/N constant 1 Methods of data collection labelled diagram or correct symbols including: 1 • labelled (d.c.) power supply • switch in series with power supply, resistor and coil • complete workable circuit circuit diagram to measure R, e.g. ammeter and voltmeter correctly positioned or R connected to ohmmeter with no other 1 connections (not ohmmeter in main circuit) method to determine t (of a few milliseconds) e.g. use (storage) oscilloscope or current/voltage sensor connected to 1 datalogger/computer method to determine A, e.g. micrometer/calipers to determine diameter of coil and A = πd2/4 1 Method of analysis plot a graph of t against 1 / R 1 (allow log t against log R) relationship valid if a straight line passing through the origin is produced 1 (allow gradient = –1 for graph of log t against log R) gradient×L 1 K = . AN2 © UCLES 2021 Page 6 of 10 1 Additional detail including safety considerations 6 D1 open switch/switch off (high voltage) circuit before changing the resistor/touching components or ensure no bare wires/use shrouded connectors D2 wear (insulating) gloves to prevent electric shock/electrocution D3 keep A and L constant D4 use ruler/calipers to measure L D5 repeat measurements of diameter in different directions/at points along the coil and average D6 method to determine R e.g. R = V / I linked to correct circuit diagram for ammeter/voltmeter method or measure resistance using ohmmeter D7 repeat experiment for each value of R and average t D8 method to determine t: use of time-base from oscilloscope explained or use of time axis of output from data logger/computer explained D9 use smaller values of R to increase I D10 reduce L or increase N or increase A to increase t © UCLES 2021 Page 7 of 10

2(a) 1 1 gradient = uA 1 y-intercept = u 2(b) 1 1 (M + m) / g / s cm–1 v 380 0.226 or 0.2262 480 0.255 or 0.2551 580 0.294 or 0.2941 680 0.331 or 0.3311 830 0.388 or 0.3876 930 0.429 or 0.4292 1 Values of (M + m) and as shown above. v Absolute uncertainties in (M + m) from ± (19 or 20) to ± (46.5 or 47 or 50). 1 2(c)(i) Six points plotted correctly. 1 Must be accurate to the nearest half a small square. Diameter of points must be less than half a small square. Error bars in (M + m) plotted correctly. 1 All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2021 Page 8 of 10 2(c)(ii) Line of best fit drawn covers all points. 1 Points must be balanced. Do not allow line from top point to bottom point. Line must pass between (425, 0.240) and (440, 0.240) and between (850, 0.400) and (865, 0.400). Worst acceptable line drawn (steepest or shallowest possible line that passes through all error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into Δy / Δx. 1 Distance between data points must be at least half the length of the drawn line. Gradient of worst acceptable line determined. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = (y-intercept of line of best fit – y-intercept of worst acceptable line) or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not allow ECF from false origin method. 2(d)(i) u determined using y-intercept and u and A given to two or three significant figures. 1 1 u = y-intercept A determined using gradient with correct substitution and units with correct power of ten for u and A. 1 y-intercept 1 A= or A= gradient u× gradient © UCLES 2021 Page 9 of 10 2(d)(ii) Percentage uncertainty in A determined, e.g. 1 Δgradient Δy-intercept percentage uncertainty in A=  +   gradient y-intercept  or Δu clearly determined using the value of u and Δgradient Δu percentage uncertainty in A=  +  ×100  gradient u  or correct substitution for max/min methods e.g. 1 maxA= minu×min gradient 1 minA= maxu×max gradient 2(e) Value of m determined from (d)(i) or (c)(iii) and (c)(iv), with correct number substitution and correct power of ten. 1 A×u ( ) m= − 330+A 2 or 0.5−y-intercept m = −330 gradient © UCLES 2021 Page 10 of 10

1 Defining the problem diameter/d is the independent variable and frequency/f is the dependent variable or vary d and measure f 1 keep L constant or length (of tube) constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • tube supported • (loud)speaker positioned in line with the tube • (loud)speaker labelled labelled microphone, positioned outside tube in line with tube, connected to labelled oscilloscope or correct circuit symbol 1 adjust/change frequency until maximum amplitude detected 1 use calipers to measure d 1 © UCLES 2021 Page 6 of 11 1 Method of analysis 1 1 1 plot a graph of against d or d against f f (Do not accept logarithmic graphs.) 1 1 1 for against d for d against f f v = gradient × k or 2L or v = y-intercept v = − gradient×2L y-intercept 1 1 1 for against d for d against f f k = gradient × v or or 2L k = gradient×2L k =− y-intercept y-intercept © UCLES 2021 Page 7 of 11 1 Additional detail including safety considerations 6 D1 wear ear defenders (to prevent damage to hearing/to avoid loud sounds) or use a low volume to prevent damage to hearing/to avoid loud sounds D2 use a rule to measure L D3 increase frequency from a low frequency to the first maximum amplitude D4 method to determine f at maximum amplitude, e.g. increase frequency to f, then continue increasing frequency, and then decrease frequency until value of f determined D5 method to determine period from oscilloscope, e.g. no. of divisions × time-base D6 for frequency/time period determined by oscilloscope, f = 1 / T D7 repeat measurements of d and average in different directions/positions or along the tube D8 perform experiment in a quiet room D9 signal generator connected to (loud)speaker in diagram D10 relationship valid if a straight line produced (Do not accept through the origin.) © UCLES 2021 Page 8 of 11

2(a) t 1 gradient = – C y-intercept = ln E 2(b) 1 (R + R ) / kΩ 1 1 2 / 10–6 Ω R +R 1 2 55 (± 3) 18 or 18.2 ± 0.9 69 (± 3 or 4) 14 or 14.5 ± 0.7 90 (± 4 or 5) 11 or 11.1 ± 0.6 80 (± 4) 13 or 12.5 ± 0.6 101 (± 5) 9.9 or 9.90 or 9.901 ± 0.5 115 (± 6) 8.7 or 8.70 or 8.696 ± 0.4 1 Values of (R + R ) and correct as shown above. 1 2 R +R 1 2 1 1 Absolute uncertainties in from ± 0.9 or ± 1 to ± 0.4 or ± 0.5. R +R 1 2 2(c)(i) Six points plotted correctly. 1 Must be accurate to half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. R +R 1 2 All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2021 Page 9 of 11 2(c)(ii) Line of best fit drawn. 1 Points must be balanced. Do not accept line from top point to bottom point. Line must pass between (10.2, 1.10) and (10.8, 1.10) and between (16.7, 0.40) and (17.2, 0.40). Worst acceptable line drawn (steepest or shallowest possible line that passes through all error bars). 1 All error bars must be plotted. 2(c)(iii) Negative gradient determined with clear substitution of data points into Δy/Δx. 1 Distance between data points must be at least half the length of the drawn line. Gradient of worst acceptable line determined. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of point on line into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution of point on line into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. 2(d)(i) C determined using gradient and C and E both given to two or three significant figures. 1 t 60 C =− =− gradient (c)(iii) E determined using y-intercept and C and E both given with correct SI unit. 1 E =ey-intercept unit of C: F or C V–1 or s Ω–1 unit of E: V © UCLES 2021 Page 10 of 11 2(d)(ii) Percentage uncertainty determined with method shown. 1  1 Δgradient percentage uncertainty =  +  ×100 60 gradient  Clear substitution must be shown for maximum/minimum methods. 2(e) (R + R ) determined to at least two significant figures from (d)(i) or (c)(iii) and (c)(iv) with correct substitution including 1 1 2 signs and correct power of ten(s). Do not accept ECF for POT from (c)(iii), (c)(iv) or (d). ( R +R )=− t × 1 = − 60 × 1 1 2 C V C lnV −lnE ln E or ( R +R )= gradient = (c)(iii) 1 2 ln5.0−y-intercept 1.61−(c)(iv) © UCLES 2021 Page 11 of 11

1 Defining the problem θ is the independent variable and x is the dependent variable or vary θ and measure x 1 keep (angle) β constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • spring attached at both ends e.g. one end connected to a clamp and stand • strip free to move • at least two labels from: clamp, stand, wire, strip, spring, bench (Do not accept extra masses added to strip.) use a rule to measure L and d 1 use a protractor to measure θ 1 or use a rule to measure appropriate distances to determine θ by trigonometry methods measure original length of spring and new length of spring using rule/calipers 1 Method of analysis plot a graph of x against cos θ or cos θ against x 1 (Allow log x against log (cos θ).) relationship is valid if a straight line passing through the origin is produced 1 (Allow straight line with gradient = 1 for log-log graph.) for x against cos θ for cos θ against x 1 W = gradient×2kdsinβ or W = 2kdsinβ L gradient×L © UCLES 2021 Page 6 of 10 1 Additional detail including safety considerations 6 D1 wear goggles to prevent spring/wire/strip entering into eyes or (retort) stand used to support spring is clamped to bench D2 keep distance d constant D3 description of (separate) experiment to determine k, e.g. weigh mass and measure extension D4 k = weight / extension or mg / extension or gradient of weight–extension graph for candidate’s workable (separate) experiment D5 method to prevent strip at point P sliding, e.g. use adhesive putty/hinge (Do not accept methods that prevent rotation at point P.) D6 use fiducial markers on spring at both ends or measure length of spring on both sides and average D7 method to attach wire to strip, e.g. wire wrapped around the strip/(strong) tape/drill hole and tie wire D8 determine x by subtracting original length of spring from new length D9 adjust support of spring to keep β constant D10 protractor correctly positioned on diagram to measure θ or correct trigonometric relationship given for θ © UCLES 2021 Page 7 of 10

2(a) gradient = –μ 1 y-intercept = ln R 0 2(b) 1 average t / mm ln (R / s–1) 0.16 ± 0.03 3.865 or 3.8649 0.25 ± 0.03 3.784 or 3.7842 0.42 ± 0.03 3.643 or 3.6428 0.56 ± 0.02 3.535 or 3.5351 0.66 ± 0.02 3.456 or 3.4563 0.76 ± 0.02 3.391 or 3.3911 Values of average t and ln R correct as shown above. Absolute uncertainties in average t correct as shown above. 1 2(c)(i) Six points plotted correctly. 1 Must be accurate to nearest half a small square. Diameter of points must be less than half a small square. Error bars in average t plotted correctly. 1 All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2021 Page 8 of 10 2(c)(ii) Line of best fit drawn. 1 Points must be balanced. Do not accept line from top point to bottom point. Line must pass between (0.22, 3.80) and (0.24, 3.80) and between (0.60, 3.50) and (0.62, 3.50). Worst acceptable line drawn (steepest or shallowest possible line that passes through all error bars). 1 All error bars must be plotted. 2(c)(iii) Negative gradient determined with clear substitution of data points into Δy / Δx. 1 Distance between data points must be at least half the length of the drawn line. Gradient of worst acceptable line determined. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of point on line into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution of point on line into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. © UCLES 2021 Page 9 of 10 2(d) μ = – gradient value 1 Do not accept negative values (from a negative gradient). R determined using y-intercept and μ and R both given with valid SI unit. 1 0 0 R =ey-intercept 0 unit of μ: mm–1 unit of R : s–1 0 absolute uncertainty in μ = absolute uncertainty in gradient 1 and absolute uncertainty in R = ey−intercept of WAL −R 0 0 Correct substitution of numbers must be seen. 2(e) Value of t determined to two or three significant figures from (d) or (c)(iii) and (c)(iv) with correct substitution and correct 1 power of ten(s). Do not accept ECF for POT from (c)(iii), (c)(iv) or (d). lnR −lnR ln20−lnR t = 0 = 0 −μ −μ or ln20−y-intercept 2.996−(c)(iv) t = = gradient (c)(iii) © UCLES 2021 Page 10 of 10

1 Defining the problem diameter/d is the independent variable and frequency/f is the dependent variable or vary d and measure f 1 keep L constant or length (of tube) constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • tube supported • (loud)speaker positioned in line with the tube • (loud)speaker labelled labelled microphone, positioned outside tube in line with tube, connected to labelled oscilloscope or correct circuit symbol 1 adjust/change frequency until maximum amplitude detected 1 use calipers to measure d 1 © UCLES 2021 Page 6 of 11 1 Method of analysis 1 1 1 plot a graph of against d or d against f f (Do not accept logarithmic graphs.) 1 1 1 for against d for d against f f v = gradient × k or 2L or v = y-intercept v = − gradient×2L y-intercept 1 1 1 for against d for d against f f k = gradient × v or or 2L k = gradient×2L k =− y-intercept y-intercept © UCLES 2021 Page 7 of 11 1 Additional detail including safety considerations 6 D1 wear ear defenders (to prevent damage to hearing/to avoid loud sounds) or use a low volume to prevent damage to hearing/to avoid loud sounds D2 use a rule to measure L D3 increase frequency from a low frequency to the first maximum amplitude D4 method to determine f at maximum amplitude, e.g. increase frequency to f, then continue increasing frequency, and then decrease frequency until value of f determined D5 method to determine period from oscilloscope, e.g. no. of divisions × time-base D6 for frequency/time period determined by oscilloscope, f = 1 / T D7 repeat measurements of d and average in different directions/positions or along the tube D8 perform experiment in a quiet room D9 signal generator connected to (loud)speaker in diagram D10 relationship valid if a straight line produced (Do not accept through the origin.) © UCLES 2021 Page 8 of 11

2(a) t 1 gradient = – C y-intercept = ln E 2(b) 1 (R + R ) / kΩ 1 1 2 / 10–6 Ω R +R 1 2 55 (± 3) 18 or 18.2 ± 0.9 69 (± 3 or 4) 14 or 14.5 ± 0.7 90 (± 4 or 5) 11 or 11.1 ± 0.6 80 (± 4) 13 or 12.5 ± 0.6 101 (± 5) 9.9 or 9.90 or 9.901 ± 0.5 115 (± 6) 8.7 or 8.70 or 8.696 ± 0.4 1 Values of (R + R ) and correct as shown above. 1 2 R +R 1 2 1 1 Absolute uncertainties in from ± 0.9 or ± 1 to ± 0.4 or ± 0.5. R +R 1 2 2(c)(i) Six points plotted correctly. 1 Must be accurate to half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. R +R 1 2 All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2021 Page 9 of 11 2(c)(ii) Line of best fit drawn. 1 Points must be balanced. Do not accept line from top point to bottom point. Line must pass between (10.2, 1.10) and (10.8, 1.10) and between (16.7, 0.40) and (17.2, 0.40). Worst acceptable line drawn (steepest or shallowest possible line that passes through all error bars). 1 All error bars must be plotted. 2(c)(iii) Negative gradient determined with clear substitution of data points into Δy/Δx. 1 Distance between data points must be at least half the length of the drawn line. Gradient of worst acceptable line determined. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of point on line into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution of point on line into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. 2(d)(i) C determined using gradient and C and E both given to two or three significant figures. 1 t 60 C =− =− gradient (c)(iii) E determined using y-intercept and C and E both given with correct SI unit. 1 E =ey-intercept unit of C: F or C V–1 or s Ω–1 unit of E: V © UCLES 2021 Page 10 of 11 2(d)(ii) Percentage uncertainty determined with method shown. 1  1 Δgradient percentage uncertainty =  +  ×100 60 gradient  Clear substitution must be shown for maximum/minimum methods. 2(e) (R + R ) determined to at least two significant figures from (d)(i) or (c)(iii) and (c)(iv) with correct substitution including 1 1 2 signs and correct power of ten(s). Do not accept ECF for POT from (c)(iii), (c)(iv) or (d). ( R +R )=− t × 1 = − 60 × 1 1 2 C V C lnV −lnE ln E or ( R +R )= gradient = (c)(iii) 1 2 ln5.0−y-intercept 1.61−(c)(iv) © UCLES 2021 Page 11 of 11

