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Learn/ A Level Mathematics/ Past Papers/ 2023/ Oct/Nov 2023 /Paper 1 (Variant 2)

Pure Mathematics 1 — Variant 2

A Level Mathematics · 9709/12 · Oct/Nov 2023

Question Paper

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Mark Scheme

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Questions

6 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]
The coefficient of x3x^{3} in the expansion of (3+2ax)5(3 + 2ax)^{5} is six times the coefficient of x2x^{2} in the expansion of (2+ax)6(2 + ax)^{6}. Find the value of the constant aa.
2 [2]
Find the exact solution of the equation 16π+tan⁡−1(4x)=−cos⁡−1(123)\frac{1}{6}\pi + \tan^{-1}(4x) = -\cos^{-1}(\frac{1}{2}\sqrt{3}).
3
The equation of a curve is such that dydx=12x+72x4\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{2}x + \frac{72}{x^{4}}. The curve passes through the point P(2,8)P(2, 8).
3(a) [2]
Find the equation of the normal to the curve at PP.
3(b) [4]
Find the equation of the curve.
5
The first, second and third terms of a geometric progression are sin⁡θ\sin\theta, cos⁡θ\cos\theta and 2−sin⁡θ2 - \sin\theta respectively, where θ\theta radians is an acute angle.
5(a) [3]
Find the value of θ\theta.
5(b) [3]
Using this value of θ\theta, find the sum of the first 1010 terms of the progression. Give the answer in the form bc−1\frac{b}{\sqrt{c} - 1}, where bb and cc are integers to be found.
7
7(a) [1]
Verify the identity (2x−1)(4x2+2x−1)≡8x3−4x+1(2x - 1)(4x^{2} + 2x - 1) \equiv 8x^{3} - 4x + 1.
7(b) [3]
Prove the identity tan⁡2θ+1tan⁡2θ−1≡11−2cos⁡2θ\frac{\tan^{2}\theta + 1}{\tan^{2}\theta - 1} \equiv \frac{1}{1 - 2\cos^{2}\theta}.
7(c) [5]
Using the results of (a) and (b), solve the equation tan⁡2θ+1tan⁡2θ−1=4cos⁡θ\frac{\tan^{2}\theta + 1}{\tan^{2}\theta - 1} = 4\cos\theta, for 0∘⩽θ⩽180∘0^\circ \leqslant \theta \leqslant 180^\circ.
10
The equation of a curve is y=f(x)y = f(x), where f(x)=(4x−3)53−203xf(x) = (4x - 3)^{\frac{5}{3}} - \frac{20}{3}x.
10(a) [6]
Find the xx-coordinates of the stationary points of the curve and determine their nature.
10(b) [1]
State the set of values for which the function ff is increasing.

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