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This document appears to be an exam paper for a Cambridge International AS & A Level Mathematics exam. It contains instructions for candidates on how to complete the exam, as well as two multi-part math problems. The first problem asks students to find the equation of a curve given its derivative and that it passes through the point (8,5). The second problem asks students to find the set of values of c for which a given quadratic curve and linear line intersect at two distinct points. The document provides space for students to show their work and solutions to each problem. It also provides information such as the total marks, number of pages, and exam code at the top and bottom.

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106 views388 pages

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Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*6482658238*

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 February/March 2022

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INSTRUCTIONS
³ Answer all questions.
³ Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
³ Write your name, centre number and candidate number in the boxes at the top of the page.
³ Write your answer to each question in the space provided.
³ Do not use an erasable pen or correction fluid.
³ Do not write on any bar codes.
³ 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.
³ You should use a calculator where appropriate.
³ You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
³ 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
³ The total mark for this paper is 75.
³ The number of marks for each question or part question is shown in brackets [ ].

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

JC22 03_9709_12/FP
© UCLES 2022 [Turn over
2

BLANK PAGE

© UCLES 2022 9709/12/F/M/22

− 13 1
1 A curve with equation y = f x is such that f ′ x = 2x − x 3 . It is given that f 8 = 5.

Find f x. [4]

................................................................................................................................................................

© UCLES 2022 9709/12/F/M/22 [Turn over

2 A curve has equation y = x2 + 2cx + 4 and a straight line has equation y = 4x + c, where c is a constant.

Find the set of values of c for which the curve and line intersect at two distinct points. [5]

................................................................................................................................................................

© UCLES 2022 9709/12/F/M/22

3 Find the term independent of x in each of the following expansions.

@ A
2 6
(a) 3x + 2 [3]
x

........................................................................................................................................................

@ A
2 6
(b) 3x + 2 1 − x3 [3]
x

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22 [Turn over

4 The first term of a geometric progression and the first term of an arithmetic progression are both equal
to a.

The third term of the geometric progression is equal to the second term of the arithmetic progression.

The fifth term of the geometric progression is equal to the sixth term of the arithmetic progression.

Given that the terms are all positive and not all equal, find the sum of the first twenty terms of the
arithmetic progression in terms of a. [6]

................................................................................................................................................................

© UCLES 2022 9709/12/F/M/22

5 (a) Express 2x2 − 8x + 14 in the form 2 x − a2 + b. [2]

........................................................................................................................................................

The functions f and g are defined by

f x = x2 for x ∈ >,
g x = 2x2 − 8x + 14 for x ∈ >.

(b) Describe fully a sequence of transformations that maps the graph of y = f x onto the graph of
y = g x, making clear the order in which the transformations are applied. [4]

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22 [Turn over

6
y

y = 3x − 20
x + 12 + y − 22 = 85

A
C
x
O

The circle with equation x + 12 + y − 22 = 85 and the straight line with equation y = 3x − 20 are
shown in the diagram. The line intersects the circle at A and B, and the centre of the circle is at C.

(a) Find, by calculation, the coordinates of A and B. [4]

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22

(b) Find an equation of the circle which has its centre at C and for which the line with equation
y = 3x − 20 is a tangent to the circle. [4]

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22 [Turn over

sin 1 + 2 cos 1 sin 1 − 2 cos 1 4

7 (a) Show that − 2
. [4]
cos 1 − 2 sin 1 cos 1 + 2 sin 1 5 cos 1 − 4

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22

sin 1 + 2 cos 1 sin 1 − 2 cos 1

(b) Hence solve the equation − = 5 for 0Å < 1 < 180Å. [3]
cos 1 − 2 sin 1 cos 1 + 2 sin 1

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22 [Turn over

8
y

A B

x
O

x − 22 + y2 = 8

The diagram shows the circle with equation x − 22 + y2 = 8. The chord AB of the circle intersects
the positive y-axis at A and is parallel to the x-axis.

(a) Find, by calculation, the coordinates of A and B. [3]

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22

(b) Find the volume of revolution when the shaded segment, bounded by the circle and the chord
AB, is rotated through 360Å about the x-axis. [5]

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22 [Turn over

9 Functions f, g and h are defined as follows:

1
f : x → x − 4x 2 + 1 for x ≥ 0,
2
g : x → mx + n for x ≥ −2, where m and n are constants,
1
h : x → x 2 − 2 for x ≥ 0.

(a) Solve the equation f x = 0, giving your solutions in the form x = a + b c, where a, b and c are
integers. [4]

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22

(b) Given that f x gh x, find the values of m and n. [4]

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22 [Turn over

D
E

A 5 C 8 B

The diagram shows a circle with centre A of radius 5 cm and a circle with centre B of radius 8 cm.
The circles touch at the point C so that ACB is a straight line. The tangent at the point D on the
smaller circle intersects the larger circle at E and passes through B.

(a) Find the perimeter of the shaded region. [5]

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22

........................................................................................................................................................

(b) Find the area of the shaded region. [3]

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22 [Turn over

11 It is given that a curve has equation y = k 3x − k−1 + 3x, where k is a constant.

(a) Find, in terms of k, the values of x at which there is a stationary point. [4]

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22

The function f has a stationary value at x = a and is defined by

f x = 4 3x − 4−1 + 3x for x ≥ 32 .

(b) Find the value of a and determine the nature of the stationary value. [3]

........................................................................................................................................................

(c) The function g is defined by g x = − 3x + 1−1 + 3x for x ≥ 0.

Determine, making your reasoning clear, whether g is an increasing function, a decreasing

function or neither. [2]

........................................................................................................................................................

© UCLES 2022 9709/12/F/M/22

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.

........................................................................................................................................................................

To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge
Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download
at www.cambridgeinternational.org after the live examination series.

Cambridge Assessment International Education is part of Cambridge Assessment. Cambridge Assessment is the brand name of the University of Cambridge
Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge.

© UCLES 2022 9709/12/F/M/22

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*0332655407*

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 May/June 2022

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 75.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 20 pages.

JC22 06_9709_12/2R
© UCLES 2022 [Turn over
2

1 The coefficient of x4 in the expansion of 3 + x5 is equal to the coefficient of x2 in the expansion of
@ A
a 6
2x + .
x

Find the value of the positive constant a. [4]

................................................................................................................................................................

© UCLES 2022 9709/12/M/J/22

2 The second and third terms of a geometric progression are 10 and 8 respectively.

Find the sum to infinity. [4]

................................................................................................................................................................

© UCLES 2022 9709/12/M/J/22 [Turn over

dy 1 −1
3 The equation of a curve is such that = 3 4x − 7 2 − 4x 2 . It is given that the curve passes through
dx
the point 4, 25 .

Find the equation of the curve. [4]

................................................................................................................................................................

© UCLES 2022 9709/12/M/J/22

4 The first, second and third terms of an arithmetic progression are k, 6k and k + 6 respectively.

(a) Find the value of the constant k. [2]

........................................................................................................................................................

(b) Find the sum of the first 30 terms of the progression. [3]

........................................................................................................................................................

© UCLES 2022 9709/12/M/J/22 [Turn over

5 The equation of a curve is y = 4x2 − kx + 12 k2 and the equation of a line is y = x − a, where k and a are
constants.

(a) Given that the curve and the line intersect at the points with x-coordinates 0 and 34 , find the values
of k and a. [4]

........................................................................................................................................................

© UCLES 2022 9709/12/M/J/22

(b) Given instead that a = − 72 , find the values of k for which the line is a tangent to the curve. [5]

........................................................................................................................................................

© UCLES 2022 9709/12/M/J/22 [Turn over

6
y y = 2x + 2
1
y = 5x 2

x
O

1
The diagram shows the curve with equation y = 5x 2 and the line with equation y = 2x + 2.

Find the exact area of the shaded region which is bounded by the line and the curve. [5]

................................................................................................................................................................

