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Learn/ A Level Mathematics/ Past Papers/ 2024/ Feb/March 2024 /Paper 2 (Variant 2)

Pure Mathematics 2 — Variant 2

A Level Mathematics · 9709/22 · Feb/March 2024

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 [4]
Use logarithms to solve the equation 34x+3=52x+73^{4x+3} = 5^{2x+7}. Give your answer correct to 3 significant figures.
2
2(a) [2]
Sketch the graph of y=∣3x−7∣y = |3x - 7|, stating the coordinates of the points where the graph meets the axes.
2(b) [2]
Hence find the set of values of the constant kk for which the equation ∣3x−7∣=k(x−4)|3x - 7| = k(x - 4) has exactly two real roots.
3
The polynomial p(x)p(x) is defined by p(x)=6x3+ax2+3x−10,p(x) = 6x^{3} + ax^{2} + 3x - 10, where aa is a constant. It is given that (2x−1)(2x - 1) is a factor of p(x)p(x).
3(a) [5]
Find the value of aa and hence factorise p(x)p(x) completely.
3(b) [2]
Solve the equation p(cosec⁡θ)=0p(\operatorname{cosec}\theta) = 0 for −90∘<θ<90∘-90^\circ < \theta < 90^\circ.
7
7(a) [4]
Prove that sin⁡2θ(acot⁡θ+btan⁡θ)≡a+b+(a−b)cos⁡2θ,\sin 2\theta (a\cot\theta + b\tan\theta) \equiv a + b + (a - b)\cos 2\theta, where aa and bb are constants.
7(b) [3]
Find the exact value of ∫112π16πsin⁡2θ(5cot⁡θ+3tan⁡θ) dθ\int_{\frac{1}{12}\pi}^{\frac{1}{6}\pi} \sin 2\theta (5\cot\theta + 3\tan\theta) \, \mathrm{d}\theta.
7(c) [3]
Solve the equation sin⁡23α(2cot⁡13α+7tan⁡13α)=11\sin\frac{2}{3}\alpha (2\cot\frac{1}{3}\alpha + 7\tan\frac{1}{3}\alpha) = 11 for −π<α<π-\pi < \alpha < \pi.

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