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Learn/ A Level Mathematics/ Past Papers/ 2022/ May/June 2022 /Paper 2 (Variant 3)

Pure Mathematics 2 — Variant 3

A Level Mathematics · 9709/23 · May/June 2022

Question Paper

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Mark Scheme

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Questions

6 of 8 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 [3]
Given that y=ln⁡xx2y = \frac{\ln x}{x^{2}}, find the exact value of dydx\frac{\mathrm{d}y}{\mathrm{d}x} when x=ex = \mathrm{e}.
2
2(a) [2]
Sketch, on the same diagram, the graphs of y=∣2x−9∣y = |2x - 9| and y=5x−3y = 5x - 3.
2(b) [2]
Solve the equation ∣2x−9∣=5x−3|2x - 9| = 5x - 3.
3 [5]
A curve has equation e2xcos⁡2y+sin⁡y=1\mathrm{e}^{2x} \cos 2y + \sin y = 1. Find the exact gradient of the curve at the point (0,16π)(0, \frac{1}{6}\pi).
4
4(a) [4]
Use the trapezium rule with three intervals to show that the value of ∫14ln⁡x dx\int_{1}^{4} \ln x \, \mathrm{d}x is approximately ln⁡12\ln 12.
4(b) [2]
Use a graph of y=ln⁡xy = \ln x to show that ln⁡12\ln 12 is an under-estimate of the true value of ∫14ln⁡x dx\int_{1}^{4} \ln x \, \mathrm{d}x.
5
The polynomial p(x)p(x) is defined by p(x)=2x3+ax2−3x−4,p(x) = 2x^{3} + ax^{2} - 3x - 4, where aa is a constant. It is given that (x−4)(x - 4) is a factor of p(x)p(x).
5(a) [4]
Find the value of aa and hence factorise p(x)\mathrm{p}(x).
5(b) [3]
Show that the equation p(e3y)=0\mathrm{p}(\mathrm{e}^{3y}) = 0 has only one real root and find its exact value.
8
8(a) [6]
Express 3sin⁡2θsec⁡θ+10cos⁡(θ−30∘)3\sin 2\theta \sec \theta + 10\cos(\theta - 30^\circ) in the form Rsin⁡(θ+α)R\sin(\theta + \alpha) where R>0R > 0 and 0∘<α<90∘0^\circ < \alpha < 90^\circ. Give the value of α\alpha correct to 2 decimal places.
8(b) [3]
Hence solve the equation 3sin⁡4βsec⁡2β+10cos⁡(2β−30∘)=23\sin 4\beta \sec 2\beta + 10\cos(2\beta - 30^\circ) = 2 for 0∘<β<90∘0^\circ < \beta < 90^\circ.

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