© UCLES 2022 9709/12/M/J/22

................................................................................................................................................................

© UCLES 2022 9709/12/M/J/22 [Turn over

7
B
2 cm
A

10 cm

1
6π
O P C

The diagram shows a sector OBAC of a circle with centre O and radius 10 cm. The point P lies on
OC and BP is perpendicular to OC. Angle AOC = 16 π and the length of the arc AB is 2 cm.

(a) Find the angle BOC. [2]

........................................................................................................................................................

© UCLES 2022 9709/12/M/J/22

(b) Hence find the area of the shaded region BPC giving your answer correct to 3 significant figures.
[4]

........................................................................................................................................................

© UCLES 2022 9709/12/M/J/22 [Turn over

8 The equation of a circle is x2 + y2 + ax + by − 12 = 0. The points A 1, 1 and B 2, −6 lie on the

circle.

(a) Find the values of a and b and hence find the coordinates of the centre of the circle. [4]

........................................................................................................................................................

© UCLES 2022 9709/12/M/J/22

(b) Find the equation of the tangent to the circle at the point A, giving your answer in the form
px + qy = k, where p, q and k are integers. [4]

........................................................................................................................................................

© UCLES 2022 9709/12/M/J/22 [Turn over

14
1
9 The equation of a curve is y = 3x + 1 − 4 3x + 1 2 for x > − 13 .

dy d2 y
(a) Find and 2 . [3]
dx dx

........................................................................................................................................................

© UCLES 2022 9709/12/M/J/22

(b) Find the coordinates of the stationary point of the curve and determine its nature. [4]

........................................................................................................................................................

© UCLES 2022 9709/12/M/J/22 [Turn over

10 Functions f and g are defined as follows:

2x + 1
f x = for x ≠ 12 ,
2x − 1
g x = x2 + 4 for x ∈ >.

(a) y

x
−6 −4 −2 0 2 4 6

−2

The diagram shows part of the graph of y = f x.

State the domain of f −1 . [1]

........................................................................................................................................................

(b) Find an expression for f −1 x. [3]

........................................................................................................................................................

(c) Find gf −1 3. [2]

........................................................................................................................................................

© UCLES 2022 9709/12/M/J/22

(d) Explain why g−1 x cannot be found. [1]

........................................................................................................................................................

2 2x + 1
(e) Show that 1 + can be expressed as . Hence find the area of the triangle enclosed
2x − 1 2x − 1
by the tangent to the curve y = f x at the point where x = 1 and the x- and y-axes. [6]

........................................................................................................................................................

© UCLES 2022 9709/12/M/J/22 [Turn over

11 The function f is given by f x = 4 cos4 x + cos2 x − k for 0 ≤ x ≤ 2π, where k is a constant.

(a) Given that k = 3, find the exact solutions of the equation f x = 0. [5]

........................................................................................................................................................

© UCLES 2022 9709/12/M/J/22

(b) Use the quadratic formula to show that, when k > 5, the equation f x = 0 has no solutions. [5]

........................................................................................................................................................

© UCLES 2022 9709/12/M/J/22

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.

........................................................................................................................................................................

© UCLES 2022 9709/12/M/J/22

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*8426767911*

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 October/November 2022

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 75.
³ The number of marks for each question or part question is shown in brackets [ ].

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

JC22 11_9709_12/RP
© UCLES 2022 [Turn over
2

BLANK PAGE

© UCLES 2022 9709/12/O/N/22

1 Points A and B have coordinates 5, 2 and 10, −1 respectively.

(a) Find the equation of the perpendicular bisector of AB. [3]

........................................................................................................................................................

(b) Find the equation of the circle with centre A which passes through B. [3]

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22 [Turn over

2 The first, second and third terms of an arithmetic progression are a, 2a and a2 respectively, where a
is a positive constant.

Find the sum of the first 50 terms of the progression. [5]

................................................................................................................................................................

© UCLES 2022 9709/12/O/N/22

3 (a) Find the set of values of k for which the equation 8x2 + kx + 2 = 0 has no real roots. [2]

........................................................................................................................................................

(b) Solve the equation 8 cos2 1 − 10 cos 1 + 2 = 0 for 0Å ≤ 1 ≤ 180Å. [3]

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22 [Turn over

4 A geometric progression is such that the third term is 1764 and the sum of the second and third terms
is 3444.

Find the 50th term. [4]

................................................................................................................................................................

© UCLES 2022 9709/12/O/N/22

5 The graph with equation y = f x is transformed to the graph @ Awith equation y = g x by a stretch in
0
the x-direction with factor 0.5, followed by a translation of .
1

(a) The diagram below shows the graph of y = f x.

On the diagram sketch the graph of y = g x. [3]

x
−10 −8 −6 −4 −2 0 2 4 6 8 10

−2

(b) Find an expression for g x in terms of f x. [2]

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22 [Turn over

6 The equation of a curve is y = 4x2 + 20x + 6.

(a) Express the equation in the form y = a x + b2 + c, where a, b and c are constants. [3]

........................................................................................................................................................

(b) Hence solve the equation 4x2 + 20x + 6 = 45. [3]

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22

(c) Sketch the graph of y = 4x2 + 20x + 6 showing the coordinates of the stationary point. You are
not required to indicate where the curve crosses the x- and y-axes. [3]

© UCLES 2022 9709/12/O/N/22 [Turn over

sin 1 cos 1 tan2 1 + 1

7 (a) Prove the identity + . [3]
sin 1 + cos 1 sin 1 − cos 1 tan2 1 − 1

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22

sin 1 cos 1
(b) Hence find the exact solutions of the equation + = 2 for 0 ≤ 1 ≤ π.
sin 1 + cos 1 sin 1 − cos 1
[4]

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22 [Turn over

dy 1 −1
8 The equation of a curve is such that = 3x 2 − 3x 2 . The curve passes through the point 3, 5.
dx

(a) Find the equation of the curve. [4]

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22

(b) Find the x-coordinate of the stationary point. [2]

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22 [Turn over

9 Functions f and g are defined by

1
f x = x + for x > 0,
x
g x = ax + 1 for x ∈ >,

where a is a constant.

(a) Find an expression for gf x. [1]

........................................................................................................................................................

(b) Given that gf 2 = 11, find the value of a. [2]

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22

It is given instead that a = 5.

(d) Find and simplify an expression for g−1 f x. [3]

........................................................................................................................................................

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

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22 [Turn over

10
R

O
2
3π
2.5 m P
5
6π
2.24 m
A B

The diagram shows a cross-section RASB of the body of an aircraft. The cross-section consists of
a sector OARB of a circle of radius 2.5 m, with centre O, a sector PASB of another circle of radius
2.24 m with centre P and a quadrilateral OAPB. Angle AOB = 23 π and angle APB = 56 π.

(a) Find the perimeter of the cross-section RASB, giving your answer correct to 2 decimal places.
[3]

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22

(b) Find the difference in area of the two triangles AOB and APB, giving your answer correct to
2 decimal places. [2]

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22 [Turn over

11 (a) Find the coordinates of the minimum point of the curve y = 94 x2 − 12x + 18. [3]

........................................................................................................................................................

5
y = 18 − 38 x 2

y = 49 x2 − 12x + 18

x
O

5
The diagram shows the curves with equations y = 94 x2 − 12x + 18 and y = 18 − 38 x 2 . The curves
intersect at the points 0, 18 and 4, 6.

(b) Find the area of the shaded region. [5]

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22

........................................................................................................................................................

5
(c) A point P is moving along the curve y = 18 − 38 x 2 in such a way that the x-coordinate of P is
increasing at a constant rate of 2 units per second.

Find the rate at which the y-coordinate of P is changing when x = 4. [3]

........................................................................................................................................................

