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Learn/ A Level Mathematics/ Past Papers/ 2022/ May/June 2022 /Paper 3 (Variant 1)

Pure Mathematics 3 — Variant 1

A Level Mathematics · 9709/31 · May/June 2022

Question Paper

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Mark Scheme

PDF · 355 KB · 17 pages

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Questions

6 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 [4]
Solve the equation 2(32x−1)=4x+12(3^{2x-1}) = 4^{x+1}, giving your answer correct to 2 decimal places.
2
2(a) [4]
Expand (2−x2)−2(2 - x^{2})^{-2} in ascending powers of xx, up to and including the term in x4x^{4}, simplifying the coefficients.
2(b) [1]
State the set of values of xx for which the expansion is valid.
3 [6]
Solve the equation 2cot⁡2x+3cot⁡x=52\cot 2x + 3\cot x = 5, for 0∘<x<180∘0^\circ < x < 180^\circ.
4 [7]
The variables xx and yy satisfy the differential equation dydx=xy1+x2\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{xy}{1 + x^{2}}, and y=2y = 2 when x=0x = 0. Solve the differential equation, obtaining a simplified expression for yy in terms of xx.
5
The polynomial ax3−10x2+bx+8ax^{3} - 10x^{2} + bx + 8, where aa and bb are constants, is denoted by p(x)\mathrm{p}(x). It is given that (x−2)(x - 2) is a factor of both p(x)\mathrm{p}(x) and p′(x)\mathrm{p}'(x).
5(a) [5]
Find the values of aa and bb.
5(b) [3]
When aa and bb have these values, factorise p(x)\mathrm{p}(x) completely.
6
Let I=∫0327(9+x2)2 dxI = \int_{0}^{3} \frac{27}{(9 + x^{2})^{2}} \, \mathrm{d}x.
6(a) [4]
Using the substitution x=3tan⁡θx = 3\tan\theta, show that I=∫014πcos⁡2θ dθI = \int_{0}^{\frac{1}{4}\pi} \cos^{2}\theta \, \mathrm{d}\theta.
6(b) [4]
Hence find the exact value of II.

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