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Learn/ A Level Mathematics/ Past Papers/ 2021/ Feb/March 2021 /Paper 2 (Variant 2)

Pure Mathematics 2 — Variant 2

A Level Mathematics · 9709/22 · Feb/March 2021

Question Paper

PDF · 110 KB · 16 pages

Mark Scheme

PDF · 287 KB · 12 pages

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Questions

6 of 7 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
1(a) [2]
Sketch, on the same diagram, the graphs of y=∣3x−5∣y = |3x - 5| and y=x+2y = x + 2.
1(b) [3]
Solve the equation ∣3x−5∣=x+2|3x - 5| = x + 2.
2 [5]
Solve the equation sec⁡2θcot⁡θ=8\sec^{2}\theta \cot\theta = 8 for 0<θ<π0 < \theta < \pi.
3 [5]
The parametric equations of a curve are x=e2tcos⁡4tx = e^{2t}\cos 4t, y=3sin⁡2ty = 3\sin 2t. Find the gradient of the curve at the point for which t=0t = 0.
5
5(a) [3]
Given that 2ln⁡(x+1)+ln⁡x=ln⁡(x+9)2\ln(x + 1) + \ln x = \ln(x + 9), show that x=9x+2x = \sqrt{\frac{9}{x + 2}}.
5(b) [2]
It is given that the equation x=9x+2x = \sqrt{\frac{9}{x + 2}} has a single root. Show by calculation that this root lies between 1.51.5 and 2.02.0.
5(c) [3]
Use an iterative formula, based on the equation in part (b), to find the root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
6
The polynomial p(x)p(x) is defined by p(x)=x3+ax+b,p(x) = x^{3} + ax + b, where aa and bb are constants. It is given that (x+2)(x + 2) is a factor of p(x)p(x) and that the remainder is 55 when p(x)p(x) is divided by (x−3)(x - 3).
6(a) [5]
Find the values of aa and bb.
6(b) [5]
Hence find the exact root of the equation p(e2y)=0p(e^{2y}) = 0.
7
7(a) [3]
Express 53cos⁡x+5sin⁡x5\sqrt{3}\cos x + 5\sin x in the form Rcos⁡(x−α)R\cos(x - \alpha), where R>0R > 0 and 0<α<12π0 < \alpha < \frac{1}{2}\pi.
7(b) [3]
As xx varies, find the least possible value of 4+53cos⁡x+5sin⁡x,4 + 5\sqrt{3}\cos x + 5\sin x, and determine the corresponding value of xx where −π<x<π-\pi < x < \pi.
7(c) [3]
Find ∫1(53cos⁡3θ+5sin⁡3θ)2 dθ\int \frac{1}{(5\sqrt{3}\cos 3\theta + 5\sin 3\theta)^{2}} \, \mathrm{d}\theta.

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