© UCLES 2022 9709/12/O/N/22

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.

........................................................................................................................................................................

© UCLES 2022 9709/12/O/N/22

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*5712505207*

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 February/March 2021

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 75.
³ The number of marks for each question or part question is shown in brackets [ ].

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

JC21 03_9709_12/RP
© UCLES 2021 [Turn over
2

BLANK PAGE

© UCLES 2021 9709/12/F/M/21

1 (a) Find the first three terms in the expansion, in ascending powers of x, of 1 + x5 . [1]

........................................................................................................................................................

(b) Find the first three terms in the expansion, in ascending powers of x, of 1 − 2x6 . [2]

........................................................................................................................................................

© UCLES 2021 9709/12/F/M/21 [Turn over

2 By using a suitable substitution, solve the equation

4
2x − 32 − − 3 = 0. [4]
2x − 32

................................................................................................................................................................

© UCLES 2021 9709/12/F/M/21

tan 1 + 2 sin 1
3 Solve the equation = 3 for 0Å < 1 < 180Å. [4]
tan 1 − 2 sin 1

................................................................................................................................................................

© UCLES 2021 9709/12/F/M/21 [Turn over

4 A line has equation y = 3x + k and a curve has equation y = x2 + kx + 6, where k is a constant.

Find the set of values of k for which the line and curve have two distinct points of intersection. [5]

................................................................................................................................................................

© UCLES 2021 9709/12/F/M/21

5
y

2
y = f x
1

x
O 1 2 3 4 5 6

In the diagram, the graph of y = f x is shown with solid lines. The graph shown with broken lines is
a transformation of y = f x.

(a) Describe fully the two single transformations of y = f x that have been combined to give the
resulting transformation. [4]

........................................................................................................................................................

(b) State in terms of y, f and x, the equation of the graph shown with broken lines. [2]

........................................................................................................................................................

© UCLES 2021 9709/12/F/M/21 [Turn over

dy 6
6 A curve is such that = and A 1, −3 lies on the curve. A point is moving along the curve
dx 3x − 23
and at A the y-coordinate of the point is increasing at 3 units per second.

(a) Find the rate of increase at A of the x-coordinate of the point. [3]

........................................................................................................................................................

© UCLES 2021 9709/12/F/M/21

(b) Find the equation of the curve. [4]

........................................................................................................................................................

© UCLES 2021 9709/12/F/M/21 [Turn over

7 Functions f and g are defined as follows:

f : x → x2 + 2x + 3 for x ≤ −1,
g : x → 2x + 1 for x ≥ −1.

(a) Express f x in the form x + a2 + b and state the range of f. [3]

........................................................................................................................................................

© UCLES 2021 9709/12/F/M/21

(b) Find an expression for f −1 x. [2]

........................................................................................................................................................

(c) Solve the equation gf x = 13. [3]

........................................................................................................................................................

© UCLES 2021 9709/12/F/M/21 [Turn over

8 The points A 7, 1, B 7, 9 and C 1, 9 are on the circumference of a circle.

(a) Find an equation of the circle. [5]

........................................................................................................................................................

© UCLES 2021 9709/12/F/M/21

........................................................................................................................................................

(b) Find an equation of the tangent to the circle at B. [2]

........................................................................................................................................................

© UCLES 2021 9709/12/F/M/21 [Turn over

9 The first term of a progression is cos 1, where 0 < 1 < 12 π.

1
(a) For the case where the progression is geometric, the sum to infinity is .
cos 1
(i) Show that the second term is cos 1 sin2 1. [3]

................................................................................................................................................

(ii) Find the sum of the first 12 terms when 1 = 13 π, giving your answer correct to 4 significant
figures. [2]

................................................................................................................................................

© UCLES 2021 9709/12/F/M/21

(b) For the case where the progression is arithmetic, the first two terms are again cos 1 and cos 1 sin2 1
respectively.

Find the 85th term when 1 = 13 π. [4]

........................................................................................................................................................

© UCLES 2021 9709/12/F/M/21 [Turn over

10
A

ka ka
a

E
D

B C

The diagram shows a sector ABC which is part of a circle of radius a. The points D and E lie on AB
and AC respectively and are such that AD = AE = ka, where k < 1. The line DE divides the sector
into two regions which are equal in area.

(a) For the case where angle BAC = 16 π radians, find k correct to 4 significant figures. [5]

........................................................................................................................................................

© UCLES 2021 9709/12/F/M/21

1
(b) For the general case in which angle BAC = 1 radians, where 0 < 1 < 12 π, it is given that > 1.
sin 1

Find the set of possible values of k. [3]

........................................................................................................................................................

© UCLES 2021 9709/12/F/M/21 [Turn over

11
y

A
x
O

−1 −3
The diagram shows the curve with equation y = 9 x 2 − 4x 2 . The curve crosses the x-axis at the
point A.

(a) Find the x-coordinate of A. [2]

........................................................................................................................................................

(b) Find the equation of the tangent to the curve at A. [4]

........................................................................................................................................................

© UCLES 2021 9709/12/F/M/21

........................................................................................................................................................

(d) Find the area of the region bounded by the curve, the x-axis and the line x = 9. [4]

........................................................................................................................................................

© UCLES 2021 9709/12/F/M/21

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.

........................................................................................................................................................................

Cambridge Assessment International Education is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of
Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge.

© UCLES 2021 9709/12/F/M/21

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*2355526103*

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 May/June 2021

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 75.
³ The number of marks for each question or part question is shown in brackets [ ].

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

JC21 06_9709_12/RP
© UCLES 2021 [Turn over
2

1 (a) Express 16x2 − 24x + 10 in the form 4x + a2 + b. [2]

........................................................................................................................................................

(b) It is given that the equation 16x2 − 24x + 10 = k, where k is a constant, has exactly one root.

Find the value of this root. [2]

........................................................................................................................................................

© UCLES 2021 9709/12/M/J/21

2 (a) The graph of y = f x is transformed to the graph of y = 2f x − 1.

Describe fully the two single transformations which have been combined to give the resulting
transformation. [3]

........................................................................................................................................................

(b) The curve y = sin 2x − 5x is reflected in the y-axis and then stretched by scale factor 13 in the
x-direction.

Write down the equation of the transformed curve. [2]

........................................................................................................................................................

© UCLES 2021 9709/12/M/J/21 [Turn over

4

3 The equation of a curve is y = x − 3 x + 1 + 3. The following points lie on the curve. Non-exact
values are rounded to 4 decimal places.
A 2, k B 2.9, 2.8025 C 2.99, 2.9800 D 2.999, 2.9980 E 3, 3

(a) Find k, giving your answer correct to 4 decimal places. [1]

........................................................................................................................................................

(b) Find the gradient of AE, giving your answer correct to 4 decimal places. [1]

........................................................................................................................................................

The gradients of BE, CE and DE , rounded to 4 decimal places, are 1.9748, 1.9975 and 1.9997
respectively.

(c) State, giving a reason for your answer, what the values of the four gradients suggest about the
gradient of the curve at the point E. [2]

........................................................................................................................................................

© UCLES 2021 9709/12/M/J/21

5
@ A
10 3 1
4 The coefficient of x in the expansion of 4x + is p. The coefficient of in the expansion of
x x
@ A5
k
2x + 2 is q.
x
Given that p = 6q, find the possible values of k. [5]

................................................................................................................................................................

© UCLES 2021 9709/12/M/J/21 [Turn over

5 The function f is defined by f x = 2x2 + 3 for x ≥ 0.

(a) Find and simplify an expression for ff x. [2]

........................................................................................................................................................

(b) Solve the equation ff x = 34x2 + 19. [4]

........................................................................................................................................................

© UCLES 2021 9709/12/M/J/21

6 Points A and B have coordinates 8, 3 and p, q respectively. The equation of the perpendicular
bisector of AB is y = −2x + 4.

