A Level Maths: Pure 3 chapter 8: Differential equations MCQs with answers
By Team Megalecture · Updated 4 Nov 2026
24 exam-style questions on Differential equations, each with the answer and a short explanation. Ten of them are below to try. The chapter test draws ten fresh questions each time and is marked the moment you finish, with your score by topic.
Try each one, then open the answer. These ten stay the same; the test draws new ones every time.
1. Find the general solution of the differential equation dy/dx = y2 sin x.
A
y = 1/(A − cos x)
B
y = 1/(A + cos x)
C
y = Ae−cos x
D
y = ∛(A − 3 cos x)
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Answer: B. ∫ y−2 dy = ∫ sin x dx gives −1/y = −cos x + c, so 1/y = cos x + A and y = 1/(A + cos x). Writing ∫ sin x dx = cos x gives the wrong sign; ∛(A − 3 cos x) comes from multiplying by y2 instead of dividing.
2. The area A of a circular oil patch increases at a rate inversely proportional to its radius r. Which differential equation connects r and the time t, where c is a positive constant?
A
dr/dt = c/r
B
dr/dt = cr
C
dr/dt = c/√r
D
dr/dt = c/r2
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Answer: D. dA/dt = k/r. Since A = πr2, dA/dt = 2πr × dr/dt, so 2πr dr/dt = k/r and dr/dt = k/(2πr2), which is c/r2. dr/dt = c/r mixes up the rate of change of the area with that of the radius.
3. The rate of change of a quantity x with respect to time t is proportional to the square root of x and inversely proportional to t. Which equation models this, where k is a constant?
A
dx/dt = kt√x
B
dx/dt = k/(t√x)
C
dx/dt = k√x/t
D
dx/dt = kt/√x
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Answer: C. "Proportional to √x" puts √x in the numerator, and "inversely proportional to t" puts t in the denominator, so dx/dt = k√x/t.
4. A cup of tea cools in a room at 20 °C. The rate of decrease of its temperature θ °C is proportional to the difference between θ and 20. Which equation models this, where k is a positive constant?
A
dθ/dt = −k(θ − 20)
B
dθ/dt = k(θ − 20)
C
dθ/dt = −kθ
D
dθ/dt = −k/(θ − 20)
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Answer: A. The difference is θ − 20, and θ is decreasing while θ > 20, so dθ/dt must be negative: dθ/dt = −k(θ − 20). Without the minus sign the model says the tea gets hotter; −kθ ignores the room temperature.
5. Water is poured at 200 cm3 per second into a tank with vertical sides and a horizontal base of area 50 cm2. At the same time water leaks out at 10h cm3 per second, where h cm is the depth. Find the differential equation for h.
A
dh/dt = 200 − 10h
B
dh/dt = (20 − h)/5
C
dh/dt = (20 + h)/5
D
dh/dt = 500(20 − h)
Show answer
Answer: B. The volume is V = 50h, so dV/dt = 50 dh/dt. Also dV/dt = 200 − 10h (rate in minus rate out). So 50 dh/dt = 200 − 10h and dh/dt = (20 − h)/5. 200 − 10h is the rate of change of the volume, not of the depth.
6. The rate of increase of a population P at time t is proportional to P. Which differential equation models this, where k is a positive constant?
A
dP/dt = k/P
B
dP/dt = kt
C
dP/dt = kP
D
P = kt
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Answer: C. "Rate of increase of P" is dP/dt, and "proportional to P" means it equals kP. dP/dt = kt would make the rate depend on the time, not on the size of the population.
7. A rumour spreads through a school of 1000 students. The rate at which the number x who have heard it increases is proportional to the product of the number who have heard it and the number who have not. Which equation models this, where k is a positive constant?
A
dx/dt = k(1000 − x)
B
dx/dt = kx(1000 − x)
C
dx/dt = kx/(1000 − x)
D
dx/dt = kx(x − 1000)
Show answer
Answer: B. The number who have not heard is 1000 − x, so the product is x(1000 − x) and dx/dt = kx(1000 − x). With x(x − 1000) the rate would be negative for 0 < x < 1000, so the number who had heard would fall.
8. The mass m milligrams of a drug in the blood satisfies dm/dt = −0.2m, where t is the time in hours. How long does it take for the mass to halve?
A
2.50 hours
B
3.47 hours
C
0.139 hours
D
1.51 hours
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Answer: B. m = m0e−0.2t, and halving needs e−0.2t = ½, so t = (ln 2)/0.2 = 5 ln 2 = 3.47 hours, whatever the starting mass. 2.50 is 0.5 ÷ 0.2, treating the fall as linear; 0.139 multiplies ln 2 by 0.2 instead of dividing; 1.51 uses log10.
9. The temperature θ °C of a liquid t minutes after it is removed from the heat is given by θ = 20 + 60e−0.05t. Find the time taken for the temperature to fall to 50 °C.
A
9.40 minutes
B
13.9 minutes
C
3.65 minutes
D
6.02 minutes
Show answer
Answer: B. 60e−0.05t = 30, so e−0.05t = ½ and t = 20 ln 2 = 13.9 minutes. Solving 80e−0.05t = 50 ignores how the 20 enters the model and gives 9.40; forgetting to subtract the 20 at all gives 3.65; using log10 instead of ln gives 6.02.
10. A population P is modelled by dP/dt = 0.1P(1 − P/500), with P = 100 when t = 0. According to the model, what happens to P in the long term?
A
P increases and approaches 500
B
P increases without limit
C
P decreases and approaches 0
D
P increases and approaches 250
Show answer
Answer: A. For 0 < P < 500 the right-hand side is positive, so P increases. As P gets close to 500 the factor (1 − P/500) tends to 0, so the growth slows and P approaches 500 without passing it. 250 is where the rate of growth is greatest, not the limit.
24, each with the answer and a short explanation, across 1 topics: Differential equations. Every chapter test draws ten of them, so you can take it many times.
Is the A Level Maths: Pure 3 chapter 8 test free?
Yes. The test, the answers and the explanations are free and need no sign-up. Sign in only if you want your scores kept across devices.
How is the chapter test marked?
The moment you press Finish: your score, the right answer and an explanation for every question, and the ones you got wrong saved in My mistakes to retry.
