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Learn/ A Level Mathematics/ Past Papers/ 2023/ Feb/March 2023 /Paper 3 (Variant 2)

Pure Mathematics 3 — Variant 2

A Level Mathematics · 9709/32 · Feb/March 2023

Question Paper

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Mark Scheme

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Questions

8 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]
It is given that x=ln⁡(2y−3)−ln⁡(y+4)x = \ln(2y - 3) - \ln(y + 4). Express yy in terms of xx.
2
2(a) [3]
On an Argand diagram, shade the region whose points represent complex numbers zz satisfying the inequalities −13π⩽arg⁡(z−1−2i)⩽13π-\frac{1}{3}\pi \leqslant \arg(z - 1 - 2i) \leqslant \frac{1}{3}\pi and Re⁡z⩽3\operatorname{Re} z \leqslant 3.
2(b) [2]
Calculate the least value of arg⁡z\arg z for points in the region from (a). Give your answer in radians correct to 3 decimal places.
3 [5]
The polynomial 2x4+ax3+bx−12x^{4} + ax^{3} + bx - 1, where aa and bb are constants, is denoted by p(x)p(x). When p(x)p(x) is divided by x2−x+1x^{2} - x + 1 the remainder is 3x+23x + 2. Find the values of aa and bb.
4 [5]
Solve the equation 5z1+2i−zz∗+30+10i=0\frac{5z}{1 + 2i} - zz^{*} + 30 + 10i = 0, giving your answers in the form x+iyx + iy, where xx and yy are real.
5
The parametric equations of a curve are x=te2tx = t e^{2t}, y=t2+t+3y = t^{2} + t + 3.
5(a) [3]
Show that dydx=e−2t\frac{\mathrm{d}y}{\mathrm{d}x} = e^{-2t}.
5(b) [3]
Hence show that the normal to the curve, where t=−1t = -1, passes through the point (0,3−1e4)(0, 3 - \frac{1}{e^{4}}).
6
6(a) [3]
Express 5sin⁡θ+12cos⁡θ5\sin\theta + 12\cos\theta in the form Rcos⁡(θ−α)R\cos(\theta - \alpha), where R>0R > 0 and 0<α<12π0 < \alpha < \frac{1}{2}\pi.
6(b) [4]
Hence solve the equation 5sin⁡2x+12cos⁡2x=65\sin 2x + 12\cos 2x = 6 for 0⩽x⩽π0 \leqslant x \leqslant \pi.
9 [7]
The variables xx and yy satisfy the differential equation dydx=e3ysin⁡22x\frac{\mathrm{d}y}{\mathrm{d}x} = e^{3y}\sin^{2}2x. It is given that y=0y = 0 when x=0x = 0. Solve the differential equation and find the value of yy when x=12x = \frac{1}{2}.
11
Let f(x)=5x2+x+11(4+x2)(1+x)f(x) = \frac{5x^{2} + x + 11}{(4 + x^{2})(1 + x)}.
11(a) [5]
Express f(x)f(x) in partial fractions.
11(b) [5]
Hence show that ∫02f(x) dx=ln⁡54−18π\int_{0}^{2} f(x) \, \mathrm{d}x = \ln 54 - \frac{1}{8}\pi.

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