Find the values of p and q. [4]

................................................................................................................................................................

© UCLES 2021 9709/12/M/J/21 [Turn over

7 The point A has coordinates 1, 5 and the line l has gradient − 23 and passes through A. A circle has

centre 5, 11 and radius 52.

(a) Show that l is the tangent to the circle at A. [2]

........................................................................................................................................................

(b) Find the equation of the other circle of radius 52 for which l is also the tangent at A. [3]

........................................................................................................................................................

© UCLES 2021 9709/12/M/J/21

8 The first, second and third terms of an arithmetic progression are a, 32 a and b respectively, where
a and b are positive constants. The first, second and third terms of a geometric progression are
a, 18 and b + 3 respectively.

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

........................................................................................................................................................

(b) Find the sum of the first 20 terms of the arithmetic progression. [3]

........................................................................................................................................................

© UCLES 2021 9709/12/M/J/21 [Turn over

9
y

y2 = x − 2

x
0 5

The diagram shows part of the curve with equation y2 = x − 2 and the lines x = 5 and y = 1. The
shaded region enclosed by the curve and the lines is rotated through 360Å about the x-axis.

Find the volume obtained. [6]

................................................................................................................................................................

© UCLES 2021 9709/12/M/J/21

................................................................................................................................................................

© UCLES 2021 9709/12/M/J/21 [Turn over

1 + sin x 1 − sin x 4 tan x

10 (a) Prove the identity − . [4]
1 − sin x 1 + sin x cos x

........................................................................................................................................................

© UCLES 2021 9709/12/M/J/21

1 + sin x 1 − sin x
(b) Hence solve the equation − = 8 tan x for 0 ≤ x ≤ 12 π. [3]
1 − sin x 1 + sin x

........................................................................................................................................................

© UCLES 2021 9709/12/M/J/21 [Turn over

dy
11 The gradient of a curve is given by = 6 3x − 53 − kx2 , where k is a constant. The curve has a
dx
stationary point at 2, −3.5.

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

........................................................................................................................................................

(b) Find the equation of the curve. [4]

........................................................................................................................................................

© UCLES 2021 9709/12/M/J/21

d2 y
(c) Find . [2]
dx2

........................................................................................................................................................

(d) Determine the nature of the stationary point at 2, −3.5. [2]

........................................................................................................................................................

© UCLES 2021 9709/12/M/J/21 [Turn over

12
Q
P
A B

F C

E D

The diagram shows a cross-section of seven cylindrical pipes, each of radius 20 cm, held together by a
thin rope which is wrapped tightly around the pipes. The centres of the six outer pipes are A, B, C, D,
E and F. Points P and Q are situated where straight sections of the rope meet the pipe with centre A.

(a) Show that angle PAQ = 13 π radians. [2]

........................................................................................................................................................

(b) Find the length of the rope. [4]

........................................................................................................................................................

© UCLES 2021 9709/12/M/J/21

17

(c) Find the area of the hexagon ABCDEF, giving your answer in terms of 3. [2]

........................................................................................................................................................

(d) Find the area of the complete region enclosed by the rope. [3]

........................................................................................................................................................

© UCLES 2021 9709/12/M/J/21

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.

........................................................................................................................................................................

© UCLES 2021 9709/12/M/J/21

BLANK PAGE

© UCLES 2021 9709/12/M/J/21

BLANK PAGE

© UCLES 2021 9709/12/M/J/21

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*2952888595*

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 October/November 2021

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 75.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 20 pages.

JC21 11_9709_12/2R
© UCLES 2021 [Turn over
2

3
1 Solve the equation 2 cos 1 = 7 − for −90Å < 1 < 90Å. [4]
cos 1

................................................................................................................................................................

© UCLES 2021 9709/12/O/N/21

2 The graph of y = f x is transformed to the graph of y = f 2x − 3.

(a) Describe fully the two single transformations that have been combined to give the resulting
transformation. [3]

........................................................................................................................................................

The point P 5, 6 lies on the transformed curve y = f 2x − 3.

(b) State the coordinates of the corresponding point on the original curve y = f x. [2]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21 [Turn over

3 The function f is defined as follows:

x+3
f x = for x > 1.
x−1

(a) Find the value of ff 5. [2]

........................................................................................................................................................

(b) Find an expression for f −1 x. [3]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21

dy 8
4 A curve is such that = 2
. The curve passes through the point 2, 5 23 .
dx 3x + 2

Find the equation of the curve. [4]

................................................................................................................................................................

© UCLES 2021 9709/12/O/N/21 [Turn over

6

5 The first, third and fifth terms of an arithmetic progression are 2 cos x, −6 3 sin x and 10 cos x
respectively, where 21 π < x < π.

(a) Find the exact value of x. [3]

........................................................................................................................................................

(b) Hence find the exact sum of the first 25 terms of the progression. [3]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21

6 The second term of a geometric progression is 54 and the sum to infinity of the progression is 243.
The common ratio is greater than 21 .

Find the tenth term, giving your answer in exact form. [5]

................................................................................................................................................................

© UCLES 2021 9709/12/O/N/21 [Turn over

15 c m P B
Q
A

9 cm

15
cm

In the diagram the lengths of AB and AC are both 15 cm. The point P is the foot of the perpendicular
from C to AB. The length CP = 9 cm. An arc of a circle with centre B passes through C and meets
AB at Q.

(a) Show that angle ABC = 1.25 radians, correct to 3 significant figures. [2]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21

(b) Calculate the area of the shaded region which is bounded by the arc CQ and the lines CP and
PQ. [4]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21 [Turn over

8 (a) It is given that in the expansion of 4 + 2x 2 − ax5 , the coefficient of x2 is −15.

Find the possible values of a. [4]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21

(b) It is given instead that in the expansion of 4 + 2x 2 − ax5 , the coefficient of x2 is k. It is also
given that there is only one value of a which leads to this value of k.

Find the values of k and a. [4]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21 [Turn over

9 The volume V m3 of a large circular mound of iron ore of radius r m is modelled by the equation
V= 3 r − 1 3 − 1 for r ≥ 2. Iron ore is added to the mound at a constant rate of 1.5 m3 per second.
2 2

(a) Find the rate at which the radius of the mound is increasing at the instant when the radius is 5.5 m.
[3]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21

(b) Find the volume of the mound at the instant when the radius is increasing at 0.1 m per second.
[3]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21 [Turn over

k
10 The function f is defined by f x = x2 + + 2 for x > 0.
x
(a) Given that the curve with equation y = f x has a stationary point when x = 2, find k. [3]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21

(b) Determine the nature of the stationary point. [2]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21 [Turn over

11
y
1
y = 12 x + 7
10 − 1
x − 2 3
A 3, 65

x
O 5
2

7 − 1
The diagram shows the line x = 52 , part of the curve y = 12 x + 10 1
and the normal to the curve
x − 2 3

at the point A 3, 65 .

(a) Find the x-coordinate of the point where the normal to the curve meets the x-axis. [5]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21

(b) Find the area of the shaded region, giving your answer correct to 2 decimal places. [6]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21 [Turn over

12
y

B
P

A
x
O

The diagram shows the circle with equation x2 + y2 − 6x + 4y − 27 = 0 and the tangent to the circle at
the point P 5, 4.

(a) The tangent to the circle at P meets the x-axis at A and the y-axis at B.

Find the area of triangle OAB, where O is the origin. [5]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21

(b) Points Q and R also lie on the circle, such that PQR is an equilateral triangle.

Find the exact area of triangle PQR. [3]

........................................................................................................................................................

© UCLES 2021 9709/12/O/N/21

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.

........................................................................................................................................................................

© UCLES 2021 9709/12/O/N/21

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*1939478240*

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 February/March 2020

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 75.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 20 pages. Blank pages are indicated.

