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Learn/ A Level Mathematics/ Past Papers/ 2023/ Oct/Nov 2023 /Paper 3 (Variant 2)

Pure Mathematics 3 — Variant 2

A Level Mathematics · 9709/32 · Oct/Nov 2023

Question Paper

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Mark Scheme

PDF · 568 KB · 24 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
1(a) [1]
Sketch the graph of y=∣4x−2∣y = |4x - 2|.
1(b) [4]
Solve the inequality 1+3x<∣4x−2∣1 + 3x < |4x - 2|.
2 [4]
The parametric equations of a curve are x=(ln⁡t)2x = (\ln t)^{2}, y=e2−t2y = e^{2-t^{2}}, for t>0t > 0. Find the gradient of the curve at the point where t=et = \mathrm{e}, simplifying your answer.
3 [5]
The polynomial 2x3+ax2−11x+b2x^{3} + ax^{2} - 11x + b is denoted by p(x)p(x). It is given that p(x)p(x) is divisible by (2x−1)(2x - 1) and that when p(x)p(x) is divided by (x+1)(x + 1) the remainder is 1212. Find the values of aa and bb.
4
4(a) [4]
On a sketch of an Argand diagram, shade the region whose points represent complex numbers zz satisfying the inequalities ∣z−4−3i∣⩽2|z - 4 - 3\mathrm{i}| \leqslant 2 and Re⁡z⩽3\operatorname{Re} z \leqslant 3.
4(b) [2]
Find the greatest value of arg⁡z\arg z for points in this region.
5 [6]
Find the exact value of ∫06x(x+1)x2+4 dx\int_{0}^{6} \frac{x(x+1)}{x^{2}+4} \, \mathrm{d}x.
6
6(a) [2]
By sketching a suitable pair of graphs, show that the equation cot⁡x=2−cos⁡x\cot x = 2 - \cos x has one root in the interval 0<x⩽12π0 < x \leqslant \frac{1}{2}\pi.
6(b) [2]
Show by calculation that this root lies between 0.60.6 and 0.80.8.
6(c) [3]
Use the iterative formula xn+1=tan⁡−1(12−cos⁡xn)x_{n+1} = \tan^{-1}(\frac{1}{2-\cos x_{n}}) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
7
7(a) [3]
By expressing 3θ3\theta as 2θ+θ2\theta + \theta, prove the identity cos⁡3θ≡4cos⁡3θ−3cos⁡θ\cos 3\theta \equiv 4\cos^{3}\theta - 3\cos\theta.
7(b) [5]
Hence solve the equation cos⁡3θ+cos⁡θcos⁡2θ=cos⁡2θ\cos 3\theta + \cos\theta\cos 2\theta = \cos^{2}\theta for 0∘⩽θ⩽180∘0^\circ \leqslant \theta \leqslant 180^\circ.
8
It is given that 2+3aia+2i=λ(2−i)\frac{2 + 3a\mathrm{i}}{a + 2\mathrm{i}} = \lambda(2 - \mathrm{i}), where aa and λ\lambda are real constants.
8(a) [4]
Show that 3a2+4a−4=03a^{2} + 4a - 4 = 0.
8(b) [3]
Hence find the possible values of aa and the corresponding values of λ\lambda.
10
The equations of the lines ll and mm are given by l:r=(3−21)+λ(112)l: \mathbf{r} = \begin{pmatrix} 3 \\ -2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix} and m:r=(6−36)+μ(−24c)m: \mathbf{r} = \begin{pmatrix} 6 \\ -3 \\ 6 \end{pmatrix} + \mu \begin{pmatrix} -2 \\ 4 \\ c \end{pmatrix}, where cc is a positive constant. It is given that the angle between ll and mm is 60∘60^\circ.
10(a) [4]
Find the value of cc.
10(b) [5]
Show that the length of the perpendicular from (6,−3,6)(6, -3, 6) to ll is 11\sqrt{11}.
11
The variables xx and yy satisfy the differential equation x2dydx+y2+y=0x^{2}\frac{\mathrm{d}y}{\mathrm{d}x} + y^{2} + y = 0. It is given that x=1x = 1 when y=1y = 1.
11(a) [8]
Solve the differential equation to obtain an expression for yy in terms of xx.
11(b) [1]
State what happens to the value of yy when xx tends to infinity. Give your answer in an exact form.

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