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Learn/ A Level Mathematics/ Past Papers/ 2022/ May/June 2022 /Paper 1 (Variant 2)

Pure Mathematics 1 — Variant 2

A Level Mathematics · 9709/12 · May/June 2022

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 coefficient of x4x^{4} in the expansion of (3+x)5(3 + x)^{5} is equal to the coefficient of x2x^{2} in the expansion of (2x+ax)6\left(2x + \frac{a}{x}\right)^{6}. Find the value of the positive constant aa.
2 [4]
The second and third terms of a geometric progression are 10 and 8 respectively. Find the sum to infinity.
3 [4]
The equation of a curve is such that dydx=3(4x−7)12−4x−12\frac{\mathrm{d}y}{\mathrm{d}x} = 3(4x - 7)^{\frac{1}{2}} - 4x^{-\frac{1}{2}}. It is given that the curve passes through the point (4,52)\left(4, \frac{5}{2}\right). Find the equation of the curve.
4
The first, second and third terms of an arithmetic progression are kk, 6k6k and k+6k + 6 respectively.
4(a) [2]
Find the value of the constant kk.
4(b) [3]
Find the sum of the first 30 terms of the progression.
5
The equation of a curve is y=4x2−kx+12k2y = 4x^{2} - kx + \frac{1}{2}k^{2} and the equation of a line is y=x−ay = x - a, where kk and aa are constants.
5(a) [4]
Given that the curve and the line intersect at the points with xx-coordinates 0 and 34\frac{3}{4}, find the values of kk and aa.
5(b) [5]
Given instead that a=−72a = -\frac{7}{2}, find the values of kk for which the line is a tangent to the curve.
8
The equation of a circle is x2+y2+ax+by−12=0x^{2} + y^{2} + ax + by - 12 = 0. The points A(1,1)A(1, 1) and B(2,−6)B(2, -6) lie on the circle.
8(a) [4]
Find the values of aa and bb and hence find the coordinates of the centre of the circle.
8(b) [4]
Find the equation of the tangent to the circle at the point AA, giving your answer in the form px+qy=kpx + qy = k, where pp, qq and kk are integers.
9
The equation of a curve is y=3x+1−4(3x+1)12y = 3x + 1 - 4(3x + 1)^{\frac{1}{2}} for x>−13x > -\frac{1}{3}.
9(a) [3]
Find dydx\frac{\mathrm{d}y}{\mathrm{d}x} and d2ydx2\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}.
9(b) [4]
Find the coordinates of the stationary point of the curve and determine its nature.
11
The function f is given by f(x)=4cos⁡4x+cos⁡2x−k\mathrm{f}(x) = 4\cos^{4}x + \cos^{2}x - k for 0⩽x⩽2π0 \leqslant x \leqslant 2\pi, where kk is a constant.
11(a) [5]
Given that k=3k = 3, find the exact solutions of the equation f(x)=0\mathrm{f}(x) = 0.
11(b) [5]
Use the quadratic formula to show that, when k>5k > 5, the equation f(x)=0\mathrm{f}(x) = 0 has no solutions.

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