JC20 03_9709_12/FP
© UCLES 2020 [Turn over
2

1
1 The function f is defined by f x = + x2 for x < −1.
3x + 2

Determine whether f is an increasing function, a decreasing function or neither. [3]

................................................................................................................................................................

© UCLES 2020 9709/12/F/M/20

3

2 The graph of y = f x is transformed to the graph of y = 1 + f 12 x .

Describe fully the two single transformations which have been combined to give the resulting
transformation. [4]

................................................................................................................................................................

© UCLES 2020 9709/12/F/M/20 [Turn over

3
y

y = x2 + 1

x
O

The diagram shows part of the curve with equation y = x2 + 1. The shaded region enclosed by the
curve, the y-axis and the line y = 5 is rotated through 360Å about the y-axis.

Find the volume obtained. [4]

................................................................................................................................................................

© UCLES 2020 9709/12/F/M/20

4 A curve has equation y = x2 − 2x − 3. A point is moving along the curve in such a way that at P the
y-coordinate is increasing at 4 units per second and the x-coordinate is increasing at 6 units per second.

Find the x-coordinate of P. [4]

................................................................................................................................................................

© UCLES 2020 9709/12/F/M/20 [Turn over

5 Solve the equation

tan 1 + 3 sin 1 + 2
=2
tan 1 − 3 sin 1 + 1
for 0Å ≤ 1 ≤ 90Å. [5]

................................................................................................................................................................

© UCLES 2020 9709/12/F/M/20

7
@ A
1 a 5
6 The coefficient of in the expansion of 2x + 2 is 720.
x x
(a) Find the possible values of the constant a. [3]

........................................................................................................................................................

1
(b) Hence find the coefficient of in the expansion. [2]
x7

........................................................................................................................................................

© UCLES 2020 9709/12/F/M/20 [Turn over

7
A

0.8 rad
O B
C
6 cm

The diagram shows a sector AOB which is part of a circle with centre O and radius 6 cm and with
angle AOB = 0.8 radians. The point C on OB is such that AC is perpendicular to OB. The arc CD is
part of a circle with centre O, where D lies on OA.

Find the area of the shaded region. [6]

................................................................................................................................................................

© UCLES 2020 9709/12/F/M/20

8 A woman’s basic salary for her first year with a particular company is $30 000 and at the end of the
year she also gets a bonus of $600.

(a) For her first year, express her bonus as a percentage of her basic salary. [1]

........................................................................................................................................................

At the end of each complete year, the woman’s basic salary will increase by 3% and her bonus will
increase by $100.

(b) Express the bonus she will be paid at the end of her 24th year as a percentage of the basic salary
paid during that year. [5]

........................................................................................................................................................

© UCLES 2020 9709/12/F/M/20 [Turn over

9 (a) Express 2x2 + 12x + 11 in the form 2 x + a2 + b, where a and b are constants. [2]

........................................................................................................................................................

The function f is defined by f x = 2x2 + 12x + 11 for x ≤ −4.

(b) Find an expression for f −1 x and state the domain of f −1 . [3]

........................................................................................................................................................

© UCLES 2020 9709/12/F/M/20

The function g is defined by g x = 2x − 3 for x ≤ k.

........................................................................................................................................................

(d) State the largest value of k possible for the composition fg to be defined. [1]

........................................................................................................................................................

© UCLES 2020 9709/12/F/M/20 [Turn over

dy 1
10 The gradient of a curve at the point x, y is given by = 2 x + 3 2 − x. The curve has a stationary
dx
point at a, 14, where a is a positive constant.

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

........................................................................................................................................................

(b) Determine the nature of the stationary point. [3]

........................................................................................................................................................

© UCLES 2020 9709/12/F/M/20

(c) Find the equation of the curve. [4]

........................................................................................................................................................

© UCLES 2020 9709/12/F/M/20 [Turn over

11 (a) Solve the equation 3 tan2 x − 5 tan x − 2 = 0 for 0Å ≤ x ≤ 180Å. [4]

........................................................................................................................................................

(b) Find the set of values of k for which the equation 3 tan2 x − 5 tan x + k = 0 has no solutions. [2]

........................................................................................................................................................

© UCLES 2020 9709/12/F/M/20

(c) For the equation 3 tan2 x − 5 tan x + k = 0, state the value of k for which there are three solutions
in the interval 0Å ≤ x ≤ 180Å, and find these solutions. [3]

........................................................................................................................................................

© UCLES 2020 9709/12/F/M/20 [Turn over

12 A diameter of a circle C1 has end-points at −3, −5 and 7, 3.

(a) Find an equation of the circle C1 . [3]

........................................................................................................................................................

C2
R
C1

x
O

@ A
8
The circle C1 is translated by to give circle C2 , as shown in the diagram.
4

(b) Find an equation of the circle C2 . [2]

........................................................................................................................................................

© UCLES 2020 9709/12/F/M/20

The two circles intersect at points R and S.

........................................................................................................................................................

(d) Hence show that the x-coordinates of R and S satisfy the equation 5x2 − 60x + 159 = 0. [2]

........................................................................................................................................................

© UCLES 2020 9709/12/F/M/20

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.

........................................................................................................................................................................

© UCLES 2020 9709/12/F/M/20

BLANK PAGE

© UCLES 2020 9709/12/F/M/20

BLANK PAGE

© UCLES 2020 9709/12/F/M/20

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*5874701744*

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 May/June 2020

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 75.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 16 pages. Blank pages are indicated.

JC20 06_9709_12/2R
© UCLES 2020 [Turn over
2
@ A
2 2 6
1 (a) Find the coefficient of x in the expansion of x − . [2]
x

........................................................................................................................................................

@ A
2 6
(b) Find the coefficient of x2 in the expansion of 2 + 3x2 x − . [3]
x

........................................................................................................................................................

© UCLES 2020 9709/12/M/J/20

2 (a) Express the equation 3 cos 1 = 8 tan 1 as a quadratic equation in sin 1. [3]

........................................................................................................................................................

(b) Hence find the acute angle, in degrees, for which 3 cos 1 = 8 tan 1. [2]

........................................................................................................................................................

© UCLES 2020 9709/12/M/J/20 [Turn over

3 A weather balloon in the shape of a sphere is being inflated by a pump. The volume of the balloon is
increasing at a constant rate of 600 cm3 per second. The balloon was empty at the start of pumping.

(a) Find the radius of the balloon after 30 seconds. [2]

........................................................................................................................................................

(b) Find the rate of increase of the radius after 30 seconds. [3]

........................................................................................................................................................

© UCLES 2020 9709/12/M/J/20

4 The nth term of an arithmetic progression is 12 3n − 15.

Find the value of n for which the sum of the first n terms is 84. [5]

................................................................................................................................................................

© UCLES 2020 9709/12/M/J/20 [Turn over

5 The function f is defined for x ∈ > by

f : x → a − 2x,
where a is a constant.

(a) Express ff x and f −1 x in terms of a and x. [4]

........................................................................................................................................................

(b) Given that ff x = f −1 x, find x in terms of a. [2]

........................................................................................................................................................

© UCLES 2020 9709/12/M/J/20

6 The equation of a curve is y = 2x2 + kx + k − 1, where k is a constant.

(a) Given that the line y = 2x + 3 is a tangent to the curve, find the value of k. [3]

........................................................................................................................................................

It is now given that k = 2.

(b) Express the equation of the curve in the form y = 2 x + a2 + b, where a and b are constants, and
hence state the coordinates of the vertex of the curve. [3]

........................................................................................................................................................

© UCLES 2020 9709/12/M/J/20 [Turn over

7
A
r

C
r
1
6 π rad
O B
2r

In the diagram, OAB is a sector of a circle with centre O and radius 2r, and angle AOB = 16 π radians.
The point C is the midpoint of OA.

