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Learn/ A Level Mathematics/ Past Papers/ 2021/ Oct/Nov 2021 /Paper 2 (Variant 1)

Pure Mathematics 2 — Variant 1

A Level Mathematics · 9709/21 · Oct/Nov 2021

Question Paper

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Mark Scheme

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Questions

5 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 [4]
Find the exact value of ∫−12(4e2x−2e−x) dx\displaystyle \int_{-1}^{2} (4e^{2x} - 2e^{-x}) \, \mathrm{d}x.
2
2(a) [2]
Sketch, on the same diagram, the graphs of y=3xy = 3x and y=∣x−3∣y = |x - 3|.
2(b) [3]
Find the coordinates of the point where the two graphs intersect.
2(c) [1]
Deduce the solution of the inequality 3x<∣x−3∣3x < |x - 3|.
4
The curve with equation y=xe2x+5e−xy = xe^{2x} + 5e^{-x} has a minimum point MM.
4(a) [5]
Show that the xx-coordinate of MM satisfies the equation x=13ln⁡5−13ln⁡(1+2x)x = \frac{1}{3}\ln 5 - \frac{1}{3}\ln(1 + 2x).
4(b) [3]
Use an iterative formula, based on the equation in part (a), to find the xx-coordinate of MM correct to 3 significant figures. Use an initial value of 0.350.35 and give the result of each iteration to 5 significant figures.
6
The polynomials f(x)f(x) and g(x)g(x) are defined by f(x)=4x3+ax2+8x+15f(x) = 4x^{3} + ax^{2} + 8x + 15 and g(x)=x2+bx+18g(x) = x^{2} + bx + 18, where aa and bb are constants.
6(a) [2]
Given that (x+3)(x + 3) is a factor of f(x)f(x), find the value of aa.
6(b) [2]
Given that the remainder is 4040 when g(x)g(x) is divided by (x−2)(x - 2), find the value of bb.
6(c) [3]
When aa and bb have these values, factorise f(x)−g(x)f(x) - g(x) completely.
6(d) [3]
Hence solve the equation f(cosec⁡θ)−g(cosec⁡θ)=0f(\operatorname{cosec}\theta) - g(\operatorname{cosec}\theta) = 0 for 0<θ<2π0 < \theta < 2\pi.
7
7(a) [3]
By first expanding cos⁡(2θ+θ)\cos(2\theta + \theta), show that cos⁡3θ≡4cos⁡3θ−3cos⁡θ\cos 3\theta \equiv 4\cos^{3}\theta - 3\cos\theta.
7(b) [2]
Find the exact value of 2cos⁡3(518π)−32cos⁡(518π)2\cos^{3}(\frac{5}{18}\pi) - \frac{3}{2}\cos(\frac{5}{18}\pi).
7(c) [4]
Find ∫(12cos⁡3x−4cos⁡33x) dx\displaystyle\int (12\cos^{3}x - 4\cos^{3} 3x) \, \mathrm{d}x.

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