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Learn/ A Level Mathematics/ Past Papers/ 2021/ May/June 2021 /Paper 1 (Variant 3)

Pure Mathematics 1 — Variant 3

A Level Mathematics · 9709/13 · May/June 2021

Question Paper

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Mark Scheme

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Questions

9 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 [3]
A curve with equation y=f(x)y = f(x) is such that f′(x)=6x2−8x2f'(x) = 6x^{2} - \frac{8}{x^{2}}. It is given that the curve passes through the point (2,7)(2, 7). Find f(x)f(x).
2 [4]
The function ff is defined by f(x)=13(2x−1)32−2xf(x) = \frac{1}{3}(2x - 1)^{\frac{3}{2}} - 2x for 12<x<a\frac{1}{2} < x < a. It is given that ff is a decreasing function. Find the maximum possible value of the constant aa.
3 [6]
A line with equation y=mx−6y = mx - 6 is a tangent to the curve with equation y=x2−4x+3y = x^{2} - 4x + 3. Find the possible values of the constant mm, and the corresponding coordinates of the points at which the line touches the curve.
4
4(a) [2]
Show that the equation tan⁡x+sin⁡xtan⁡x−sin⁡x=k,\frac{\tan x + \sin x}{\tan x - \sin x} = k, where kk is a constant, may be expressed as 1+cos⁡x1−cos⁡x=k.\frac{1 + \cos x}{1 - \cos x} = k.
4(b) [2]
Hence express cos⁡x\cos x in terms of kk.
4(c) [2]
Hence solve the equation tan⁡x+sin⁡xtan⁡x−sin⁡x=4\frac{\tan x + \sin x}{\tan x - \sin x} = 4 for −π<x<π-\pi < x < \pi.
6
Functions ff and gg are both defined for x∈Rx \in \mathbb{R} and are given by f(x)=x2−2x+5,f(x) = x^{2} - 2x + 5, g(x)=x2+4x+13.g(x) = x^{2} + 4x + 13.
6(a) [4]
By first expressing each of f(x)f(x) and g(x)g(x) in completed square form, express g(x)g(x) in the form f(x+p)+q,f(x + p) + q, where pp and qq are constants.
6(b) [2]
Describe fully the transformation which transforms the graph of y=f(x)y = f(x) to the graph of y=g(x)y = g(x).
7
7(a) [2]
Write down the first four terms of the expansion, in ascending powers of xx, of (a−x)6(a - x)^{6}.
7(b) [5]
Given that the coefficient of x2x^{2} in the expansion of (1+2ax)(a−x)6(1 + \frac{2}{ax})(a - x)^{6} is −20-20, find in exact form the possible values of the constant aa.
8
Functions ff and gg are defined as follows: f:x↦x2−1f : x \mapsto x^{2} - 1 for x<0,x < 0, g:x↦12x+1g : x \mapsto \frac{1}{2x + 1} for x<−12.x < -\frac{1}{2}.
8(a) [4]
Solve the equation fg(x)=3fg(x) = 3.
8(b) [3]
Find an expression for (fg)−1(x)(fg)^{-1}(x).
9
9(a) [3]
A geometric progression is such that the second term is equal to 24\% of the sum to infinity. Find the possible values of the common ratio.
9(b) [6]
An arithmetic progression PP has first term aa and common difference dd. An arithmetic progression QQ has first term 2(a+1)2(a + 1) and common difference (d+1)(d + 1). It is given that 5th term of P12th term of Q=13\frac{\text{5th term of } P}{\text{12th term of } Q} = \frac{1}{3} and Sum of first 5 terms of PSum of first 5 terms of Q=23.\frac{\text{Sum of first 5 terms of } P}{\text{Sum of first 5 terms of } Q} = \frac{2}{3}. Find the value of aa and the value of dd.
10
Points A(−2,3),B(3,0)A(-2, 3), B(3, 0) and C(6,5)C(6, 5) lie on the circumference of a circle with centre DD.
10(a) [2]
Show that angle ABC=90∘ABC = 90^\circ.
10(b) [1]
Hence state the coordinates of DD.
10(c) [2]
Find an equation of the circle. The point EE lies on the circumference of the circle such that BEBE is a diameter.
10(d) [5]
Find an equation of the tangent to the circle at EE.

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