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Learn/ A Level Mathematics/ Past Papers/ 2023/ Oct/Nov 2023 /Paper 1 (Variant 1)

Pure Mathematics 1 — Variant 1

A Level Mathematics · 9709/11 · Oct/Nov 2023

Question Paper

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Mark Scheme

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Questions

5 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
1(a) [2]
Expand (1+3x)6(1 + 3x)^{6} in ascending powers of xx up to, and including, the term in x2x^{2}.
1(b) [2]
Hence find the coefficient of x2x^{2} in the expansion of (1−7x+x2)(1+3x)6(1 - 7x + x^{2})(1 + 3x)^{6}.
4
The transformation R denotes a reflection in the xx-axis and the transformation T denotes a translation of (3−1)\begin{pmatrix} 3 \\ -1 \end{pmatrix}.
4(a) [2]
Find the equation, y=g(x)y = \mathrm{g}(x), of the curve with equation y=x2y = x^{2} after it has been transformed by the sequence of transformations R followed by T.
4(b) [2]
Find the equation, y=h(x)y = \mathrm{h}(x), of the curve with equation y=x2y = x^{2} after it has been transformed by the sequence of transformations T followed by R.
4(c) [2]
State fully the transformation that maps the curve y=g(x)y = \mathrm{g}(x) onto the curve y=h(x)y = \mathrm{h}(x).
5
5(a) [3]
Show that the equation 4sin⁡x+5tan⁡x+2sin⁡x=04\sin x + \frac{5}{\tan x} + \frac{2}{\sin x} = 0 may be expressed in the form acos⁡2x+bcos⁡x+c=0a\cos^{2}x + b\cos x + c = 0, where aa, bb and cc are integers to be found.
5(b) [3]
Hence solve the equation 4sin⁡x+5tan⁡x+2sin⁡x=04\sin x + \frac{5}{\tan x} + \frac{2}{\sin x} = 0 for 0∘⩽x⩽360∘0^\circ \leqslant x \leqslant 360^\circ.
7 [7]
The sum of the first two terms of a geometric progression is 1515 and the sum to infinity is 1257\frac{125}{7}. The common ratio of the progression is negative. Find the third term of the progression.
10
A curve has a stationary point at (2,−10)(2, -10) and is such that d2ydx2=6x\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}} = 6x.
10(a) [3]
Find dydx\frac{\mathrm{d}y}{\mathrm{d}x}.
10(b) [3]
Find the equation of the curve.
10(c) [3]
Find the coordinates of the other stationary point and determine its nature.
10(d) [2]
Find the equation of the tangent to the curve at the point where the curve crosses the yy-axis.

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