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Learn/ A Level Mathematics/ Past Papers/ 2022/ Oct/Nov 2022 /Paper 1 (Variant 2)

Pure Mathematics 1 — Variant 2

A Level Mathematics · 9709/12 · Oct/Nov 2022

Question Paper

PDF · 130 KB · 20 pages

Mark Scheme

PDF · 472 KB · 18 pages

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Questions

8 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
Points AA and BB have coordinates (5,2)(5, 2) and (10,−1)(10, -1) respectively.
1(a) [3]
Find the equation of the perpendicular bisector of ABAB.
1(b) [3]
Find the equation of the circle with centre AA which passes through BB.
2 [5]
The first, second and third terms of an arithmetic progression are aa, 2a2a and a2a^{2} respectively, where aa is a positive constant. Find the sum of the first 50 terms of the progression.
3
3(a) [2]
Find the set of values of kk for which the equation 8x2+kx+2=08x^{2} + kx + 2 = 0 has no real roots.
3(b) [3]
Solve the equation 8cos⁡2θ−10cos⁡θ+2=08\cos^{2}\theta - 10\cos\theta + 2 = 0 for 0∘⩽θ⩽180∘0^\circ \leqslant \theta \leqslant 180^\circ.
4 [4]
A geometric progression is such that the third term is 17641764 and the sum of the second and third terms is 34443444. Find the 50th term.
6
The equation of a curve is y=4x2+20x+6y = 4x^{2} + 20x + 6.
6(a) [3]
Express the equation in the form y=a(x+b)2+cy = a(x + b)^{2} + c, where aa, bb and cc are constants.
6(b) [3]
Hence solve the equation 4x2+20x+6=454x^{2} + 20x + 6 = 45.
6(c) [3]
Sketch the graph of y=4x2+20x+6y = 4x^{2} + 20x + 6 showing the coordinates of the stationary point. You are not required to indicate where the curve crosses the xx- and yy-axes.
7
7(a) [3]
Prove the identity sin⁡θsin⁡θ+cos⁡θ+cos⁡θsin⁡θ−cos⁡θ≡tan⁡2θ+1tan⁡2θ−1\frac{\sin\theta}{\sin\theta + \cos\theta} + \frac{\cos\theta}{\sin\theta - \cos\theta} \equiv \frac{\tan^{2}\theta + 1}{\tan^{2}\theta - 1}.
7(b) [4]
Hence find the exact solutions of the equation sin⁡θsin⁡θ+cos⁡θ+cos⁡θsin⁡θ−cos⁡θ=2\frac{\sin\theta}{\sin\theta + \cos\theta} + \frac{\cos\theta}{\sin\theta - \cos\theta} = 2 for 0⩽θ⩽π0 \leqslant \theta \leqslant \pi.
8
The equation of a curve is such that dydx=3x12−3x−12\frac{\mathrm{d}y}{\mathrm{d}x} = 3x^{\frac{1}{2}} - 3x^{-\frac{1}{2}}. The curve passes through the point (3,5)(3, 5).
8(a) [4]
Find the equation of the curve.
8(b) [2]
Find the xx-coordinate of the stationary point.
8(c) [1]
State the set of values of xx for which yy increases as xx increases.
9
Functions f and g are defined by f(x)=x+1xf(x) = x + \frac{1}{x} for x>0x > 0, g(x)=ax+1g(x) = ax + 1 for x∈Rx \in \mathbb{R}, where aa is a constant.
9(a) [1]
Find an expression for gf(x)gf(x).
9(b) [2]
Given that gf(2)=11gf(2) = 11, find the value of aa.
9(c) [1]
Given that the graph of y=f(x)y = f(x) has a minimum point when x=1x = 1, explain whether or not f has an inverse. It is given instead that a=5a = 5.
9(d) [3]
Find and simplify an expression for g−1f(x)g^{-1}f(x).
9(e) [1]
Explain why the composite function fg cannot be formed.

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