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Learn/ A Level Mathematics/ Past Papers/ 2020/ Oct/Nov 2020 /Paper 3 (Variant 1)

Pure Mathematics 3 — Variant 1

A Level Mathematics · 9709/31 · Oct/Nov 2020

Question Paper

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Mark Scheme

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Questions

10 of 11 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 2−5x>2∣x−3∣2 - 5x > 2|x - 3|.
2 [4]
On a sketch of an Argand diagram, shade the region whose points represent complex numbers zz satisfying the inequalities ∣z∣⩾2|z| \geqslant 2 and ∣z−1+i∣⩽1|z - 1 + i| \leqslant 1.
3 [5]
The parametric equations of a curve are x=3−cos⁡2θx = 3 - \cos 2\theta, y=2θ+sin⁡2θy = 2\theta + \sin 2\theta, for 0<θ<12π0 < \theta < \frac{1}{2}\pi. Show that dydx=cot⁡θ\frac{\mathrm{d}y}{\mathrm{d}x} = \cot\theta.
4 [6]
Solve the equation log⁡10(2x+1)=2log⁡10(x+1)−1\log_{10}(2x + 1) = 2\log_{10}(x + 1) - 1. Give your answers correct to 3 decimal places.
5
5(a) [2]
By sketching a suitable pair of graphs, show that the equation cosec⁡x=1+e−12x\operatorname{cosec} x = 1 + e^{-\frac{1}{2}x} has exactly two roots in the interval 0<x<π0 < x < \pi.
5(b) [3]
The sequence of values given by the iterative formula xn+1=π−sin⁡−1(1e−12xn+1)x_{n+1} = \pi - \sin^{-1}(\frac{1}{e^{-\frac{1}{2}x_{n}} + 1}), with initial value x1=2x_{1} = 2, converges to one of these roots. Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
6
6(a) [3]
Express 6cos⁡θ+3sin⁡θ\sqrt{6}\cos\theta + 3\sin\theta in the form Rcos⁡(θ−α)R\cos(\theta - \alpha), where R>0R > 0 and 0∘<α<90∘0^\circ < \alpha < 90^\circ. State the exact value of RR and give α\alpha correct to 2 decimal places.
6(b) [4]
Hence solve the equation 6cos⁡13x+3sin⁡13x=2.5\sqrt{6}\cos\frac{1}{3}x + 3\sin\frac{1}{3}x = 2.5, for 0∘<x<360∘0^\circ < x < 360^\circ.
7
7(a) [3]
Verify that −1+5i-1 + \sqrt{5}i is a root of the equation 2x3+x2+6x−18=02x^{3} + x^{2} + 6x - 18 = 0.
7(b) [4]
Find the other roots of this equation.
8 [6]
The coordinates (x,y)(x, y) of a general point of a curve satisfy the differential equation xdydx=(1−2x2)yx\frac{\mathrm{d}y}{\mathrm{d}x} = (1 - 2x^{2})y, for x>0x > 0. It is given that y=1y = 1 when x=1x = 1. Solve the differential equation, obtaining an expression for yy in terms of xx.
9
Let f(x)=8+5x+12x2(1−x)(2+3x)2f(x) = \frac{8 + 5x + 12x^{2}}{(1 - x)(2 + 3x)^{2}}.
9(a) [5]
Express f(x)f(x) in partial fractions.
9(b) [5]
Hence obtain the expansion of f(x)f(x) in ascending powers of xx, up to and including the term in x2x^{2}.
11
Two lines have equations r=i+2j+k+λ(ai+2j−k)\mathbf{r} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(a\mathbf{i} + 2\mathbf{j} - \mathbf{k}) and r=2i+j−k+μ(2i−j+k)\mathbf{r} = 2\mathbf{i} + \mathbf{j} - \mathbf{k} + \mu(2\mathbf{i} - \mathbf{j} + \mathbf{k}), where aa is a constant.
11(a) [5]
Given that the two lines intersect, find the value of aa and the position vector of the point of intersection.
11(b) [6]
Given instead that the acute angle between the directions of the two lines is cos⁡−1(16)\cos^{-1}(\frac{1}{6}), find the two possible values of aa.

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