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Learn/ A Level Mathematics/ Past Papers/ 2021/ Feb/March 2021 /Paper 3 (Variant 2)

Pure Mathematics 3 — Variant 2

A Level Mathematics · 9709/32 · Feb/March 2021

Question Paper

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Mark Scheme

PDF · 313 KB · 15 pages

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Questions

9 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⁡(x3−3)=3ln⁡x−ln⁡3\ln(x^{3} - 3) = 3\ln x - \ln 3. Give your answer correct to 3 significant figures.
2 [5]
The polynomial ax3+5x2−4x+bax^{3} + 5x^{2} - 4x + b, where aa and bb are constants, is denoted by p(x)\mathrm{p}(x). It is given that (x+2)(x + 2) is a factor of p(x)\mathrm{p}(x) and that when p(x)\mathrm{p}(x) is divided by (x+1)(x + 1) the remainder is 22. Find the values of aa and bb.
3 [6]
By first expressing the equation tan⁡(x+45∘)=2cot⁡x+1\tan(x + 45^\circ) = 2\cot x + 1 as a quadratic equation in tan⁡x\tan x, solve the equation for 0∘<x<180∘0^\circ < x < 180^\circ.
4
The variables xx and yy satisfy the differential equation (1−cos⁡x)dydx=ysin⁡x(1 - \cos x)\frac{\mathrm{d}y}{\mathrm{d}x} = y \sin x. It is given that y=4y = 4 when x=πx = \pi.
4(a) [6]
Solve the differential equation, obtaining an expression for yy in terms of xx.
4(b) [1]
Sketch the graph of yy against xx for 0<x<2π0 < x < 2\pi.
5
5(a) [3]
Express 7sin⁡x+2cos⁡x\sqrt{7}\sin x + 2\cos x in the form Rsin⁡(x+α)R\sin(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.
5(b) [5]
Hence solve the equation 7sin⁡2θ+2cos⁡2θ=1\sqrt{7}\sin 2\theta + 2\cos 2\theta = 1, for 0∘<θ<180∘0^\circ < \theta < 180^\circ.
6
Let f(x)=5a(2x−a)(3a−x)f(x) = \frac{5a}{(2x - a)(3a - x)}, where aa is a positive constant.
6(a) [3]
Express f(x)f(x) in partial fractions.
6(b) [4]
Hence show that ∫a2af(x) dx=ln⁡6\int_{a}^{2a} f(x) \, \mathrm{d}x = \ln 6.
7
Two lines have equations r=(132)+s(2−13)\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} + s\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix} and r=(214)+t(1−14)\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + t\begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}.
7(a) [5]
Show that the lines are skew.
7(b) [3]
Find the acute angle between the directions of the two lines.
8
The complex numbers uu and vv are defined by u=−4+2iu = -4 + 2i and v=3+iv = 3 + i.
8(a) [3]
Find uv\frac{u}{v} in the form x+iyx + iy, where xx and yy are real.
8(b) [2]
Hence express uv\frac{u}{v} in the form reiθre^{i\theta}, where rr and θ\theta are exact. In an Argand diagram, with origin OO, the points AA, BB and CC represent the complex numbers uu, vv and 2u+v2u + v respectively.
8(c) [2]
State fully the geometrical relationship between OAOA and BCBC.
8(d) [2]
Prove that angle AOB=34πAOB = \frac{3}{4}\pi.
9
Let f(x)=e2x+1e2x−1f(x) = \frac{e^{2x} + 1}{e^{2x} - 1}, for x>0x > 0.
9(a) [2]
The equation x=f(x)x = f(x) has one root, denoted by aa. Verify by calculation that aa lies between 11 and 1.51.5.
9(b) [3]
Use an iterative formula based on the equation in part (a) to determine aa correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
9(c) [6]
Find f′(x)f'(x). Hence find the exact value of xx for which f′(x)=−8f'(x) = -8.

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