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Learn/ A Level Mathematics/ Past Papers/ 2023/ Feb/March 2023 /Paper 1 (Variant 2)

Pure Mathematics 1 — Variant 2

A Level Mathematics · 9709/12 · Feb/March 2023

Question Paper

PDF · 130 KB · 20 pages

Mark Scheme

PDF · 369 KB · 15 pages

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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 [4]
A line has equation y=3x−2ky = 3x - 2k and a curve has equation y=x2−kx+2y = x^{2} - kx + 2, where kk is a constant. Show that the line and the curve meet for all values of kk.
2 [4]
A function f is defined by f(x)=x2−2x+5f(x) = x^{2} - 2x + 5 for x∈Rx \in \mathbb{R}. A sequence of transformations is applied in the following order to the graph of y=f(x)y = f(x) to give the graph of y=g(x)y = g(x). Stretch parallel to the xx-axis with scale factor 12\frac{1}{2} Reflection in the yy-axis Stretch parallel to the yy-axis with scale factor 33 Find g(x)g(x), giving your answer in the form ax2+bx+cax^{2} + bx + c, where aa, bb and cc are constants.
3 [4]
A curve has equation y=160(3x+1)2y = \frac{1}{60}(3x + 1)^{2} and a point is moving along the curve. Find the xx-coordinate of the point on the curve at which the xx- and yy-coordinates are increasing at the same rate.
4
The circumference round the trunk of a large tree is measured and found to be 5.005.00 m. After one year the circumference is measured again and found to be 5.025.02 m.
4(a) [2]
Given that the circumferences at yearly intervals form an arithmetic progression, find the circumference 2020 years after the first measurement.
4(b) [3]
Given instead that the circumferences at yearly intervals form a geometric progression, find the circumference 2020 years after the first measurement.
5 [6]
Points A(7,12)A(7, 12) and BB lie on a circle with centre (−2,5)(-2, 5). The line ABAB has equation y=−2x+26y = -2x + 26. Find the coordinates of BB.
6 [6]
In the expansion of (xa+ax2)7(\frac{x}{a} + \frac{a}{x^{2}})^{7}, it is given that the coefficient of x4the coefficient of x=3\frac{\text{the coefficient of } x^{4}}{\text{the coefficient of } x} = 3. Find the possible values of the constant aa.
7
7(a) [5]
By first obtaining a quadratic equation in cos⁡θ\cos \theta, solve the equation tan⁡θsin⁡θ=1\tan \theta \sin \theta = 1 for 0∘<θ<360∘0^\circ < \theta < 360^\circ.
7(b) [3]
Show that tan⁡θsin⁡θ−sin⁡θtan⁡θ≡tan⁡θsin⁡θ\frac{\tan \theta}{\sin \theta} - \frac{\sin \theta}{\tan \theta} \equiv \tan \theta \sin \theta.
10
At the point (4,−1)(4, -1) on a curve, the gradient of the curve is −32-\frac{3}{2}. It is given that dydx=x−12+k\frac{\mathrm{d}y}{\mathrm{d}x} = x^{-\frac{1}{2}} + k, where kk is a constant.
10(a) [1]
Show that k=−2k = -2.
10(b) [4]
Find the equation of the curve.
10(c) [3]
Find the coordinates of the stationary point.
10(d) [2]
Determine the nature of the stationary point.

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