A Level · Mathematics 9709 · P3

A Level Mathematics Paper 3 · Pure Mathematics 3 practice papers

75 marks · 1h 50m. Each generated paper follows the 9709 Paper 3 · Pure Mathematics 3 format, with the same number and style of questions as the real exam, and comes with a Cambridge-style mark scheme.

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Sample Paper 3 · Pure Mathematics 3 questions

Question 1Structured · 12 marksA* challenge
(a)

Find the exact value of ∫01 ln(1 + x)1 + x2 dx.

[6]
(b)

Find the exact value of ∫0π x sin x1 + cos2 x dx.

[6]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)π8 ln 2
x = tan θ giving ∫0π/4 ln(1 + tan θ) dθ M1
correct limits and dx = sec2 θ dθ cancelling A1
θ = π4 − φ: 1 + tan(π4 − φ) = 21 + tan φ M1
I = ∫0π/4 (ln 2 − ln(1 + tan φ)) dφ A1
2I = π4 ln 2 M1
I = π8 ln 2 A1
6
(b)π24
substitution x = π − u M1
integrand becomes (π − u) sin u1 + cos2 u A1
2I = π∫0π sin x1 + cos2 x dx M1
substitution v = cos x giving ∫−11 11 + v2 dv M1
= π2 A1
I = π24 A1
6
Question 2Structured · 12 marksA* challenge

In this question m is a real constant.

(a)

Solve the inequality |2x − 3| < 1 − ½x.

[3]
(b)

Find the set of values of m for which the equation |2x − 3| = m|x| + 1 has exactly two distinct real roots.

[6]
(c)

Find the value of m for which the equation has two roots whose product is −1, and find these roots exactly.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)43 < x < 85
solves 3 − 2x = 1 − ½x or 2x − 3 = 1 − ½x M1
critical values 43 and 85 A1
43 < x < 85 A1
3
(b)−23 < m < 2 or m > 2
considers the three intervals x < 0, 0 ≤ x < 32, x ≥ 32 (or graphs of y = |2x − 3| and y = m|x| + 1) M1
x = 22 − m, valid (negative) only when m > 2 A1
x = 2m + 2, valid when m > −23 A1
x = 42 − m, valid when −23 ≤ m < 2 A1
m = 2 gives only x = ½; m = −23 gives only x = 32 M1
−23 < m < 2 or m > 2 A1
6
(c)m = 2√2; x = √2 − 1 and −(√2 + 1)
product negative so m > 2: 2m + 2 × 22 − m = −1 M1
m2 = 8, m = 2√2 A1
x = √2 − 1 and −1 − √2 A1
3
Question 3Structured · 3 marks

The equation x3 − 4x − 2 = 0 has a root α. Use the iterative formula xn+1 = ∛(4xn + 2) 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 = 2.214 (iterates 2.15443, 2.19791, 2.20985, 2.21310…)
Uses iteration correctly at least once M1
Iterates to sufficient accuracy M1
2.214 A1
3
Question 4Structured · 5 marks

The complex numbers u and w are given by u = −5 + 2i and w = −3 + 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)7 − 26i
Expand with i2 = −1 M1
Correct answer A1
2
(b)|u| = 5.39, arg u = 2.76
Modulus B1
Argument method in correct quadrant M1
2.76 A1
3
Question 5Structured · 5 marks

Express 7x2 + 2x + 3(x + 1)(x2 + 1) in partial fractions.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
4x + 1 + 3x − 1x2 + 1
correct form A/(x + 1) + (Bx + C)/(x2 + 1) B1
equate numerators/substitute x = −1 M1
A = 4 A1
B = 3 A1
C = −1 A1
5
Question 6Structured · 4 marks

Find the exact value of ∫0π6 sin2(3x) dx.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
π12
sin2 = ½(1 − cos 2θ) M1
½x − 112 sin 6x A1
limits M1
π12 A1
4

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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