1 Defining the problem 1 θ is the independent variable and v is the dependent variable, or vary θ and measure v. Keep d constant 1 Methods of data collection 1 Labelled diagram of workable experiment including: • sheet supported by stand / jack • light gate positioned at X • support, light gate and X labelled. Light gate connected to timer / datalogger. 1 Measure length (L) (of card) interrupted by beam for single light gate. 1 Method to measure θ, e.g. use protractor 1 or Method to determine θ , e.g. use a rule(r) to measure two appropriate distances to use in a trigonometrical ratio Method of Analysis 1 Plots a graph of v2on y-axis and sin θ on x-axis. Allow other valid graphs, e.g. sin θ against v2 Do not accept log graphs. gradient 1 p= for v2 against sin θ 2d or 1 p= for sin θ against v2 2d×gradient © UCLES 2022 Page 6 of 10 1 m×y −intercept 1 q =− for v2 against sin θ 2Bd or mp×y −intercept m×y −intercept q = = B 2dB×gradient for sin θ against v2 Additional detail including safety considerations 6 Any six from: D1 Method to stop the trolley once the trolley passes X, e.g. place a block / stop on the bench near the end of the sheet Ignore trolley falls D2 Keep B and m constant D3 Use a rule(r) to measure d D4 Method to keep d constant, e.g. mark distance d on the sheet or the starting position of the trolley on the sheet D5 Method to measure mass of trolley (and magnet), e.g. use balance or use newton meter to measure weight and divide by g and Measure B using a (calibrated) Hall probe D6 Additional detail on use of Hall probe, e.g. adjust probe until maximum value or measure B using Hall probe first in one direction, then in the opposite direction and average D7 Determine v (the velocity at X) from L / t (for a single light gate) D8 Additional detail on measuring θ, e.g. protractor drawn in correct position on diagram, or additional detail on determining θ , e.g. relationship between measured lengths and θ © UCLES 2022 Page 7 of 10 1 D9 Relationship valid if a straight line is produced (not passing through the origin) D10 Repeat experiment for each θ and average v. Question Answer Marks

2(a) 1 1 Gradient = 2πfC 2(b) 1 1 / 10–3 Ω–1 tan θ R 83 or 83.3 6.17 or 6.174 63 or 62.5 4.51 or 4.511 45 or 45.5 3.27 or 3.271 30 or 30.3 2.16 or 2.164 26 or 25.6 1.86 or 1.857 23 or 23.3 1.68 or 1.684 1 1 Absolute uncertainties in R from ± 4 to ± 1 © UCLES 2022 Page 8 of 10 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. R All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Points must be balanced. Do not accept line from top plot to bottom plot. Line must pass between (33.5, 2.5) and (35.0, 2.5) and (74.0, 5.5) and (76.0, 5.5) Worst acceptable line drawn. 1 Steepest or shallowest possible line that passes through all the error bars. All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into Δy/Δx; distance between data points must be greater than half 1 the length of the drawn line. Gradient determined of WAL with clear substitution of data points into Δy/Δx; 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(d) 99 ± 2 (Hz) 1 2(e)(i) C determined using gradient and C given to two or three significant figures. 1 1 1 C = = 2πf ×gradient 2π×(d)×(c)(iii) C determined using gradient with correct SI unit and power of ten for C: F or s Ω–1 1 © UCLES 2022 Page 9 of 10 2(e)(ii) Percentage uncertainty in C determined with method shown. 1  Δf Δgradient %uncertainty =  +  ×100  f gradient  OR Correct substitution for max/min methods 1 maxC = 2π×minf ×mingradient 1 minC = 2π×maxf ×maxgradient 2(f) R determined to at least two significant figures with appropriate power of ten from (c)(iii) OR (d) and (e)(i) with correct 1 substitution seen. gradient (c)(iii) R = = tanθ 0.839 OR 1 1 R = = 2πfCtanθ 2π×(d)×(e)(i)×0.839 Absolute uncertainty in R determined. 1 Method must be consistent with determination of R and correct substitution must be seen. For R determined by using the gradient: Δgradient ΔR = ×R gradient OR For R determined by using (d) and (e)(i):  Δf ΔC ΔR = + ×R    f C  OR ΔR determined by max / min methods. © UCLES 2022 Page 10 of 10

1 Defining the problem d is the independent variable and V is the dependent variable or vary d and measure V 1 keep A or area (of overlap) of plates constant 1 Methods of data collection labelled diagram of workable experiment including: 1  circuit diagram with voltmeter connected in parallel with the capacitor  capacitor and voltmeter connected to the metal plates with no power supply in discharge part of the circuit  correct symbols for capacitor and voltmeter method to charge parallel plates, e.g. separate circuit diagram showing plates connected to a d.c. power supply or combined 1 circuit with switches and d.c. power supply use calipers to measure d 1 or use micrometer/calipers to measure thickness of spacers use rule(r) to measure lengths to determine A and A = length  breadth 1 Method of analysis 1 1 1 plot a graph of against d or equivalent (e.g. d against ) V V (Do not accept log graphs.) y-interceptC C 1 K  or K  gradientA gradientAW 1 y-interceptC (for d against : K  ) V A © UCLES 2022 Page 6 of 10 1 1 1 W  y-intercept 1 gradient gradientC (for d against : W  or W  ) V y-intercept AK Additional detail including safety considerations 6 D1 use gloves to prevent electric shock or do not touch metal plates to avoid shocks D2 keep the initial p.d. across plates or initial charge constant D3 method to determine the value of C, e.g. description of an experiment to measure p.d. or current against time during discharge through a resistor D4 method of operation of circuit(s) using switch(es) D5 description of method to fully discharge capacitor, e.g. between experiments, short-circuit the capacitor or use of switch in parallel with capacitor D6 repeat measurements of d at different points across plates and average D7 repeat measurements of V for same d and average V D8 bottom plate resting on insulating material or top plate supported by strings D9 use high voltage power supply to increase charge on plates or use a very small value of capacitance to increase voltmeter reading D10 relationship valid if a straight line is produced (not passing through the origin) © UCLES 2022 Page 7 of 10

2(a) gradient = n 1 y-intercept = lgSZ 2(b) 1 lg (M / 1030kg) lg (L / 1028W) 0.68 or 0.681  0.03 or 0.04 0.15 or 0.146 0.81 or 0.806  0.02 or 0.03 0.49 or 0.491 1.08 or 1.079  0.07 or 0.08 1.51 or 1.505 1.36 or 1.362  0.04 2.54 or 2.544 1.63 or 1.633  0.04 3.56 or 3.556 1.96 or 1.959  0.02 4.82 or 4.820 Values of lg (M / 1030kg) and lg (L / 1028W) correct as shown above. Absolute uncertainties in lg (M / 1030kg) correct as shown above. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in lg (M / 1030kg) plotted correctly. 1 All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Points must be balanced. Do not accept line from top point to bottom point. Line must pass between (0.92, 1.0) and (0.96, 1.0) and between (1.86, 4.5) and (1.90, 4.5) Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. © UCLES 2022 Page 8 of 10 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient of worst acceptable line determined. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not allow methods using a false origin. © UCLES 2022 Page 9 of 10 2(d) n gradient (c)(iii) and n and Z both given to two or three significant figures. 1 Value of Z determined using y-intercept. Correct method must be seen. 1 10y-intercept 1028 10(c)(iv)1028 Z   S 3.851026 or Z 10y-interceptlg S 1028 or Z 10(c)(iv)lg 3.851026 1028 Absolute uncertainty in n = absolute uncertainty in gradient 1 and  10y-intercept 10WAL y-intercept 1028 Z  S Correct substitution of numbers must be seen. 2(e) L determined from (d) or (c)(iii) and (c)(iv) with correct substitution and correct power of ten(s). 1 Do not accept incorrect POT for n or Z. L = 3.85  1026  (d)  3.0(c)(iii) or lgL(c)(iii)lg3.0y-intercept © UCLES 2022 Page 10 of 10

1 Defining the problem L is the independent variable and I is the dependent variable or vary L and measure I 1 keep E constant 1 Methods of data collection labelled diagram of workable experiment including: 1  circuit diagram with power supply connected to ends C  ammeter in series with power supply and conductors  correct symbol for ammeter and power supply circuit diagram with voltmeter correctly positioned to measure E across the power supply 1 use a rule(r) to measure L and x 1 use a micrometer/calipers to measure y 1 © UCLES 2022 Page 6 of 11 1 Method of analysis 1 1 1 plot a graph of against L or equivalent (e.g. L against ) I I (Do not accept log graphs.) gradientAE 1 P 

2 1 AE (for L against : P  ) I 2gradient y-interceptEy2 1 Q  x 1 y-intercept2Py2 y-interceptEy2 (for L against : Q  or Q  ) I Ax gradientx © UCLES 2022 Page 7 of 11 9702/52 Cambridge International AS & A Level – Mark Scheme May/June 2022 PUBLISHED Question Answer Marks 1 Additional detail including safety considerations 6 D1 do not touch/use (heat resistant) gloves to avoid hot conductors/metal bar or use a protective resistor/small e.m.f. to reduce the current or switch off when not in use/when moving bar D2 keep A and y constant D3 keep x constant D4 use of micrometer/calipers to measure diameter of conductor and A = d2 / 4. D5 repeat measurements of diameter along conductors/different (perpendicular) directions/different points and average or repeat measurements of y in different (perpendicular) directions/different points/along bar and average D6 method to ensure that L is the same for each conductor, e.g. check both lengths D7 method to determine L e.g. measure to edge and add y / 2 or method to determine x e.g. measure between the conductors and add diameter D8 method to keep x constant with reason, e.g. adhesive/plasticine/blocks (one either side of each conductor) to prevent cylindrical conductors from moving D9 method of ensuring good electrical contact, e.g. clean metal bar/cylindrical conductors or use of solder or crocodile clips to connect circuit to the conductors D10 relationship valid if a straight line is produced (not passing through the origin) © UCLES 2022 Page 8 of 11 2(a) gradient = a 1 y-intercept = lgSK 2(b) 1 lg (T / days) lg (L / 1030W) 1.34 or 1.342 0.46 or 0.462  0.03 1.51 or 1.505 0.69 or 0.690  0.02 1.62 or 1.623 0.84 or 0.839  0.01 1.73 or 1.732 0.99 or 0.991  0.01 1.89 or 1.892 1.20 or 1.204  0.05 or 0.06 1.99 or 1.987 1.32 or 1.322  0.04 Values of lg (T / days) and lg (L / 1030W) correct as shown above. Absolute uncertainties in lg (L / 1030W) correct as shown above. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in lg (L / 1030W) plotted correctly. 1 All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Points must be balanced. Do not accept line from top point to bottom point. Line must pass between (1.43, 0.60) and (1.45, 0.60) and between (1.84, 1.15) and (1.86, 1.15). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. © UCLES 2022 Page 9 of 11 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient of worst acceptable line determined. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not allow methods using a false origin. © UCLES 2022 Page 10 of 11 2(d) a = gradient = (c)(iii) and a and K both given to two or three significant figures. 1 Value of K determined using y-intercept. Correct method must be seen. 1 10y-intercept 1030 10(c)(iv)1030 K   S 3.851026 or K 10y-interceptlg S 1030 or K 10(c)(iv)lg 3.851026 1030 absolute uncertainty in a = absolute uncertainty in gradient 1 and  10y-intercept 10WAL y-intercept 1030 K  S Correct substitution of numbers must be seen. 2(e) L determined from (d) or (c)(iii) and (c)(iv) with correct substitution and correct power of ten(s). 1 Do not accept incorrect POT for a or K. L = 3.85  1026  (d)  5.0(c)(iii) or lgL(c)(iii)lg5.0y-intercept © UCLES 2022 Page 11 of 11

1 Defining the problem d is the independent variable and V is the dependent variable or vary d and measure V 1 keep A or area (of overlap) of plates constant 1 Methods of data collection labelled diagram of workable experiment including: 1  circuit diagram with voltmeter connected in parallel with the capacitor  capacitor and voltmeter connected to the metal plates with no power supply in discharge part of the circuit  correct symbols for capacitor and voltmeter method to charge parallel plates, e.g. separate circuit diagram showing plates connected to a d.c. power supply or combined 1 circuit with switches and d.c. power supply use calipers to measure d 1 or use micrometer/calipers to measure thickness of spacers use rule(r) to measure lengths to determine A and A = length  breadth 1 Method of analysis 1 1 1 plot a graph of against d or equivalent (e.g. d against ) V V (Do not accept log graphs.) y-interceptC C 1 K  or K  gradientA gradientAW 1 y-interceptC (for d against : K  ) V A © UCLES 2022 Page 6 of 10 1 1 1 W  y-intercept 1 gradient gradientC (for d against : W  or W  ) V y-intercept AK Additional detail including safety considerations 6 D1 use gloves to prevent electric shock or do not touch metal plates to avoid shocks D2 keep the initial p.d. across plates or initial charge constant D3 method to determine the value of C, e.g. description of an experiment to measure p.d. or current against time during discharge through a resistor D4 method of operation of circuit(s) using switch(es) D5 description of method to fully discharge capacitor, e.g. between experiments, short-circuit the capacitor or use of switch in parallel with capacitor D6 repeat measurements of d at different points across plates and average D7 repeat measurements of V for same d and average V D8 bottom plate resting on insulating material or top plate supported by strings D9 use high voltage power supply to increase charge on plates or use a very small value of capacitance to increase voltmeter reading D10 relationship valid if a straight line is produced (not passing through the origin) © UCLES 2022 Page 7 of 10

2(a) gradient = n 1 y-intercept = lgSZ 2(b) 1 lg (M / 1030kg) lg (L / 1028W) 0.68 or 0.681  0.03 or 0.04 0.15 or 0.146 0.81 or 0.806  0.02 or 0.03 0.49 or 0.491 1.08 or 1.079  0.07 or 0.08 1.51 or 1.505 1.36 or 1.362  0.04 2.54 or 2.544 1.63 or 1.633  0.04 3.56 or 3.556 1.96 or 1.959  0.02 4.82 or 4.820 Values of lg (M / 1030kg) and lg (L / 1028W) correct as shown above. Absolute uncertainties in lg (M / 1030kg) correct as shown above. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in lg (M / 1030kg) plotted correctly. 1 All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Points must be balanced. Do not accept line from top point to bottom point. Line must pass between (0.92, 1.0) and (0.96, 1.0) and between (1.86, 4.5) and (1.90, 4.5) Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. © UCLES 2022 Page 8 of 10 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient of worst acceptable line determined. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not allow methods using a false origin. © UCLES 2022 Page 9 of 10 2(d) n gradient (c)(iii) and n and Z both given to two or three significant figures. 1 Value of Z determined using y-intercept. Correct method must be seen. 1 10y-intercept 1028 10(c)(iv)1028 Z   S 3.851026 or Z 10y-interceptlg S 1028 or Z 10(c)(iv)lg 3.851026 1028 Absolute uncertainty in n = absolute uncertainty in gradient 1 and  10y-intercept 10WAL y-intercept 1028 Z  S Correct substitution of numbers must be seen. 2(e) L determined from (d) or (c)(iii) and (c)(iv) with correct substitution and correct power of ten(s). 1 Do not accept incorrect POT for n or Z. L = 3.85  1026  (d)  3.0(c)(iii) or lgL(c)(iii)lg3.0y-intercept © UCLES 2022 Page 10 of 10

