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Learn/ A Level Mathematics/ Past Papers/ 2023/ May/June 2023 /Paper 1 (Variant 3)

Pure Mathematics 1 — Variant 3

A Level Mathematics · 9709/13 · May/June 2023

Question Paper

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Mark Scheme

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Questions

5 of 10 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.

2 [4]
The function ff is defined for x∈Rx \in \mathbb{R} by f(x)=x2−6x+cf(x) = x^{2} - 6x + c, where cc is a constant. It is given that f(x)>2f(x) > 2 for all values of xx. Find the set of possible values of cc.
3
3(a) [2]
Give the complete expansion of (x+2x)5(x + \frac{2}{x})^{5}.
3(b) [4]
In the expansion of (a+bx2)(x+2x)5(a + bx^{2})(x + \frac{2}{x})^{5}, the coefficient of xx is zero and the coefficient of 1x\frac{1}{x} is 8080. Find the values of the constants aa and bb.
4
4(a) [3]
Show that the equation 3tan⁡2x−3sin⁡2x−4=03\tan^{2}x - 3\sin^{2}x - 4 = 0 may be expressed in the form acos⁡4x+bcos⁡2x+c=0a\cos^{4}x + b\cos^{2}x + c = 0, where aa, bb and cc are constants to be found.
4(b) [4]
Hence solve the equation 3tan⁡2x−3sin⁡2x−4=03\tan^{2}x - 3\sin^{2}x - 4 = 0 for 0∘⩽x⩽180∘0^\circ \leqslant x \leqslant 180^\circ.
5
A circle has equation (x−1)2+(y+4)2=40(x - 1)^{2} + (y + 4)^{2} = 40. A line with equation y=x−9y = x - 9 intersects the circle at points AA and BB.
5(a) [4]
Find the coordinates of the two points of intersection.
5(b) [3]
Find an equation of the circle with diameter ABAB.
8
A progression has first term aa and second term a2a+2\frac{a^{2}}{a + 2}, where aa is a positive constant.
8(a) [5]
For the case where the progression is geometric and the sum to infinity is 264264, find the value of aa.
8(b) [5]
For the case where the progression is arithmetic and a=6a = 6, determine the least value of nn required for the sum of the first nn terms to be less than −480-480.

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