(a) Show that the exact length of BC is r 5 − 2 3. [2]

........................................................................................................................................................

© UCLES 2020 9709/12/M/J/20

(b) Find the exact perimeter of the shaded region. [2]

........................................................................................................................................................

(c) Find the exact area of the shaded region. [3]

........................................................................................................................................................

© UCLES 2020 9709/12/M/J/20 [Turn over

8
y

1, 6

3, 2
1, 2 6
y=
x

x
O

6
The diagram shows part of the curve y = . The points 1, 6 and 3, 2 lie on the curve. The shaded
x
region is bounded by the curve and the lines y = 2 and x = 1.

(a) Find the volume generated when the shaded region is rotated through 360Å about the y-axis. [5]

........................................................................................................................................................

© UCLES 2020 9709/12/M/J/20

(b) The tangent to the curve at a point X is parallel to the line y + 2x = 0. Show that X lies on the
line y = 2x. [3]

........................................................................................................................................................

© UCLES 2020 9709/12/M/J/20 [Turn over

9 Functions f and g are such that

f x = 2 − 3 sin 2x for 0 ≤ x ≤ π,
g x = −2f x for 0 ≤ x ≤ π.

(a) State the ranges of f and g. [3]

........................................................................................................................................................

The diagram below shows the graph of y = f x.

y = f x

x
O π

(b) Sketch, on this diagram, the graph of y = g x. [2]

The function h is such that

h x = g x + π for − π ≤ x ≤ 0.

........................................................................................................................................................

© UCLES 2020 9709/12/M/J/20

10 The equation of a curve is y = 54x − 2x − 73 .

dy d2 y
(a) Find and 2 . [4]
dx dx

........................................................................................................................................................

(b) Find the coordinates of each of the stationary points on the curve. [3]

........................................................................................................................................................

(c) Determine the nature of each of the stationary points. [2]

........................................................................................................................................................

© UCLES 2020 9709/12/M/J/20 [Turn over

11 The equation of a circle with centre C is x2 + y2 − 8x + 4y − 5 = 0.

(a) Find the radius of the circle and the coordinates of C. [3]

........................................................................................................................................................

The point P 1, 2 lies on the circle.

(b) Show that the equation of the tangent to the circle at P is 4y = 3x + 5. [3]

........................................................................................................................................................

© UCLES 2020 9709/12/M/J/20

The point Q also lies on the circle and PQ is parallel to the x-axis.

(c) Write down the coordinates of Q. [2]

........................................................................................................................................................

The tangents to the circle at P and Q meet at T .

(d) Find the coordinates of T . [3]

........................................................................................................................................................

© UCLES 2020 9709/12/M/J/20

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.

........................................................................................................................................................................

© UCLES 2020 9709/12/M/J/20

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*8279021580*

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 October/November 2020

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 75.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 20 pages. Blank pages are indicated.

JC20 11_9709_12/2R
© UCLES 2020 [Turn over
2

BLANK PAGE

© UCLES 2020 9709/12/O/N/20

1 The coefficient of x3 in the expansion of 1 + kx 1 − 2x5 is 20.

Find the value of the constant k. [4]

................................................................................................................................................................

© UCLES 2020 9709/12/O/N/20 [Turn over

2 The first, second and third terms of a geometric progression are 2p + 6, −2p and p + 2 respectively,
where p is positive.

Find the sum to infinity of the progression. [5]

................................................................................................................................................................

© UCLES 2020 9709/12/O/N/20

3 The equation of a curve is y = 2x2 + m 2x + 1, where m is a constant, and the equation of a line is
y = 6x + 4.

Show that, for all values of m, the line intersects the curve at two distinct points. [5]

................................................................................................................................................................

© UCLES 2020 9709/12/O/N/20 [Turn over

4 The sum, Sn , of the first n terms of an arithmetic progression is given by

Sn = n2 + 4n.
The kth term in the progression is greater than 200.

Find the smallest possible value of k. [5]

................................................................................................................................................................

© UCLES 2020 9709/12/O/N/20

5 Functions f and g are defined by

f x = 4x − 2, for x ∈ >,
4
g x = , for x ∈ >, x ≠ −1.
x+1

(a) Find the value of fg 7. [1]

........................................................................................................................................................

(b) Find the values of x for which f −1 x = g−1 x. [5]

........................................................................................................................................................

© UCLES 2020 9709/12/O/N/20 [Turn over

8
@ A@ A
1 1 1
6 (a) Prove the identity − tan x +1 . [4]
cos x sin x tan x

........................................................................................................................................................

@ A@ A
1 1
(b) Hence solve the equation − tan x + 1 = 2 tan2 x for 0Å ≤ x ≤ 180Å. [2]
cos x sin x

........................................................................................................................................................

© UCLES 2020 9709/12/O/N/20

− 12 −3
7 The point 4, 7 lies on the curve y = f x and it is given that f ′ x = 6x − 4x 2 .

(a) A point moves along the curve in such a way that the x-coordinate is increasing at a constant rate
of 0.12 units per second.

Find the rate of increase of the y-coordinate when x = 4. [3]

........................................................................................................................................................

(b) Find the equation of the curve. [4]

........................................................................................................................................................

© UCLES 2020 9709/12/O/N/20 [Turn over

8
D
A C
1 rad

r cm r cm

In the diagram, ABC is an isosceles triangle with AB = BC = r cm and angle BAC = 1 radians. The
point D lies on AC and ABD is a sector of a circle with centre A.

(a) Express the area of the shaded region in terms of r and 1. [3]

........................................................................................................................................................

© UCLES 2020 9709/12/O/N/20

(b) In the case where r = 10 and 1 = 0.6, find the perimeter of the shaded region. [4]

........................................................................................................................................................

© UCLES 2020 9709/12/O/N/20 [Turn over

9 A circle has centre at the point B 5, 1. The point A −1, −2 lies on the circle.

(a) Find the equation of the circle. [3]

........................................................................................................................................................

Point C is such that AC is a diameter of the circle. Point D has coordinates 5, 16.

(b) Show that DC is a tangent to the circle. [4]

........................................................................................................................................................

© UCLES 2020 9709/12/O/N/20

The other tangent from D to the circle touches the circle at E.

(c) Find the coordinates of E. [2]

........................................................................................................................................................

© UCLES 2020 9709/12/O/N/20 [Turn over

10
y

x
O M

2
The diagram shows part of the curve y = − x and its minimum point M , which lies on the
3 − 2x2
x-axis.

dy d2 y
(a) Find expressions for , and Ó y dx. [6]
dx dx2

........................................................................................................................................................

© UCLES 2020 9709/12/O/N/20

(b) Find, by calculation, the x-coordinate of M . [2]

........................................................................................................................................................

© UCLES 2020 9709/12/O/N/20 [Turn over

11 A curve has equation y = 3 cos 2x + 2 for 0 ≤ x ≤ π.

(a) State the greatest and least values of y. [2]

........................................................................................................................................................

(b) Sketch the graph of y = 3 cos 2x + 2 for 0 ≤ x ≤ π. [2]

(c) By considering the straight line y = kx, where k is a constant, state the number of solutions of the
equation 3 cos 2x + 2 = kx for 0 ≤ x ≤ π in each of the following cases.
(i) k = −3 [1]

................................................................................................................................................

(ii) k = 1 [1]

................................................................................................................................................

(iii) k = 3 [1]

................................................................................................................................................

© UCLES 2020 9709/12/O/N/20

Functions f, g and h are defined for x ∈ > by

f x = 3 cos 2x + 2,
g x = f 2x + 4,

h x = 2f x + 12 π .

(d) Describe fully a sequence of transformations that maps the graph of y = f x on to y = g x. [2]

........................................................................................................................................................