1 Defining the problem A is the independent variable and s is the dependent variable or vary A and measure s 1 keep B and t constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • pin / rod through hole • supported by a stand • sheet able to oscillate freely • at least one label from copper/sheet, hole, clamp, stand, rod, pin. drawn clamped rule(r) parallel to the direction of the oscillations (by eye) (to measure s) 1 use rule(r) to measure lengths to determine A 1 and A = length  breadth use of micrometer to measure t 1 Method of Analysis plot a graph of ln s against A or equivalent 1 relationship valid if a straight line (with y-intercept = ln s ) is produced 1 0 gradient 1 K =− Bt 1 (K =− for A against ln s) Btgradient © UCLES 2022 Page 5 of 9 1 Additional detail including safety considerations 6 D1 use of cushion/sand box in case sheet falls or use gloves to protect hands from cuts / sharp edges D2 keep (initial) distance between (copper) sheet and (poles of) magnet constant or keep (initial) distance between (copper) sheet and coil(s) constant D3 keep s constant 0 D4 method to ensure s is constant, e.g. initially line up (corner of) plate with fiducial marker / vertical pin to keep s 0 0 constant D5 method to determine s using video camera: • rule(r) in a position to measure s in the diagram • video camera shown in diagram or description of use of video camera • playback video recording by frame by frame / slow motion (to measure s) D6 repeat measurements of t in different positions and average t D7 measure B/magnetic flux density using a (calibrated) Hall probe D8 additional detail on use of Hall probe, e.g. adjust probe until maximum value or measure B using Hall probe first in one direction and then in the opposite direction and average D9 drawn method to create a magnetic field perpendicular to the area of the sheet, e.g. pair of magnets/horseshoe magnet/pair of coils connected to a (d.c.) supply D10 repeat experiment for each A and average s © UCLES 2022 Page 6 of 9

2(a) YZd2 1 gradient = 4 2(b) 1 1 / 10–3 –1 R 45 or 45.5 37 or 37.0 30 or 30.3 26 or 25.6 21 or 21.3 19 or 18.5 1 1 Absolute uncertainties in from ± 2 to ± 0.9 or ± 1. R 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. R All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Points must be balanced. Do not accept line from top point to bottom point. Line must pass between (22.0, 30.0) and (23.0, 30.0) and (40.5, 65.0) and (42.0, 65.0). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. © UCLES 2022 Page 7 of 9 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient of worst acceptable line determined with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(d) 0.261 ± 0.003 (mm) 1 2(e)(i)  determined using gradient and  given to two or three significant figures. 1 YZd2 2222(d)2 = = 4gradient 4(c)(iii)  determined using gradient and given with correct SI unit ( m) and correct power of ten 1 2(e)(ii) percentage uncertainty in : 1  2d gradient  percentage uncertainty= + +0.05+0.05100  d gradient  or correct substitution for max/min methods (1.0522)(1.0522)(d +d)2 max= 4mingradient (0.9522)(0.9522)(d −d)2 min= 4maxgradient © UCLES 2022 Page 8 of 9 2(f) R determined to at least two significant figures from (c)(iii) or (d) and (e)(i) with correct substitution seen. 1 gradient R = 0.950 or YZd2 2222(d)2 R = = 4L 4(e)(i)0.950 Absolute uncertainty in R determined. 1 Method must be consistent with determination of R and correct substitution must be seen. for R determined using the gradient: gradient R = R gradient or for R determined using (d) and (e)(i):  2d   R= + +0.05+0.05 R  d   or correct substitution for max/min methods: (1.0522)(1.0522)(d +d)2 maxR = 4min0.950 (0.9522)(0.9522)(d −d)2 min R = 4max0.950 © UCLES 2022 Page 9 of 9

1 Defining the problem z is the independent variable and t is the dependent variable or vary z and measure t 1 keep B and A constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • pin/rod though hole • supported by a stand • sheet able to oscillate freely • at least one label from copper/sheet, hole, clamp stand, rod, pin use of stop-watch/timer to measure t (from release to stopping) 1 or use of stop-watch/timer to measure time for the sheet (to stop) oscillating use of micrometer to measure z 1 use of rule(r) to measure lengths to determine A 1 and A = length  breadth Method of Analysis plot a graph of lg t against lg z or equivalent (e.g. ln t against ln z) 1 q=gradient 1 K = AB10y-intercept 1 (K = ABey-intercept for ln t against ln z) © UCLES 2022 Page 5 of 9 1 Additional detail including safety considerations 6 D1 use of cushion / sand box in case sheet falls or use gloves to protect hands from cuts / sharp edges D2 keep (initial) distance between (copper) sheet and (poles of) magnet constant or keep (initial) distance between (copper) sheet and coil(s) constant D3 keep initial displacement (of copper sheet) constant D4 method to ensure initial displacement (of copper sheet) is constant, e.g. initially line up (corner of) plate with fiducial marker/vertical pin  K  D5 relationship valid if a straight line (with y-intercept = log   ) is produced AB D6 repeat measurements of z in different positions and average z D7 measure B / magnetic flux density using a (calibrated) Hall probe D8 additional detail on use of Hall probe, e.g. adjust (position of) probe until maximum value or measure B using Hall probe first in one direction and then in the opposite direction and average D9 drawn method to create a magnetic field perpendicular to the area of the sheet, e.g. pair of magnets/horseshoe magnet/pair of coils connected to a (d.c.) supply D10 repeat experiment for each z and average t D11 method to determine , e.g. measure mass with balance and volume = Az and density = mass / volume © UCLES 2022 Page 6 of 9

2(a) c 1 gradient = 2h c y-intercept = − 4h 2(b) 1 T / ms f / Hz 7.0 or 7.00 ± 1 140 or 143 ± (10–30) 2.9 or 2.90 ± 0.2 340 or 345 ± (20–30) 1.8 or 1.80 ± 0.1 560 or 556 ± 30 1.4 or 1.35 ± 0.1 710 or 714 ± (40–60) or 740 or 741 1.1 or 1.05 ± 0.1 910 or 909 ± (80–100) or 950or952 0.88 or 0.880 ± 0.02 1100 or 1140 ± (30–60) Values of T and f correct as shown above. Absolute uncertainties in T and f correct as shown above. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in f plotted correctly. 1 All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2022 Page 7 of 9 2(c)(ii) Straight line of best fit drawn. 1 Points must be balanced. Do not accept line from top point to bottom point. Line must pass between (2.20, 400) and (2.40, 400) and (5.20, 1000) and (5.60, 1000). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient of worst acceptable line determined with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(d) 83.2 ± 0.3 (cm) 1 2(e)(i) c determined using gradient and c given to two or three significant figures. 1 c = 2  h  gradient = 2  (d)  (c)(iii) c determined using gradient and given with correct SI unit and correct power of ten: m s–1 or cm s–1. 1 2(e)(ii) Percentage uncertainty in c from (c)(iii) and (d) with method shown. 1  h gradient percentage uncertainty= + 100  h gradient  or correct substitution for max/min methods: max c = 2  max h  max gradient min c = 2  min h  min gradient © UCLES 2022 Page 8 of 9 2(f) h determined to at least two significant figures from (e)(i) with correct substitution. 1 3(e)(i) h= 4130 Absolute uncertainty in h determined. Correct substitution must be seen. 1  f c  5 c h= + h= + h      f c  130 c  or correct substitution for max/min methods: 3 max c 3 max(e)(i) maxh= = 4min f 4125 3min c 3min(e)(i) minh= = 4max f 4135 © UCLES 2022 Page 9 of 9

1 Defining the problem A is the independent variable and s is the dependent variable or vary A and measure s 1 keep B and t constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • pin / rod through hole • supported by a stand • sheet able to oscillate freely • at least one label from copper/sheet, hole, clamp, stand, rod, pin. drawn clamped rule(r) parallel to the direction of the oscillations (by eye) (to measure s) 1 use rule(r) to measure lengths to determine A 1 and A = length  breadth use of micrometer to measure t 1 Method of Analysis plot a graph of ln s against A or equivalent 1 relationship valid if a straight line (with y-intercept = ln s ) is produced 1 0 gradient 1 K =− Bt 1 (K =− for A against ln s) Btgradient © UCLES 2022 Page 5 of 9 1 Additional detail including safety considerations 6 D1 use of cushion/sand box in case sheet falls or use gloves to protect hands from cuts / sharp edges D2 keep (initial) distance between (copper) sheet and (poles of) magnet constant or keep (initial) distance between (copper) sheet and coil(s) constant D3 keep s constant 0 D4 method to ensure s is constant, e.g. initially line up (corner of) plate with fiducial marker / vertical pin to keep s 0 0 constant D5 method to determine s using video camera: • rule(r) in a position to measure s in the diagram • video camera shown in diagram or description of use of video camera • playback video recording by frame by frame / slow motion (to measure s) D6 repeat measurements of t in different positions and average t D7 measure B/magnetic flux density using a (calibrated) Hall probe D8 additional detail on use of Hall probe, e.g. adjust probe until maximum value or measure B using Hall probe first in one direction and then in the opposite direction and average D9 drawn method to create a magnetic field perpendicular to the area of the sheet, e.g. pair of magnets/horseshoe magnet/pair of coils connected to a (d.c.) supply D10 repeat experiment for each A and average s © UCLES 2022 Page 6 of 9

2(a) YZd2 1 gradient = 4 2(b) 1 1 / 10–3 –1 R 45 or 45.5 37 or 37.0 30 or 30.3 26 or 25.6 21 or 21.3 19 or 18.5 1 1 Absolute uncertainties in from ± 2 to ± 0.9 or ± 1. R 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. R All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Points must be balanced. Do not accept line from top point to bottom point. Line must pass between (22.0, 30.0) and (23.0, 30.0) and (40.5, 65.0) and (42.0, 65.0). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. © UCLES 2022 Page 7 of 9 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient of worst acceptable line determined with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(d) 0.261 ± 0.003 (mm) 1 2(e)(i)  determined using gradient and  given to two or three significant figures. 1 YZd2 2222(d)2 = = 4gradient 4(c)(iii)  determined using gradient and given with correct SI unit ( m) and correct power of ten 1 2(e)(ii) percentage uncertainty in : 1  2d gradient  percentage uncertainty= + +0.05+0.05100  d gradient  or correct substitution for max/min methods (1.0522)(1.0522)(d +d)2 max= 4mingradient (0.9522)(0.9522)(d −d)2 min= 4maxgradient © UCLES 2022 Page 8 of 9 2(f) R determined to at least two significant figures from (c)(iii) or (d) and (e)(i) with correct substitution seen. 1 gradient R = 0.950 or YZd2 2222(d)2 R = = 4L 4(e)(i)0.950 Absolute uncertainty in R determined. 1 Method must be consistent with determination of R and correct substitution must be seen. for R determined using the gradient: gradient R = R gradient or for R determined using (d) and (e)(i):  2d   R= + +0.05+0.05 R  d   or correct substitution for max/min methods: (1.0522)(1.0522)(d +d)2 maxR = 4min0.950 (0.9522)(0.9522)(d −d)2 min R = 4max0.950 © UCLES 2022 Page 9 of 9

1 Defining the problem f is the independent variable and Q is the dependent variable, or vary f and measure Q. 1 Keep h constant 1 Methods of data collection Labelled diagram of workable experiment including: 1 • fan is positioned in line with the turbine so that blades of both fan and turbine overlap • base of fan on same bench as turbine • fan labelled and one other label from bench, (wind) turbine, cable, pump, pipe, liquid Labelled apparatus showing workable method to collect the liquid from the top of the pipe, e.g. hose / pipe / tube connected 1 to top of pipe with the other end over a beaker / measuring cylinder below top of pipe. At least one label related to collection of liquid. Use of stop-watch / timer to measure time to collect liquid or to measure time for blades to rotate. 1 Use of (top pan) balance to measure mass of liquid leaving the pipe. 1 Method of Analysis Plots a graph of Q against f3or equivalent (e.g. f3 against Q). 1 Do not accept logarithmic graphs. C =ghy-intercept(for f3 against Q: C =−Dy-intercept) 1 D=ghgradient 1 gh (for f3 against Q: D = ) gradient © UCLES 2023 Page 5 of 9 1 Additional detail including safety considerations 6 Any six from: D1 Precaution with reason linked to prevent liquid spilling (on bench / floor) e.g. use of large bucket / bowl / tray to contain any spilled liquid or Precaution with reason linked to prevent air / dust particles in eye, e.g. use of goggles or Precaution with reason linked to turbine falling, e.g. clamp turbine to bench. D2 Use rule to measure h. D3 Method to determine mass of liquid, e.g. mass of beaker + liquid – mass of empty beaker or mass of container / pipe before – mass of container / pipe after. D4 Method to determine f, e.g. measure time t for many rotations / revolutions N and period T = t /N and f = 1/T or measure time t for many rotations / revolutions N and f = N/t or video rotating blades, playback frame by frame and use a time stamp to determine period T and f = 1/T. D5 Mark one of the blades to assist in counting number of rotations. D6 Method to vary f, e.g. change speed of fan / change distance between fan and blades / vary current in fan. D7 Wait for steady air flow before starting timing and / or collecting liquid. mass (of liquid) D8 Q = time (to collect liquid) or method and explanation to reduce uncertainty in Q, e.g. use large value of time or mass of liquid collected. D9 Repeat measurements of Q for the same value of f and average Q. © UCLES 2023 Page 6 of 9 1 D10 Relationship valid if a straight line is produced (not passing through the origin). Do not accept passing through the origin. Question Answer Marks

