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Learn/ A Level Mathematics/ Past Papers/ 2022/ Feb/March 2022 /Paper 3 (Variant 2)

Pure Mathematics 3 — Variant 2

A Level Mathematics · 9709/32 · Feb/March 2022

Question Paper

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Mark Scheme

PDF · 361 KB · 19 pages

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Questions

10 of 11 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 inequality ∣2x+3∣>3∣x+2∣|2x + 3| > 3|x + 2|.
2 [4]
On a sketch of an Argand diagram, shade the region whose points represent complex numbers zz satisfying the inequalities ∣z+2−3i∣⩽2|z + 2 - 3i| \leqslant 2 and arg⁡z⩽34π\arg z \leqslant \frac{3}{4}\pi.
3 [5]
The variables xx and yy satisfy the equation xny2=Cx^{n}y^{2} = C, where nn and CC are constants. The graph of ln⁡y\ln y against ln⁡x\ln x is a straight line passing through the points (0.31,1.21)(0.31, 1.21) and (1.06,0.91)(1.06, 0.91), as shown in the diagram. Find the value of nn and find the value of CC correct to 2 decimal places.
4 [5]
The parametric equations of a curve are x=1−cos⁡θx = 1 - \cos\theta, y=cos⁡θ−14cos⁡2θy = \cos\theta - \frac{1}{4}\cos 2\theta. Show that dydx=−2sin⁡2(12θ)\frac{\mathrm{d}y}{\mathrm{d}x} = -2\sin^{2}(\frac{1}{2}\theta).
5 [6]
The angles α\alpha and β\beta lie between 0∘0^\circ and 180∘180^\circ and are such that tan⁡(α+β)=2\tan(\alpha + \beta) = 2 and tan⁡α=3tan⁡β\tan \alpha = 3\tan \beta. Find the possible values of α\alpha and β\beta.
6 [6]
Find the complex numbers ww which satisfy the equation w2+2iw∗=1w^{2} + 2iw^{*} = 1 and are such that Re w⩽0\mathrm{Re}\, w \leqslant 0. Give your answers in the form x+iyx + iy, where xx and yy are real.
7
7(a) [2]
By sketching a suitable pair of graphs, show that the equation 4−x2=sec⁡12x4 - x^{2} = \sec\frac{1}{2}x has exactly one root in the interval 0⩽x<π0 \leqslant x < \pi.
7(b) [2]
Verify by calculation that this root lies between 11 and 22.
7(c) [3]
Use the iterative formula xn+1=4−sec⁡12xnx_{n+1} = \sqrt{4 - \sec\frac{1}{2}x_{n}} to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
8
8(a) [3]
Find the quotient and remainder when 8x3+4x2+2x+78x^{3} + 4x^{2} + 2x + 7 is divided by 4x2+14x^{2} + 1.
8(b) [5]
Hence find the exact value of ∫0128x3+4x2+2x+74x2+1 dx\int_{0}^{\frac{1}{2}} \frac{8x^{3} + 4x^{2} + 2x + 7}{4x^{2} + 1} \, \mathrm{d}x.
9 [9]
The variables xx and yy satisfy the differential equation (x+1)(3x+1)dydx=y(x + 1)(3x + 1)\frac{\mathrm{d}y}{\mathrm{d}x} = y, and it is given that y=1y = 1 when x=1x = 1. Solve the differential equation and find the exact value of yy when x=3x = 3, giving your answer in a simplified form.
10
The points AA and BB have position vectors 2i+j+k2\mathbf{i} + \mathbf{j} + \mathbf{k} and i−2j+2k\mathbf{i} - 2\mathbf{j} + 2\mathbf{k} respectively. The line ll has vector equation r=i+2j−3k+μ(i−3j−2k)\mathbf{r} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} + \mu(\mathbf{i} - 3\mathbf{j} - 2\mathbf{k}).
10(a) [3]
Find a vector equation for the line through AA and BB.
10(b) [3]
Find the acute angle between the directions of ABAB and ll, giving your answer in degrees.
10(c) [4]
Show that the line through AA and BB does not intersect the line ll.

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