AS Level · Mathematics 9709 · Topic

Pure Mathematics 2: AS Level Mathematics questions and answers

10 free AS Level Mathematics Pure Mathematics 2 questions written in the style of Cambridge Mathematics papers, each with its answer and mark scheme. 4 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. Work through the questions below, then generate a full Pure Mathematics 2 paper with a fresh set every time.

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Pure Mathematics 2 sub-topics

AlgebraLogarithmic & exponential functionsTrigonometryDifferentiationIntegrationNumerical solution of equations

Pure Mathematics 2 practice questions

Question 1Structured · 5 marksA* challenge
(a)

Use the trapezium rule with 2 intervals to estimate ∫02 √(1 + x3) dx. Give your answer correct to 3 decimal places.

[3]
(b)

Explain, with reference to the shape of the curve, whether your answer to (a) is an over-estimate or an under-estimate.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)3.414
h = 1 B1
correct trapezium formula with y-values M1
3.414 A1
3
(b)the curve is convex (concave upwards) on this interval, so the chords lie above the curve
curve concave up / convex B1
trapezia lie above the curve B1
2
Question 2Structured · 10 marksA* challenge

The polynomial p(x) is defined by p(x) = 2x3 + ax2 + bx + 2, where a and b are constants. It is given that (x − 2) is a factor of p(x) and that the remainder is −6 when p(x) is divided by (x + 1).

(a)

Find the values of a and b.

[5]
(b)

Hence factorise p(x) completely.

[2]
(c)

Hence solve p(3y) = 0, giving your answers correct to 3 significant figures where appropriate.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)a = −5, b = 1
p(2) = 0 substituted M1
p(−1) = −6 substituted M1
two correct linear equations A1
solve simultaneously M1
a = −5, b = 1 A1
5
(b)(x − 2)(2x + 1)(x − 1)
divide by (x − 2) to obtain the quadratic factor M1
complete factorisation A1
2
(c)y = 0.631, y = 0 (3y > 0 so negative root rejected)
3y = positive root(s) M1
logarithms used correctly M1
0.631, 0, and negative root rejected A1
3
Question 3Structured · 6 marksA* challenge

A curve has equation y = ln xx3 for x > 0.

(a)

Find the exact coordinates of the stationary point.

[4]
(b)

Determine the nature of the stationary point.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)(e1/3, 13e)
quotient rule M1
1 − 3ln xx4 A1
ln x = 13 M1
(e1/3, 13e) A1
4
(b)maximum: gradient changes from positive to negative
valid method (sign of derivative either side or second derivative) M1
maximum A1
2
Question 4Structured · 5 marksA* challenge
(a)

Show that cos 2θ + 3 sin θ = 2 can be written as 2 sin2 θ − 3 sin θ + 1 = 0.

[2]
(b)

Hence solve the equation for 0° ≤ θ ≤ 360°.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)cos 2θ = 1 − 2 sin2 θ
correct double-angle identity M1
rearranged to the given form A1
2
(b)30°, 90°, 150°
factorise: (2 sin θ − 1)(sin θ − 1) = 0 M1
sin θ = ½ or 1 A1
30°, 90°, 150° only A1
3
Question 5Structured · 4 marks
(a)

Solve the inequality |5x − 2| < |x + 2|.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)0 < x < 1
square both sides or consider cases M1
(5x − 2)2 < (x + 2)2 A1
critical values 0 and 1 A1
0 < x < 1 A1
4
Question 6Structured · 3 marks

The equation x3 − 7x − 3 = 0 has a root α. Use the iterative formula xn+1 = ∛(7xn + 3) 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 = 2.838 (iterates 2.57128, 2.75888, 2.81523, 2.83172…)
Uses iteration correctly at least once M1
Iterates to sufficient accuracy M1
2.838 A1
3
Question 7Structured · 4 marks
(a)

Prove the identity 1 − cos 2θsin 2θ ≡ tan θ.

[3]
(b)

Hence find the exact value of 1 − cos(π/3)sin(π/3).

[1]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)shown
1 − cos 2θ = 2 sin2θ M1
sin 2θ = 2 sin θ cos θ M1
simplify to tan θ A1
3
(b)1√3
tan π6 = 1√3 B1
1
Question 8Structured · 3 marks

Solve 2x + 1 = 5x, giving your answer correct to 3 significant figures.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
0.756
take logs M1
x(ln 5 − ln 2) = ln 2 A1
0.756 A1
3
Question 9Structured · 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 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 = 330.4°, 103.3°
sin(θ + α) = c/R M1
One correct angle A1
Second correct angle A1
3
Question 10Structured · 4 marks

Solve the equation 22x − 17(2x) + 16 = 0.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
x = 0, x = 4
u = 2x M1
u2 − 17u + 16 = 0 A1
u = 1, u = 16 A1
x = 0, x = 4 A1
4

Other AS Level Mathematics topics

Pure Mathematics 1MechanicsProbability & Statistics 1

Frequently asked questions

What is in AS Level Mathematics Pure Mathematics 2?

In the 9709 syllabus, Pure Mathematics 2 covers Algebra, Logarithmic & exponential functions, Trigonometry, Differentiation, Integration, Numerical solution of equations.

Where can I get Pure Mathematics 2 questions with mark schemes?

This page has 10 Pure Mathematics 2 questions with answers. For more, the free generator on mentorriyaz.in builds a new Pure Mathematics 2 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