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Cambridge International AS & A Level: Mathematics 9709/15

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104 views20 pages

Cambridge International AS & A Level: Mathematics 9709/15

maths paper 1 china variant
Copyright
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Cambridge International AS & A Level

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MATHEMATICS 9709/15
Paper 1 Pure Mathematics 1 May/June 2025

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INSTRUCTIONS
● Answer all questions.
● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
● Write your name, centre number and candidate number in the boxes at the top of the page.
● Write your answer to each question in the space provided.
● Do not use an erasable pen or correction fluid.
● Do not write on any bar codes.
● If additional space is needed, you should use the lined page at the end of this booklet; the question
number or numbers must be clearly shown.
● You should use a calculator where appropriate.
● You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
● Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in
degrees, unless a different level of accuracy is specified in the question.

INFORMATION
● The total mark for this paper is 75.
● The number of marks for each question or part question is shown in brackets [ ].

This document has 20 pages. Any blank pages are indicated.

DC (PQ/FC) 357567/1
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dy
1 The equation of a curve is such that = 12 (2x - 5) 2 + 8x . It is given that the curve passes through the
dx
point (2, 4).

Find an equation of the curve. [4]

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2 In the expansion of (3 + ax) 5 + (6 - x) 4 , the coefficient of x 2 is six times the coefficient of x.

Find the possible values of the constant a. [5]

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3 (a) Use completing the square to find the exact solutions of the equation 4x 2 - 4x - 1 = 0 . [2]

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(b) Hence solve the equation 4 tan i = 4 + for 0° 1 i 1 180° . [3]
tan i

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4
y
1
y = x2 -
x2

O 2 x
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The diagram shows part of the curve y = x 2 - 2 . The shaded region is bounded by the curve, the line
x = 2 and the x-axis. x

Find the volume formed when the shaded region is rotated through 360° about the x-axis, giving your
answer correct to 2 decimal places. [5]

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5
D

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A 5 cm C 5 cm B

The diagram shows a sector ABD of a circle with centre A and radius 10 cm. The perpendicular bisector
of AB passes through D.

(a) Find the perimeter of the shaded region BCD, giving your answer correct to 1 decimal place. [4]

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(b) Find the area of the shaded region BCD, giving your answer correct to 1 decimal place. [2]

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6 Each year, on her birthday, Ananya receives some money from each of her parents.

On Ananya’s first birthday, her father gives her $10. Every subsequent year, her father gives her $5
more than he gave her the previous year.

On Ananya’s first birthday, her mother also gives her $10. Every subsequent year, her mother gives her
20% more than she gave her the previous year.

(a) Show that on Ananya’s eleventh birthday she receives more from her mother than from her father.
[3]

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(b) Find the total amount of money Ananya receives up to and including her eighteenth birthday. [5]

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7 In the parallelogram ABCD, the coordinates of A are (3, 7), the coordinates of B are (6, p) and the
coordinates of D are (1, p). It is given that the gradient of AB is - 23 .

(a) Find the value of p. [2]

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(b) Find the coordinates of C. [2]

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(c) Find the area of the triangle formed by the perpendicular bisector of AB and the x- and y-axes. [5]

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8 The equation of a curve is y = x 3 + ax 2 + bx + 5 . The curve has a stationary point at (1, 9).

(a) Find the values of the constants a and b. [5]

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(b) Find the coordinates of the other stationary point. [3]

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(c) A point P is moving along part of the curve in such a way that the y-coordinate of P is increasing
at a constant rate of 6 units per second.

Find the rate at which the x-coordinate of P is increasing when x = 5. [3]

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9 Functions f and g are defined as follows.

f (x) = cos x for 0 G x G r


g (x) = 3 cos (x - r) + 2 for r G x G 2r

(a) Describe fully the transformations that have been combined to transform the graph of y = f (x) to
the graph of y = g (x) . [4]

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(b) On the given axes, sketch the graphs of y = f (x) and y = g (x) . [4]

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(c) Find g -1 f `13 rj. [4]

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(d) Explain why the composite function fg cannot be formed. [1]

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10 The equation of a circle is x 2 + y 2 + 4x - 8y - 12 = 0 .

(a) Find an equation of the tangent to the circle at the point (2, 8), giving your answer in the form
ax + by + c = 0 . [4]

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(b) Given that the line x + 3y = k does not intersect the circle, show that k 2 - 20k - 220 2 0 . [5]

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18
,  ,

Additional page

If you use the following page to complete the answer to any question, the question number must be clearly
shown.

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