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Learn/ A Level Mathematics/ Past Papers/ 2022/ May/June 2022 /Paper 1 (Variant 3)

Pure Mathematics 1 — Variant 3

A Level Mathematics · 9709/13 · May/June 2022

Question Paper

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Mark Scheme

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Questions

6 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]
The coefficient of x3x^{3} in the expansion of (p+1px)4(p + \frac{1}{p}x)^{4} is 144144. Find the possible values of the constant pp.
3
An arithmetic progression has first term 44 and common difference dd. The sum of the first nn terms of the progression is 58635863.
3(a) [1]
Show that (n−1)d=11726n−8(n - 1)d = \frac{11726}{n} - 8.
3(b) [4]
Given that the nnth term is 139139, find the values of nn and dd, giving the value of dd as a fraction.
4
4(a) [3]
The curve with equation y=x2+2x−5y = x^{2} + 2x - 5 is translated by (−13)\begin{pmatrix} -1 \\ 3 \end{pmatrix}. Find the equation of the translated curve, giving your answer in the form y=ax2+bx+cy = ax^{2} + bx + c.
4(b) [2]
The curve with equation y=x2+2x−5y = x^{2} + 2x - 5 is transformed to a curve with equation y=4x2+4x−5y = 4x^{2} + 4x - 5. Describe fully the single transformation that has been applied.
5
5(a) [4]
Solve the equation 6y+2y−7=06\sqrt{y} + \frac{2}{\sqrt{y}} - 7 = 0.
5(b) [3]
Hence solve the equation 6tan⁡x+2tan⁡x−7=06\sqrt{\tan x} + \frac{2}{\sqrt{\tan x}} - 7 = 0 for 0∘⩽x⩽360∘0^\circ \leqslant x \leqslant 360^\circ.
10
The function ff is defined by f(x)=(4x+2)−2f(x) = (4x + 2)^{-2} for x>−12x > -\frac{1}{2}.
10(a) [4]
Find ∫1∞f(x) dx\int_{1}^{\infty} f(x) \, \mathrm{d}x. A point is moving along the curve y=f(x)y = f(x) in such a way that, as it passes through the point AA, its yy-coordinate is decreasing at the rate of kk units per second and its xx-coordinate is increasing at the rate of kk units per second.
10(b) [6]
Find the coordinates of AA.
11
The point PP lies on the line with equation y=mx+cy = mx + c, where mm and cc are positive constants. A curve has equation y=−mxy = -\frac{m}{x}. There is a single point PP on the curve such that the straight line is a tangent to the curve at PP.
11(a) [6]
Find the coordinates of PP, giving the yy-coordinate in terms of mm. The normal to the curve at PP intersects the curve again at the point QQ.
11(b) [4]
Find the coordinates of QQ in terms of mm.

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