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/ 2024/ Feb/March 2024 /Paper 3 (Variant 2)

Pure Mathematics 3 — Variant 2

A Level Mathematics · 9709/32 · Feb/March 2024

Question Paper

PDF · 3.2 MB · 20 pages

Mark Scheme

PDF · 449 KB · 18 pages

This browser cannot show the PDF inline.

Open the question paper · Open the mark scheme

Questions

7 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 [3]
Find the quotient and remainder when x4−3x3+9x2−12x+27x^{4} - 3x^{3} + 9x^{2} - 12x + 27 is divided by x2+5x^{2} + 5.
2
2(a) [4]
Find the coefficient of x2x^{2} in the expansion of (2x−5)4−x(2x - 5)\sqrt{4 - x}.
2(b) [1]
State the set of values of xx for which the expansion in part (a) is valid.
4 [4]
The positive numbers pp and qq are such that ln⁡(pq)=a\ln\left( \dfrac{p}{q} \right) = a and ln⁡(q2p)=b\ln(q^{2}p) = b. Express ln⁡(p7q)\ln(p^{7}q) in terms of aa and bb.
5
5(a) [4]
On a sketch of an Argand diagram, shade the region whose points represent complex numbers zz satisfying the inequalities ∣z−4−2i∣⩽3|z - 4 - 2\text{i}| \leqslant 3 and ∣z∣⩾∣10−z∣|z| \geqslant |10 - z|.
5(b) [2]
Find the greatest value of arg⁡z\arg z for points in this region.
9
Relative to the origin OO, the position vectors of the points AA, BB and CC are given by OA→=5i−2j+k\overrightarrow{OA} = 5\mathbf{i} - 2\mathbf{j} + \mathbf{k}, OB→=8i+2j−6k\overrightarrow{OB} = 8\mathbf{i} + 2\mathbf{j} - 6\mathbf{k} and OC→=3i+4j−7k\overrightarrow{OC} = 3\mathbf{i} + 4\mathbf{j} - 7\mathbf{k}.
9(a) [4]
Show that OABCOABC is a rectangle.
9(b) [4]
Use a scalar product to find the acute angle between the diagonals of OABCOABC.
10
Let f(x)=36a2(2a+x)(2a−x)(5a−2x)\mathrm{f}(x) = \dfrac{36a^{2}}{(2a + x)(2a - x)(5a - 2x)}, where aa is a positive constant.
10(a) [5]
Express f(x)\mathrm{f}(x) in partial fractions.
10(b) [5]
Hence find the exact value of ∫−aaf(x) dx\int_{-a}^{a} \mathrm{f}(x) \, \mathrm{d}x, giving your answer in the form pln⁡q+rln⁡sp\ln q + r\ln s where pp and rr are integers and qq and ss are prime numbers.
11 [9]
The variables yy and θ\theta satisfy the differential equation (1+y)(1+cos⁡2θ)dydθ=e3y(1 + y)(1 + \cos 2\theta)\dfrac{\mathrm{d}y}{\mathrm{d}\theta} = \mathrm{e}^{3y}. It is given that y=0y = 0 when θ=14π\theta = \dfrac{1}{4}\pi. Solve the differential equation and find the exact value of tan⁡θ\tan\theta when y=1y = 1.

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