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Learn/ A Level Mathematics/ Past Papers/ 2023/ May/June 2023 /Paper 1 (Variant 2)

Pure Mathematics 1 — Variant 2

A Level Mathematics · 9709/12 · May/June 2023

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]
The equation of a curve is such that dydx=4(x−3)3\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{4}{(x - 3)^{3}} for x>3x > 3. The curve passes through the point (4,5)(4, 5). Find the equation of the curve.
2 [4]
The coefficient of x4x^{4} in the expansion of (x+a)6(x + a)^{6} is pp and the coefficient of x2x^{2} in the expansion of (ax+3)4(ax + 3)^{4} is qq. It is given that p+q=276p + q = 276. Find the possible values of the constant aa.
3
3(a) [2]
Express 4x2−24x+p4x^{2} - 24x + p in the form a(x+b)2+ca(x + b)^{2} + c, where aa and bb are integers and cc is to be given in terms of the constant pp.
3(b) [1]
Hence or otherwise find the set of values of pp for which the equation 4x2−24x+p=04x^{2} - 24x + p = 0 has no real roots.
4 [3]
Solve the equation 8x6+215x3−27=08x^{6} + 215x^{3} - 27 = 0.
9
The second term of a geometric progression is 1616 and the sum to infinity is 100100.
9(a) [4]
Find the two possible values of the first term.
9(b) [4]
Show that the nnth term of one of the two possible geometric progressions is equal to 4n−24^{n-2} multiplied by the nnth term of the other geometric progression.
10
The equation of a circle is (x−a)2+(y−3)2=20(x - a)^{2} + (y - 3)^{2} = 20. The line y=12x+6y = \frac{1}{2}x + 6 is a tangent to the circle at the point PP.
10(a) [5]
Show that one possible value of aa is 44 and find the other possible value.
10(b) [4]
For a=4a = 4, find the equation of the normal to the circle at PP.
10(c) [4]
For a=4a = 4, find the equations of the two tangents to the circle which are parallel to the normal found in (b).
11
The equation of a curve is y=k4x+1−x+5y = k\sqrt{4x + 1} - x + 5, where kk is a positive constant.
11(a) [2]
Find dydx\frac{\mathrm{d}y}{\mathrm{d}x}.
11(b) [2]
Find the xx-coordinate of the stationary point in terms of kk.
11(c) [4]
Given that k=10.5k = 10.5, find the equation of the normal to the curve at the point where the tangent to the curve makes an angle of tan⁡−1(2)\tan^{-1}(2) with the positive xx-axis.

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