Note that these solutions are not yet complete. I will work on them as and when I can but it might take several weeks before I can finish posting them.

Firstly, I would like to caution students regarding the use of solutions when preparing for exams. They are certainly a double-edged sword and since you are planning to invest an enormous amount of time, effort and money into getting a degree, it would be a good idea to use solutions properly from now on. In all cases you should only consult a solution once you think your answer is correct or if you have spent a significant amount of time trying to reach an answer. Coming up with your own conclusion after several minutes of independent thought is far more worthwhile than learning directly from the solution itself. You should also make sure you are critical of all solutions presented to you. Not only will this help your understanding, but solutions may have errors. Therefore if you find that you can't reconcile a solution with yours then take the time to get in touch with the author to discuss the discrepancy. For example, please do contact me if you have spotted a mistake below or would like further guidance on any of the questions.

For the 2020 solutions I have decided to revert to fuller explanations since the questions are more challenging than in previous years. I have also included the time taken next to each question. Note that there is wide variability in the time taken to solve each question. It is definitely a good strategy to be well rested for the exam with no cramming the night before! I will firstly set out a series of hints. Try to do the questions from these alone and if you are still having difficulty then scroll down to the time-saving solutions.

Hints

Section 1

Part A

1. Write down the energy supplied and calculate the energy retained using $mc\Delta T$ then work out the difference. Time taken 0:37.

2. Form two simultaneous equations and solve Time taken 0:44.

3. Record how many alpha decays must have occurred then work out how much the proton number needs to increase by to reach lead. Time taken 0:30.

4. $5$ can be achieved by adding $3$ and $2$ in either order. Time taken 0:32.

5. Use $P=IV$ to work out the power supplied and received, dividing the latter by the former to calculate a percentage. Time taken 1:07.

6. Record the gradients of the lines by inspection. Time taken 0:22.

7. Apply $F=BIl$ and Fleming's left hand rule, taking care with the length over which the current flows. Time taken 1:22.

8. Save time by defining $y=\frac{x}{4}+3$ and solve the resulting quadratic. Time taken 1:28.

9. Apply conservation of momentum to work out the new speed and then the kinetic energies. Time taken 0:38.

10. Note that the constant term cancels. Expand in one step then complete the square. Time taken 0:51.

11. Use the equations of constant acceleration to find $a$ then multiply by $m$. Time taken 3:23.

12. Start with $N=\frac{k}{t^3}$. Leave $k$ as a product to aid cancellation. Time taken 0:52.

13. Try to find the wavelength from the information given without too much delay. The wave travels $vt$ but the distance between $P$ and $Q$ needs to be taken into account. Then apply $v=f\lambda$. Time taken 1:58.

14. Define the correct original price to be (say) $X$. Use percentage multipliers to work out the incorrect original price in terms of the correct original price, subtract $X$ and equate the result to $15$. Time taken 5:55 (several numerical/conceptual slips).

15. Use energy at the start - energy at the end = energy wasted. Time taken 1:29.

16. Visualise $SQ$ by inspection and apply trigonometry to the rightmost triangle. Time taken 0:47.

17. The energy at the start is elastic potential which is converted to gravitational potential and kinetic energy. Time taken 1:38.

18. Draw a rough diagram and visualise the two possible squares (whose orientations will be at $45^{\circ}$ to one another). Time taken 1:15.

19. Starting from force equals rate of change of momentum, use the product rule and rearrange for $\frac{dv}{dt}$. Time taken 1:16.

20. Write the right-hand term of the quadratic formula and equate to $3$. Time taken 1:09.

Part B

21. The energy at the start is kinetic and gravitational potential, changing to kinetic energy and work done against friction. Time taken 0:49.

22. Substitute $1$ and $2$ in and solve the resulting equations. Time taken 0:40.

23. The pressure is the same at both pistons and is equal to force/area. Note that the area has gone up by a factor of 9. Apply work done = force $\times$ distance to work out how the distance moved by piston $Y$ should relate to that travelled by piston $X$. Time taken 0:59.

