StudentAid
HomeLearn
9th Class10th Class11th Class12th ClassCambridge past papersBrowse everything
Attendance
Class attendanceVisitor register
For schoolsFor driversFor parentsBook a demo
Learn/ A Level Mathematics/ Past Papers/ 2022/ Oct/Nov 2022 /Paper 3 (Variant 2)

Pure Mathematics 3 — Variant 2

A Level Mathematics · 9709/32 · Oct/Nov 2022

Question Paper

PDF · 126 KB · 20 pages

Mark Scheme

PDF · 495 KB · 20 pages

This browser cannot show the PDF inline.

Open the question paper · Open the mark scheme

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 [4]
Solve the equation 23x−1=5(31−x)2^{3x-1} = 5(3^{1-x}). Give your answer in the form ln⁡aln⁡b\frac{\ln a}{\ln b} where aa and bb are integers.
2
The polynomial 2x3−x2+a2x^{3} - x^{2} + a, where aa is a constant, is denoted by p(x)\mathrm{p}(x). It is given that (2x+3)(2x + 3) is a factor of p(x)\mathrm{p}(x).
2(a) [2]
Find the value of aa.
2(b) [4]
When aa has this value, solve the inequality p(x)<0\mathrm{p}(x) < 0.
3 [6]
The equation of a curve is y=sin⁡xsin⁡2xy = \sin x \sin 2x. The curve has a stationary point in the interval 0<x<12π0 < x < \frac{1}{2}\pi. Find the xx-coordinate of this point, giving your answer correct to 3 significant figures.
4
4(a) [3]
Express 4cos⁡x−sin⁡x4\cos x - \sin x in the form Rcos⁡(x+α)R\cos(x + \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.
4(b) [5]
Hence solve the equation 4cos⁡2x−sin⁡2x=34\cos 2x - \sin 2x = 3 for 0∘<x<180∘0^\circ < x < 180^\circ.
5
5(a) [3]
Solve the equation z2−6iz−12=0z^{2} - 6iz - 12 = 0, giving the answers in the form x+iyx + iy, where xx and yy are real and exact.
5(b) [1]
On a sketch of an Argand diagram with origin OO, show points AA and BB representing the roots of the equation in part (a).
5(c) [3]
Find the exact modulus and argument of each root.
5(d) [1]
Hence show that the triangle OABOAB is equilateral.
6
Relative to the origin OO, the points AA, BB and CC have position vectors given by OA→=(131)\overrightarrow{OA} = \begin{pmatrix} 1 \\ 3 \\ 1 \end{pmatrix}, OB→=(312)\overrightarrow{OB} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} and OC→=(53−2)\overrightarrow{OC} = \begin{pmatrix} 5 \\ 3 \\ -2 \end{pmatrix}.
6(a) [4]
Using a scalar product, find the cosine of angle BACBAC.
6(b) [4]
Hence find the area of triangle ABCABC. Give your answer in a simplified exact form.
7
The variables xx and θ\theta satisfy the differential equation xsin⁡2θdxdθ=tan⁡2θ−2cot⁡θx \sin^{2}\theta \frac{\mathrm{d}x}{\mathrm{d}\theta} = \tan^{2}\theta - 2\cot\theta, for 0<θ<12π0 < \theta < \frac{1}{2}\pi and x>0x > 0. It is given that x=2x = 2 when θ=14π\theta = \frac{1}{4}\pi.
7(a) [1]
Show that ddθ(cot⁡2θ)=−2cot⁡θsin⁡2θ\frac{\mathrm{d}}{\mathrm{d}\theta}(\cot^{2}\theta) = -\frac{2\cot\theta}{\sin^{2}\theta}. (You may assume without proof that the derivative of cot⁡θ\cot\theta with respect to θ\theta is −cosec⁡2θ-\operatorname{cosec}^{2}\theta.)
7(b) [7]
Solve the differential equation and find the value of xx when θ=16π\theta = \frac{1}{6}\pi.
10
Let f(x)=4−x+x2(1+x)(2+x2)\mathrm{f}(x) = \frac{4 - x + x^{2}}{(1 + x)(2 + x^{2})}.
10(a) [5]
Express f(x)\mathrm{f}(x) in partial fractions.
10(b) [5]
Find the exact value of ∫04f(x) dx\int_{0}^{4} \mathrm{f}(x) \, \mathrm{d}x. Give your answer as a single logarithm.

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