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Learn/ A Level Mathematics/ Past Papers/ 2020/ Oct/Nov 2020 /Paper 1 (Variant 1)

Pure Mathematics 1 — Variant 1

A Level Mathematics · 9709/11 · Oct/Nov 2020

Question Paper

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Mark Scheme

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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 [3]
Find the set of values of mm for which the line with equation y=mx−3y = mx - 3 and the curve with equation y=2x2+5y = 2x^{2} + 5 do not meet.
2 [4]
The equation of a curve is such that dydx=1(x−3)2+x\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{(x - 3)^{2}} + x. It is given that the curve passes through the point (2,7)(2, 7). Find the equation of the curve.
3 [3]
Air is being pumped into a balloon in the shape of a sphere so that its volume is increasing at a constant rate of 5050 cm3s−1\text{cm}^{3}\text{s}^{-1}. Find the rate at which the radius of the balloon is increasing when the radius is 1010 cm.
5
In the expansion of (2x2+ax)6\left(2x^{2} + \frac{a}{x}\right)^{6}, the coefficients of x6x^{6} and x3x^{3} are equal.
5(a) [4]
Find the value of the non-zero constant aa.
5(b) [1]
Find the coefficient of x6x^{6} in the expansion of (1−x3)(2x2+ax)6(1 - x^{3})\left(2x^{2} + \frac{a}{x}\right)^{6}.
6 [5]
The equation of a curve is y=2+25−x2y = 2 + \sqrt{25 - x^{2}}. Find the coordinates of the point on the curve at which the gradient is 43\frac{4}{3}.
7
7(a) [3]
Show that sin⁡θ1−sin⁡θ−sin⁡θ1+sin⁡θ≡2tan⁡2θ\frac{\sin\theta}{1 - \sin\theta} - \frac{\sin\theta}{1 + \sin\theta} \equiv 2\tan^{2}\theta.
7(b) [3]
Hence solve the equation sin⁡θ1−sin⁡θ−sin⁡θ1+sin⁡θ=8\frac{\sin\theta}{1 - \sin\theta} - \frac{\sin\theta}{1 + \sin\theta} = 8, for 0∘<θ<180∘0^\circ < \theta < 180^\circ.
8
A geometric progression has first term aa, common ratio rr and sum to infinity SS. A second geometric progression has first term aa, common ratio RR and sum to infinity 2S2S.
8(a) [3]
Show that r=2R−1r = 2R - 1. It is now given that the 3rd term of the first progression is equal to the 2nd term of the second progression.
8(b) [4]
Express SS in terms of aa.
11
The functions ff and gg are defined by f(x)=x2+3f(x) = x^{2} + 3 for x>0x > 0, g(x)=2x+1g(x) = 2x + 1 for x>−12x > -\frac{1}{2}.
11(a) [1]
Find an expression for fg(x)fg(x).
11(b) [4]
Find an expression for (fg)−1(x)(fg)^{-1}(x) and state the domain of (fg)−1(fg)^{-1}.
11(c) [4]
Solve the equation fg(x)−3=gf(x)fg(x) - 3 = gf(x).

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