24. I sketched this but there was no need in retrospect as one of the functions is clearly beneath the other for the whole integral. Time taken 1:26 (inefficient method and numerical slips).

25. Determine the frequency from the graph, apply $c=f\lambda$ and note that the speed in water is greater than that in air which means the wave will be more 'stretched out' . Time taken 1:22.

26. Imagine what will happen to a line with gradient greater than $1$ - its gradient after the reflection will be between $0$ and $1$. Time taken 0:08.

27. Try not to be overfaced by the text and apply $F=BIl$ and moment $=$ force $\times$ perpendicular distance from pivot. Time taken 1:37.

28. .Time taken 1:24.

29. . Time taken 1:38.

30. W. Time taken 2:22.

31. . Time taken 1:27.

32. . Time taken 3:27.

33. . Time taken 2:48.

34. . Time taken 1:23.

35. . Time taken 1:47.

36. . Time taken not recorded.

37. . Time taken not recorded.

38. . Time taken 1:30.

39. . Time taken not recorded.

40. . Time taken 1:10.

Section 2

1. . Time taken 2:07.

2. . Time taken 0:46.

3. . Time taken 1:05.

4. . Time taken 3:57.

5. . Time taken 6:17.

6. . Time taken 2:40.

7. . Time taken 2:08.

8. . Time taken 0:44.

9. . Time taken 1:21.

10. . Time taken 1:14.

11. . Time taken 6:22.

12. . Time taken 2:06.

13. . Time taken 3:02.

14. . Time taken 0:48.

15. . Time taken 1:05.

16. . Time taken 2:40.

17. . Time taken 1:24.

18. . Time taken 2:10.

19. . Time taken 0:58.

20. . Time taken 10:13.

Full solutions

Section 1

Part A

1. Write down the energy supplied and calculate the energy retained using $mc\Delta T$ then work out the difference. Time taken 0:37.

2. Form two simultaneous equations and solve Time taken 0:44.

3. Record how many alpha decays must have occurred then work out how much the proton number needs to increase by to reach lead. Time taken 0:30.

4. $5$ can be achieved by adding $3$ and $2$ in either order. Time taken 0:32.

5. Use $P=IV$ to work out the power supplied and received, dividing the latter by the former to calculate a percentage. Time taken 1:07.

6. Record the gradients of the lines by inspection. Time taken 0:22.

7. Apply $F=BIl$ and Fleming's left hand rule, taking care with the length over which the current flows. Time taken 1:22.

8. Save time by defining $y=\frac{x}{4}+3$ and solve the resulting quadratic. Time taken 1:28.

9. Apply conservation of momentum to work out the new speed and then the kinetic energies. Time taken 0:38.

10. Note that the constant term cancels. Expand in one step then complete the square. Time taken 0:51.

11. Use the equations of constant acceleration to find $a$ then multiply by $m$. Time taken 3:23.

12. Start with $N=\frac{k}{t^3}$. Leave $k$ as a product to aid cancellation. Time taken 0:52.

13. Try to find the wavelength from the information given without too much delay. The wave travels $vt$ but the distance between $P$ and $Q$ needs to be taken into account. Then apply $v=f\lambda$. Time taken 1:58.

14. Define the correct original price to be (say) $X$. Use percentage multipliers to work out the incorrect original price in terms of the correct original price, subtract $X$ and equate the result to $15$. Time taken 5:55 (several numerical/conceptual slips).

15. Use energy at the start - energy at the end = energy wasted. Time taken 1:29.

16. Visualise $SQ$ by inspection and apply trigonometry to the rightmost triangle. Time taken 0:47.

17. The energy at the start is elastic potential which is converted to gravitational potential and kinetic energy. Time taken 1:38.

18. Draw a rough diagram and visualise the two possible squares (whose orientations will be at $45^{\circ}$ to one another). Time taken 1:15.

19. Starting from force equals rate of change of momentum, use the product rule and rearrange for $\frac{dv}{dt}$. Time taken 1:16.

20. Write the right-hand term of the quadratic formula and equate to $3$. Time taken 1:09.

Part B

21. The energy at the start is kinetic and gravitational potential, changing to kinetic energy and work done against friction. Time taken 0:49.