(e) Describe fully a sequence of transformations that maps the graph of y = f x on to y = h x. [2]

........................................................................................................................................................

© UCLES 2020 9709/12/O/N/20

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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Cambridge Assessment International Education
Cambridge International Advanced Subsidiary and Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*8809618584*

NUMBER NUMBER

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) February/March 2019
1 hour 45 minutes
Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

Write your centre number, candidate number and name in the spaces at the top of this page.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
DO NOT WRITE IN ANY BARCODES.

Answer all the questions in the space provided. If additional space is required, you should use the lined page
at the end of this booklet. The question number(s) must be clearly shown.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.

This document consists of 19 printed pages and 1 blank page.

JC19 03_9709_12/RP
© UCLES 2019 [Turn over
2

BLANK PAGE

© UCLES 2019 9709/12/F/M/19

1 The coefficient of x3 in the expansion of 1 − px5 is −2160. Find the value of the constant p. [3]

................................................................................................................................................................

© UCLES 2019 9709/12/F/M/19 [Turn over

2 A curve with equation y = f x passes through the points 0, 2 and 3, −1. It is given that
f ′ x = kx2 − 2x, where k is a constant. Find the value of k. [5]

................................................................................................................................................................

© UCLES 2019 9709/12/F/M/19

3
C

7 8

Y X
A B

In the diagram, CXD is a semicircle of radius 7 cm with centre A and diameter CD. The straight line
YABX is perpendicular to CD, and the arc CYD is part of a circle with centre B and radius 8 cm. Find
the total area of the region enclosed by the two arcs. [6]

................................................................................................................................................................

© UCLES 2019 9709/12/F/M/19 [Turn over

4 A curve has equation y = 2x − 1−1 + 2x.

dy d2 y
(i) Find and 2 . [3]
dx dx

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19

(ii) Find the x-coordinates of the stationary points and, showing all necessary working, determine
the nature of each stationary point. [4]

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19 [Turn over

5 Two vectors, u and v, are such that

` a ` a
q 8
u= 2 and v= q−1 ,
6 q2 − 7
where q is a constant.

(i) Find the values of q for which u is perpendicular to v. [3]

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19

(ii) Find the angle between u and v when q = 0. [4]

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19 [Turn over

6 (i) The first and second terms of a geometric progression are p and 2p respectively, where p is a
positive constant. The sum of the first n terms is greater than 1000p. Show that 2n > 1001. [2]

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19

(ii) In another case, p and 2p are the first and second terms respectively of an arithmetic progression.
The nth term is 336 and the sum of the first n terms is 7224. Write down two equations in n and
p and hence find the values of n and p. [5]

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19 [Turn over

7 (a) Solve the equation 3 sin2 21 + 8 cos 21 = 0 for 0Å ≤ 1 ≤ 180Å. [5]

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19

(b)
y

y = a + tan bx
ï3

x
− 16 0 O

The diagram shows part of the graph of y = a + tan bx, where x is measured in radians and a and

b are constants. The curve intersects the x-axis at − 16 0 , 0 and the y-axis at 0, ï3. Find the
values of a and b. [3]

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19 [Turn over

8 (i) Express x2 − 4x + 7 in the form x + a2 + b. [2]

........................................................................................................................................................

The function f is defined by f x = x2 − 4x + 7 for x < k, where k is a constant.

(ii) State the largest value of k for which f is a decreasing function. [1]

........................................................................................................................................................

The value of k is now given to be 1.

(iii) Find an expression for f −1 x and state the domain of f −1 . [3]

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19

2
(iv) The function g is defined by g x = for x > 1. Find an expression for gf x and state the
x−1
range of gf. [4]

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19 [Turn over

9
y
y= x3 + x2
P

x
O 3

The diagram shows part of the curve with equation y = x3 + x2 . The shaded region is bounded by
the curve, the x-axis and the line x = 3.

(i) Find, showing all necessary working, the volume obtained when the shaded region is rotated
through 360Å about the x-axis. [4]

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19

(ii) P is the point on the curve with x-coordinate 3. Find the y-coordinate of the point where the
normal to the curve at P crosses the y-axis. [6]

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19 [Turn over

10
y

1
y = 4x 2

x
O

1
The diagram shows the curve with equation y = 4x 2 .

(i) The straight line with equation y = x + 3 intersects the curve at points A and B. Find the length
of AB. [6]

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19

(ii) The tangent to the curve at a point T is parallel to AB. Find the coordinates of T . [3]

........................................................................................................................................................

(iii) Find the coordinates of the point of intersection of the normal to the curve at T with the line AB.
[3]

........................................................................................................................................................

© UCLES 2019 9709/12/F/M/19

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.

........................................................................................................................................................................

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge Assessment
International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at
www.cambridgeinternational.org after the live examination series.
Cambridge Assessment International Education is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of
Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge.

© UCLES 2019 9709/12/F/M/19

Cambridge Assessment International Education
Cambridge International Advanced Subsidiary and Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*6611123410*

NUMBER NUMBER

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) May/June 2019
1 hour 45 minutes
Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

This document consists of 18 printed pages and 2 blank pages.

JC19 06_9709_12/RP
© UCLES 2019 [Turn over
2
A5 @
2
1 Find the coefficient of x in the expansion of − 3x . [3]
x

................................................................................................................................................................

© UCLES 2019 9709/12/M/J/19

2 Two points A and B have coordinates 1, 3 and 9, −1 respectively. The perpendicular bisector of
AB intersects the y-axis at the point C. Find the coordinates of C. [5]

................................................................................................................................................................

© UCLES 2019 9709/12/M/J/19 [Turn over

dy 4
3 A curve is such that = x3 − 2 . The point P 2, 9 lies on the curve.
dx x
(i) A point moves on the curve in such a way that the x-coordinate is decreasing at a constant rate
of 0.05 units per second. Find the rate of change of the y-coordinate when the point is at P. [2]

........................................................................................................................................................

(ii) Find the equation of the curve. [3]

........................................................................................................................................................

© UCLES 2019 9709/12/M/J/19

4 Angle x is such that sin x = a + b and cos x = a − b, where a and b are constants.

(i) Show that a2 + b2 has a constant value for all values of x. [3]

........................................................................................................................................................

(ii) In the case where tan x = 2, express a in terms of b. [2]

........................................................................................................................................................

© UCLES 2019 9709/12/M/J/19 [Turn over

1 rad
A O r B

The diagram shows a semicircle with diameter AB, centre O and radius r. The point C lies on the
circumference and angle AOC = 1 radians. The perimeter of sector BOC is twice the perimeter of
sector AOC. Find the value of 1 correct to 2 significant figures. [5]

................................................................................................................................................................

© UCLES 2019 9709/12/M/J/19

3x
6 The equation of a curve is y = 3 cos 2x and the equation of a line is 2y + = 5.
0
(i) State the smallest and largest values of y for both the curve and the line for 0 ≤ x ≤ 20. [3]

........................................................................................................................................................

3x
(ii) Sketch, on the same diagram, the graphs of y = 3 cos 2x and 2y + = 5 for 0 ≤ x ≤ 20. [3]
0

3x
(iii) State the number of solutions of the equation 6 cos 2x = 5 − for 0 ≤ x ≤ 20. [1]
0

........................................................................................................................................................

© UCLES 2019 9709/12/M/J/19 [Turn over

7 Functions f and g are defined by

f : x → 3x − 2, x ∈ >,
2x + 3
g : x → , x ∈ >, x ≠ 1.
x−1

(i) Obtain expressions for f −1 x and g−1 x, stating the value of x for which g−1 x is not defined.
[4]

........................................................................................................................................................

© UCLES 2019 9709/12/M/J/19

(ii) Solve the equation fg x = 73 . [3]

........................................................................................................................................................

