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Learn/ A Level Mathematics/ Past Papers/ 2021/ Oct/Nov 2021 /Paper 3 (Variant 1)

Pure Mathematics 3 — Variant 1

A Level Mathematics · 9709/31 · Oct/Nov 2021

Question Paper

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Mark Scheme

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Questions

8 of 10 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]
Solve the equation 4∣5x−1∣=5x4|5^{x} - 1| = 5^{x}, giving your answers correct to 3 decimal places.
2
2(a) [3]
Express 5sin⁡x−3cos⁡x5\sin x - 3\cos x in the form Rsin⁡(x−α)R\sin(x - \alpha), where R>0R > 0 and 0<α<12π0 < \alpha < \frac{1}{2}\pi. Give the exact value of RR and give α\alpha correct to 2 decimal places.
2(b) [2]
Hence state the greatest and least possible values of (5sin⁡x−3cos⁡x)2(5\sin x - 3\cos x)^{2}.
3
The curve with equation y=xe1−2xy = x\mathrm{e}^{1-2x} has one stationary point.
3(a) [4]
Find the coordinates of this point.
3(b) [2]
Determine whether the stationary point is a maximum or a minimum.
4 [6]
Using the substitution u=xu = \sqrt{x}, find the exact value of ∫3∞1(x+1)x dx\int_{3}^{\infty} \frac{1}{(x+1)\sqrt{x}} \, \mathrm{d}x.
5
5(a) [3]
Show that the equation cot⁡2θ+cot⁡θ=2\cot 2\theta + \cot \theta = 2 can be expressed as a quadratic equation in tan⁡θ\tan \theta.
5(b) [3]
Hence solve the equation cot⁡2θ+cot⁡θ=2\cot 2\theta + \cot \theta = 2, for 0<θ<π0 < \theta < \pi, giving your answers correct to 3 decimal places.
6 [6]
When (a+bx)1+4x(a + bx)\sqrt{1 + 4x}, where aa and bb are constants, is expanded in ascending powers of xx, the coefficients of xx and x2x^{2} are 33 and −6-6 respectively. Find the values of aa and bb.
8
The constant aa is such that ∫1aln⁡xx dx=6\int_{1}^{a} \frac{\ln x}{\sqrt{x}} \, \mathrm{d}x = 6.
8(a) [5]
Show that a=exp⁡(1a+2)a = \exp(\frac{1}{\sqrt{a}} + 2). [exp⁡(x)\exp(x) is an alternative notation for ex\mathrm{e}^{x}.]
8(b) [2]
Verify by calculation that aa lies between 99 and 1111.
8(c) [3]
Use an iterative formula based on the equation in part (a) to determine aa correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
9
Two lines ll and mm have equations r=3i+2j+5k+s(4i−j+3k)\mathbf{r} = 3\mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + s(4\mathbf{i} - \mathbf{j} + 3\mathbf{k}) and r=i−j−2k+t(−i+2j+2k)\mathbf{r} = \mathbf{i} - \mathbf{j} - 2\mathbf{k} + t(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}) respectively.
9(a) [2]
Show that ll and mm are perpendicular.
9(b) [5]
Show that ll and mm intersect and state the position vector of the point of intersection.
9(c) [4]
Show that the length of the perpendicular from the origin to the line mm is 135\frac{1}{3}\sqrt{5}.

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