22. Substitute $1$ and $2$ in and solve the resulting equations. Time taken 0:40.

23. The pressure is the same at both pistons and is equal to force/area. Note that the area has gone up by a factor of 9. Apply work done = force $\times$ distance to work out how the distance moved by piston $Y$ should relate to that travelled by piston $X$. Time taken 0:59.

24. I sketched this but there was no need in retrospect as one of the functions is clearly beneath the other for the whole integral. Time taken 1:26 (inefficient method and numerical slips).

25. Determine the frequency from the graph, apply $c=f\lambda$ and note that the speed in water is greater than that in air which means the wave will be more 'stretched out' . Time taken 1:22.

26. Imagine what will happen to a line with gradient greater than $1$ - its gradient after the reflection will be between $0$ and $1$. Time taken 0:08.

27. Try not to be overfaced by the text and apply $F=BIl$ and moment $=$ force $\times$ perpendicular distance from pivot. Time taken 1:37.

28. .Time taken 1:24.

29. . Time taken 1:38.

30. W. Time taken 2:22.

31. . Time taken 1:27.

32. . Time taken 3:27.

33. . Time taken 2:48.

34. . Time taken 1:23.

35. . Time taken 1:47.

36. . Time taken not recorded.

37. . Time taken not recorded.

38. . Time taken 1:30.

39. . Time taken not recorded.

40. . Time taken 1:10.

Section 2

1. . Time taken 2:07.

2. . Time taken 0:46.

3. . Time taken 1:05.

4. . Time taken 3:57.

5. . Time taken 6:17.

6. . Time taken 2:40.

7. . Time taken 2:08.

8. . Time taken 0:44.

9. . Time taken 1:21.

10. . Time taken 1:14.

11. . Time taken 6:22.

12. . Time taken 2:06.

13. . Time taken 3:02.

14. . Time taken 0:48.

15. . Time taken 1:05.

16. . Time taken 2:40.

17. . Time taken 1:24.

18. . Time taken 2:10.

19. . Time taken 0:58.

20. . Time taken 10:13.

Firstly, I would like to caution students regarding the use of solutions when preparing for exams. They are certainly a double-edged sword and since you are planning to invest an enormous amount of time, effort and money into getting a degree, it would be a good idea to use solutions properly from now on. In all cases you should only consult a solution once you think your answer is correct or if you have spent a significant amount of time trying to reach an answer. Coming up with your own conclusion after several minutes of independent thought is far more worthwhile than learning directly from the solution itself. You should also make sure you are critical of all solutions presented to you. Not only will this help your understanding, but solutions may have errors. Therefore if you find that you can't reconcile a solution with yours then take the time to get in touch with the author to discuss the discrepancy. For example, please do contact me if you have spotted a mistake below or would like further guidance on any of the questions.

For the 2020 solutions I have decided to revert to fuller explanations since the questions are more challenging than in previous years. I have also included the time taken next to each question. Note that there is wide variability in the time taken to solve each question. It is definitely a good strategy to be well rested for the exam with no cramming the night before! I will firstly set out a series of hints. Try to do the questions from these alone and if you are still having difficulty then scroll down to the time-saving solutions.

Hints

Section 1

Part A

1. Write down the energy supplied and calculate the energy retained using $mc\Delta T$ then work out the difference. Time taken 0:37.

2. Form two simultaneous equations and solve Time taken 0:44.

3. Record how many alpha decays must have occurred then work out how much the proton number needs to increase by to reach lead. Time taken 0:30.

4. $5$ can be achieved by adding $3$ and $2$ in either order. Time taken 0:32.

5. Use $P=IV$ to work out the power supplied and received, dividing the latter by the former to calculate a percentage. Time taken 1:07.

6. Record the gradients of the lines by inspection. Time taken 0:22.

7. Apply $F=BIl$ and Fleming's left hand rule, taking care with the length over which the current flows. Time taken 1:22.

