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Learn/ A Level Mathematics/ Past Papers/ 2021/ Feb/March 2021 /Paper 1 (Variant 2)

Pure Mathematics 1 — Variant 2

A Level Mathematics · 9709/12 · Feb/March 2021

Question Paper

PDF · 124 KB · 20 pages

Mark Scheme

PDF · 308 KB · 14 pages

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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
1(a) [1]
Find the first three terms in the expansion, in ascending powers of xx, of (1+x)5(1 + x)^{5}.
1(b) [2]
Find the first three terms in the expansion, in ascending powers of xx, of (1−2x)6(1 - 2x)^{6}.
1(c) [2]
Hence find the coefficient of x2x^{2} in the expansion of (1+x)5(1−2x)6(1 + x)^{5}(1 - 2x)^{6}.
2 [4]
By using a suitable substitution, solve the equation (2x−3)2−4(2x−3)2−3=0(2x - 3)^{2} - \frac{4}{(2x - 3)^{2}} - 3 = 0.
3 [4]
Solve the equation tan⁡θ+2sin⁡θtan⁡θ−2sin⁡θ=3\frac{\tan \theta + 2 \sin \theta}{\tan \theta - 2 \sin \theta} = 3 for 0∘<θ<180∘0^\circ < \theta < 180^\circ.
4 [5]
A line has equation y=3x+ky = 3x + k and a curve has equation y=x2+kx+6y = x^{2} + kx + 6, where kk is a constant. Find the set of values of kk for which the line and curve have two distinct points of intersection.
6
A curve is such that dydx=6(3x−2)3\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{6}{(3x - 2)^{3}} and A(1,−3)A(1, -3) lies on the curve. A point is moving along the curve and at AA the yy-coordinate of the point is increasing at 3 units per second.
6(a) [3]
Find the rate of increase at AA of the xx-coordinate of the point.
6(b) [4]
Find the equation of the curve.
7
Functions f and g are defined as follows: f:x↦x2+2x+3\mathrm{f} : x \mapsto x^{2} + 2x + 3 for x⩽−1x \leqslant -1, g:x↦2x+1\mathrm{g} : x \mapsto 2x + 1 for x⩾−1x \geqslant -1.
7(a) [3]
Express f(x)\mathrm{f}(x) in the form (x+a)2+b(x + a)^{2} + b and state the range of f.
7(b) [2]
Find an expression for f−1(x)\mathrm{f}^{-1}(x).
7(c) [3]
Solve the equation gf(x)=13\mathrm{gf}(x) = 13.
8
The points A(7,1)A(7, 1), B(7,9)B(7, 9) and C(1,9)C(1, 9) are on the circumference of a circle.
8(a) [5]
Find an equation of the circle.
8(b) [2]
Find an equation of the tangent to the circle at BB.
9
The first term of a progression is cos⁡θ\cos \theta, where 0<θ<12π0 < \theta < \frac{1}{2}\pi.
9(a)
For the case where the progression is geometric, the sum to infinity is 1cos⁡θ\frac{1}{\cos \theta}.
9(a)(i) [3]
Show that the second term is cos⁡θsin⁡2θ\cos \theta \sin^{2} \theta.
9(a)(ii) [2]
Find the sum of the first 12 terms when θ=13π\theta = \frac{1}{3}\pi, giving your answer correct to 4 significant figures.
9(b) [4]
For the case where the progression is arithmetic, the first two terms are again cos⁡θ\cos \theta and cos⁡θsin⁡2θ\cos \theta \sin^{2} \theta respectively. Find the 85th term when θ=13π\theta = \frac{1}{3}\pi.

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