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

Pure Mathematics 3 — Variant 3

A Level Mathematics · 9709/33 · May/June 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]
Expand (1+3x)23(1 + 3x)^{\frac{2}{3}} in ascending powers of xx, up to and including the term in x3x^{3}, simplifying the coefficients.
2 [5]
Solve the equation 4x=3+4−x4^{x} = 3 + 4^{-x}. Give your answer correct to 3 decimal places.
3
The parametric equations of a curve are x=t+ln⁡(t+2)x = t + \ln(t + 2), y=(t−1)e−2ty = (t - 1)\mathrm{e}^{-2t}, where t>−2t > -2.
3(a) [5]
Express dydx\frac{\mathrm{d}y}{\mathrm{d}x} in terms of tt, simplifying your answer.
3(b) [2]
Find the exact yy-coordinate of the stationary point of the curve.
4
Let f(x)=15−6x(1+2x)(4−x)\mathrm{f}(x) = \frac{15 - 6x}{(1 + 2x)(4 - x)}.
4(a) [3]
Express f(x)\mathrm{f}(x) in partial fractions.
4(b) [4]
Hence find ∫12f(x) dx\int_{1}^{2} \mathrm{f}(x) \, \mathrm{d}x, giving your answer in the form ln⁡(ab)\ln\left(\frac{a}{b}\right), where aa and bb are integers.
5
5(a) [4]
By first expanding tan⁡(2θ+2θ)\tan(2\theta + 2\theta), show that the equation tan⁡4θ=12tan⁡θ\tan 4\theta = \frac{1}{2}\tan\theta may be expressed as tan⁡4θ+2tan⁡2θ−7=0\tan^{4}\theta + 2\tan^{2}\theta - 7 = 0.
5(b) [3]
Hence solve the equation tan⁡4θ=12tan⁡θ\tan 4\theta = \frac{1}{2}\tan\theta, for 0∘<θ<180∘0^\circ < \theta < 180^\circ.
6
6(a) [2]
By sketching a suitable pair of graphs, show that the equation cot⁡12x=1+e−x\cot \frac{1}{2}x = 1 + \mathrm{e}^{-x} has exactly one root in the interval 0<x⩽π0 < x \leqslant \pi.
6(b) [2]
Verify by calculation that this root lies between 1 and 1.5.
6(c) [3]
Use the iterative formula xn+1=2tan⁡−1(11+e−xn)x_{n+1} = 2\tan^{-1}\left(\frac{1}{1 + \mathrm{e}^{-x_{n}}}\right) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
9
The quadrilateral ABCDABCD is a trapezium in which ABAB and DCDC are parallel. With respect to the origin OO, the position vectors of AA, BB and CC are given by OA→=−i+2j+3k\overrightarrow{OA} = -\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}, OB→=i+3j+k\overrightarrow{OB} = \mathbf{i} + 3\mathbf{j} + \mathbf{k} and OC→=2i+2j−3k\overrightarrow{OC} = 2\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}.
9(a) [3]
Given that DC→=3AB→\overrightarrow{DC} = 3\overrightarrow{AB}, find the position vector of DD.
9(b) [1]
State a vector equation for the line through AA and BB.
9(c) [5]
Find the distance between the parallel sides and hence find the area of the trapezium.
10
10(a) [3]
Verify that −1+2i-1 + \sqrt{2}i is a root of the equation z4+3z2+2z+12=0z^{4} + 3z^{2} + 2z + 12 = 0.
10(b) [7]
Find the other roots of this equation.

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