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

Pure Mathematics 3 — Variant 3

A Level Mathematics · 9709/33 · Oct/Nov 2023

Question Paper

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Mark Scheme

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Questions

9 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]
Find the set of values of xx satisfying the inequality ∣2x+1−2∣<0.5\left| 2^{x+1} - 2 \right| < 0.5, giving your answer to 3 significant figures.
2 [5]
On an Argand diagram, shade the region whose points represent complex numbers zz satisfying the inequalities ∣z−1+2i∣⩽∣z∣|z - 1 + 2\mathrm{i}| \leqslant |z| and ∣z−2∣⩽1|z - 2| \leqslant 1.
3 [5]
The polynomial 2x3+ax2+bx+62x^{3} + ax^{2} + bx + 6, where aa and bb are constants, is denoted by p(x)\mathrm{p}(x). When p(x)\mathrm{p}(x) is divided by (x+2)(x + 2) the remainder is −38-38 and when p(x)\mathrm{p}(x) is divided by (2x−1)(2x - 1) the remainder is 192\frac{19}{2}. Find the values of aa and bb.
4 [5]
Solve the quadratic equation (3+i)w2−2w+3−i=0(3 + \mathrm{i})w^{2} - 2w + 3 - \mathrm{i} = 0, giving your answers in the form x+iyx + \mathrm{i}y, where xx and yy are real.
5 [6]
Find the exact coordinates of the stationary points of the curve y=e3x2−11−x2y = \dfrac{\mathrm{e}^{3x^{2}-1}}{1 - x^{2}}.
6
6(a) [3]
Show that the equation cot⁡2θ+2cos⁡2θ=4\cot^{2}\theta + 2\cos 2\theta = 4 can be written in the form 4sin⁡4θ+3sin⁡2θ−1=04\sin^{4}\theta + 3\sin^{2}\theta - 1 = 0.
6(b) [3]
Hence solve the equation cot⁡2θ+2cos⁡2θ=4\cot^{2}\theta + 2\cos 2\theta = 4, for 0∘<θ<360∘0^\circ < \theta < 360^\circ.
7
The equation of a curve is x3+y2+3x2+3y=4x^{3} + y^{2} + 3x^{2} + 3y = 4.
7(a) [3]
Show that dydx=−3x2+6x2y+3\dfrac{\mathrm{d}y}{\mathrm{d}x} = -\dfrac{3x^{2} + 6x}{2y + 3}.
7(b) [5]
Hence find the coordinates of the points on the curve at which the tangent is parallel to the xx-axis.
8 [7]
The variables xx and yy satisfy the differential equation e4xdydx=cos⁡23y.\mathrm{e}^{4x} \dfrac{\mathrm{d}y}{\mathrm{d}x} = \cos^{2} 3y. It is given that y=0y = 0 when x=2x = 2. Solve the differential equation, obtaining an expression for yy in terms of xx.
9
Let f(x)=17x2−7x+16(2+3x2)(2−x)\mathrm{f}(x) = \dfrac{17x^{2} - 7x + 16}{(2 + 3x^{2})(2 - x)}.
9(a) [5]
Express f(x)\mathrm{f}(x) in partial fractions.
9(b) [5]
Hence obtain the expansion of f(x)\mathrm{f}(x) in ascending powers of xx, up to and including the term in x3x^{3}.
9(c) [1]
State the set of values of xx for which the expansion in (b) is valid. Give your answer in an exact form.

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