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Learn/ A Level Mathematics/ Past Papers/ 2023/ Feb/March 2023 /Paper 2 (Variant 2)

Pure Mathematics 2 — Variant 2

A Level Mathematics · 9709/22 · Feb/March 2023

Question Paper

PDF · 110 KB · 16 pages

Mark Scheme

PDF · 281 KB · 11 pages

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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]
Find the exact value of ∫012π2tan⁡2(12x) dx\int_{0}^{\frac{1}{2}\pi} 2\tan^{2}(\frac{1}{2}x) \, \mathrm{d}x.
2 [5]
Solve the equation tan⁡(θ−60∘)=3cot⁡θ\tan(\theta - 60^\circ) = 3\cot\theta for −90∘<θ<90∘-90^{\circ} < \theta < 90^{\circ}.
4
4(a) [2]
Sketch, on the same diagram, the graphs of y=∣2x−11∣y = |2x - 11| and y=3x−3y = 3x - 3.
4(b) [3]
Solve the inequality ∣2x−11∣<3x−3|2x - 11| < 3x - 3.
4(c) [2]
Find the smallest integer NN satisfying the inequality ∣2ln⁡N−11∣<3ln⁡N−3|2\ln N - 11| < 3\ln N - 3.
5
It is given that ∫1a(41+2x+3x) dx=ln⁡10\int_{1}^{a} ( \frac{4}{1+2x} + \frac{3}{x} ) \, \mathrm{d}x = \ln 10, where aa is a constant greater than 1.
5(a) [5]
Show that a=90(1+2a)−23a = \sqrt[3]{90(1+2a)^{-2}}.
5(b) [3]
Use an iterative formula, based on the equation in (a), to find the value of aa correct to 3 significant figures. Use an initial value of 1.7 and give the result of each iteration to 5 significant figures.

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