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Learn/ A Level Mathematics/ Past Papers/ 2024/ Feb/March 2024 /Paper 1 (Variant 2)

Pure Mathematics 1 — Variant 2

A Level Mathematics · 9709/12 · Feb/March 2024

Question Paper

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Mark Scheme

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Questions

7 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]
Find the exact value of ∫3∞2x2 dx\int_{3}^{\infty} \frac{2}{x^{2}} \, \mathrm{d}x.
3 [5]
A curve is such that dydx=3(4x+5)12\frac{\mathrm{d}y}{\mathrm{d}x} = 3(4x + 5)^{\frac{1}{2}}. It is given that the points (1,9)(1, 9) and (5,a)(5, a) lie on the curve. Find the value of aa.
5 [6]
A curve has the equation y=32x2−5y = \frac{3}{2x^{2} - 5}. Find the equation of the normal to the curve at the point (2,1)(2, 1), giving your answer in the form ax+by+c=0ax + by + c = 0, where aa, bb and cc are integers.
6 [5]
It is given that the coefficient of x3x^{3} in the expansion of (2+ax)4(5−ax)(2 + ax)^{4}(5 - ax) is 432. Find the value of the constant aa.
7
The straight line y=x+5y = x + 5 meets the curve 2x2+3y2=k2x^{2} + 3y^{2} = k at a single point PP.
7(a) [4]
Find the value of the constant kk.
7(b) [2]
Find the coordinates of PP.
8
8(a) [4]
An arithmetic progression is such that its first term is 6 and its tenth term is 19.5. Find the sum of the first 100 terms of this arithmetic progression.
8(b) [4]
A geometric progression a1a_{1}, a2a_{2}, a3a_{3}, ... is such that a1=24a_{1} = 24 and the common ratio is 12\frac{1}{2}. The sum to infinity of this geometric progression is denoted by SS. The sum to infinity of the even-numbered terms (i.e. a2a_{2}, a4a_{4}, a6a_{6}, ...) is denoted by SES_{E}. Find the values of SS and SES_{E}.
9
The functions f and g are defined for all real values of xx by f(x)=(3x−2)2+k\mathrm{f}(x) = (3x - 2)^{2} + k and g(x)=5x−1\mathrm{g}(x) = 5x - 1, where kk is a constant.
9(a) [4]
Given that the range of the function gf is gf(x)⩾39\mathrm{gf}(x) \geqslant 39, find the value of kk.
9(b) [2]
For this value of kk, determine the range of the function fg.
9(c) [3]
The function h is defined for all real values of xx and is such that gh(x)=35x+19\mathrm{gh}(x) = 35x + 19. Find an expression for g−1(x)\mathrm{g}^{-1}(x) and hence, or otherwise, find an expression for h(x)\mathrm{h}(x).

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