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Learn/ A Level Mathematics/ Past Papers/ 2021/ May/June 2021 /Paper 2 (Variant 3)

Pure Mathematics 2 — Variant 3

A Level Mathematics · 9709/23 · May/June 2021

Question Paper

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Mark Scheme

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Questions

5 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
1(a) [3]
Solve the equation ln⁡(2+x)−ln⁡x=2ln⁡3\ln(2 + x) - \ln x = 2 \ln 3.
1(b) [2]
Hence solve the equation ln⁡(2+cot⁡y)−ln⁡(cot⁡y)=2ln⁡3\ln(2 + \cot y) - \ln(\cot y) = 2 \ln 3 for 0<y<12π0 < y < \frac{1}{2}\pi. Give your answer correct to 4 significant figures.
2 [5]
The solutions of the equation 5∣x∣=5−2x5|x| = 5 - 2x are x=ax = a and x=bx = b, where a<ba < b. Find the value of ∣3a−1∣+∣7b−1∣|3a - 1| + |7b - 1|.
3 [6]
Solve the equation sin⁡(2θ+30∘)=5cos⁡(2θ+60∘)\sin(2\theta + 30^\circ) = 5\cos(2\theta + 60^\circ) for 0∘<θ<180∘0^\circ < \theta < 180^\circ.
4
4(a) [3]
Find the exact value of ∫026e2x+1 dx\int_{0}^{2} 6\mathrm{e}^{2x+1} \, \mathrm{d}x.
4(b) [5]
Find ∫(tan⁡2x+4sin⁡22x) dx\int (\tan^{2}x + 4\sin^{2}2x) \, \mathrm{d}x.
5
5(a) [3]
Find the quotient when x4−32x+55x^{4} - 32x + 55 is divided by (x−2)2(x - 2)^{2} and show that the remainder is 77.
5(b) [2]
Factorise x4−32x+48x^{4} - 32x + 48.
5(c) [2]
Hence solve the equation e−12y−32e−3y+48=0\mathrm{e}^{-12y} - 32\mathrm{e}^{-3y} + 48 = 0, giving your answer in an exact form.

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