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Learn/ A Level Mathematics/ Past Papers/ 2020/ May/June 2020 /Paper 1 (Variant 2)

Pure Mathematics 1 — Variant 2

A Level Mathematics · 9709/12 · May/June 2020

Question Paper

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Mark Scheme

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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
1(a) [2]
Find the coefficient of x2x^{2} in the expansion of (x−2x)6(x - \frac{2}{x})^{6}.
1(b) [3]
Find the coefficient of x2x^{2} in the expansion of (2+3x2)(x−2x)6(2 + 3x^{2})(x - \frac{2}{x})^{6}.
2
2(a) [3]
Express the equation 3cos⁡θ=8tan⁡θ3\cos\theta = 8\tan\theta as a quadratic equation in sin⁡θ\sin\theta.
2(b) [2]
Hence find the acute angle, in degrees, for which 3cos⁡θ=8tan⁡θ3\cos\theta = 8\tan\theta.
3
A weather balloon in the shape of a sphere is being inflated by a pump. The volume of the balloon is increasing at a constant rate of 600600 cm3\text{cm}^{3} per second. The balloon was empty at the start of pumping.
3(a) [2]
Find the radius of the balloon after 30 seconds.
3(b) [3]
Find the rate of increase of the radius after 30 seconds.
4 [5]
The nnth term of an arithmetic progression is 12(3n−15)\frac{1}{2}(3n - 15). Find the value of nn for which the sum of the first nn terms is 8484.
5
The function f is defined for x∈Rx \in \mathbb{R} by f:x↦a−2x,\mathrm{f} : x \mapsto a - 2x, where aa is a constant.
5(a) [4]
Express ff(x)\mathrm{ff}(x) and f−1(x)\mathrm{f}^{-1}(x) in terms of aa and xx.
5(b) [2]
Given that ff(x)=f−1(x)\mathrm{ff}(x) = \mathrm{f}^{-1}(x), find xx in terms of aa.
6
The equation of a curve is y=2x2+kx+k−1y = 2x^{2} + kx + k - 1, where kk is a constant.
6(a) [3]
Given that the line y=2x+3y = 2x + 3 is a tangent to the curve, find the value of kk. It is now given that k=2k = 2.
6(b) [3]
Express the equation of the curve in the form y=2(x+a)2+by = 2(x + a)^{2} + b, where aa and bb are constants, and hence state the coordinates of the vertex of the curve.
10
The equation of a curve is y=54x−(2x−7)3y = 54x - (2x - 7)^{3}.
10(a) [4]
Find dydx\frac{\mathrm{d}y}{\mathrm{d}x} and d2ydx2\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}.
10(b) [3]
Find the coordinates of each of the stationary points on the curve.
10(c) [2]
Determine the nature of each of the stationary points.
11
The equation of a circle with centre CC is x2+y2−8x+4y−5=0x^{2} + y^{2} - 8x + 4y - 5 = 0.
11(a) [3]
Find the radius of the circle and the coordinates of CC. The point P(1,2)P(1, 2) lies on the circle.
11(b) [3]
Show that the equation of the tangent to the circle at PP is 4y=3x+54y = 3x + 5. The point QQ also lies on the circle and PQPQ is parallel to the xx-axis.
11(c) [2]
Write down the coordinates of QQ. The tangents to the circle at PP and QQ meet at TT.
11(d) [3]
Find the coordinates of TT.

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