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Learn/ A Level Mathematics/ Past Papers/ 2023/ Oct/Nov 2023 /Paper 3 (Variant 1)

Pure Mathematics 3 — Variant 1

A Level Mathematics · 9709/31 · Oct/Nov 2023

Question Paper

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Mark Scheme

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Questions

4 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 [5]
Find the exact coordinates of the points on the curve y=x21−3xy = \frac{x^{2}}{1 - 3x} at which the gradient of the tangent is equal to 8.
2 [4]
On an Argand diagram, shade the region whose points represent complex numbers zz satisfying the inequalities ∣z−2i∣⩽∣z+2−i∣|z - 2i| \leqslant |z + 2 - i| and 0⩽arg⁡(z+1)⩽14π0 \leqslant \arg(z + 1) \leqslant \frac{1}{4}\pi.
4
The complex number uu is defined by u=3+2ia−5iu = \frac{3 + 2i}{a - 5i}, where aa is real.
4(a) [3]
Express uu in the Cartesian form x+iyx + iy, where xx and yy are in terms of aa.
4(b) [2]
Given that arg⁡u=14π\arg u = \frac{1}{4}\pi, find the value of aa.
10
Let f(x)=24x+13(1−2x)(2+x)2\mathrm{f}(x) = \frac{24x + 13}{(1 - 2x)(2 + x)^{2}}.
10(a) [5]
Express f(x)\mathrm{f}(x) in partial fractions.
10(b) [5]
Hence obtain the expansion of f(x)\mathrm{f}(x) in ascending powers of xx, up to and including the term in x2x^{2}.
10(c) [1]
State the set of values of xx for which the expansion in (b) is valid.

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