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Learn/ A Level Mathematics/ Past Papers/ 2021/ Oct/Nov 2021 /Paper 1 (Variant 3)

Pure Mathematics 1 — Variant 3

A Level Mathematics · 9709/13 · Oct/Nov 2021

Question Paper

PDF · 128 KB · 20 pages

Mark Scheme

PDF · 320 KB · 15 pages

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Questions

4 of 10 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 graph of y=f(x)y = \mathrm{f}(x) is transformed to the graph of y=3−f(x)y = 3 - \mathrm{f}(x). Describe fully, in the correct order, the two transformations that have been combined.
2
2(a) [1]
Find the first three terms, in ascending powers of xx, in the expansion of (1+ax)6(1 + ax)^{6}.
2(b) [4]
Given that the coefficient of x2x^{2} in the expansion of (1−3x)(1+ax)6(1 - 3x)(1 + ax)^{6} is −3-3, find the possible values of the constant aa.
3
3(a) [2]
Express 5y2−30y+505y^{2} - 30y + 50 in the form 5(y+a)2+b5(y + a)^{2} + b, where aa and bb are constants.
3(b) [3]
The function f\mathrm{f} is defined by f(x)=x5−10x3+50x\mathrm{f}(x) = x^{5} - 10x^{3} + 50x for x∈Rx \in \mathbb{R}. Determine whether f\mathrm{f} is an increasing function, a decreasing function or neither.
7
7(a) [4]
Show that the equation tan⁡x+cos⁡xtan⁡x−cos⁡x=k\frac{\tan x + \cos x}{\tan x - \cos x} = k, where kk is a constant, can be expressed as (k+1)sin⁡2x+(k−1)sin⁡x−(k+1)=0(k + 1)\sin^{2}x + (k - 1)\sin x - (k + 1) = 0.
7(b) [4]
Hence solve the equation tan⁡x+cos⁡xtan⁡x−cos⁡x=4\frac{\tan x + \cos x}{\tan x - \cos x} = 4 for 0∘⩽x⩽360∘0^\circ \leqslant x \leqslant 360^\circ.

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