2(a) v 1 gradient = 4 y-intercept = −k 2(b) 2 1/f / 10–3 Hz–1 d / cm 0.67 or 0.667 24.7  0.2 0.48 or 0.476 17.4  0.2 0.36 or 0.357 12.7  0.3 0.24 or 0.244 8.4  0.3 0.19 or 0.192 6.6  0.4 0.13 or 0.132 4.6  0.4 First mark: values of 1 / f and d correct as shown. Second mark: uncertainties in d correct as shown. 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in d / cm plotted correctly. 1 All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2023 Page 7 of 9 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top plot to bottom plot. Points must be balanced. Line must pass between (0.170, 6.0) and (0.185, 6.0) and between (0.590, 22.0) and (0.610, 22.0) Worst acceptable line drawn. 1 Steepest or shallowest possible line that passes through all the error bars. All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x; distance between data points must be greater than 1 half the length of the drawn line. Gradient determined of worst acceptable line uncertainty = (gradient of line of best fit – gradient of worst acceptable line) 1 or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and x into y = mx + c. 1 Expect y-intercept to be negative. y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line, or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) © UCLES 2023 Page 8 of 9 2(d) v determined using gradient and v and k given to 2 or 3 sf. 1 v =4gradient=4(c)(iii) k determined using y-intercept and units for v and k 1 k =−y-intercept=−(c)(iv) Units: v: m s–1, cm s–1 k: m, cm Absolute uncertainties in v and k. 1  gradient v: v with correct substitution or v: 4  uncertainty in gradient gradient and k: uncertainty in y-intercept 2(e) f determined to a minimum of 2 significant figures from (c)(iii) and (c)(iv) OR (d) with correct substitution and correct 1 powers of ten used for all quantities. v f = 4(d +k) or gradient f = d −(y-intercept) © UCLES 2023 Page 9 of 9

1 Defining the problem  is the independent variable and t is the dependent variable or vary  and measure t 1 keep d constant 1 Methods of data collection labelled diagram of workable experiment including: 1  plane supported by stand / support  stand/support on bench/floor/horizontal surface  minimum of two labels from cube, cylinder, (inclined) plane, method of support, bench/floor/horizontal surface, pulley, string diagram showing method to measure d, e.g. 1 clamped vertical rule near cylinder or drawn rule on plane used to measure d or distance d marked on plane and rule used to determine d use a protractor to measure θ 1 or use a rule(r) to measure appropriate lengths for a trigonometric calculation use a timer/stop-watch to measure t or light gates connected to a timer to measure t 1 © UCLES 2023 Page 6 of 12 1 Method of Analysis 1 1 1 plot a graph of against sin  or equivalent (e.g. sin  against ) t2 t2 Do not accept logarithms. 2dABgradient 1 H  A 1 2dAB (for sin  against : H  ) t2 Agradient 2dABy-intercept 1 K  A 1 (for sin  against : K Hy-intercept) t2 © UCLES 2023 Page 7 of 12 1 Additional detail including safety considerations 6 D1 Safety precaution linked to falling cylinder, e.g. use of cushion/sand box to collect cylinder/prevent damage to cylinder/floor/bench/injury D2 protractor correctly positioned on diagram or appropriate trigonometric relationship for marked lengths D3 keep A and B constant D4 use a (top-pan) balance to measure A and B D5 correct positioning of light gates to determine t, e.g. two light gates either end of distance d, connected to a timer or correct position of video camera with timer in frame of the video to determine t D6 method to release cylinder/cube, e.g. cube held by set square, set square moved to release cube. D7 reasoned method to keep d constant as θ changes, e.g. (when measuring d by position of cylinder) adjust the length of the string or adjust the position of vertical marks or adjust the position of the vertical rule or initial position of the cube or (when measuring d by position of cube) use fixed marks on the plane or ruler placed on the plane with d measured between the marks D8 method to increase t for cylinder to fall, e.g. use large d to increase t D9 repeat measurements of t for the same θ and average t D10 relationship valid if a straight line is produced (not passing through the origin) Do not accept straight line passing through the origin. © UCLES 2023 Page 8 of 12

2(a) V  1 gradient = Rln  V   0 2(b) 1 C / 10–4 F T / s 0.89 or 0.892 13.7  0.8 1.3 or 1.32 20.4  0.7 1.6 or 1.58 24.3  0.6 1.0 or 1.03 16.1  0.8 1.2 or 1.18 18.3  0.7 2.1 or 2.08 31.5  0.6 Values of C and T correct as shown above. Absolute uncertainties in T correct as shown above. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in T plotted correctly. 1 All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2023 Page 9 of 12 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Points must be balanced. Line must pass between (1.42, 22.0) and (1.45, 22.0) and between (1.95, 30.0) and (2.00, 30.0) Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient of worst acceptable line determined with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(d) – 0.69 or – 0.693 and  0.06 1 2(e)(i) R determined using gradient and R given to 2 or 3 significant figures. 1 gradient (c)(iii) R   V  (d) ln  V  0  R correctly determined using gradient and SI unit with correct power of ten for R (e.g. ). 1 © UCLES 2023 Page 10 of 12 2(e)(ii) Percentage uncertainty in R with method shown. 1   V   ln   percentage uncertainty      V 0     gradient 100   V  gradient  ln    V    0   or Correct substitution for max/min methods. © UCLES 2023 Page 11 of 12 2(f) C determined to a minimum of 2 significant figures from (c)(iii) or (d) and (e)(i) with correct substitution. 1 T 60.0 C   gradient gradient or T 60.0 C   V  (e)(i)(d) Rln  V  0  Absolute uncertainty in C determined with correct method used: 1 Using gradient to determine C:  gradient C  C  gradient  Allow using R to determine C:   V   ln   C      V 0     (e)(ii) C   V  100  ln    V    0   © UCLES 2023 Page 12 of 12

1 Defining the problem R is the independent variable and E is the dependent variable or vary R and measure E 1 keep V constant 1 Methods of data collection labelled diagram of workable experiment including: 1  coil P placed close to coil Q  separate workable circuit for coil Q  (a.c.) voltmeter or oscilloscope connected across coil Q (Do not accept a power supply connected to coil Q.) a.c. power supply/signal generator connected to resistor and coil P in series 1 workable circuit with power supply and (a.c.) voltmeter/oscilloscope in parallel with resistor and coil P or across terminals of 1 power supply/signal generator method to determine R, e.g. measure current in R and p.d. across R and use R = V /I or measure R using an ohmmeter 1 R © UCLES 2023 Page 6 of 12 1 Method of Analysis 1 1 1 plot a graph of against R or equivalent (e.g. R against ) E E Do not accept logarithms. 1 1 M  2fV gradient 1 gradient (for R against : M  ) E 2fV k 2fVMy-intercept 1 or y-intercept k  gradient 1 (for R against : k =  y-intercept) E © UCLES 2023 Page 7 of 12 1 Additional detail including safety considerations 6 D1 precaution linked to hot coil (P) / hot resistor, e.g. use of (heat-proof) gloves, wait until circuit cools down or precaution linked to shocks from high voltages e.g. use of (insulating) gloves or switch off supply before touching the circuit (to change R) D2 keep the number of turns on (both) coils constant D3 keep f constant D4 keep distance between the coils constant D5 method to keep distance between the coils constant, e.g. fix/clamp coils to bench D6 method to measure f, e.g. read from signal generator or use of oscilloscope D7 method to determine f from oscilloscope, e.g. period from oscilloscope T = time-base  horizontal distance and f = 1/T D8 method to determine V or E from oscilloscope, e.g. V = y-gain  vertical distance D9 method to increase E e.g. use iron core/more turns on coil Q/high frequency/high p.d. (across R and coil P) D10 relationship valid if a straight line is produced (not passing through the origin) Do not accept straight line passing through the origin. © UCLES 2023 Page 8 of 12

2(a) Yk 1 gradient = p YkZ y-intercept = p 2(b) 1 V / 10–5 m3 absolute uncertainty 3.81 or 3.815  0.03 3.99 or 3.986  0.03 4.16 or 4.163  0.04 4.33 or 4.335  0.04 4.48 or 4.481  0.04 4.65 or 4.652  0.04 Values of V correct as shown above. Absolute uncertainties in V correct as shown above. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in V plotted correctly. 1 All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2023 Page 9 of 12 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Points must be balanced. Line must pass between (27.5, 3.90) and (29.5, 3.90) and between (82.0, 4.60) and (84.0, 4.60). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient of worst acceptable line determined with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and y into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. © UCLES 2023 Page 10 of 12 2(d)(i) Y determined using gradient and Y and Z given to 2 or 3 significant figures. 1 pgradient Y  7.31881027 gradient k Z determined using y-intercept and Y and Z given with SI units. 1 py-intercept y-intercept Z  or Z  Yk gradient Units: Y: no unit Z: °C 2(d)(ii) Percentage uncertainty in Y with method shown. 1 p gradient percentage uncertainty   100  p gradient  or Correct substitution for max/min methods. © UCLES 2023 Page 11 of 12 2(e)  determined to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution and correct 1 powers of ten. 0.02792 0.0600 V  3.67105 4 and pV V V y-intercept  Z or  Z or  Yk gradient gradient or using h directly: d2h y-intercept pd2h d2h 4  Z or  Z or  4Yk 4gradient gradient © UCLES 2023 Page 12 of 12

1 Defining the problem  is the independent variable and t is the dependent variable or vary  and measure t 1 keep d constant 1 Methods of data collection labelled diagram of workable experiment including: 1  plane supported by stand / support  stand/support on bench/floor/horizontal surface  minimum of two labels from cube, cylinder, (inclined) plane, method of support, bench/floor/horizontal surface, pulley, string diagram showing method to measure d, e.g. 1 clamped vertical rule near cylinder or drawn rule on plane used to measure d or distance d marked on plane and rule used to determine d use a protractor to measure θ 1 or use a rule(r) to measure appropriate lengths for a trigonometric calculation use a timer/stop-watch to measure t or light gates connected to a timer to measure t 1 © UCLES 2023 Page 6 of 12 1 Method of Analysis 1 1 1 plot a graph of against sin  or equivalent (e.g. sin  against ) t2 t2 Do not accept logarithms. 2dABgradient 1 H  A 1 2dAB (for sin  against : H  ) t2 Agradient 2dABy-intercept 1 K  A 1 (for sin  against : K Hy-intercept) t2 © UCLES 2023 Page 7 of 12 1 Additional detail including safety considerations 6 D1 Safety precaution linked to falling cylinder, e.g. use of cushion/sand box to collect cylinder/prevent damage to cylinder/floor/bench/injury D2 protractor correctly positioned on diagram or appropriate trigonometric relationship for marked lengths D3 keep A and B constant D4 use a (top-pan) balance to measure A and B D5 correct positioning of light gates to determine t, e.g. two light gates either end of distance d, connected to a timer or correct position of video camera with timer in frame of the video to determine t D6 method to release cylinder/cube, e.g. cube held by set square, set square moved to release cube. D7 reasoned method to keep d constant as θ changes, e.g. (when measuring d by position of cylinder) adjust the length of the string or adjust the position of vertical marks or adjust the position of the vertical rule or initial position of the cube or (when measuring d by position of cube) use fixed marks on the plane or ruler placed on the plane with d measured between the marks D8 method to increase t for cylinder to fall, e.g. use large d to increase t D9 repeat measurements of t for the same θ and average t D10 relationship valid if a straight line is produced (not passing through the origin) Do not accept straight line passing through the origin. © UCLES 2023 Page 8 of 12

2(a) V  1 gradient = Rln  V   0 2(b) 1 C / 10–4 F T / s 0.89 or 0.892 13.7  0.8 1.3 or 1.32 20.4  0.7 1.6 or 1.58 24.3  0.6 1.0 or 1.03 16.1  0.8 1.2 or 1.18 18.3  0.7 2.1 or 2.08 31.5  0.6 Values of C and T correct as shown above. Absolute uncertainties in T correct as shown above. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in T plotted correctly. 1 All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2023 Page 9 of 12 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Points must be balanced. Line must pass between (1.42, 22.0) and (1.45, 22.0) and between (1.95, 30.0) and (2.00, 30.0) Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient of worst acceptable line determined with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(d) – 0.69 or – 0.693 and  0.06 1 2(e)(i) R determined using gradient and R given to 2 or 3 significant figures. 1 gradient (c)(iii) R   V  (d) ln  V  0  R correctly determined using gradient and SI unit with correct power of ten for R (e.g. ). 1 © UCLES 2023 Page 10 of 12 2(e)(ii) Percentage uncertainty in R with method shown. 1   V   ln   percentage uncertainty      V 0     gradient 100   V  gradient  ln    V    0   or Correct substitution for max/min methods. © UCLES 2023 Page 11 of 12 2(f) C determined to a minimum of 2 significant figures from (c)(iii) or (d) and (e)(i) with correct substitution. 1 T 60.0 C   gradient gradient or T 60.0 C   V  (e)(i)(d) Rln  V  0  Absolute uncertainty in C determined with correct method used: 1 Using gradient to determine C:  gradient C  C  gradient  Allow using R to determine C:   V   ln   C      V 0     (e)(ii) C   V  100  ln    V    0   © UCLES 2023 Page 12 of 12

1 Defining the problem. f is the independent variable and E is the dependent variable or vary f and measure E 1 keep V and R constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • coils C and D placed with their axes on a straight line • separate workable circuit for coil D • (a.c.) voltmeter or oscilloscope connected across coil D (Do not accept a power supply connected to coil D.) a.c. power supply/signal generator connected to coil C 1 workable circuit for coil C with power supply and (a.c.) voltmeter/oscilloscope in parallel with resistor and coil C 1 method to determine f, e.g. read from signal generator or use of oscilloscope 1 Method of Analysis plot a graph of lg E against lg f or equivalent (e.g. ln E against ln f) 1 q = gradient 1 R 1 p= 10y-intercept V R (for ln E against ln f: p= ey-intercept) V © UCLES 2023 Page 5 of 10 1 Additional detail including safety considerations 6 D1 precaution (to prevent burns) from hot coils/hot resistor, e.g. use gloves to handle hot coil/resistor, switch off circuit and wait for hot coil/resistor to cool D2 keep the number of turns on each coil constant D3 keep distance between the coils constant D4 workable circuit diagram to determine R. e.g. circuit with ammeter connected in series and voltmeter in parallel with resistor or resistor connected to ohmmeter only D5 determination of resistance R: potential difference across R ÷ current in R or use ohmmeter to measure R D6 method to keep distance between the coils constant, e.g. fix/clamp coils to bench D7 method to determine f from oscilloscope, e.g. period T = time-base  horizontal distance and f = 1 / T D8 method to determine V or E from oscilloscope, e.g. V or E = y-gain  vertical distance D9 method to increase E e.g. use iron core, place coils closer, increase V, decrease R pV  D10 relationship valid if a straight line is produced (passing through log )    R  Do not accept line passing through the origin. © UCLES 2023 Page 6 of 10

2(a) 2g 1 gradient = uZ 2g y-intercept = u 2(b) 1 1 1 − /cm 2 h 0.218 or 0.2182 0.237 or 0.2370 0.248 or 0.2485 0.262 or 0.2617 0.282 or 0.2817 0.313 or 0.3131 Values correct as shown above. 1 1 Uncertainties in from ± 0.001 to ± 0.003. h 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. h All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2023 Page 7 of 10 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Points must be balanced. Line must pass between (605, 0.230) and (615, 0.230) and between (845, 0.300) and (855, 0.300) Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. © UCLES 2023 Page 8 of 10 2(d)(i) u determined using y-intercept and u and Z given to 2, 3 or 4 significant figures. 1 2981 44.29 u = = y-intercept (c)(iv) Z determined using gradient with method shown and u and Z given with SI units with appropriate powers of ten. 1 2981 44.29 y-intercept (c)(iv) Z = = or Z = = u gradient u(c)(iii) gradient (c)(iii) 2(d)(ii) Percentage uncertainty in Z with method shown. 1 y-intercept gradient percentage uncertainty in Z = + 100  y-intercept gradient  or Correct substitution for u and u gradient percentage uncertainty in Z = + 100  u gradient  or Correct substitution for max/min methods. © UCLES 2023 Page 9 of 10 2(e) M determined to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution. 1  1  −y-intercept    25  M = gradient or uZ uZ M = −Z = −Z 2gh 221.5 © UCLES 2023 Page 10 of 10

