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Learn/ A Level Mathematics/ Past Papers/ 2023/ May/June 2023 /Paper 3 (Variant 2)

Pure Mathematics 3 — Variant 2

A Level Mathematics · 9709/32 · May/June 2023

Question Paper

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Mark Scheme

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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 ∣5x−3∣<2∣3x−7∣|5x - 3| < 2|3x - 7|.
2 [3]
Solve the equation ln⁡(2x2−3)=2ln⁡x−ln⁡2\ln(2x^{2} - 3) = 2\ln x - \ln 2, giving your answer in an exact form.
3
3(a) [2]
On an Argand diagram, sketch the locus of points representing complex numbers zz satisfying ∣z+3−2i∣=2|z + 3 - 2i| = 2.
3(b) [2]
Find the least value of ∣z∣|z| for points on this locus, giving your answer in an exact form.
4 [5]
Solve the equation 2cos⁡x−cos⁡12x=12\cos x - \cos \frac{1}{2}x = 1 for 0⩽x⩽2π0 \leqslant x \leqslant 2\pi.
5
The complex number 2+yi2 + yi is denoted by aa, where yy is a real number and y<0y < 0. It is given that f(a)=a3−a2−2af(a) = a^{3} - a^{2} - 2a.
5(a) [3]
Find a simplified expression for f(a)f(a) in terms of yy.
5(b) [3]
Given that Re(f(a))=−20\mathrm{Re}(f(a)) = -20, find arg⁡a\arg a.
6
The equation cot⁡12x=3x\cot \frac{1}{2}x = 3x has one root in the interval 0<x<π0 < x < \pi, denoted by α\alpha.
6(a) [2]
Show by calculation that α\alpha lies between 0.50.5 and 11.
6(b) [2]
Show that, if a sequence of positive values given by the iterative formula xn+1=13(xn+4tan⁡−1(13xn))x_{n+1} = \frac{1}{3}(x_{n} + 4\tan^{-1}(\frac{1}{3x_{n}})) converges, then it converges to α\alpha.
6(c) [3]
Use this iterative formula to calculate α\alpha correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
7
The equation of a curve is 3x2+4xy+3y2=53x^{2} + 4xy + 3y^{2} = 5.
7(a) [4]
Show that dydx=−3x+2y2x+3y\frac{\mathrm{d}y}{\mathrm{d}x} = -\frac{3x + 2y}{2x + 3y}.
7(b) [5]
Hence find the exact coordinates of the two points on the curve at which the tangent is parallel to y+2x=0y + 2x = 0.
8
8(a) [7]
The variables xx and yy satisfy the differential equation dydx=4+9y2e2x+1\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{4 + 9y^{2}}{e^{2x+1}}. It is given that y=0y = 0 when x=1x = 1. Solve the differential equation, obtaining an expression for yy in terms of xx.
8(b) [1]
State what happens to the value of yy as xx tends to infinity. Give your answer in an exact form.
9
Let f(x)=2x2+17x−17(1+2x)(2−x)2f(x) = \frac{2x^{2} + 17x - 17}{(1 + 2x)(2 - x)^{2}}.
9(a) [5]
Express f(x)f(x) in partial fractions.
9(b) [5]
Hence show that ∫01f(x) dx=52−ln⁡72\int_{0}^{1} f(x) \, \mathrm{d}x = \frac{5}{2} - \ln 72.
11
The points AA and BB have position vectors i+2j−2k\mathbf{i} + 2\mathbf{j} - 2\mathbf{k} and 2i−j+k2\mathbf{i} - \mathbf{j} + \mathbf{k} respectively. The line ll has equation r=i−j+3k+μ(2i−3j+4k)\mathbf{r} = \mathbf{i} - \mathbf{j} + 3\mathbf{k} + \mu(2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}).
11(a) [5]
Show that ll does not intersect the line passing through AA and BB.
11(b) [4]
Find the position vector of the foot of the perpendicular from AA to ll.

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