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Learn/ A Level Mathematics/ Past Papers/ 2021/ Oct/Nov 2021 /Paper 1 (Variant 2)

Pure Mathematics 1 — Variant 2

A Level Mathematics · 9709/12 · Oct/Nov 2021

Question Paper

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Mark Scheme

PDF · 399 KB · 21 pages

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Questions

8 of 12 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 equation 2cos⁡θ=7−3cos⁡θ2\cos\theta = 7 - \frac{3}{\cos\theta} for −90∘<θ<90∘-90^\circ < \theta < 90^\circ.
3
The function ff is defined as follows: f(x)=x+3x−1f(x) = \frac{x+3}{x-1} for x>1x > 1.
3(a) [2]
Find the value of ff(5)ff(5).
3(b) [3]
Find an expression for f−1(x)f^{-1}(x).
4 [4]
A curve is such that dydx=8(3x+2)2\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{8}{(3x+2)^{2}}. The curve passes through the point (2,523)(2, 5\frac{2}{3}). Find the equation of the curve.
5
The first, third and fifth terms of an arithmetic progression are 2cos⁡x2\cos x, −63sin⁡x-6\sqrt{3}\sin x and 10cos⁡x10\cos x respectively, where 12π<x<π\frac{1}{2}\pi < x < \pi.
5(a) [3]
Find the exact value of xx.
5(b) [3]
Hence find the exact sum of the first 25 terms of the progression.
6 [5]
The second term of a geometric progression is 54 and the sum to infinity of the progression is 243. The common ratio is greater than 12\frac{1}{2}. Find the tenth term, giving your answer in exact form.
8
8(a) [4]
It is given that in the expansion of (4+2x)(2−ax)5(4 + 2x)(2 - ax)^{5}, the coefficient of x2x^{2} is −15-15. Find the possible values of aa.
8(b) [4]
It is given instead that in the expansion of (4+2x)(2−ax)5(4 + 2x)(2 - ax)^{5}, the coefficient of x2x^{2} is kk. It is also given that there is only one value of aa which leads to this value of kk. Find the values of kk and aa.
9
The volume VV m3\text{m}^{3} of a large circular mound of iron ore of radius rr m is modelled by the equation V=32(r−12)3−1V = \frac{3}{2}(r - \frac{1}{2})^{3} - 1 for r⩾2r \geqslant 2. Iron ore is added to the mound at a constant rate of 1.51.5 m3\text{m}^{3} per second.
9(a) [3]
Find the rate at which the radius of the mound is increasing at the instant when the radius is 5.55.5 m.
9(b) [3]
Find the volume of the mound at the instant when the radius is increasing at 0.10.1 m per second.
10
The function ff is defined by f(x)=x2+kx+2f(x) = x^{2} + \frac{k}{x} + 2 for x>0x > 0.
10(a) [3]
Given that the curve with equation y=f(x)y = f(x) has a stationary point when x=2x = 2, find kk.
10(b) [2]
Determine the nature of the stationary point.
10(c) [2]
Given that this is the only stationary point of the curve, find the range of ff.

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