1 Defining the problem V is the independent variable and z is the dependent variable or vary V and measure z 1 keep h constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • pulley supported by stand • stand placed on surface/bench/floor • minimum of two labels from stand, beaker, oil, surface/bench/floor, pulley, string use (metre) rule to measure h or (metre) rule correctly positioned with h marked on diagram 1 use measuring cylinder to measure V 1 timing method to measure time t of fall of beaker to determine z 1 e.g. use timer/stopwatch or use light gate(s) connected to a timer/data logger © UCLES 2023 Page 5 of 11 1 Method of Analysis 1 1 1 1 1 plot a graph of against or equivalent (e.g. against ) z2 V V z2 Do not accept logarithms. 1 1 b= 2hy-intercept 1 1 Mgradient gradient (for against :b= or b=− ) V z2 ah 2hy-intercept M 2My-intercept 1 a= or a= bhgradient gradient 1 1 (for against : a=−2My-intercept) V z2 © UCLES 2023 Page 6 of 11 1 Additional detail including safety considerations 6 D1 precaution linked to oil spillage, e.g. use of cushion/sand box/tray for falling beaker to land or use of bungs/lids on beakers or use foam on bench/floor or use foam to prevent rising beaker hitting pulley D2 precaution linked to oil contact with skin e.g. use gloves to avoid contact with oil D3 keep M constant D4 use a (top-pan) balance to measure M D5 method to keep h constant e.g. use a fiducial mark to release the beaker from the same position or release from the same position on the clamped rule each time D6 equation to determine z for method used, e.g. for timing h, z = 2h / t or for one light gate, z = L / t where L is the length of the interrupted beam or for two light gates, z = distance between light gates / t Do not accept h / t. D7 additional detail on diagram to measure h, e.g. clamp (metre) rule with stand on surface or use of set squares positioned on the surface to side of rule or spirit level positioned to side of rule D8 use large value of h to increase time of fall of beaker D9 repeat measurements of z for the same V and average z  1  D10 relationship valid if a straight line is produced (passing through )   2bh Do not accept line passing through the origin. © UCLES 2023 Page 7 of 11

2(a) t 1 gradient = − R y-intercept = ln I R 0 2(b) 1 1 / C / 104 F–1 ln (V / V) 0.91 or 0.909 0.896 or 0.8961 0.76 or 0.758 1.012 or 1.0116 0.63 or 0.633 1.115 or 1.1151 0.61 or 0.606 1.131 or 1.1314 0.48 or 0.482 1.253 or 1.2528 0.36 or 0.357 1.348 or 1.3481 Values correct as shown above. Uncertainties in ln (V / V) from ± 0.021 or ± 0.020 to ± 0.010 or ± 0.013 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in ln (V / V) plotted correctly. 1 All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2023 Page 8 of 11 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Points must be balanced. Line must pass between (0.820, 0.95) and (0.845, 0.95) and between (0.400, 1.30) and (0.425, 1.30) Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient must be negative. Gradient determined of worst acceptable line. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and x into y = mx + c. 1 © UCLES 2023 Page 9 of 11 2(d)(i) R determined using gradient. 1 30.0 30.0 R =− = gradient (c)(iii) I determined using y-intercept with method shown. 1 0 ey-intercept e(c)(iv) I = = 0 R (d)(i) R and I determined correctly using gradient and y-intercept 1 0 and R and I given to 2 or 3 significant figures 0 and R and I given with SI units with appropriate powers of ten. 0 Units: R:  or s F-1 I : A or V F s–1 or V  –1 0 2(d)(ii) Percentage uncertainty in R with method shown. 1 t gradient percentage uncertainty in R= + 100  t gradient  or Correct substitution for max/min methods. © UCLES 2023 Page 10 of 11 2(e) C determined to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitutions. 1 gradient gradient C = or C =− lnV −y-intercept y-intercept−lnV or t t C =− or C = R(lnV −lnI R) R(lnI R −lnV) 0 0 © UCLES 2023 Page 11 of 11

1 Defining the problem. f is the independent variable and E is the dependent variable or vary f and measure E 1 keep V and R constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • coils C and D placed with their axes on a straight line • separate workable circuit for coil D • (a.c.) voltmeter or oscilloscope connected across coil D (Do not accept a power supply connected to coil D.) a.c. power supply/signal generator connected to coil C 1 workable circuit for coil C with power supply and (a.c.) voltmeter/oscilloscope in parallel with resistor and coil C 1 method to determine f, e.g. read from signal generator or use of oscilloscope 1 Method of Analysis plot a graph of lg E against lg f or equivalent (e.g. ln E against ln f) 1 q = gradient 1 R 1 p= 10y-intercept V R (for ln E against ln f: p= ey-intercept) V © UCLES 2023 Page 5 of 10 1 Additional detail including safety considerations 6 D1 precaution (to prevent burns) from hot coils/hot resistor, e.g. use gloves to handle hot coil/resistor, switch off circuit and wait for hot coil/resistor to cool D2 keep the number of turns on each coil constant D3 keep distance between the coils constant D4 workable circuit diagram to determine R. e.g. circuit with ammeter connected in series and voltmeter in parallel with resistor or resistor connected to ohmmeter only D5 determination of resistance R: potential difference across R ÷ current in R or use ohmmeter to measure R D6 method to keep distance between the coils constant, e.g. fix/clamp coils to bench D7 method to determine f from oscilloscope, e.g. period T = time-base  horizontal distance and f = 1 / T D8 method to determine V or E from oscilloscope, e.g. V or E = y-gain  vertical distance D9 method to increase E e.g. use iron core, place coils closer, increase V, decrease R pV  D10 relationship valid if a straight line is produced (passing through log )    R  Do not accept line passing through the origin. © UCLES 2023 Page 6 of 10

2(a) 2g 1 gradient = uZ 2g y-intercept = u 2(b) 1 1 1 − /cm 2 h 0.218 or 0.2182 0.237 or 0.2370 0.248 or 0.2485 0.262 or 0.2617 0.282 or 0.2817 0.313 or 0.3131 Values correct as shown above. 1 1 Uncertainties in from ± 0.001 to ± 0.003. h 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. h All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © UCLES 2023 Page 7 of 10 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Points must be balanced. Line must pass between (605, 0.230) and (615, 0.230) and between (845, 0.300) and (855, 0.300) Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. © UCLES 2023 Page 8 of 10 2(d)(i) u determined using y-intercept and u and Z given to 2, 3 or 4 significant figures. 1 2981 44.29 u = = y-intercept (c)(iv) Z determined using gradient with method shown and u and Z given with SI units with appropriate powers of ten. 1 2981 44.29 y-intercept (c)(iv) Z = = or Z = = u gradient u(c)(iii) gradient (c)(iii) 2(d)(ii) Percentage uncertainty in Z with method shown. 1 y-intercept gradient percentage uncertainty in Z = + 100  y-intercept gradient  or Correct substitution for u and u gradient percentage uncertainty in Z = + 100  u gradient  or Correct substitution for max/min methods. © UCLES 2023 Page 9 of 10 2(e) M determined to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution. 1  1  −y-intercept    25  M = gradient or uZ uZ M = −Z = −Z 2gh 221.5 © UCLES 2023 Page 10 of 10

1 Defining the problem L is the independent variable and f is the dependent variable, or vary L and measure f. 1 Keep  constant 1 Methods of data collection Labelled diagram of workable experiment including: 1 • rod supported by string / elastic bands from a clamp • clamp attached to stand, with stand on bench • two labels from stand, clamp, hammer, microphone, rod, string. Diagram showing labelled microphone connected to labelled oscilloscope. 1 Method to measure L, e.g. use a metre rule 1 Method to measure mass (m) (of metal rod), e.g. use a (top-pan) balance 1 Method of Analysis 1 1 Plots a graph of log f against log L or equivalent e.g. log f against log L n = − gradient 1 1 (for log f against log : n = gradient) L E =4102y-intercept 1 1 (for lg f vs lg : E =4102y-intercept) L (for ln f against ln L; E =4e2y-intercept) © Cambridge University Press & Assessment 2024 Page 5 of 9 1 Additional detail including safety considerations 6 Any six from: D1 Precaution linked to falling rod, e.g. sand tray / cushion (in case rod falls) OR gently hit rod prevent rod falling D2 Method to determine area of rod (A) e.g. measure diameter (d) of rod using a micrometer / calipers D3 Repeat measurements of diameter along the length of rod / around the rod and average diameter m d2 D4 Method to determine ρ from experimental method, e.g. = and A = AL 4 4m or = d2L d m or r = and =

2 r2L D5 Perform experiment in a quiet room D6 Reasoned method to prevent rod hitting microphone, e.g. have a gap between rod and microphone / gently hit rod or method to obtain measurable signal from the microphone, e.g. use a cone to increase the sound detected by the microphone D7 Method to determine frequency from oscilloscope, e.g. T = time-base  (horizontal) length (of one wave) and f = 1/T D8 Method to reduce uncertainties e.g. use large values of L to reduce (percentage) uncertainty in L or adjust time-base to display as few waves as possible or Z waves on oscilloscope and divide time by Z or wait for the wave(form) / frequency to stabilise (and reach resonance) D9 Repeat measurements of f for each value of L and average f © Cambridge University Press & Assessment 2024 Page 6 of 9 2(a) 3 1 Gradient = E 4Z y-intercept = E 2(b) 1 1 / A−1 I 4440 or 4444 5410 or 5405 6250 or 6250 7140 or 7143 8000 or 8000 8700 or 8696 1 1 Uncertainties in I From  90–110 to  360–400 © Cambridge University Press & Assessment 2024 Page 7 of 9 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. I All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top plot to bottom plot. Points must be balanced. Line must pass between (1.8, 5000) and (2.1, 5000) and between (7.2, 8500) and (7.5, 8500) Worst acceptable line drawn. 1 Steepest or shallowest possible line that passes through all the error bars. All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x; distance between data points must be greater than 1 half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x; 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and x into y = mx + c 1 y-intercept of worst acceptable line determined by substitution into y = mx + c 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line, or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) © Cambridge University Press & Assessment 2024 Page 8 of 9 2(d)(i) E determined using gradient and 1 E and Z given to 2, 3 or 4 sf. 3 E = gradient Z determined using y-intercept and 1 E and Z given with SI units with correct powers of ten Ey-intercept 3  y-intercept Z = or Z = 4 4  gradient Unit of E: V or A  Unit of Z:  2(d)(ii) Percentage uncertainty in Z with method shown. 1  gradient y-intercept %uncertainty= +   gradient y-intercept  or Correct substitution for max/min methods. 2(e) R determined to a minimum of 2sf from (c)(iii) and (c)(iv) or (d)(i) with correct substitution and correct powers of ten. 1 0.1 mA = 0.1  10–3 A and 1 − y-intercept 0.1010−3 R = or gradient E 4Z R = − 30.1010−3 3 © Cambridge University Press & Assessment 2024 Page 9 of 9

1 Defining the problem t is the independent variable and T is the dependent variable or vary t and measure T 1 C C keep T constant 1 R Methods of data collection labelled diagram of workable experiment including: 1  solid cylinder cooling  insulation surrounding all of the cylinder  thermometer touching cylinder inside insulation  insulation and thermometer labelled method to heat the cylinder uniformly, e.g. place in oven/immerse in hot water or diagram showing cylinder in oven or hot 1 water method to determine time t, e.g. stopwatch or temperature sensor connected to a data logger 1 method to measure L e.g. use a ruler/calipers/micrometer 1 and method to measure d e.g. use calipers/micrometer Method of Analysis plot a graph of ln (T – T ) against t or equivalent 1 C R mcgradient 1 U  A Z = ey-intercept 1 © Cambridge University Press & Assessment 2024 Page 5 of 9 1 Additional detail including safety considerations 6 D1 precaution to prevent burns or use of hot cylinder / oven / hot water e.g. use of gloves, use of tongs D2 keep thickness of the insulating material constant (for each T ) C D3 method to measure m, e.g. use a (top-pan) balance D4 for water bath/oven methods, wait for initial temperature of the cylinder to become uniform or constant throughout the cylinder d2 d2  D5 (surface) AdL or dL2 

2 4   D6 repeat measurements of d along the length of the cylinder / in different directions and determine the average value of d D7 description of how c is determined from a separate experiment by heating the cylinder using electrical heater and E c  m D8 method of determining energy supplied to electrical heater to determine c, e.g. use of joulemeter for E or electrical method using ammeter and voltmeter to determine IVt D9 use several temperature sensors and determine the average T C D10 relationship valid if a straight line is produced (with y-intercept = ln Z) Do not accept line passing through the origin. © Cambridge University Press & Assessment 2024 Page 6 of 9 2(a) 1 1 gradient =  kf s 1 y-intercept = f s 2(b) 1 1 v / ms–1 / 10–3 Hz–1 f 3.5  0.4 1.118 or 1.1183 6.3  0.4 1.110 or 1.1096 8.7  0.5 1.101 or 1.1013 11.4  0.5 1.092 or 1.0919 13.9  0.6 1.083 or 1.0827 16.2  0.6 1.074 or 1.0739 1 Values of v and correct as shown above. f Uncertainties in v correct as shown above. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in v plotted correctly. 1 All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © Cambridge University Press & Assessment 2024 Page 7 of 9 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Line must pass between (14.5, 1.080) and (14.9, 1.080) and between (4.5, 1.115) and (4.8, 1.115). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Gradient must be negative. Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and y into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. © Cambridge University Press & Assessment 2024 Page 8 of 9 2(d)(i) f determined using y-intercept and f given to 2, 3 or 4 significant figures and k given to 2 or 3 significant figures. 1 s s 1 f  s y-intercept k determined using gradient with method shown and f and k given with SI units with appropriate powers of ten. 1 s y-intercept 1 k  or k  gradient gradientf s Units of f : Hz s Units of k: m s–1 2(d)(ii) Percentage uncertainty in k with method shown. 1 y-intercept gradient percentage uncertainty  100  y-intercept gradient  or correct substitution for max/min methods. 2(e) v determined (non-zero) to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution. 1 1 y-intercept f v  gradient or kf v k  s f © Cambridge University Press & Assessment 2024 Page 9 of 9

