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Learn/ A Level Mathematics/ Past Papers/ 2023/ May/June 2023 /Paper 1 (Variant 1)

Pure Mathematics 1 — Variant 1

A Level Mathematics · 9709/11 · May/June 2023

Question Paper

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Mark Scheme

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Questions

7 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]
Solve the equation 4sin⁡θ+tan⁡θ=04\sin\theta + \tan\theta = 0 for 0∘<θ<180∘0^\circ < \theta < 180^\circ.
2
2(a) [2]
Find the first three terms in the expansion, in ascending powers of xx, of (2+3x)4(2 + 3x)^{4}.
2(b) [2]
Find the first three terms in the expansion, in ascending powers of xx, of (1−2x)5(1 - 2x)^{5}.
2(c) [2]
Hence find the coefficient of x2x^{2} in the expansion of (2+3x)4(1−2x)5(2 + 3x)^{4}(1 - 2x)^{5}.
5 [5]
The line with equation y=kx−ky = kx - k, where kk is a positive constant, is a tangent to the curve with equation y=−12xy = -\frac{1}{2x}. Find, in either order, the value of kk and the coordinates of the point where the tangent meets the curve.
6
The first three terms of an arithmetic progression are p26\frac{p^{2}}{6}, 2p−62p - 6 and pp.
6(a) [3]
Given that the common difference of the progression is not zero, find the value of pp.
6(b) [2]
Using this value, find the sum to infinity of the geometric progression with first two terms p26\frac{p^{2}}{6} and 2p−62p - 6.
8
The functions f and g are defined as follows, where aa and bb are constants. f(x)=1+2ax−a\mathrm{f}(x) = 1 + \frac{2a}{x - a} for x>ax > a g(x)=bx−2\mathrm{g}(x) = bx - 2 for x∈Rx \in \mathbb{R}
8(a) [4]
Given that f(7)=52\mathrm{f}(7) = \frac{5}{2} and gf(5)=4\mathrm{gf}(5) = 4, find the values of aa and bb. For the rest of this question, you should use the value of aa which you found in (a).
8(b) [1]
Find the domain of f−1\mathrm{f}^{-1}.
8(c) [3]
Find an expression for f−1(x)\mathrm{f}^{-1}(x).
9
Water is poured into a tank at a constant rate of 500500 cm3\text{cm}^{3} per second. The depth of water in the tank, tt seconds after filling starts, is hh cm. When the depth of water in the tank is hh cm, the volume, VV cm3\text{cm}^{3}, of water in the tank is given by the formula V=43(25+h)3−625003V = \frac{4}{3}(25 + h)^{3} - \frac{62500}{3}.
9(a) [3]
Find the rate at which hh is increasing at the instant when h=10h = 10 cm.
9(b) [3]
At another instant, the rate at which hh is increasing is 0.0750.075 cm per second. Find the value of VV at this instant.
11
The equation of a curve is such that dydx=6x2−30x+6a\frac{\mathrm{d}y}{\mathrm{d}x} = 6x^{2} - 30x + 6a, where aa is a positive constant. The curve has a stationary point at (a,−15)(a, -15).
11(a) [2]
Find the value of aa.
11(b) [2]
Determine the nature of this stationary point.
11(c) [3]
Find the equation of the curve.
11(d) [2]
Find the coordinates of any other stationary points on the curve.

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