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Learn/ A Level Mathematics/ Past Papers/ 2022/ Oct/Nov 2022 /Paper 2 (Variant 3)

Pure Mathematics 2 — Variant 3

A Level Mathematics · 9709/23 · Oct/Nov 2022

Question Paper

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Mark Scheme

PDF · 290 KB · 12 pages

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Questions

7 of 8 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]
Solve the inequality ∣2x−5∣>x|2x - 5| > x.
2 [4]
Use logarithms to solve the equation 14e−2x=5x+114\mathrm{e}^{-2x} = 5^{x+1}, giving your answer correct to 3 significant figures.
3 [4]
It is given that sec⁡θ=17\sec\theta = \sqrt{17} where 0<θ<12π0 < \theta < \frac{1}{2}\pi. Find the exact value of tan⁡(θ+14π)\tan(\theta + \frac{1}{4}\pi).
4
4(a) [2]
By sketching a suitable pair of graphs on the same diagram, show that the equation e−12x=x5\mathrm{e}^{-\frac{1}{2}x} = x^{5} has exactly one real root.
4(b) [3]
Use the iterative formula xn+1=e−12xn5x_{n+1} = \sqrt[5]{\mathrm{e}^{-\frac{1}{2}x_{n}}} to determine the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
5 [5]
A curve has equation 4e2xy+y2=214\mathrm{e}^{2x}y + y^{2} = 21. Find the gradient of the curve at the point (0,−7)(0, -7).
6
The polynomial p(x)p(x) is defined by p(x)=12x3−9x2+8x−4p(x) = 12x^{3} - 9x^{2} + 8x - 4.
6(a) [3]
Find the quotient when p(x)p(x) is divided by (4x−3)(4x - 3) and show that the remainder is 2.
6(b) [6]
Hence find ∫212(p(x)4x−3−3x2) dx\int_{2}^{12} ( \frac{p(x)}{4x - 3} - 3x^{2} ) \, \mathrm{d}x, giving your answer in the form a+ln⁡ba + \ln b.
8
The expression f(θ)f(\theta) is defined by f(θ)=12sin⁡θcos⁡θ+16cos⁡2θf(\theta) = 12\sin\theta\cos\theta + 16\cos^{2}\theta.
8(a) [5]
Express f(θ)f(\theta) in the form Rcos⁡(2θ−α)+kR\cos(2\theta - \alpha) + k, where R>0R > 0, 0<α<12π0 < \alpha < \frac{1}{2}\pi and kk is a constant. State the values of RR and kk, and give the value of α\alpha correct to 4 significant figures.
8(b) [3]
Find the smallest positive value of θ\theta satisfying the equation f(θ)=17f(\theta) = 17.
8(c) [2]
Find ∫f(θ) dθ\int f(\theta) \, \mathrm{d}\theta.

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