1 Defining the problem t is the independent variable and s is the dependent variable or vary t and measure s 1 keep k constant 1 Methods of data collection labelled diagram of workable experiment including: 1  spring connected to magnet  vertical rule parallel to spring to determine s  rule held in position by a stand  stand resting on the bench  rule labelled and at least one other label from stand, clamp, card, (magnetic) sheet, (cylindrical) magnet, spring s = (new) length/position of spring – original length/position of spring 1 use a micrometer to measure t 1 measure B using a (calibrated) Hall probe and rotate probe until maximum value 1 or measure B using Hall probe first in one direction, then in the opposite direction and average © Cambridge University Press & Assessment 2024 Page 5 of 10 1 Method of Analysis 1 1 plot a graph of s against or equivalent t (allow lg s against lg t) relationship valid if a straight line that passes through the origin is produced 1 (for lg s against lg t: relationship valid if a straight line with gradient 1) k 1 Z  gradient ALB k10y-intercept (for lg s against lg t: Z  ) ALB © Cambridge University Press & Assessment 2024 Page 6 of 10 1 Additional detail including safety considerations 6 D1 precaution related to spring and/or magnet hitting eyes, e.g. use of goggles/use of safety screen around experiment D2 keep A, L and B constant D3 use a rule to measure L D4 micrometer/calipers to measure diameter d of the magnet and A = d2 / 4 D5 description of method to determine k, e.g. add mass to spring and k = mg / extension or use newton meter to measure force applied to spring and k = force / extension or take several readings of force and extension, plot a force–extension graph and k = gradient D6 (magnetic) sheet clamped to bench D7 use pointer(s)/marker(s) on the spring to read off values from the rule D8 method to use video recorder and replay to determine maximum length of the spring or increase s or force gradually/slowly until magnet (just) leaves the card D9 repeat measurements of t in different positions on the card and average t or repeat measurements of s for each value of t and average s D10 method to check that the spring has not exceeded the elastic limit D11 use of non-magnetic stand or named non-magnetic material for stand, e.g. wood © Cambridge University Press & Assessment 2024 Page 7 of 10

2(a) gradient = n 1 2 y-intercept = lg C 2(b) 1 lg (L / cm) lg (T / 10–5 s) 1.73 or 1.732 1.38 or 1.380  0.02 1.85 or 1.845 1.51 or 1.505  0.01 1.93 or 1.934 1.59 or 1.591  0.01 2.033 or 2.0334 1.69 or 1.690  0.02 2.146 or 2.1461 1.81 or 1.806  0.01 2.223 or 2.2227 1.87 or 1.869  0.01 Values of lg (L/ cm) and lg (T/ 10–5 s) correct as shown above. Uncertainties in lg (T/ 10–5 s) correct as shown above. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in lg T plotted correctly. 1 All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © Cambridge University Press & Assessment 2024 Page 8 of 10 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Line must pass between (1.780, 1.45) and (1.800, 1.45) and between (2.085, 1.75) and (2.100, 1.75) Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and x into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. © Cambridge University Press & Assessment 2024 Page 9 of 10 2(d) Value of n determined using gradient (n = gradient) and C given to 2 or 3 significant figures. 1 Value of C determined using y-intercept with method shown. 1 2 C  10y-intercept Absolute uncertainties in n and C. 1 uncertainty in n = uncertainty in gradient and 2 2  2 10minworst y-intercept 10maxworst y-intercept C=C or C = 10worst y-intercept 2 Clear method must be shown with C correctly evaluated. 2(e) Value of L determined (non-zero) to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d) with correct substitution 1 and correct power of ten. Units of T and either C or y-intercept must be consistent. 2 logT log C log10y-intercept logL  n n log10y-intercept L10 n or TC L n 2 © Cambridge University Press & Assessment 2024 Page 10 of 10

1 Defining the problem t is the independent variable and T is the dependent variable or vary t and measure T 1 C C keep T constant 1 R Methods of data collection labelled diagram of workable experiment including: 1  solid cylinder cooling  insulation surrounding all of the cylinder  thermometer touching cylinder inside insulation  insulation and thermometer labelled method to heat the cylinder uniformly, e.g. place in oven/immerse in hot water or diagram showing cylinder in oven or hot 1 water method to determine time t, e.g. stopwatch or temperature sensor connected to a data logger 1 method to measure L e.g. use a ruler/calipers/micrometer 1 and method to measure d e.g. use calipers/micrometer Method of Analysis plot a graph of ln (T – T ) against t or equivalent 1 C R mcgradient 1 U  A Z = ey-intercept 1 © Cambridge University Press & Assessment 2024 Page 5 of 9 1 Additional detail including safety considerations 6 D1 precaution to prevent burns or use of hot cylinder / oven / hot water e.g. use of gloves, use of tongs D2 keep thickness of the insulating material constant (for each T ) C D3 method to measure m, e.g. use a (top-pan) balance D4 for water bath/oven methods, wait for initial temperature of the cylinder to become uniform or constant throughout the cylinder d2 d2  D5 (surface) AdL or dL2 

2 4   D6 repeat measurements of d along the length of the cylinder / in different directions and determine the average value of d D7 description of how c is determined from a separate experiment by heating the cylinder using electrical heater and E c  m D8 method of determining energy supplied to electrical heater to determine c, e.g. use of joulemeter for E or electrical method using ammeter and voltmeter to determine IVt D9 use several temperature sensors and determine the average T C D10 relationship valid if a straight line is produced (with y-intercept = ln Z) Do not accept line passing through the origin. © Cambridge University Press & Assessment 2024 Page 6 of 9 2(a) 1 1 gradient =  kf s 1 y-intercept = f s 2(b) 1 1 v / ms–1 / 10–3 Hz–1 f 3.5  0.4 1.118 or 1.1183 6.3  0.4 1.110 or 1.1096 8.7  0.5 1.101 or 1.1013 11.4  0.5 1.092 or 1.0919 13.9  0.6 1.083 or 1.0827 16.2  0.6 1.074 or 1.0739 1 Values of v and correct as shown above. f Uncertainties in v correct as shown above. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in v plotted correctly. 1 All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © Cambridge University Press & Assessment 2024 Page 7 of 9 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Line must pass between (14.5, 1.080) and (14.9, 1.080) and between (4.5, 1.115) and (4.8, 1.115). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Gradient must be negative. Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and y into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. © Cambridge University Press & Assessment 2024 Page 8 of 9 2(d)(i) f determined using y-intercept and f given to 2, 3 or 4 significant figures and k given to 2 or 3 significant figures. 1 s s 1 f  s y-intercept k determined using gradient with method shown and f and k given with SI units with appropriate powers of ten. 1 s y-intercept 1 k  or k  gradient gradientf s Units of f : Hz s Units of k: m s–1 2(d)(ii) Percentage uncertainty in k with method shown. 1 y-intercept gradient percentage uncertainty  100  y-intercept gradient  or correct substitution for max/min methods. 2(e) v determined (non-zero) to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution. 1 1 y-intercept f v  gradient or kf v k  s f © Cambridge University Press & Assessment 2024 Page 9 of 9

1 Defining the problem s is the independent variable and v is the dependent variable or vary s and measure v 1 keep D constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • light gate positioned at P • light gate connected to timer / data logger • labels for light gate and P and data logger / timer and at least one other label from block, magnet(s), trolley, s and D measure D with a rule(r) and measure L with a rule(r) or calipers 1 description to determine v at P, e.g. (measure length of) card to interrupt beam 1 method to measure s, e.g. use calipers 1 © Cambridge University Press & Assessment 2024 Page 5 of 11 1 Method of Analysis 1 1 1 plot a graph of v2 against or equivalent (e.g. against v2) s4 s4 Do not accept logarithms. mgradient 1 K = 2DA2B2L2 m 1 (or K = for against v2) 2DA2B2L2gradient s4 my-intercept 1 Q=− 2D my-intercept 1 (or Q =KA2B2L2y-intercept or Q = for against v2) 2Dgradient s4 © Cambridge University Press & Assessment 2024 Page 6 of 11 1 Additional detail including safety considerations 6 D1 method to stop the trolley (after passing point P), e.g. labelled block / buffer / cushion drawn after P or place a block / buffer / cushion after P to stop the trolley D2 keep L, A, m and B constant d2 D3 use micrometer / calipers to measure diameter (d) of the magnet and A = 4 D4 method to secure block to bench, e.g. clamp block to bench or (heavy) mass on top of block or method to secure magnets, e.g. use glue to stick magnets to trolley / block D5 method to increase the accuracy of measuring s or D, e.g. use a marker to left of the trolley D6 measure B using a (calibrated) Hall probe and adjust / rotate probe until maximum value or measure B using Hall probe first in one direction, then in the opposite direction and average D7 use a (top-pan) balance to measure m D8 use of strong magnets to increase v D9 repeat measurements of v for each value of s and average v  2DQ D10 relationship valid if a straight line is produced (passing through −  )  m  Do not accept line passing through the origin. © Cambridge University Press & Assessment 2024 Page 7 of 11

2(a) 3 1 gradient = 3E −E s 2Z y-intercept = 3E −E s 2(b) 1 1 / A−1 I 5150 or 5155 5560 or 5556 5810 or 5814 6250 6670 or 6667 6940 or 6944 Values correct as shown above. 1 1 Uncertainties in from 50 or 60 to 90 or 100. I 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. I All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © Cambridge University Press & Assessment 2024 Page 8 of 11 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Points must be balanced. Line must pass between (1.63, 5400) and (1.67, 5400) and between (2.58, 6800) and (2.62, 6800). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and x into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. © Cambridge University Press & Assessment 2024 Page 9 of 11 2(d)(i) E determined using gradient 1 and E and Z given to 2 or 3 or 4 significant figures. 1 3  3+gradientE 1 E E =  +E = s = + s 3gradient s  3gradient gradient 3 1 E = +0.733 gradient Z determined using y-intercept 1 and E and Z given with SI units with correct powers of ten. (3E −E )y-intercept 3y-intercept Z = s or Z = 2 2gradient Unit of E: V Unit of Z:  2(d)(ii) Absolute uncertainty in E with method shown. 1  gradient 1  0.05 uncertainty=  +  gradient gradient 3 or correct substitution for max/min methods. © Cambridge University Press & Assessment 2024 Page 10 of 11 2(e) Value of R determined to a minimum of two significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution and 1 correct use of power of ten. 1 −y-intercept 25010−6 R = gradient or 1 2Z R = − gradient25010−6 3 or 3E −2.2 2Z R = − 325010−6 3 © Cambridge University Press & Assessment 2024 Page 11 of 11

1 Defining the problem L is the independent variable and  or temperature change/increase is the dependent variable 1 or vary L and measure  or temperature change/increase keep t constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • oil in a beaker/container (on a bench) • coil fully submerged in oil • (bulb of) thermometer in the oil • at least three labels from thermometer, coil or resistance wire, oil, beaker/container, clamp/stand, bench Do not accept other heating sources. method to determine V – diagram of workable circuit including: 1 • power supply connected to wire • voltmeter positioned to measure V across the coil measure the initial and final temperature and find the difference  1 method to determine t, e.g. use stopwatch/timer 1 and method to determine L e.g. use a rule(r) to measure L / length of wire or e.g. using number of turns and measure the diameter of the coil with rule(r) / calipers © Cambridge University Press & Assessment 2024 Page 5 of 10 1 Method of Analysis 1 1 1 plot a graph of  against or equivalent, e.g. against  L L Do not accept logarithms. 1 1 1 for  against for against  L L AtV2 AtV2gradient K = K = mgradient m 1 1 1 for  against for against  L L Z =−mKy-intercept Z = AtV2y-intercept or AtV2y-intercept Z =− gradient © Cambridge University Press & Assessment 2024 Page 6 of 10 1 Additional detail including safety considerations 6 D1 precaution linked to hot oil / beaker / wire, e.g. use of gloves to prevent burns from oil or precaution linked to spillage of oil, e.g. perform experiment in a tray D2 keep A and m and V constant d2 D3 use a micrometer to measure the diameter (d) of the wire and A = 4 D4 repeat measurements of d along the wire and average D5 method to reduce heat loss e.g. add insulation around the container / add a lid to the container D6 method to keep V constant, e.g. adjust / change a variable resistor / power supply to keep V or voltmeter reading constant D7 use a balance to determine the mass of the oil and mass of oil = mass of (beaker + oil) − mass of beaker or place beaker on balance and zero balance, then add oil and read balance D8 stir the oil for uniform temperature or keep the initial temperature (of oil) constant D9 repeat the experiment for the same value of L and average  / average temperature change  Z  D10 relationship valid if a straight line is produced (passing through  − )  mK  Do not accept line passing through the origin. D11 method to determine L accurately, e.g. measure length of unwound coil © Cambridge University Press & Assessment 2024 Page 7 of 10

2(a) gradient = Bn2 1 y-intercept = −B 2(b) 1 d2 / cm2 615 or 615.0 458 or 458.0 292 or 292.4 216 or 216.1 166 or 166.4 139 or 139.2 Values correct as shown above. Uncertainties in d2 decreasing from 10 to 4 or 5. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in d2 plotted correctly. 1 All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Points must be balanced. Line must pass between (1.90, 250) and (2.00, 250) and between (3.55, 500) and (3.65, 500). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. © Cambridge University Press & Assessment 2024 Page 8 of 10 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and x into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. 2(d)(i) B determined using y-intercept (B = – y-intercept) and B and n given to 2 or 3 or 4 significant figures. 1 n determined using gradient 1 and B and n given with SI units with correct powers of ten. gradient gradient n= or n= B −y-intercept Unit for B: cm2 No unit for n. © Cambridge University Press & Assessment 2024 Page 9 of 10 2(d)(ii) Percentage uncertainty in n determined with method shown. 1 1y-intercept gradient percentage uncertainty=  + 100 2 y-intercept gradient  or correct substitution for max/min methods. 2(e)  determined to a minimum of two significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution and correct 1 power of ten. gradient =sin−1 −y-intercept+900 or n2B =sin−1 B+900 © Cambridge University Press & Assessment 2024 Page 10 of 10

1 Defining the problem s is the independent variable and v is the dependent variable or vary s and measure v 1 keep D constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • light gate positioned at P • light gate connected to timer / data logger • labels for light gate and P and data logger / timer and at least one other label from block, magnet(s), trolley, s and D measure D with a rule(r) and measure L with a rule(r) or calipers 1 description to determine v at P, e.g. (measure length of) card to interrupt beam 1 method to measure s, e.g. use calipers 1 © Cambridge University Press & Assessment 2024 Page 5 of 11 1 Method of Analysis 1 1 1 plot a graph of v2 against or equivalent (e.g. against v2) s4 s4 Do not accept logarithms. mgradient 1 K = 2DA2B2L2 m 1 (or K = for against v2) 2DA2B2L2gradient s4 my-intercept 1 Q=− 2D my-intercept 1 (or Q =KA2B2L2y-intercept or Q = for against v2) 2Dgradient s4 © Cambridge University Press & Assessment 2024 Page 6 of 11 1 Additional detail including safety considerations 6 D1 method to stop the trolley (after passing point P), e.g. labelled block / buffer / cushion drawn after P or place a block / buffer / cushion after P to stop the trolley D2 keep L, A, m and B constant d2 D3 use micrometer / calipers to measure diameter (d) of the magnet and A = 4 D4 method to secure block to bench, e.g. clamp block to bench or (heavy) mass on top of block or method to secure magnets, e.g. use glue to stick magnets to trolley / block D5 method to increase the accuracy of measuring s or D, e.g. use a marker to left of the trolley D6 measure B using a (calibrated) Hall probe and adjust / rotate probe until maximum value or measure B using Hall probe first in one direction, then in the opposite direction and average D7 use a (top-pan) balance to measure m D8 use of strong magnets to increase v D9 repeat measurements of v for each value of s and average v  2DQ D10 relationship valid if a straight line is produced (passing through −  )  m  Do not accept line passing through the origin. © Cambridge University Press & Assessment 2024 Page 7 of 11

