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Learn/ A Level Mathematics/ Past Papers/ 2022/ Oct/Nov 2022 /Paper 3 (Variant 1)

Pure Mathematics 3 — Variant 1

A Level Mathematics · 9709/31 · Oct/Nov 2022

Question Paper

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Mark Scheme

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Questions

8 of 11 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=∣2x+1∣y = |2x + 1|.
1(b) [3]
Solve the inequality 3x+5<∣2x+1∣3x + 5 < |2x + 1|.
2 [4]
On a sketch of an Argand diagram shade the region whose points represent complex numbers zz satisfying the inequalities ∣z∣⩽3|z| \leqslant 3, Re⁡z⩾−2\operatorname{Re} z \geqslant -2 and 14π⩽arg⁡z⩽π\frac{1}{4}\pi \leqslant \arg z \leqslant \pi.
3 [4]
Solve the equation 23x−1=5(3−x)2^{3x-1} = 5(3^{-x}). Give your answer in the form ln⁡aln⁡b\frac{\ln a}{\ln b}, where aa and bb are integers.
4 [5]
Solve the equation tan⁡(x+45∘)=2cot⁡x\tan(x + 45^\circ) = 2\cot x for 0∘<x<180∘0^\circ < x < 180^\circ.
5
The complex numbers uu and ww are defined by u=2e14πiu = 2e^{\frac{1}{4}\pi i} and w=3e13πiw = 3e^{\frac{1}{3}\pi i}.
5(a) [3]
Find u2w\frac{u^{2}}{w}, giving your answer 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.
5(b) [1]
State the least positive integer nn such that both Im⁡wn=0\operatorname{Im} w^{n} = 0 and Re⁡wn>0\operatorname{Re} w^{n} > 0.
6
6(a) [4]
Prove the identity cos⁡4θ+4cos⁡2θ+3≡8cos⁡4θ\cos 4\theta + 4\cos 2\theta + 3 \equiv 8\cos^{4}\theta.
6(b) [3]
Hence solve the equation cos⁡4θ+4cos⁡2θ=4\cos 4\theta + 4\cos 2\theta = 4 for 0∘⩽θ⩽180∘0^\circ \leqslant \theta \leqslant 180^\circ.
8
In a certain chemical reaction the amount, xx grams, of a substance is increasing. The differential equation satisfied by xx and tt, the time in seconds since the reaction began, is dxdt=kxe−0.1t,\frac{\mathrm{d}x}{\mathrm{d}t} = kxe^{-0.1t}, where kk is a positive constant. It is given that x=20x = 20 at the start of the reaction.
8(a) [5]
Solve the differential equation, obtaining a relation between xx, tt and kk.
8(b) [3]
Given that x=40x = 40 when t=10t = 10, find the value of kk and find the value approached by xx as tt becomes large.
10
Let f(x)=2x2+7x+8(1+x)(2+x)2f(x) = \frac{2x^{2} + 7x + 8}{(1 + x)(2 + x)^{2}}.
10(a) [5]
Express f(x)f(x) in partial fractions.
10(b) [5]
Hence obtain the expansion of f(x)f(x) in ascending powers of xx, up to and including the term in x2x^{2}.

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