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Learn/ A Level Mathematics/ Past Papers/ 2024/ May/June 2024 /Paper 3 (Variant 2)

Pure Mathematics 3 — Variant 2

A Level Mathematics · 9709/32 · May/June 2024

Question Paper

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Mark Scheme

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Questions

8 of 10 questions shown here, checked against the paper. The others are in the PDF above — they contain diagrams or maths we could not yet verify, and we would rather show you the original than a transcription we are not sure of.

1
1(a) [1]
Sketch the graph of y=∣x−2a∣y = |x - 2a|, where aa is a positive constant.
1(b) [2]
Solve the inequality 2x−3a<∣x−2a∣2x - 3a < |x - 2a|.
2 [5]
Express 6x2−9x−162x2−5x−12\frac{6x^{2} - 9x - 16}{2x^{2} - 5x - 12} in partial fractions.
3
The variables xx and yy satisfy the equation a2y−1=bx−ya^{2y-1} = b^{x-y}, where aa and bb are constants.
3(a) [3]
Show that the graph of yy against xx is a straight line.
3(b) [2]
Given that a=b3a = b^{3}, state the equation of the straight line in the form y=px+qy = px + q, where pp and qq are rational numbers in their simplest form.
4 [6]
The equation of a curve is ye2x+y2ex=6ye^{2x} + y^{2}e^{x} = 6. Find the gradient of the curve at the point where y=1y = 1.
5
5(a) [2]
It is given that the equation e2x=5+cos⁡3xe^{2x} = 5 + \cos 3x has only one root. Show by calculation that this root lies in the interval 0.7<x<0.80.7 < x < 0.8.
5(b) [1]
Show that if a sequence of values in the interval 0.7<x<0.80.7 < x < 0.8 given by the iterative formula xn+1=12ln⁡(5+cos⁡3xn)x_{n+1} = \frac{1}{2}\ln(5 + \cos 3x_{n}) converges then it converges to the root of the equation in part (a).
5(c) [3]
Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
7
7(a) [3]
Show that cos⁡4θ−sin⁡4θ≡cos⁡2θ\cos^{4}\theta - \sin^{4}\theta \equiv \cos 2\theta.
7(b) [6]
Hence find the exact value of ∫−18π18π(cos⁡4θ−sin⁡4θ+4sin⁡2θcos⁡2θ) dθ\int_{-\frac{1}{8}\pi}^{\frac{1}{8}\pi} (\cos^{4}\theta - \sin^{4}\theta + 4\sin^{2}\theta\cos^{2}\theta) \, \mathrm{d}\theta.
9
The complex numbers zz and ω\omega are defined by z=1−iz = 1 - i and ω=−3+33i\omega = -3 + 3\sqrt{3}i.
9(a) [1]
Express zωz\omega in the form a+bia + bi, where aa and bb are real and in exact surd form.
9(b) [4]
Express zz and ω\omega in the form reiθre^{i\theta}, where r>0r > 0 and −π<θ⩽π-\pi < \theta \leqslant \pi. Give the exact values of rr and θ\theta in each case.
9(c) [2]
On an Argand diagram, the points representing ω\omega and zωz\omega are AA and BB respectively. Prove that OABOAB is an isosceles right-angled triangle, where OO is the origin.
9(d) [3]
Using your answers to part (b), prove that tan⁡512π=3+13−1\tan\frac{5}{12}\pi = \frac{\sqrt{3}+1}{\sqrt{3}-1}.
10
10(a) [2]
By writing y=sec⁡3θy = \sec^{3}\theta as 1cos⁡3θ\frac{1}{\cos^{3}\theta}, show that dydθ=3sin⁡θsec⁡4θ\frac{\mathrm{d}y}{\mathrm{d}\theta} = 3\sin\theta\sec^{4}\theta.
10(b) [8]
The variables xx and θ\theta satisfy the differential equation (x2+9)sin⁡θdθdx=(x+3)cos⁡4θ(x^{2}+9)\sin\theta\frac{\mathrm{d}\theta}{\mathrm{d}x} = (x+3)\cos^{4}\theta. It is given that x=3x = 3 when θ=13π\theta = \frac{1}{3}\pi. Solve the differential equation to find the value of cos⁡θ\cos\theta when x=0x = 0. Give your answer correct to 3 significant figures.

Published by Cambridge Assessment International Education. Reproduced for study use.