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/ 2022/ Oct/Nov 2022 /Paper 1 (Variant 1)

Pure Mathematics 1 — Variant 1

A Level Mathematics · 9709/11 · Oct/Nov 2022

Question Paper

PDF · 132 KB · 20 pages

Mark Scheme

PDF · 376 KB · 17 pages

This browser cannot show the PDF inline.

Open the question paper · Open the mark scheme

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]
Solve the equation 3x+2=2x−13x + 2 = \frac{2}{x - 1}.
2
The equation of a curve is such that dydx=12(12x−1)−4\frac{\mathrm{d}y}{\mathrm{d}x} = 12(\frac{1}{2}x - 1)^{-4}. It is given that the curve passes through the point P(6,4)P(6, 4).
2(a) [2]
Find the equation of the tangent to the curve at PP.
2(b) [4]
Find the equation of the curve.
3 [5]
A curve has equation y=ax12−2xy = ax^{\frac{1}{2}} - 2x, where x>0x > 0 and aa is a constant. The curve has a stationary point at the point PP, which has xx-coordinate 99. Find the yy-coordinate of PP.
4 [6]
The coefficient of x2x^{2} in the expansion of (1+2px)5+(1+px)6(1 + \frac{2}{p}x)^{5} + (1 + px)^{6} is 7070. Find the possible values of the constant pp.
6
6(a) [3]
Show that the equation 1sin⁡θ+cos⁡θ+1sin⁡θ−cos⁡θ=1\frac{1}{\sin\theta + \cos\theta} + \frac{1}{\sin\theta - \cos\theta} = 1 may be expressed in the form asin⁡2θ+bsin⁡θ+c=0a\sin^{2}\theta + b\sin\theta + c = 0, where aa, bb and cc are constants to be found.
6(b) [3]
Hence solve the equation 1sin⁡θ+cos⁡θ+1sin⁡θ−cos⁡θ=1\frac{1}{\sin\theta + \cos\theta} + \frac{1}{\sin\theta - \cos\theta} = 1 for 0∘⩽θ⩽360∘0^\circ \leqslant \theta \leqslant 360^\circ.
7
A tool for putting fence posts into the ground is called a 'post-rammer'. The distances in millimetres that the post sinks into the ground on each impact of the post-rammer follow a geometric progression. The first three impacts cause the post to sink into the ground by 5050 mm, 4040 mm and 3232 mm respectively.
7(a) [3]
Verify that the 9th impact is the first in which the post sinks less than 1010 mm into the ground.
7(b) [2]
Find, to the nearest millimetre, the total depth of the post in the ground after 2020 impacts.
7(c) [2]
Find the greatest total depth in the ground which could theoretically be achieved.
8
The function ff is defined by f(x)=2−34x−pf(x) = 2 - \frac{3}{4x - p} for x>p4x > \frac{p}{4}, where pp is a constant.
8(a) [3]
Find f′(x)f'(x) and hence determine whether ff is an increasing function, a decreasing function or neither.
8(b) [4]
Express f−1(x)f^{-1}(x) in the form pa−bcx−d\frac{p}{a} - \frac{b}{cx - d}, where aa, bb, cc and dd are integers.
8(c) [1]
Hence state the value of pp for which f−1(x)≡f(x)f^{-1}(x) \equiv f(x).
9
Functions ff and gg are both defined for x∈Rx \in \mathbb{R} and are given by f(x)=x2−4x+9f(x) = x^{2} - 4x + 9, g(x)=2x2+4x+12g(x) = 2x^{2} + 4x + 12.
9(a) [1]
Express f(x)f(x) in the form (x−a)2+b(x - a)^{2} + b.
9(b) [2]
Express g(x)g(x) in the form 2[(x+c)2+d]2[(x + c)^{2} + d].
9(c) [1]
Express g(x)g(x) in the form kf(x+h)kf(x + h), where kk and hh are integers.
9(d) [4]
Describe fully the two transformations that have been combined to transform the graph of y=f(x)y = f(x) to the graph of y=g(x)y = g(x).

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