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Learn/ A Level Mathematics/ Past Papers/ 2022/ Oct/Nov 2022 /Paper 1 (Variant 3)

Pure Mathematics 1 — Variant 3

A Level Mathematics · 9709/13 · Oct/Nov 2022

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]
Solve the equation 8sin⁡2θ+6cos⁡θ+1=08\sin^{2}\theta + 6\cos\theta + 1 = 0 for 0∘<θ<180∘0^\circ < \theta < 180^\circ.
2
The function f is defined by f(x)=−2x2−8x−13f(x) = -2x^{2} - 8x - 13 for x<−3x < -3.
2(a) [2]
Express f(x) in the form −2(x+a)2+b-2(x + a)^{2} + b, where aa and bb are integers.
2(b) [1]
Find the range of f.
2(c) [3]
Find an expression for f−1(x)f^{-1}(x).
3
3(a) [2]
Find the first three terms in ascending powers of xx of the expansion of (1+2x)5(1 + 2x)^{5}.
3(b) [2]
Find the first three terms in ascending powers of xx of the expansion of (1−3x)4(1 - 3x)^{4}.
3(c) [2]
Hence find the coefficient of x2x^{2} in the expansion of (1+2x)5(1−3x)4(1 + 2x)^{5}(1 - 3x)^{4}.
4 [5]
A large industrial water tank is such that, when the depth of the water in the tank is xx metres, the volume VV m3\text{m}^{3} of water in the tank is given by V=243−13(9−x)3V = 243 - \frac{1}{3}(9 - x)^{3}. Water is being pumped into the tank at a constant rate of 3.63.6 m3\text{m}^{3} per hour. Find the rate of increase of the depth of the water when the depth is 44 m, giving your answer in cm per minute.
6 [5]
It is given that α=cos⁡−1(817)\alpha = \cos^{-1}(\frac{8}{17}). Find, without using the trigonometric functions on your calculator, the exact value of 1sin⁡α+1tan⁡α\frac{1}{\sin \alpha} + \frac{1}{\tan \alpha}.
7
The curve y=f(x)y = f(x) is such that f′(x)=−3(x+2)4f'(x) = \frac{-3}{(x + 2)^{4}}.
7(a) [4]
The tangent at a point on the curve where x=ax = a has gradient −1627-\frac{16}{27}. Find the possible values of aa.
7(b) [3]
Find f(x)f(x) given that the curve passes through the point (−1,5)(-1, 5).
11
The coordinates of points AA, BB and CC are A(5,−2)A(5, -2), B(10,3)B(10, 3) and C(2p,p)C(2p, p), where pp is a constant.
11(a) [3]
Given that ACAC and BCBC are equal in length, find the value of the fraction pp.
11(b)
It is now given instead that ACAC is perpendicular to BCBC and that pp is an integer.
11(b)(i) [4]
Find the value of pp.
11(b)(ii) [4]
Find the equation of the circle which passes through AA, BB and CC, giving your answer in the form x2+y2+ax+by+c=0x^{2} + y^{2} + ax + by + c = 0, where aa, bb and cc are constants.

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