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Differential equations

Past-paper questions pack

Name
Class
Date set
Date due
Score/ 102

Questions are numbered; the mark schemes (the answer key) follow on separate pages so the question pages can be handed out alone. www.mathsaleveltutor.com

Edexcel AS and A-level Maths past-paper questions since June 2018 · 33 parts, 102 marks · mark schemes follow the questions · www.mathsaleveltutor.com · Questions and mark schemes © Pearson Education Ltd

1. June 2025 · Paper 2 · Q10(a) · 2 marks
10. Water flows at a constant rate into a large container. There is a tap at the bottom of the container. At time t hours after the tap was opened • the volume of water in the container is V m • water is flowing into the container at a constant rate of 0.45 m 3 per hour • water is leaving the container through the tap at a rate of 0.3V m 3 per hour (a) Show that V d V t d (2)
2. June 2025 · Paper 2 · Q10(b) · 5 marks
3 of water in the container, Given that when the tap was opened, there was 0.25 m (b) solve the differential equation to show that –kt V = P – Qe where P, Q and k are positive constants to be found. (5)
3. June 2025 · Paper 2 · Q10(c) · 2 marks
Given that • the capacity of the container is 2 m • the tap remains open • the water continues to flow into the tank at the same rate (c) determine whether the container will ever become full, giving a reason for your answer. (2)
4. June 2024 · Paper 1 · Q7(a) · 3 marks
7. Diagram not drawn to scale. 1.5 m L H m Figure 2 Figure 2 shows a cylindrical tank of height 1.5 m. Initially the tank is full of water. The water starts to leak from a small hole, at a point L, in the side of the tank. While the tank is leaking, the depth, H metres, of the water in the tank is modelled by the differential equation H d t O 3 0 2 0 12 e t d where t hours is the time after the leak starts. Using the model, (a) show that t H A e 0 2. B where A and B are constants to be found, (3)
5. June 2024 · Paper 1 · Q7(b) · 3 marks
(b) find the time taken for the depth of the water to decrease to 1.2 m. Give your answer in hours and minutes, to the nearest minute. (3)
6. June 2024 · Paper 1 · Q7(c) · 2 marks
In the long term, the water level in the tank falls to the same height as the hole. (c) Find, according to the model, the height of the hole from the bottom of the tank. (2)
7. June 2024 · Paper 1 · Q14(a) · 1 marks
14. A balloon is being inflated. In a simple model, • the balloon is modelled as a sphere • the rate of increase of the radius of the balloon is inversely proportional to the square root of the radius of the balloon At time t seconds, the radius of the balloon is r cm. (a) Write down a differential equation to model this situation. (1)
8. June 2024 · Paper 1 · Q14(b) · 5 marks
At the instant when t = 10 • the radius is 16 cm • the radius is increasing at a rate of 0.9 cm s –1 (b) Solve the differential equation to show that r t 5 4 (5)
9. June 2024 · Paper 1 · Q14(c) · 2 marks
(c) Hence find the radius of the balloon when t = 20 Give your answer to the nearest millimetre. (2)
10. June 2024 · Paper 1 · Q14(d) · 1 marks
(d) Suggest a limitation of the model. (1)
11. June 2024 · Paper 2 · Q12(b) · 5 marks
The volume, V microlitres, of a plant cell t hours after the plant is watered is modelled by the differential equation V d / V V t d The plant cell has an initial volume of 20 microlitres. (b) Find, according to the model, the time taken, in minutes, for the volume of the plant cell to reach 24 microlitres. (5)
12. June 2024 · Paper 2 · Q12(c) · 3 marks
(c) Show that A V kt e B where A, B and k are constants to be found. (3)
13. June 2024 · Paper 2 · Q12(d) · 2 marks
The model predicts that there is an upper limit, L microlitres, on the volume of the plant cell. (d) Find the value of L, giving a reason for your answer. (2)
14. June 2023 · Paper 2 · Q11(a) · 3 marks
11. h m 5 m 10 m 20 m Figure 1 A tank in the shape of a cuboid is being filled with water. The base of the tank measures 20 m by 10 m and the height of the tank is 5 m, as shown in Figure 1. At time t minutes after water started flowing into the tank the height of the water was h m and the volume of water in the tank was V m In a model of this situation • the sides of the tank have negligible thickness • the rate of change of V is inversely proportional to the square root of h (a) Show that h λ d = t d h where λ is a constant. (3)
15. June 2023 · Paper 2 · Q11(b) · 5 marks
Given that • initially the height of the water in the tank was 1.44 m • exactly 8 minutes after water started flowing into the tank the height of the water was 3.24 m (b) use the model to find an equation linking h with t, giving your answer in the form 2 O h At B where A and B are constants to be found. (5)
16. June 2023 · Paper 2 · Q11(c) · 2 marks
(c) Hence find the time taken, from when water started flowing into the tank, for the tank to be completely full. (2)
17. June 2022 · Paper 2 · Q14(b) · 5 marks
When chemical A and chemical B are mixed, oxygen is produced. A scientist mixed these two chemicals and measured the total volume of oxygen produced over a period of time. 3 , t hours after the chemicals were mixed, is The total volume of oxygen produced, V m modelled by the differential equation d V V t = V 0 t k d t - t + ( )( ) where k is a constant. 3 of Given that exactly 2 hours after the chemicals were mixed, a total volume of 3 m oxygen had been produced, (b) solve the differential equation to show that t - 3 2 ( ) V = t + ( ) (5)
18. June 2022 · Paper 2 · Q14(c) · 2 marks
The scientist noticed that • there was a time delay between the chemicals being mixed and oxygen being produced • there was a limit to the total volume of oxygen produced Deduce from the model (c) (i) the time delay giving your answer in minutes, (ii) the limit giving your answer in m (2)
19. Oct 2021 · Paper 2 · Q14(a) · 4 marks
14. 5 m h m T 3 m 8 m Figure 5 Water flows at a constant rate into a large tank. The tank is a cuboid, with all sides of negligible thickness. The base of the tank measures 8 m by 3 m and the height of the tank is 5 m. There is a tap at a point T at the bottom of the tank, as shown in Figure 5. At time t minutes after the tap has been opened • the depth of water in the tank is h metres • water is flowing into the tank at a constant rate of 0.48 m3 per minute • water is modelled as leaving the tank through the tap at a rate of 0.1h m3 per minute (a) Show that, according to the model, 1200 d h t = 24 – 5h d (4)
20. Oct 2021 · Paper 2 · Q14(b) · 6 marks
Given that when the tap was opened, the depth of water in the tank was 2 m, (b) show that, according to the model, h = A + B e–kt where A, B and k are constants to be found. (6)
21. Oct 2021 · Paper 2 · Q14(c) · 2 marks
Given that the tap remains open, (c) determine, according to the model, whether the tank will ever become full, giving a reason for your answer. (2)
22. Oct 2020 · Paper 1 · Q8 · 2 marks
8. A new smartphone was released by a company. The company monitored the total number of phones sold, n, at time t days after the phone was released. The company observed that, during this time, the rate of increase of n was proportional to n Use this information to write down a suitable equation for n in terms of t. (You do not need to evaluate any unknown constants in your equation.) (2)
23. Oct 2020 · Paper 1 · Q14(a) · 3 marks
14. A large spherical balloon is deflating. At time t seconds the balloon has radius r cm and volume V cm3 The volume of the balloon is modelled as decreasing at a constant rate. (a) Using this model, show that d r t = -k d r 2 where k is a positive constant. (3)
24. Oct 2020 · Paper 1 · Q14(b) · 5 marks
Given that ● the initial radius of the balloon is 40 cm ● after 5 seconds the radius of the balloon is 20 cm ● the volume of the balloon continues to decrease at a constant rate until the balloon is empty (b) solve the differential equation to find a complete equation linking r and t. (5)
25. Oct 2020 · Paper 1 · Q14(c) · 2 marks
(c) Find the limitation on the values of t for which the equation in part (b) is valid. (2)
26. June 2019 · Paper 2 · Q14(b) · 2 marks
A team of scientists is studying a species of slow growing tree. The rate of change in height of a tree in this species is modelled by the differential equation ( ) 0 25 4 t h - d h = d t where h is the height in metres and t is the time, measured in years, after the tree is planted. (b) Find, according to the model, the range in heights of trees in this species. (2)
27. June 2019 · Paper 2 · Q14(c) · 7 marks
One of these trees is one metre high when it is first planted. According to the model, (c) calculate the time this tree would take to reach a height of 12 metres, giving your answer to 3 significant figures. (7)
28. June 2018 · Paper 1 · Q10(a) · 5 marks
10. The height above ground, H metres, of a passenger on a roller coaster can be modelled by the differential equation d H H cos( . 0 25 t ) = d t where t is the time, in seconds, from the start of the ride. Given that the passenger is 5 m above the ground at the start of the ride, (a) show that H = 5e0.1 sin (0.25t) (5)
29. June 2018 · Paper 1 · Q10(b) · 1 marks
(b) State the maximum height of the passenger above the ground. (1)
30. June 2018 · Paper 1 · Q10(c) · 2 marks
The passenger reaches the maximum height, for the second time, T seconds after the start of the ride. (c) Find the value of T. (2)
31. June 2018 · Paper 2 · Q10(a) · 5 marks
10. A spherical mint of radius 5 mm is placed in the mouth and sucked. Four minutes later, the radius of the mint is 3 mm. In a simple model, the rate of decrease of the radius of the mint is inversely proportional to the square of the radius. Using this model and all the information given, (a) find an equation linking the radius of the mint and the time. (You should define the variables that you use.) (5)
32. June 2018 · Paper 2 · Q10(b) · 2 marks
(b) Hence find the total time taken for the mint to completely dissolve. Give your answer in minutes and seconds to the nearest second. (2)
33. June 2018 · Paper 2 · Q10(c) · 1 marks
(c) Suggest a limitation of the model. (1)

