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Learn/ A Level Mathematics/ Past Papers/ 2022/ May/June 2022 /Paper 2 (Variant 1)

Pure Mathematics 2 — Variant 1

A Level Mathematics · 9709/21 · May/June 2022

Question Paper

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Mark Scheme

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Questions

4 of 7 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
The variables xx and yy satisfy the equation y=42x−ay = 4^{2x-a}, where aa is an integer. As shown in the diagram, the graph of ln⁡y\ln y against xx is a straight line passing through the point (0,−20.8)(0, -20.8), where the second coordinate is given correct to 3 significant figures.
1(a) [2]
Show that the gradient of the straight line is ln⁡16\ln 16.
1(b) [2]
Determine the value of aa.
2
2(a) [3]
Express the equation 7tan⁡θ+4cot⁡θ−13sec⁡θ=07\tan\theta + 4\cot\theta - 13\sec\theta = 0 in terms of sin⁡θ\sin\theta only.
2(b) [3]
Hence solve the equation 7tan⁡θ+4cot⁡θ−13sec⁡θ=07\tan\theta + 4\cot\theta - 13\sec\theta = 0 for 0∘<θ<360∘0^\circ < \theta < 360^\circ.
3
The diagram shows the curve with equation y=3sin⁡x−3sin⁡2xy = 3\sin x - 3\sin 2x for 0⩽x⩽π0 \leqslant x \leqslant \pi. The curve meets the xx-axis at the origin and at the points with xx-coordinates aa and π\pi.
3(a) [3]
Find the exact value of aa.
3(b) [4]
Find the area of the shaded region.
6
A curve has equation y=9e2x+16ex−1y = \frac{9e^{2x} + 16}{e^{x} - 1}.
6(a) [4]
Show that the xx-coordinate of any stationary point on the curve satisfies the equation\n\nex(3ex−8)(3ex+2)=0e^{x}(3e^{x} - 8)(3e^{x} + 2) = 0.
6(b) [4]
Hence show that the curve has only one stationary point and find its exact coordinates.

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