2(a) 3 1 gradient = 3E −E s 2Z y-intercept = 3E −E s 2(b) 1 1 / A−1 I 5150 or 5155 5560 or 5556 5810 or 5814 6250 6670 or 6667 6940 or 6944 Values correct as shown above. 1 1 Uncertainties in from 50 or 60 to 90 or 100. I 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. I All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © Cambridge University Press & Assessment 2024 Page 8 of 11 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top point to bottom point. Points must be balanced. Line must pass between (1.63, 5400) and (1.67, 5400) and between (2.58, 6800) and (2.62, 6800). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and x into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. © Cambridge University Press & Assessment 2024 Page 9 of 11 2(d)(i) E determined using gradient 1 and E and Z given to 2 or 3 or 4 significant figures. 1 3  3+gradientE 1 E E =  +E = s = + s 3gradient s  3gradient gradient 3 1 E = +0.733 gradient Z determined using y-intercept 1 and E and Z given with SI units with correct powers of ten. (3E −E )y-intercept 3y-intercept Z = s or Z = 2 2gradient Unit of E: V Unit of Z:  2(d)(ii) Absolute uncertainty in E with method shown. 1  gradient 1  0.05 uncertainty=  +  gradient gradient 3 or correct substitution for max/min methods. © Cambridge University Press & Assessment 2024 Page 10 of 11 2(e) Value of R determined to a minimum of two significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution and 1 correct use of power of ten. 1 −y-intercept 25010−6 R = gradient or 1 2Z R = − gradient25010−6 3 or 3E −2.2 2Z R = − 325010−6 3 © Cambridge University Press & Assessment 2024 Page 11 of 11

1 Defining the problem Vary p and measure B OR p is the independent variable and B is the dependent variable. 1 Keep V constant or potential difference between the ends of each conductor constant. 1 Methods of data collection Labelled diagram of workable experiment including: 1 • conductors in parallel connected in series to power supply and resistor • circuit symbols for (variable) resistor and power supply • X labelled and one other label from L, P, Q, p and q. Voltmeter connected in parallel with conductors (to measure V) and conductors in parallel connected to a power supply. 1 Method to measure L and p and q e.g. use a rule / ruler / calipers. 1 Method to measure B, e.g. use a (calibrated) Hall probe and adjust / rotate probe until maximum value. 1 Method of Analysis 1 1 Plots a graph of B against or equivalent. p Do not accept logarithms. 1 1 1 B against against B p p Lgradient L Y = Y = AV AV gradient © Cambridge University Press & Assessment 2025 Page 7 of 12 1 1 1 1 B against against B p p qy-intercept Z =−qy-intercept Z = gradient OR Lqy-intercept Z = YAV Additional detail including safety considerations 6 Any six from: D1 precaution linked to high current / hot conductors, e.g. use gloves / switch off power supply when not measuring B / between measurements / allow conductors to cool D2 keep A and L and q constant d2 D3 use calipers / micrometer to measure diameter / d of conductor and A= 4 D4 repeat measurements of d in different positions and average d D5 method to determine the position of X in relation to the conductors, e.g. divide L by two to find the midpoint of P / Q and use a set square / protractor / plumb line to mark X OR divide L by two to find the midpoint of P / Q and use a grid to mark X D6 measure B (using Hall probe) first in one direction and then in the opposite direction and average B OR Measure B with current / p.d. in one direction and then in the opposite direction and average B D7 additional detail on measuring p and / or q, e.g. measure to the conductor and add on the radius © Cambridge University Press & Assessment 2025 Page 8 of 12 1 D8 description of method to keep q constant, e.g. tape / adhesive putty to fix conductor Q to the bench OR for vertical methods fix conductor Q in clamp(s) attached to stand(s) to keep q constant D9 method to keep P and Q parallel, e.g. measure the separation (between the conductors) at different points YZAV D10 relationship valid if a straight line is produced (with a y-intercept = ). Lq Do not accept passing through the origin. D11 method to keep V constant, e.g. adjust / change variable resistor / power supply to keep voltmeter reading constant. Question Answer Marks

2(a) 1 1 gradient = − K y-intercept = ln( − ) 0 R © Cambridge University Press & Assessment 2025 Page 9 of 12 2(b) ( – ) / C ln (( – ) / C) R R 56.5  1.0 4.034 or 4.0342  0.018 46.0  1.0 3.829 or 3.8286  0.022 38.5  1.0 3.651 or 3.6507  0.026 31.5  1.0 3.450 or 3.4500  0.032 26.0  1.0 3.258 or 3.2581  0.038 22.5  1.0 3.114 or 3.1135  0.044 Values of ( –  R) / C and ln (( –  R ) / C) 1 Uncertainties in ( –  R) and ln (( –  R ) / C) 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in ln (( – ) / C) plotted correctly. 1 R All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © Cambridge University Press & Assessment 2025 Page 10 of 12 2(c)(ii) Straight line of best fit drawn. 1 Do not accept line from top plot to bottom plot. Line must pass between (31.5, 3.2) and (33.0, 3.2) and between (16.0, 3.7) and (17.0, 3.7) Worst acceptable line drawn. 1 Steepest or shallowest possible line that passes through all the error bars. All error bars must be plotted. 2(c)(iii) Gradient must be negative. 1 Gradient determined with clear substitution of data into y / x; distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data into y / x; 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent unit of time into y = mx + c 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line, or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ecf from false origin method. © Cambridge University Press & Assessment 2025 Page 11 of 12 2(d)(i) K determined using gradient and 1 K and  given to 3 or 4 sf. 0 1 K =− gradient  determined using y-intercept and 1 0 K and  0 given with units with appropriate powers of ten  =ey-intercept +18.5 0 Unit of K: min or minute(s) unit of : C 0 2(d)(ii) Absolute uncertainty determined with clear method shown. 1  = ( emaxy-intercept +19 ) −(ey-intercept +18.5) 0 OR  = (ey-intercept +18.5)− ( eminy-intercept +18 ) 0 OR ( emaxy-intercept +19 ) − ( eminy-intercept +18 )  = 0 2 2(e) t determined to a minimum of 2sf from (c)(iii) and (c)(iv) OR (d)(i) with correct substitution and correct power of ten. 1 ln(25.0−18.5)−y-intercept t = gradient OR t =−K(ln(25.0−18.5)−y-intercept) OR 25.0−18.5 t =−Kln   −18.5   0 © Cambridge University Press & Assessment 2025 Page 12 of 12

1 Defining the problem vary f and measure V or f is the independent variable and V is the dependent variable 1 keep E constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • circuit with a.c. supply • oscilloscope connected in parallel with the resistor • workable circuit • oscilloscope and a.c. supply labelled labelled signal generator or variable frequency power supply connected across the terminals 1 method to determine V or E from oscilloscope, e.g. multiply amplitude / height of wave by y-gain on oscilloscope 1 method to determine f from oscilloscope, e.g. determine period T by multiplying number of divisions in 1 cycle or horizontal 1 distance in 1 cycle by the time base and f = 1/T Method of Analysis 1 1 1 plot a graph of against f or equivalent, e.g. f against V V Allow logarithms e.g. lg V against lg f. relationship valid if a straight line is produced passing through the origin 1 (for lg V against lg f: relationship valid if a straight line is produced with gradient = −1) © Cambridge University Press & Assessment 2025 Page 7 of 12 1 1 1 1 against f f against V V lES lES 1 K = gradient K =  AN2 AN2 gradient ElS (for lg V against lg f: K = 10−y-intercept). AN2 Additional detail including safety considerations 6 D1 precaution linked to hot coil or hot resistor or prevention of burns from coil or resistor, e.g. use gloves / switch off power supply when not measuring V to prevent burns from coil / resistor D2 keep N and A and l and S constant D3 method to keep S constant, e.g. switch off power supply between readings to prevent heating of resistor or to allow resistor to cool d2 D4 method to determine A, e.g. use calipers / micrometer to measure diameter (of coil) / d and A= 4 D5 repeat measurements of diameter d along the length of the coil / in different directions and determine the average value of d D6 method to determine the value of S, e.g. separate circuit diagram showing resistor connected to ohmmeter, or circuit diagram showing resistor connected to a power supply with an ammeter and voltmeter and S = V / I D7 measure l with a ruler / calipers D8 oscilloscope drawn connected across terminals / across signal generator and description to determine E D9 adjust y-gain for maximum amplitude or adjust time base for length of one wave or measure n waves and divide measured time by n © Cambridge University Press & Assessment 2025 Page 8 of 12 1 D10 method to keep E constant, e.g. check p.d. and alter supply or method to keep l constant, e.g. tape coil or method to keep A constant, e.g. wind wire on a cylinder Question Answer Marks

2(a) R 1 gradient = E Z y-intercept = E 2(b) 1 1 / 103 A–1 I 2.20 or 2.198 1.90 or 1.905 1.72 or 1.724 1.57 or 1.575 1.46 or 1.460 1.31 or 1.307 1 Values of / 103 A–1 correct as shown above. I © Cambridge University Press & Assessment 2025 Page 9 of 12 2(b) 1 1 Uncertainties in / 103 A–1 from  0.02 or  0.03 decreasing to  0.01. I 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. I All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Thickness of the line must be less than half a small square. Do not accept line from top point to bottom point. Line must pass between (0.101, 1.40) and (0.104, 1.40) and between (0.189, 2.10) and (0.194, 2.10) Worst acceptable straight line drawn (steepest or shallowest possible line that passes through all the error bars). 1 Thickness of the line must be less than half a small square. All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) © Cambridge University Press & Assessment 2025 Page 10 of 12 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and y into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. 2(d)(i) R determined using gradient and R and Z given to 2 or 3 significant figures. 1 R = gradient  5.8 Z determined using y-intercept and R and Z given with units with appropriate powers of ten. 1 Z = y-intercept  5.8 unit of R:  or V A–1 unit of Z:  or V A–1 2(d)(ii) Percentage uncertainty determined using E = 0.2 (V) with method shown. 1 E gradient R%= + 100  E gradient  or 0.2 gradient R%= + 100 5.8 gradient  © Cambridge University Press & Assessment 2025 Page 11 of 12 2(e) I determined to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution. 1 1 I = gradient +y-intercept 20 or E I =  R  +Z   20  © Cambridge University Press & Assessment 2025 Page 12 of 12

1 Defining the problem vary m and measure v or m is the independent variable and v is the dependent variable 1 keep h constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • axle resting on support(s) (on stands) • supports placed on bench • light gate (connected to timer) positioned at a distance h • light gate labelled and h indicated vertical metre rule clamped to a stand in a position close to block to measure h 1 method to determine v using an interrupt length, e.g. v = length of block / time recorded by the timer 1 method to measure m, e.g. use a (top-pan) balance 1 Method of analysis 1 1 1 plot a graph of against or equivalent v2 m Do not accept logarithms. © Cambridge University Press & Assessment 2025 Page 7 of 12 1 1 1 1 1 1 against against v2 m m v2 1 gradient P = P =− hy-intercept y-intercepth or r2zgradient P = 2Qh 1 1 1 1 1 against against v2 m m v2 r2z r2zy-intercept Q= Q=− 2Phgradient 2 or r2zy-intercept Q= 2gradient Additional detail including safety considerations 6 D1 precaution linked to falling block resulting in damage to block / bench, e.g. use a cushion / sand tray to prevent damage to bench or precaution linked to stands falling, e.g. clamp stand(s) to the bench prevent stand falling D2 keep r and z constant D3 method to determine r e.g. use calipers / ruler to measure diameter d and r = d / 2 D4 measure z with a micrometer / calipers © Cambridge University Press & Assessment 2025 Page 8 of 12 1 D5 set square correctly positioned between rule and bench to ensure that rule to measure h is vertical D6 method to keep h constant by identifying constant initial position of the bottom of the block, e.g. clamped pin / rod to indicate the starting point each time or (fiducial) marker on rule D7 description of method to ensure that axle can rotate, e.g. axle is lubricated at the supports to enable axle to rotate, axle is not fixed at the supports so axle can rotate D8 use a large length of block to reduce (percentage) uncertainty in interrupt time or increase the time light gate is interrupted D9 repeat experiment for the same value of m and determine the average v 1 D10 relationship valid if a straight line is produced (with y-intercept = ). hP Do not accept line passing through the origin. © Cambridge University Press & Assessment 2025 Page 9 of 12

2(a) C 1 gradient = EA 1 y-intercept = E 2(b) 1 1 V/ V / V–1 V 4.25 0.235 or 0.2353 3.70 0.270 or 0.2703 3.25 0.308 or 0.3077 2.90 0.345 or 0.3448 2.60 0.385 or 0.3846 2.35 0.426 or 0.4255 1 Values of V / V and / V–1 correct as shown above. V Uncertainties in V all  0.05 1 and 1 uncertainties in from  0.002 or  0.003 increasing to  0.009. V © Cambridge University Press & Assessment 2025 Page 10 of 12 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. V All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Thickness of the line must be less than half a small square. Do not accept line from top point to bottom point. Line must pass between (2.65, 0.26) and (2.80, 0.26) and between (6.30, 0.40) and (6.50, 0.40). Worst acceptable straight line drawn (steepest or shallowest possible line that passes through all the error bars). 1 Thickness of the line must be less than half a small square. All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. © Cambridge University Press & Assessment 2025 Page 11 of 12 2(d)(i) E determined using y-intercept and E and C given to 2 or 3 significant figures. 1 1 E = y-intercept C determined using gradient and E and C given with correct units with appropriate powers of ten. 1 Agradient C = or C=AEgradient y-intercept unit of E: V unit of C: F 2(d)(ii) Percentage uncertainty determined with method shown. 1 A gradient y-intercept C%= + + 100  A gradient y-intercept  or A gradient E C%= + + 100 with method to determine E shown  A gradient E  2(e) V determined to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution. 1 1 V = 10gradient+y-intercept or EA V = 10C +A © Cambridge University Press & Assessment 2025 Page 12 of 12