8. Save time by defining $y=\frac{x}{4}+3$ and solve the resulting quadratic. Time taken 1:28.

9. Apply conservation of momentum to work out the new speed and then the kinetic energies. Time taken 0:38.

10. Note that the constant term cancels. Expand in one step then complete the square. Time taken 0:51.

11. Use the equations of constant acceleration to find $a$ then multiply by $m$. Time taken 3:23.

12. Start with $N=\frac{k}{t^3}$. Leave $k$ as a product to aid cancellation. Time taken 0:52.

13. Try to find the wavelength from the information given without too much delay. The wave travels $vt$ but the distance between $P$ and $Q$ needs to be taken into account. Then apply $v=f\lambda$. Time taken 1:58.

14. Define the correct original price to be (say) $X$. Use percentage multipliers to work out the incorrect original price in terms of the correct original price, subtract $X$ and equate the result to $15$. Time taken 5:55 (several numerical/conceptual slips).

15. Use energy at the start - energy at the end = energy wasted. Time taken 1:29.

16. Visualise $SQ$ by inspection and apply trigonometry to the rightmost triangle. Time taken 0:47.

17. The energy at the start is elastic potential which is converted to gravitational potential and kinetic energy. Time taken 1:38.

18. Draw a rough diagram and visualise the two possible squares (whose orientations will be at $45^{\circ}$ to one another). Time taken 1:15.

19. Starting from force equals rate of change of momentum, use the product rule and rearrange for $\frac{dv}{dt}$. Time taken 1:16.

20. Write the right-hand term of the quadratic formula and equate to $3$. Time taken 1:09.

Part B

21. The energy at the start is kinetic and gravitational potential, changing to kinetic energy and work done against friction. Time taken 0:49.

22. Substitute $1$ and $2$ in and solve the resulting equations. Time taken 0:40.

23. The pressure is the same at both pistons and is equal to force/area. Note that the area has gone up by a factor of 9. Apply work done = force $\times$ distance to work out how the distance moved by piston $Y$ should relate to that travelled by piston $X$. Time taken 0:59.

24. I sketched this but there was no need in retrospect as one of the functions is clearly beneath the other for the whole integral. Time taken 1:26 (inefficient method and numerical slips).

25. Determine the frequency from the graph, apply $c=f\lambda$ and note that the speed in water is greater than that in air which means the wave will be more 'stretched out' . Time taken 1:22.

26. Imagine what will happen to a line with gradient greater than $1$ - its gradient after the reflection will be between $0$ and $1$. Time taken 0:08.

27. Try not to be overfaced by the text and apply $F=BIl$ and moment $=$ force $\times$ perpendicular distance from pivot. Time taken 1:37.

28. .Time taken 1:24.

29. . Time taken 1:38.

30. W. Time taken 2:22.

31. . Time taken 1:27.

32. . Time taken 3:27.

33. . Time taken 2:48.

34. . Time taken 1:23.

35. . Time taken 1:47.

36. . Time taken not recorded.

37. . Time taken not recorded.

38. . Time taken 1:30.

39. . Time taken not recorded.

40. . Time taken 1:10.

Section 2

1. . Time taken 2:07.

2. . Time taken 0:46.

3. . Time taken 1:05.

4. . Time taken 3:57.

5. . Time taken 6:17.

6. . Time taken 2:40.

7. . Time taken 2:08.

8. . Time taken 0:44.

9. . Time taken 1:21.

10. . Time taken 1:14.

11. . Time taken 6:22.

12. . Time taken 2:06.

13. . Time taken 3:02.

14. . Time taken 0:48.

15. . Time taken 1:05.

16. . Time taken 2:40.

17. . Time taken 1:24.

18. . Time taken 2:10.

19. . Time taken 0:58.

20. . Time taken 10:13.

Full solutions

Section 1

Part A

1. Write down the energy supplied and calculate the energy retained using $mc\Delta T$ then work out the difference. Time taken 0:37.

2. Form two simultaneous equations and solve Time taken 0:44.

3. Record how many alpha decays must have occurred then work out how much the proton number needs to increase by to reach lead. Time taken 0:30.

4. $5$ can be achieved by adding $3$ and $2$ in either order. Time taken 0:32.

