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Cambridge International AS & A Level

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MATHEMATICS 9709/11
Paper 1 Pure Mathematics 1 October/November 2024

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 (DE/CT) 337381/4
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1 In the expansion of bkx + l , where k is a positive constant, the term independent of x is equal to 150.
x
Find the value of k and hence determine the coefficient of x 2 in the expansion. [4]

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a
2 The curve y = x 2 - has a stationary point at (-3, b).
x
Find the values of the constants a and b. [4]

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B C

15 cm 15 cm
A D

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The diagram shows a sector of a circle, centre O, where OB = OC = 15 cm . The size of angle BOC is
2
5 r radians. Points A and D on the lines OB and OC respectively are joined by an arc AD of a circle
with centre O. The shaded region is bounded by the arcs AD and BC and by the straight lines AB and
DC. It is given that the area of the shaded region is 209 2
5 r cm .

Find the perimeter of the shaded region. Give your answer in terms of r . [5]

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4 Show that the curve with equation x 2 - 3xy - 40 = 0 and the line with equation 3x + y + k = 0 meet for
all values of the constant k. [5]

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dy
5 The equation of a curve is such that = 4x - 3 x + 1.
dx
11
(a) Find the x-coordinate of the point on the curve at which the gradient is 2 . [3]

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(b) Given that the curve passes through the point (4, 11), find the equation of the curve. [4]

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6 Circles C1 and C2 have equations

x 2 + y 2 + 6x - 10y + 18 = 0 and (x - 9) 2 + (y + 4) 2 - 64 = 0

respectively.

(a) Find the distance between the centres of the circles. [4]

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P and Q are points on C1 and C2 respectively. The distance between P and Q is denoted by d.

(b) Find the greatest and least possible values of d. [3]

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y

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O 7 x
2

12
The diagram shows part of the curve with equation y = 3 . The point A on the curve has
2x + 1
coordinates b72 , 6l.

(a) Find the equation of the tangent to the curve at A. Give your answer in the form y = mx + c . [4]

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(b) Find the area of the region bounded by the curve and the lines x = 0 , x = 2 and y = 0 . [4]

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8 (a) It is given that b is an angle between 90° and 180° such that sin b = a .

Express tan 2 b - 3 sin b cos b in terms of a. [3]

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(b) Solve the equation sin 2 i + 2 cos 2 i = 4 sin i + 3 for 0° 1 i 1 360° . [5]

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9 The equation of a curve is y = 4 + 5x + 6x 2 - 3x 3.

(a) Find the set of values of x for which y decreases as x increases. [4]

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(b) It is given that y = 9x + k is a tangent to the curve.

Find the value of the constant k. [4]

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10 An arithmetic progression has first term 5 and common difference d, where d 2 0 . The second, fifth
and eleventh terms of the arithmetic progression, in that order, are the first three terms of a geometric
progression.

(a) Find the value of d. [3]

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(b) The sum of the first 77 terms of the arithmetic progression is denoted by S77 . The sum of the first
10 terms of the geometric progression is denoted by G 10 .

Find the value of S77 - G 10 . [5]

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11 The function f is defined by f (x) = 3 + 6x - 2x 2 for x ! R .

(a) Express f (x) in the form a - b (x - c) 2 , where a, b and c are constants, and state the range
of f. [3]

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(b) The graph of y = f (x) is transformed to the graph of y = h (x) by a reflection in one of the axes
followed by a translation. It is given that the graph of y = h (x) has a minimum point at the origin.

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Give details of the reflection and translation involved. [2]

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The function g is defined by g (x) = 3 + 6x - 2x 2 for x G 0 .

(c) Sketch the graph of y = g (x) and explain why g is a one-one function. You are not required to find
the coordinates of any intersections with the axes. [2]
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(d) Sketch the graph of y = g -1 (x) on your diagram in (c), and find an expression for g -1 (x) .
You should label the two graphs in your diagram appropriately and show any relevant mirror line.
[4]

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 ,

Additional page

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

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