A Level · Mathematics 9709 · Topic

Pure Mathematics 3: A Level Mathematics questions and answers

10 free A Level Mathematics Pure Mathematics 3 questions written in the style of Cambridge Mathematics papers, each with its answer and mark scheme. 3 of them are A* challenge questions, built to catch the mistakes strong students still make.

This topic covers Algebra, Logarithmic & exponential functions, Trigonometry, Differentiation, Integration, Numerical solution of equations, Vectors, Differential equations, Complex numbers. Work through the questions below, then generate a full Pure Mathematics 3 paper with a fresh set every time.

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Pure Mathematics 3 sub-topics

AlgebraLogarithmic & exponential functionsTrigonometryDifferentiationIntegrationNumerical solution of equationsVectorsDifferential equationsComplex numbers

Pure Mathematics 3 practice questions

Question 1Structured · 10 marksA* challenge
(a)

Show that ∫1e x3 ln x dx = 3e4 + 116.

[5]
(b)

Express 9x + 14(x + 1)(x + 2) in partial fractions.

[2]
(c)

Hence find ∫01 9x + 14(x + 1)(x + 2) dx, giving your answer in the form a ln 2 + b ln 3.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)integration by parts with u = ln x
u = ln x, dv = xn chosen M1
x44 ln x − ∫ x34 dx A1
integrate the second term M1
limits substituted correctly M1
given answer obtained with no errors A1
5
(b)5x + 1 + 4x + 2
correct method M1
A = 5, B = 4 A1
2
(c)ln 2 + 4ln 3
integrate to ln terms M1
substitute limits M1
ln 2 + 4ln 3 A1
3
Question 2Structured · 7 marksA* challenge

The number of fish, N, in a lake satisfies dNdt = 0.05N(1 − N1000), where t is in weeks. Initially N = 100.

Solve the differential equation to find N in terms of t, and find the time when N = 500, giving your answer correct to 3 significant figures.

[7]
Show answers and mark scheme
PartAnswer and marking pointsMarks
N = 10001 + 9e−0.05t; t = 43.9 weeks
separate variables M1
partial fractions M1
integrate A1
use initial condition M1
N = 10001 + 9e−0.05t oe A1
substitute N = 500 M1
43.9 (allow exact 20 ln 9) A1
7
Question 3Structured · 6 marksA* challenge
(a)

Prove the identity sin 2θ1 + cos 2θ ≡ tan θ.

[3]
(b)

Hence find the exact value of ∫0π/3 sin 2θ1 + cos 2θ dθ.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)2 sin θ cos θ ÷ 2 cos2 θ
sin 2θ = 2 sin θ cos θ B1
1 + cos 2θ = 2cos2 θ B1
cancelled to tan θ A1
3
(b)ln 2
∫ tan θ dθ = −ln|cos θ| M1
correct limits M1
ln 2 A1
3
Question 4Structured · 4 marks

A curve has equation x2 + 3xy + y2 = 5. Find the gradient of the curve at the point (1, 1).

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
−1
implicit differentiation M1
3y + 3xdydx product rule A1
2x + 3y + (3x + 2y)dydx = 0 A1
−1 A1
4
Question 5Structured · 6 marks
(a)

Express 3 sin θ + 4 cos θ in the form R sin(θ + α), where R > 0 and 0° < α < 90°. Give α correct to 2 decimal places.

[3]
(b)

Hence solve 3 sin θ + 4 cos θ = 2.1 for 0° < θ < 360°.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)R = 5, α = 53.13°
R correct B1
tan α = 4/3 M1
α = 53.13 A1
3
(b)θ = 331.7°, 102.0°
sin(θ + α) = c/R M1
One correct angle A1
Second correct angle A1
3
Question 6Structured · 5 marks

The complex numbers u and w are given by u = 2 + 3i and w = −4 − 4i.

(a)

Find uw in the form x + iy.

[2]
(b)

Find the modulus and argument of u. Give the argument in radians correct to 3 significant figures.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)4 − 20i
Expand with i2 = −1 M1
Correct answer A1
2
(b)|u| = 3.61, arg u = 0.983
Modulus B1
Argument method in correct quadrant M1
0.983 A1
3
Question 7Structured · 3 marks

Find the exact value of ∫ from 0 to 2 of e4x dx.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
14(e8 − 1)
14e4x M1
Correct limits M1
14(e8 − 1) A1
3
Question 8Structured · 4 marks

Find the angle between the vectors 3i + j − 4k and 4i − 2j + 2k.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
85.4°
Scalar product = 2 M1
Magnitudes M1
cos θ = a·b ÷ (|a||b|) M1
85.4 A1
4
Question 9Structured · 3 marks

The equation x3 − 7x − 7 = 0 has a root α. Use the iterative formula xn+1 = ∛(7xn + 7) with x1 = 2 to find α correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
α = 3.049 (iterates 2.75892, 2.97432, 3.03008, 3.04418…)
Uses iteration correctly at least once M1
Iterates to sufficient accuracy M1
3.049 A1
3
Question 10Structured · 4 marks

Solve cos 2θ = cos θ for 0° < θ < 360°.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
120°, 240°
double angle formula M1
2 cos2θ − cos θ − 1 = 0 A1
cos θ = 1 or −½ A1
120°, 240° A1
4

Other A Level Mathematics topics

MechanicsProbability & Statistics 1Probability & Statistics 2

Frequently asked questions

What is in A Level Mathematics Pure Mathematics 3?

In the 9709 syllabus, Pure Mathematics 3 covers Algebra, Logarithmic & exponential functions, Trigonometry, Differentiation, Integration, Numerical solution of equations, Vectors, Differential equations, Complex numbers.

Where can I get Pure Mathematics 3 questions with mark schemes?

This page has 10 Pure Mathematics 3 questions with answers. For more, the free generator on mentorriyaz.in builds a new Pure Mathematics 3 paper with a full mark scheme each time you click.

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Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in