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Learn/ A Level Mathematics/ Past Papers/ 2022/ May/June 2022 /Paper 1 (Variant 1)

Pure Mathematics 1 — Variant 1

A Level Mathematics · 9709/11 · May/June 2022

Question Paper

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Mark Scheme

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Questions

8 of 10 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
1(a) [2]
Express x2−8x+11x^{2} - 8x + 11 in the form (x+p)2+q(x + p)^{2} + q where pp and qq are constants.
1(b) [2]
Hence find the exact solutions of the equation x2−8x+11=1x^{2} - 8x + 11 = 1.
2 [5]
The thirteenth term of an arithmetic progression is 1212 and the sum of the first 3030 terms is −15-15. Find the sum of the first 5050 terms of the progression.
3
The coefficient of x4x^{4} in the expansion of (2x2+k2x)5(2x^{2} + \frac{k^{2}}{x})^{5} is aa. The coefficient of x2x^{2} in the expansion of (2kx−1)4(2kx - 1)^{4} is bb.
3(a) [3]
Find aa and bb in terms of the constant kk.
3(b) [3]
Given that a+b=216a + b = 216, find the possible values of kk.
4
4(a) [4]
Prove the identity sin⁡3θsin⁡θ−1−sin⁡2θ1+sin⁡θ≡−tan⁡2θ(1+sin⁡2θ)\frac{\sin^{3}\theta}{\sin\theta - 1} - \frac{\sin^{2}\theta}{1 + \sin\theta} \equiv -\tan^{2}\theta(1 + \sin^{2}\theta).
4(b) [2]
Hence solve the equation sin⁡3θsin⁡θ−1−sin⁡2θ1+sin⁡θ=tan⁡2θ(1−sin⁡2θ)\frac{\sin^{3}\theta}{\sin\theta - 1} - \frac{\sin^{2}\theta}{1 + \sin\theta} = \tan^{2}\theta(1 - \sin^{2}\theta) for 0<θ<2π0 < \theta < 2\pi.
6
The function ff is defined as follows: f(x)=x2−4x2+4f(x) = \frac{x^{2} - 4}{x^{2} + 4} for x>2x > 2.
6(a) [3]
Find an expression for f−1(x)f^{-1}(x).
6(b) [4]
Show that 1−8x2+41 - \frac{8}{x^{2} + 4} can be expressed as x2−4x2+4\frac{x^{2} - 4}{x^{2} + 4} and hence state the range of ff.
6(c) [1]
Explain why the composite function ffff cannot be formed.
8
8(a) [5]
The curve y=sin⁡xy = \sin x is transformed to the curve y=4sin⁡(12x−30∘)y = 4\sin(\frac{1}{2}x - 30^\circ). Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations are applied.
8(b) [3]
Find the exact solutions of the equation 4sin⁡(12x−30∘)=224\sin(\frac{1}{2}x - 30^\circ) = 2\sqrt{2} for 0∘⩽x⩽360∘0^\circ \leqslant x \leqslant 360^\circ.
9
The equation of a circle is x2+y2+6x−2y−26=0x^{2} + y^{2} + 6x - 2y - 26 = 0.
9(a) [4]
Find the coordinates of the centre of the circle and the radius. Hence find the coordinates of the lowest point on the circle.
9(b) [6]
Find the set of values of the constant kk for which the line with equation y=kx−5y = kx - 5 intersects the circle at two distinct points.
10
The equation of a curve is such that d2ydx2=6x2−4x3\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}} = 6x^{2} - \frac{4}{x^{3}}. The curve has a stationary point at (−1,92)(-1, \frac{9}{2}).
10(a) [1]
Determine the nature of the stationary point at (−1,92)(-1, \frac{9}{2}).
10(b) [5]
Find the equation of the curve.
10(c) [3]
Show that the curve has no other stationary points.
10(d) [3]
A point AA is moving along the curve and the yy-coordinate of AA is increasing at a rate of 55 units per second. Find the rate of increase of the xx-coordinate of AA at the point where x=1x = 1.

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