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

Pure Mathematics 1 — Variant 1

A Level Mathematics · 9709/11 · 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 [4]
The sum of the first nine terms of an arithmetic progression is 117117. The sum of the next four terms is 9191. Find the first term and the common difference of the progression.
2 [5]
The coefficient of 1x\frac{1}{x} in the expansion of (kx+1x)5+(1−2x)8\left(kx + \frac{1}{x}\right)^{5} + \left(1 - \frac{2}{x}\right)^{8} is 7474. Find the value of the positive constant kk.
3
Each year the selling price of a diamond necklace increases by 5%5\% of the price the year before. The selling price of the necklace in the year 20002000 was $36 000\$36\,000.
3(a) [3]
Write down an expression for the selling price of the necklace nn years later and hence find the selling price in 20082008.
3(b) [2]
The company that makes the necklace only sells one each year. Find the total amount of money obtained in the ten-year period starting in the year 20002000.
5
The equation of a line is y=mx+cy = mx + c, where mm and cc are constants, and the equation of a curve is xy=16xy = 16.
5(a) [3]
Given that the line is a tangent to the curve, express mm in terms of cc.
5(b) [3]
Given instead that m=−4m = -4, find the set of values of cc for which the line intersects the curve at two distinct points.
6
Functions f\mathrm{f} and g\mathrm{g} are defined for x∈Rx \in \mathbb{R} by f:x↦12x−a,\mathrm{f} : x \mapsto \frac{1}{2}x - a, g:x↦3x+b,\mathrm{g} : x \mapsto 3x + b, where aa and bb are constants.
6(a) [4]
Given that gg(2)=10\mathrm{gg}(2) = 10 and f−1(2)=14\mathrm{f}^{-1}(2) = 14, find the values of aa and bb.
6(b) [2]
Using these values of aa and bb, find an expression for gf(x)\mathrm{gf}(x) in the form cx+dcx + d, where cc and dd are constants.
7
7(a) [3]
Prove the identity 1+sin⁡θcos⁡θ+cos⁡θ1+sin⁡θ≡2cos⁡θ\frac{1 + \sin\theta}{\cos\theta} + \frac{\cos\theta}{1 + \sin\theta} \equiv \frac{2}{\cos\theta}.
7(b) [3]
Hence solve the equation 1+sin⁡θcos⁡θ+cos⁡θ1+sin⁡θ=3sin⁡θ\frac{1 + \sin\theta}{\cos\theta} + \frac{\cos\theta}{1 + \sin\theta} = \frac{3}{\sin\theta}, for 0⩽θ⩽2π0 \leqslant \theta \leqslant 2\pi.
9
The equation of a curve is y=(3−2x)3+24xy = (3 - 2x)^{3} + 24x.
9(a) [4]
Find expressions for dydx\frac{\mathrm{d}y}{\mathrm{d}x} and d2ydx2\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}.
9(b) [3]
Find the coordinates of each of the stationary points on the curve.
9(c) [2]
Determine the nature of each stationary point.
10
The coordinates of the points AA and BB are (−1,−2)(-1, -2) and (7,4)(7, 4) respectively.
10(a) [4]
Find the equation of the circle, CC, for which ABAB is a diameter.
10(b) [4]
Find the equation of the tangent, TT, to circle CC at the point BB.
10(c) [3]
Find the equation of the circle which is the reflection of circle CC in the line TT.

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