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

Pure Mathematics 3 — Variant 2

A Level Mathematics · 9709/32 · Oct/Nov 2020

Question Paper

PDF · 125 KB · 20 pages

Mark Scheme

PDF · 328 KB · 18 pages

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Questions

8 of 10 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 [3]
Solve the equation ln⁡(1+e−3x)=2\ln(1 + e^{-3x}) = 2. Give the answer correct to 3 decimal places.
2
2(a) [4]
Expand 1+6x3\sqrt[3]{1 + 6x} in ascending powers of xx, up to and including the term in x3x^{3}, simplifying the coefficients.
2(b) [1]
State the set of values of xx for which the expansion is valid.
3
The variables xx and yy satisfy the relation 2y=31−2x2^{y} = 3^{1-2x}.
3(a) [3]
By taking logarithms, show that the graph of yy against xx is a straight line. State the exact value of the gradient of this line.
3(b) [2]
Find the exact xx-coordinate of the point of intersection of this line with the line y=3xy = 3x. Give your answer in the form ln⁡aln⁡b\frac{\ln a}{\ln b}, where aa and bb are integers.
4
4(a) [3]
Show that the equation tan⁡(θ+60∘)=2cot⁡θ\tan(\theta + 60^\circ) = 2\cot\theta can be written in the form tan⁡2θ+33tan⁡θ−2=0\tan^{2}\theta + 3\sqrt{3}\tan\theta - 2 = 0.
4(b) [3]
Hence solve the equation tan⁡(θ+60∘)=2cot⁡θ\tan(\theta + 60^\circ) = 2\cot\theta, for 0∘<θ<180∘0^\circ < \theta < 180^\circ.
6
The complex number uu is defined by u=7+i1−iu = \frac{7 + i}{1 - i}.
6(a) [3]
Express uu in the form x+iyx + iy, where xx and yy are real.
6(b) [2]
Show on a sketch of an Argand diagram the points AA, BB and CC representing uu, 7+i7 + i and 1−i1 - i respectively.
6(c) [3]
By considering the arguments of 7+i7 + i and 1−i1 - i, show that tan⁡−1(43)=tan⁡−1(17)+14π\tan^{-1}\left(\frac{4}{3}\right) = \tan^{-1}\left(\frac{1}{7}\right) + \frac{1}{4}\pi.
7
The variables xx and tt satisfy the differential equation e3tdxdt=cos⁡22xe^{3t}\frac{\mathrm{d}x}{\mathrm{d}t} = \cos^{2}2x, for t⩾0t \geqslant 0. It is given that x=0x = 0 when t=0t = 0.
7(a) [7]
Solve the differential equation and obtain an expression for xx in terms of tt.
7(b) [1]
State what happens to the value of xx when tt tends to infinity.
8
With respect to the origin OO, the position vectors of the points AA, BB, CC and DD are given by OA→=(215)\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ 5 \end{pmatrix}, OB→=(4−11)\overrightarrow{OB} = \begin{pmatrix} 4 \\ -1 \\ 1 \end{pmatrix}, OC→=(112)\overrightarrow{OC} = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix} and OD→=(323)\overrightarrow{OD} = \begin{pmatrix} 3 \\ 2 \\ 3 \end{pmatrix}.
8(a) [3]
Show that AB=2CDAB = 2CD.
8(b) [3]
Find the angle between the directions of AB→\overrightarrow{AB} and CD→\overrightarrow{CD}.
8(c) [4]
Show that the line through AA and BB does not intersect the line through CC and DD.
9
Let f(x)=7x+18(3x+2)(x2+4)f(x) = \frac{7x + 18}{(3x + 2)(x^{2} + 4)}.
9(a) [5]
Express f(x)f(x) in partial fractions.
9(b) [6]
Hence find the exact value of ∫02f(x) dx\int_{0}^{2} f(x) \, \mathrm{d}x.

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