5. Use $P=IV$ to work out the power supplied and received, dividing the latter by the former to calculate a percentage. Time taken 1:07.

6. Record the gradients of the lines by inspection. Time taken 0:22.

7. Apply $F=BIl$ and Fleming's left hand rule, taking care with the length over which the current flows. Time taken 1:22.

8. Save time by defining $y=\frac{x}{4}+3$ and solve the resulting quadratic. Time taken 1:28.

9. Apply conservation of momentum to work out the new speed and then the kinetic energies. Time taken 0:38.

10. Note that the constant term cancels. Expand in one step then complete the square. Time taken 0:51.

11. Use the equations of constant acceleration to find $a$ then multiply by $m$. Time taken 3:23.

12. Start with $N=\frac{k}{t^3}$. Leave $k$ as a product to aid cancellation. Time taken 0:52.

13. Try to find the wavelength from the information given without too much delay. The wave travels $vt$ but the distance between $P$ and $Q$ needs to be taken into account. Then apply $v=f\lambda$. Time taken 1:58.

14. Define the correct original price to be (say) $X$. Use percentage multipliers to work out the incorrect original price in terms of the correct original price, subtract $X$ and equate the result to $15$. Time taken 5:55 (several numerical/conceptual slips).

15. Use energy at the start - energy at the end = energy wasted. Time taken 1:29.

16. Visualise $SQ$ by inspection and apply trigonometry to the rightmost triangle. Time taken 0:47.

17. The energy at the start is elastic potential which is converted to gravitational potential and kinetic energy. Time taken 1:38.

18. Draw a rough diagram and visualise the two possible squares (whose orientations will be at $45^{\circ}$ to one another). Time taken 1:15.

19. Starting from force equals rate of change of momentum, use the product rule and rearrange for $\frac{dv}{dt}$. Time taken 1:16.

20. Write the right-hand term of the quadratic formula and equate to $3$. Time taken 1:09.

Part B

21. The energy at the start is kinetic and gravitational potential, changing to kinetic energy and work done against friction. Time taken 0:49.

22. Substitute $1$ and $2$ in and solve the resulting equations. Time taken 0:40.

23. The pressure is the same at both pistons and is equal to force/area. Note that the area has gone up by a factor of 9. Apply work done = force $\times$ distance to work out how the distance moved by piston $Y$ should relate to that travelled by piston $X$. Time taken 0:59.

24. I sketched this but there was no need in retrospect as one of the functions is clearly beneath the other for the whole integral. Time taken 1:26 (inefficient method and numerical slips).

25. Determine the frequency from the graph, apply $c=f\lambda$ and note that the speed in water is greater than that in air which means the wave will be more 'stretched out' . Time taken 1:22.

26. Imagine what will happen to a line with gradient greater than $1$ - its gradient after the reflection will be between $0$ and $1$. Time taken 0:08.

27. Try not to be overfaced by the text and apply $F=BIl$ and moment $=$ force $\times$ perpendicular distance from pivot. Time taken 1:37.

28. .Time taken 1:24.

29. . Time taken 1:38.

30. W. Time taken 2:22.

31. . Time taken 1:27.

32. . Time taken 3:27.

33. . Time taken 2:48.

34. . Time taken 1:23.

35. . Time taken 1:47.

36. . Time taken not recorded.

37. . Time taken not recorded.

38. . Time taken 1:30.

39. . Time taken not recorded.

40. . Time taken 1:10.

Section 2

1. . Time taken 2:07.

2. . Time taken 0:46.

3. . Time taken 1:05.

4. . Time taken 3:57.

5. . Time taken 6:17.

6. . Time taken 2:40.

7. . Time taken 2:08.

8. . Time taken 0:44.

9. . Time taken 1:21.

10. . Time taken 1:14.

11. . Time taken 6:22.

12. . Time taken 2:06.

13. . Time taken 3:02.

14. . Time taken 0:48.

15. . Time taken 1:05.

16. . Time taken 2:40.

17. . Time taken 1:24.

18. . Time taken 2:10.

19. . Time taken 0:58.

20. . Time taken 10:13.