Mark schemes

Each mark scheme shows the whole question; the parts on this topic are the ones listed above.

June 2025 · Paper 2 · Q10
Mark scheme June 2025 Paper 2 Q10(c)
June 2024 · Paper 1 · Q7
Mark scheme June 2024 Paper 1 Q7(c)
June 2024 · Paper 1 · Q14
Mark scheme June 2024 Paper 1 Q14(d)
June 2024 · Paper 2 · Q12
Mark scheme June 2024 Paper 2 Q12(d)
June 2023 · Paper 2 · Q11
Mark scheme June 2023 Paper 2 Q11(c)
June 2022 · Paper 2 · Q14
Mark scheme June 2022 Paper 2 Q14(c)
Oct 2021 · Paper 2 · Q14
Mark scheme Oct 2021 Paper 2 Q14(c)
Oct 2020 · Paper 1 · Q8
Mark scheme Oct 2020 Paper 1 Q8
Oct 2020 · Paper 1 · Q14
Mark scheme Oct 2020 Paper 1 Q14(c)
June 2019 · Paper 2 · Q14
Mark scheme June 2019 Paper 2 Q14(c)
June 2018 · Paper 1 · Q10
Mark scheme June 2018 Paper 1 Q10(c)
June 2018 · Paper 2 · Q10
Mark scheme June 2018 Paper 2 Q10(c)