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Learn/ A Level Mathematics/ Past Papers/ 2020/ May/June 2020 /Paper 2 (Variant 3)

Pure Mathematics 2 — Variant 3

A Level Mathematics · 9709/23 · May/June 2020

Question Paper

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Mark Scheme

PDF · 290 KB · 12 pages

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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 2y=93x2^{y} = 9^{3x}, use logarithms to show that y=kxy = kx and find the value of kk correct to 3 significant figures.
2 [5]
Find the exact coordinates of the stationary point on the curve with equation y=5xe12xy = 5xe^{\frac{1}{2}x}.
3 [5]
The equation of a curve is cos⁡3x+5sin⁡y=3\cos 3x + 5\sin y = 3. Find the gradient of the curve at the point (19π,16π)(\frac{1}{9}\pi, \frac{1}{6}\pi).
5
5(a) [2]
Sketch, on the same diagram, the graphs of y=∣2x−3∣y = |2x - 3| and y=3x+5y = 3x + 5.
5(b) [3]
Solve the inequality 3x+5<∣2x−3∣3x + 5 < |2x - 3|.
6
The polynomial p(x)\mathrm{p}(x) is defined by p(x)=6x3+ax2−4x−3,\mathrm{p}(x) = 6x^{3} + ax^{2} - 4x - 3, where aa is a constant. It is given that (x+3)(x + 3) is a factor of p(x)\mathrm{p}(x).
6(a) [2]
Find the value of aa.
6(b) [3]
Using this value of aa, factorise p(x)\mathrm{p}(x) completely.
6(c) [3]
Hence solve the equation p(cosec⁡θ)=0\mathrm{p}(\operatorname{cosec}\theta) = 0 for 0∘<θ<360∘0^\circ < \theta < 360^\circ.
8
8(a) [2]
Show that 3sin⁡2θcot⁡θ≡6cos⁡2θ3\sin 2\theta \cot\theta \equiv 6\cos^{2}\theta.
8(b) [3]
Solve the equation 3sin⁡2θcot⁡θ=53\sin 2\theta \cot\theta = 5 for 0<θ<π0 < \theta < \pi.
8(c) [5]
Find the exact value of ∫14π12π3sin⁡xcot⁡12x dx\int_{\frac{1}{4}\pi}^{\frac{1}{2}\pi} 3\sin x \cot \frac{1}{2}x \, \mathrm{d}x.

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