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Learn/ A Level Mathematics/ Past Papers/ 2021/ May/June 2021 /Paper 1 (Variant 1)

Pure Mathematics 1 — Variant 1

A Level Mathematics · 9709/11 · May/June 2021

Question Paper

PDF · 123 KB · 20 pages

Mark Scheme

PDF · 400 KB · 17 pages

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Questions

4 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.

3
3(a) [3]
Find the first three terms in the expansion of (3−2x)5(3 - 2x)^{5} in ascending powers of xx.
3(b) [3]
Hence find the coefficient of x2x^{2} in the expansion of (4+x)2(3−2x)5(4 + x)^{2}(3 - 2x)^{5}.
7
7(a) [2]
Prove the identity 1−2sin⁡2θ1−sin⁡2θ≡1−tan⁡2θ\frac{1 - 2\sin^{2}\theta}{1 - \sin^{2}\theta} \equiv 1 - \tan^{2}\theta.
7(b) [3]
Hence solve the equation 1−2sin⁡2θ1−sin⁡2θ=2tan⁡4θ\frac{1 - 2\sin^{2}\theta}{1 - \sin^{2}\theta} = 2\tan^{4}\theta for 0∘⩽θ⩽180∘0^\circ \leqslant \theta \leqslant 180^\circ.
10
The equation of a circle is x2+y2−4x+6y−77=0x^{2} + y^{2} - 4x + 6y - 77 = 0.
10(a) [2]
Find the xx-coordinates of the points AA and BB where the circle intersects the xx-axis.
10(b) [6]
Find the point of intersection of the tangents to the circle at AA and BB.
11
The equation of a curve is y=23x+4−xy = 2\sqrt{3x + 4} - x.
11(a) [5]
Find the equation of the normal to the curve at the point (4,4)(4, 4), giving your answer in the form y=mx+cy = mx + c.
11(b) [3]
Find the coordinates of the stationary point.
11(c) [2]
Determine the nature of the stationary point.
11(d) [4]
Find the exact area of the region bounded by the curve, the xx-axis and the lines x=0x = 0 and x=4x = 4.

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