1 Defining the problem vary f and measure V or f is the independent variable and V is the dependent variable 1 keep E constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • circuit with a.c. supply • oscilloscope connected in parallel with the resistor • workable circuit • oscilloscope and a.c. supply labelled labelled signal generator or variable frequency power supply connected across the terminals 1 method to determine V or E from oscilloscope, e.g. multiply amplitude / height of wave by y-gain on oscilloscope 1 method to determine f from oscilloscope, e.g. determine period T by multiplying number of divisions in 1 cycle or horizontal 1 distance in 1 cycle by the time base and f = 1/T Method of Analysis 1 1 1 plot a graph of against f or equivalent, e.g. f against V V Allow logarithms e.g. lg V against lg f. relationship valid if a straight line is produced passing through the origin 1 (for lg V against lg f: relationship valid if a straight line is produced with gradient = −1) © Cambridge University Press & Assessment 2025 Page 7 of 12 1 1 1 1 against f f against V V lES lES 1 K = gradient K =  AN2 AN2 gradient ElS (for lg V against lg f: K = 10−y-intercept). AN2 Additional detail including safety considerations 6 D1 precaution linked to hot coil or hot resistor or prevention of burns from coil or resistor, e.g. use gloves / switch off power supply when not measuring V to prevent burns from coil / resistor D2 keep N and A and l and S constant D3 method to keep S constant, e.g. switch off power supply between readings to prevent heating of resistor or to allow resistor to cool d2 D4 method to determine A, e.g. use calipers / micrometer to measure diameter (of coil) / d and A= 4 D5 repeat measurements of diameter d along the length of the coil / in different directions and determine the average value of d D6 method to determine the value of S, e.g. separate circuit diagram showing resistor connected to ohmmeter, or circuit diagram showing resistor connected to a power supply with an ammeter and voltmeter and S = V / I D7 measure l with a ruler / calipers D8 oscilloscope drawn connected across terminals / across signal generator and description to determine E D9 adjust y-gain for maximum amplitude or adjust time base for length of one wave or measure n waves and divide measured time by n © Cambridge University Press & Assessment 2025 Page 8 of 12 1 D10 method to keep E constant, e.g. check p.d. and alter supply or method to keep l constant, e.g. tape coil or method to keep A constant, e.g. wind wire on a cylinder Question Answer Marks

2(a) R 1 gradient = E Z y-intercept = E 2(b) 1 1 / 103 A–1 I 2.20 or 2.198 1.90 or 1.905 1.72 or 1.724 1.57 or 1.575 1.46 or 1.460 1.31 or 1.307 1 Values of / 103 A–1 correct as shown above. I © Cambridge University Press & Assessment 2025 Page 9 of 12 2(b) 1 1 Uncertainties in / 103 A–1 from  0.02 or  0.03 decreasing to  0.01. I 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. 1 1 Error bars in plotted correctly. I All error bars to be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Thickness of the line must be less than half a small square. Do not accept line from top point to bottom point. Line must pass between (0.101, 1.40) and (0.104, 1.40) and between (0.189, 2.10) and (0.194, 2.10) Worst acceptable straight line drawn (steepest or shallowest possible line that passes through all the error bars). 1 Thickness of the line must be less than half a small square. All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) © Cambridge University Press & Assessment 2025 Page 10 of 12 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and y into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. 2(d)(i) R determined using gradient and R and Z given to 2 or 3 significant figures. 1 R = gradient  5.8 Z determined using y-intercept and R and Z given with units with appropriate powers of ten. 1 Z = y-intercept  5.8 unit of R:  or V A–1 unit of Z:  or V A–1 2(d)(ii) Percentage uncertainty determined using E = 0.2 (V) with method shown. 1 E gradient R%= + 100  E gradient  or 0.2 gradient R%= + 100 5.8 gradient  © Cambridge University Press & Assessment 2025 Page 11 of 12 2(e) I determined to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d)(i) with correct substitution. 1 1 I = gradient +y-intercept 20 or E I =  R  +Z   20  © Cambridge University Press & Assessment 2025 Page 12 of 12

1 Defining the problem vary r and measure v or r is the independent variable and v is the dependent variable 1 keep x constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • one end of spring resting against block clamped to bench using G-clamp • light gate positioned at P • light gate connected to timer • apparatus shown on bench • labels for light gate and P and at least one other label from bench, block, stand, spring, ball, timer method to determine r, e.g. use calipers or micrometer to measure diameter d and r = d / 2 1 description of method to determine v, use diameter of ball (to interrupt beam) ÷ measured time at light gate positioned at P 1 instrument to determine x, e.g. rule(r) or calipers 1 Method of Analysis plot a graph of (2 lg v) or (lg v2) against lg r 1 or plot a graph of (lg v) against (lg r) or equivalent, e.g. (ln v) against (ln r) n=−gradient for (2 lg v) or (lg v2) against lg r 1 or n=−2gradient for (lg v) against lg r © Cambridge University Press & Assessment 2025 Page 7 of 11 1 10y-intercept 1 Y = for (2 lg v) or (lg v2) against lg r kx2 or 102y-intercept Y = for (lg v) against lg r kx2 Additional detail including safety considerations 6 D1 precaution to prevent ball leaving bench, e.g. screens around apparatus / cushions on bench (to stop the ball) D2 keep k and  constant D3 description of method to determine k, e.g. add mass to spring and k = mg / extension or use newton meter to measure force applied to spring and k = force / extension or take several readings of force and extension, plot a force–extension graph and k = gradient m D4 description of experimental method to determine , e.g. measure mass of ball using a balance and = 4 r3 3 D5 repeat measurements of diameter or d in different directions and determine the average value of d D6 method to keep x constant, e.g. use a pin / ruler / card to indicate the starting point each time to keep x constant D7 x = original length of spring – compressed length of spring D8 adjust (vertical) position of light gate so that the diameter of (each) ball cuts the beam D9 repeat experiment for the same value of r and determine the average v © Cambridge University Press & Assessment 2025 Page 8 of 11 1 Ykx2  1 Ykx2  D10 relationship valid if a straight line is produced (with y-intercept = lg  or lg ).  2      Do not accept line through the origin. Question Answer Marks

2(a) gradient = H 1 0 y-intercept =  0 2(b) 1 d / 1015s c (1.6 or 1.60)  0.40 (3.47 or 3.467)  0.40 (4.83 or 4.833)  0.40 (6.00 or 6.000)  0.40 (9.50 or 9.500)  0.40 (12.5 or 12.50)  0.40 d Values of correct as shown above. c d 1 Uncertainties in correct as shown above. c © Cambridge University Press & Assessment 2025 Page 9 of 11 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. d 1 Error bars in plotted correctly. c All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Thickness of the line must be less than half a small square. Do not accept line from top point to bottom point. Line must pass between (2.5, 660.0) and (2.9, 660.0) and between (11.2, 676.0) and (11.6, 676.0). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 Thickness of the line must be less than half a small square. All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and x and y into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. © Cambridge University Press & Assessment 2025 Page 10 of 11 2(d)  determined using y-intercept and  given to 3 or 4 significant figures and H given to 2, 3 or 4 significant figures. 1 0 0  = y-intercept 0 H determined using gradient and  and H given with SI units with appropriate powers of ten. 1 0 gradient gradient H = or H = y-intercept  0 Unit of : m, nm, m 0 Unit of H: s−1 2(e) Value of T determined to a minimum of two significant figures from (d) and correct power of ten. 1 1 T = H Absolute uncertainty determined with correct substitution. 1 y-intercept gradient T = + T  y-intercept gradient  or  max  maxy-intercept T = 0 −T or T = −T min gradient  min gradient  or  min  miny-intercept T = 0 −T or T = −T max gradient  max gradient  © Cambridge University Press & Assessment 2025 Page 11 of 11

1 Defining the problem vary v and measure I or v is the independent variable and I is the dependent variable 1 keep A and R constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • fan positioned in line with the turbine so that blades of both fan and turbine overlap • fan on bench • fan labelled and one other label from bench, turbine, blade(s) (of turbine), terminals, L workable circuit diagram showing resistor connected to an ammeter in series with the terminals of the turbine using correct 1 circuit symbols method to vary v, e.g. change speed of fan / change distance between fan and blades / vary current in or p.d. across fan 1 method to determine temperature T, e.g. use a thermometer 1 Method of Analysis plot a graph of I2 against v3 or equivalent (e.g. lg I against lgv or 2 lnI against lnv) 1 relationship valid if a straight line is produced passing through the origin 1 (For lg I against lg v: relationship valid if a straight line with gradient = 1.5 is produced) 2TRgradient 1 Q= AP 2TR102y-intercept (For lg I against lg v: Q = ) AP © Cambridge University Press & Assessment 2025 Page 7 of 11 1 Additional detail including safety considerations 6 D1 precaution with reason linked to (moving) fan blades / turbine blades, e.g. keep away from the fan to avoid (moving) blades or use a screen around the fan / turbine to avoid (moving) blades or precaution with reason linked to prevent air / dust particles in eye, e.g. use goggles to avoid air stream (into eye) D2 clamp turbine / fan to bench D3 keep P and T constant D4 T = t + 273 D5 method to determine A: use a rule(r) / calipers to measure L and A = L2 D6 repeat measurements of L in different positions / different blades and average D7 method to measure v, e.g. use an anemometer or air speed meter or method to measure P, e.g. use a manometer or barometer or pressure gauge D8 wait for steady / constant air flow / movement of blades / current D9 method to determine R, e.g.: separate circuit showing ohmmeter connected to R only or terminals of turbine connected correctly to resistor and ammeter and voltmeter across resistor and R = V / I or separate workable circuit with power supply resistor, ammeter and voltmeter across R and R = V / I D10 method to check temperature / pressure is constant, e.g. measure temperature / pressure several times / before and after © Cambridge University Press & Assessment 2025 Page 8 of 11

2(a) gradient = n 1 2 y-intercept = lg k 2(b) 1 lg (r / 108 m) lg (T / 103 s) 0.152 or 0.1523 (1.72 or 1.716)  0.04 0.270 or 0.2695 (1.91 or 1.908)  0.03 0.377 or 0.3766 (2.08 or 2.079 or 2.0792)  0.04 0.470 or 0.4698 (2.23 or 2.230 or 2.2304)  0.03 0.576 or 0.5763 (2.38 or 2.380 or 2.3802)  0.04 0.723 or 0.7226 (2.59 or 2.591 or 2.5911)  0.03 Values of lg (r / 108 m) and lg (T / 103 s) correct as shown above. Uncertainties in lg (T / 103 s) correct as shown above. 1 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. Error bars in lg (T / 103 s) plotted correctly. 1 All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. © Cambridge University Press & Assessment 2025 Page 9 of 11 2(c)(ii) Straight line of best fit drawn. 1 Thickness of the line must be less than half a small square. Do not accept line from top point to bottom point. Line must pass between (0.19, 1.80) and (0.21, 1.80) and between (0.645, 2.50) and (0.66, 2.50). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 Thickness of the line must be less than half a small square. All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent powers of ten into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. © Cambridge University Press & Assessment 2025 Page 10 of 11 2(d) Value of n determined using gradient and n and k given to 2 or 3 significant figures. 1 n=gradient=(c)(iii) Value of k determined using y-intercept. 1 Correct method must be seen. 2 2 k = = 10y-intercept 10(c)(iv) Absolute uncertainties in n and k determined. 1 Absolute uncertainty in n = absolute uncertainty in gradient and 2 2 k = − 10y-intercept 10WAL y-intercept Correct method must be seen. 2(e) r determined to a minimum of 2 significant figures from (c)(iii) and (c)(iv) or (d) with correct substitution and correct power 1 of ten. lg1380−y-intercept r =10 gradient 108 or k1380 r = n 108 2π © Cambridge University Press & Assessment 2025 Page 11 of 11

1 Defining the problem vary r and measure v or r is the independent variable and v is the dependent variable 1 keep x constant 1 Methods of data collection labelled diagram of workable experiment including: 1 • one end of spring resting against block clamped to bench using G-clamp • light gate positioned at P • light gate connected to timer • apparatus shown on bench • labels for light gate and P and at least one other label from bench, block, stand, spring, ball, timer method to determine r, e.g. use calipers or micrometer to measure diameter d and r = d / 2 1 description of method to determine v, use diameter of ball (to interrupt beam) ÷ measured time at light gate positioned at P 1 instrument to determine x, e.g. rule(r) or calipers 1 Method of Analysis plot a graph of (2 lg v) or (lg v2) against lg r 1 or plot a graph of (lg v) against (lg r) or equivalent, e.g. (ln v) against (ln r) n=−gradient for (2 lg v) or (lg v2) against lg r 1 or n=−2gradient for (lg v) against lg r © Cambridge University Press & Assessment 2025 Page 7 of 11 1 10y-intercept 1 Y = for (2 lg v) or (lg v2) against lg r kx2 or 102y-intercept Y = for (lg v) against lg r kx2 Additional detail including safety considerations 6 D1 precaution to prevent ball leaving bench, e.g. screens around apparatus / cushions on bench (to stop the ball) D2 keep k and  constant D3 description of method to determine k, e.g. add mass to spring and k = mg / extension or use newton meter to measure force applied to spring and k = force / extension or take several readings of force and extension, plot a force–extension graph and k = gradient m D4 description of experimental method to determine , e.g. measure mass of ball using a balance and = 4 r3 3 D5 repeat measurements of diameter or d in different directions and determine the average value of d D6 method to keep x constant, e.g. use a pin / ruler / card to indicate the starting point each time to keep x constant D7 x = original length of spring – compressed length of spring D8 adjust (vertical) position of light gate so that the diameter of (each) ball cuts the beam D9 repeat experiment for the same value of r and determine the average v © Cambridge University Press & Assessment 2025 Page 8 of 11 1 Ykx2  1 Ykx2  D10 relationship valid if a straight line is produced (with y-intercept = lg  or lg ).  2      Do not accept line through the origin. Question Answer Marks

2(a) gradient = H 1 0 y-intercept =  0 2(b) 1 d / 1015s c (1.6 or 1.60)  0.40 (3.47 or 3.467)  0.40 (4.83 or 4.833)  0.40 (6.00 or 6.000)  0.40 (9.50 or 9.500)  0.40 (12.5 or 12.50)  0.40 d Values of correct as shown above. c d 1 Uncertainties in correct as shown above. c © Cambridge University Press & Assessment 2025 Page 9 of 11 2(c)(i) Six points from (b) plotted correctly. 1 Must be within half a small square. Diameter of points must be less than half a small square. d 1 Error bars in plotted correctly. c All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical. 2(c)(ii) Straight line of best fit drawn. 1 Thickness of the line must be less than half a small square. Do not accept line from top point to bottom point. Line must pass between (2.5, 660.0) and (2.9, 660.0) and between (11.2, 676.0) and (11.6, 676.0). Worst acceptable line drawn (steepest or shallowest possible line that passes through all the error bars). 1 Thickness of the line must be less than half a small square. All error bars must be plotted. 2(c)(iii) Gradient determined with clear substitution of data points into y / x. 1 Distance between data points must be greater than half the length of the drawn line. Gradient determined of worst acceptable line with clear substitution of data points into y / x. 1 uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient) 2(c)(iv) y-intercept determined by substitution of correct point with consistent power of ten in m and x and y into y = mx + c. 1 y-intercept of worst acceptable line determined by substitution into y = mx + c. 1 uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method. © Cambridge University Press & Assessment 2025 Page 10 of 11 2(d)  determined using y-intercept and  given to 3 or 4 significant figures and H given to 2, 3 or 4 significant figures. 1 0 0  = y-intercept 0 H determined using gradient and  and H given with SI units with appropriate powers of ten. 1 0 gradient gradient H = or H = y-intercept  0 Unit of : m, nm, m 0 Unit of H: s−1 2(e) Value of T determined to a minimum of two significant figures from (d) and correct power of ten. 1 1 T = H Absolute uncertainty determined with correct substitution. 1 y-intercept gradient T = + T  y-intercept gradient  or  max  maxy-intercept T = 0 −T or T = −T min gradient  min gradient  or  min  miny-intercept T = 0 −T or T = −T max gradient  max gradient  © Cambridge University Press & Assessment 2025 Page 11 of 11