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Learn/ A Level Mathematics/ Past Papers/ 2020/ Oct/Nov 2020 /Paper 2 (Variant 3)

Pure Mathematics 2 — Variant 3

A Level Mathematics · 9709/23 · Oct/Nov 2020

Question Paper

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Mark Scheme

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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]
Given that ln⁡(2x+1)−ln⁡(x−3)=2\ln(2x + 1) - \ln(x - 3) = 2, find xx in terms of e\mathrm{e}.
2 [5]
The polynomial p(x)p(x) is defined by p(x)=x3+ax2+bx+16p(x) = x^{3} + ax^{2} + bx + 16, 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 7272 when p(x)p(x) is divided by (x−2)(x - 2). Find the values of aa and bb.
4
4(a) [3]
Solve the equation ∣2x−5∣=∣x+6∣|2x - 5| = |x + 6|.
4(b) [2]
Hence find the value of yy such that ∣21−y−5∣=∣2−y+6∣|2^{1-y} - 5| = |2^{-y} + 6|. Give your answer correct to 3 significant figures.
5
The sequence of values given by the iterative formula xn+1=6+8xn8+xn2x_{n+1} = \frac{6 + 8x_{n}}{8 + x_{n}^{2}} with initial value x1=2x_{1} = 2 converges to α\alpha.
5(a) [3]
Use the iterative formula to find the value of α\alpha correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
5(b) [2]
State an equation satisfied by α\alpha and hence determine the exact value of α\alpha.
6
It is given that 3sin⁡2θ=cos⁡θ3\sin 2\theta = \cos\theta where θ\theta is an angle such that 0∘<θ<90∘0^\circ < \theta < 90^\circ.
6(a) [2]
Find the exact value of sin⁡θ\sin\theta.
6(b) [2]
Find the exact value of sec⁡θ\sec\theta.
6(c) [2]
Find the exact value of cos⁡2θ\cos 2\theta.
7
A curve is defined by the parametric equations x=3t−2sin⁡tx = 3t - 2\sin t, y=5t+4cos⁡ty = 5t + 4\cos t, where 0⩽t⩽2π0 \leqslant t \leqslant 2\pi. At each of the points PP and QQ on the curve, the gradient of the curve is 52\frac{5}{2}.
7(a) [3]
Show that the values of tt at PP and QQ satisfy the equation 10cos⁡t−8sin⁡t=510\cos t - 8\sin t = 5.
7(b) [3]
Express 10cos⁡t−8sin⁡t10\cos t - 8\sin t in the form Rcos⁡(t+α)R\cos(t + \alpha), where R>0R > 0 and 0<α<12π0 < \alpha < \frac{1}{2}\pi. Give the exact value of RR and the value of α\alpha correct to 3 significant figures.
7(c) [4]
Hence find the values of tt at the points PP and QQ.
8
A curve has equation y=f(x)y = f(x) where f(x)=4x3+8x−42x−1f(x) = \frac{4x^{3} + 8x - 4}{2x - 1}.
8(a) [5]
Find an expression for dydx\frac{\mathrm{d}y}{\mathrm{d}x} and hence find the coordinates of each of the stationary points of the curve y=f(x)y = f(x).
8(b) [5]
Divide 4x3+8x−44x^{3} + 8x - 4 by (2x−1)(2x - 1), and hence find ∫f(x) dx\int f(x) \, \mathrm{d}x.

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