© UCLES 2019 9709/12/M/J/19 [Turn over

8 The position vectors of points A and B, relative to an origin O, are given by

` a ` a
−−→ 6 −−→ 3
OA = −2 and OB = k ,
−6 −3
where k is a constant.

(i) Find the value of k for which angle AOB is 90Å. [2]

........................................................................................................................................................

(ii) Find the values of k for which the lengths of OA and OB are equal. [2]

........................................................................................................................................................

© UCLES 2019 9709/12/M/J/19

11
−−→ −−→
The point C is such that AC = 2CB.
−−→
(iii) In the case where k = 4, find the unit vector in the direction of OC. [4]

........................................................................................................................................................

© UCLES 2019 9709/12/M/J/19 [Turn over

9 The curve C1 has equation y = x2 − 4x + 7. The curve C2 has equation y2 = 4x + k, where k is a

constant. The tangent to C1 at the point where x = 3 is also the tangent to C2 at the point P. Find the
value of k and the coordinates of P. [8]

................................................................................................................................................................

© UCLES 2019 9709/12/M/J/19

................................................................................................................................................................

© UCLES 2019 9709/12/M/J/19 [Turn over

10 (a) In an arithmetic progression, the sum of the first ten terms is equal to the sum of the next five
terms. The first term is a.
(i) Show that the common difference of the progression is 31 a. [4]

................................................................................................................................................

(ii) Given that the tenth term is 36 more than the fourth term, find the value of a. [2]

................................................................................................................................................

© UCLES 2019 9709/12/M/J/19

(b) The sum to infinity of a geometric progression is 9 times the sum of the first four terms. Given
that the first term is 12, find the value of the fifth term. [4]

........................................................................................................................................................

© UCLES 2019 9709/12/M/J/19 [Turn over

11
y

9
M y= 4x + 1 +
4x + 1

x
O

9
The diagram shows part of the curve y = 4x + 1 + and the minimum point M .
4x + 1

dy
(i) Find expressions for and Ó y dx. [6]
dx

........................................................................................................................................................

© UCLES 2019 9709/12/M/J/19

(ii) Find the coordinates of M . [3]

........................................................................................................................................................

The shaded region is bounded by the curve, the y-axis and the line through M parallel to the x-axis.

(iii) Find, showing all necessary working, the area of the shaded region. [3]

........................................................................................................................................................

© UCLES 2019 9709/12/M/J/19

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.

........................................................................................................................................................................

© UCLES 2019 9709/12/M/J/19

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© UCLES 2019 9709/12/M/J/19

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© UCLES 2019 9709/12/M/J/19

Cambridge Assessment International Education
Cambridge International Advanced Subsidiary and Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*9810374883*

NUMBER NUMBER

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) October/November 2019
1 hour 45 minutes
Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

This document consists of 19 printed pages and 1 blank page.

JC19 11_9709_12/RP
© UCLES 2019 [Turn over
2

BLANK PAGE

© UCLES 2019 9709/12/O/N/19

3
@ A
x 6
1 The coefficient of x in the expansion of 4 + ax 1 +
2
is 3. Find the value of the constant a. [4]
2

................................................................................................................................................................

© UCLES 2019 9709/12/O/N/19 [Turn over

2 The point M is the mid-point of the line joining the points 3, 7 and −1, 1. Find the equation of the
x y
line through M which is parallel to the line + = 1. [4]
3 2

................................................................................................................................................................

© UCLES 2019 9709/12/O/N/19

dy k
3 A curve is such that = , where k is a constant. The points P 1, −1 and Q 4, 4 lie on the
dx x
curve. Find the equation of the curve. [4]

................................................................................................................................................................

© UCLES 2019 9709/12/O/N/19 [Turn over

4
A

r cm

O 21 rad T

The diagram shows a circle with centre O and radius r cm. Points A and B lie on the circle and
angle AOB = 21 radians. The tangents to the circle at A and B meet at T .

(i) Express the perimeter of the shaded region in terms of r and 1. [3]

........................................................................................................................................................

© UCLES 2019 9709/12/O/N/19

(ii) In the case where r = 5 and 1 = 1.2, find the area of the shaded region. [4]

........................................................................................................................................................

© UCLES 2019 9709/12/O/N/19 [Turn over

15 cm h cm

The diagram shows a solid cone which has a slant height of 15 cm and a vertical height of h cm.

(i) Show that the volume, V cm3, of the cone is given by V = 13 0 225h − h3 . [2]

[The volume of a cone of radius r and vertical height h is 31 0r2 h.]

........................................................................................................................................................

© UCLES 2019 9709/12/O/N/19

(ii) Given that h can vary, find the value of h for which V has a stationary value. Determine, showing
all necessary working, the nature of this stationary value. [5]

........................................................................................................................................................

© UCLES 2019 9709/12/O/N/19 [Turn over

6 (a) Given that x > 0, find the two smallest values of x, in radians, for which 3 tan 2x + 1 = 1. Show
all necessary working. [4]

........................................................................................................................................................

© UCLES 2019 9709/12/O/N/19

(b) The function f : x → 3 cos2 x − 2 sin2 x is defined for 0 ≤ x ≤ 0.

(i) Express f x in the form a cos2 x + b, where a and b are constants. [1]

................................................................................................................................................

(ii) Find the range of f. [2]

................................................................................................................................................

© UCLES 2019 9709/12/O/N/19 [Turn over

7
G Q
F

D
E

P
C
B
k j

O i A

The diagram shows a three-dimensional shape OABCDEFG. The base OABC and the upper surface
DEFG are identical horizontal rectangles. The parallelograms OAED and CBFG both lie in vertical
planes. Points P and Q are the mid-points of OD and GF respectively. Unit vectors i and j are parallel
−−→ −−→
to OA and OC respectively and the unit vector k is vertically upwards. The position vectors of A, C
−−→ −−→ −−→
and D are given by OA = 6i, OC = 8j and OD = 2i + 10k.
−−→ −−→
(i) Express each of the vectors PB and PQ in terms of i, j and k. [4]

........................................................................................................................................................

© UCLES 2019 9709/12/O/N/19

(ii) Determine whether P is nearer to Q or to B. [2]

........................................................................................................................................................

(iii) Use a scalar product to find angle BPQ. [3]

........................................................................................................................................................

© UCLES 2019 9709/12/O/N/19 [Turn over

8 (a) Over a 21-day period an athlete prepares for a marathon by increasing the distance she runs each
day by 1.2 km. On the first day she runs 13 km.
(i) Find the distance she runs on the last day of the 21-day period. [1]

................................................................................................................................................

(ii) Find the total distance she runs in the 21-day period. [2]

................................................................................................................................................

© UCLES 2019 9709/12/O/N/19

(b) The first, second and third terms of a geometric progression are x, x − 3 and x − 5 respectively.
(i) Find the value of x. [2]

................................................................................................................................................

(ii) Find the fourth term of the progression. [2]

................................................................................................................................................

(iii) Find the sum to infinity of the progression. [2]

................................................................................................................................................

© UCLES 2019 9709/12/O/N/19 [Turn over

9 Functions f and g are defined by

f x = 2x2 + 8x + 1 for x ∈ >,

g x = 2x − k for x ∈ >,

where k is a constant.

(i) Find the value of k for which the line y = g x is a tangent to the curve y = f x. [3]

........................................................................................................................................................

(ii) In the case where k = −9, find the set of values of x for which f x < g x. [3]

........................................................................................................................................................

© UCLES 2019 9709/12/O/N/19

(iii) In the case where k = −1, find g−1 f x and solve the equation g−1 f x = 0. [3]

........................................................................................................................................................

(iv) Express f x in the form 2 x + a2 + b, where a and b are constants, and hence state the least
value of f x. [3]

........................................................................................................................................................

© UCLES 2019 9709/12/O/N/19 [Turn over

10
y
4
y= 1−
2x + 12
B
x
O A

4
The diagram shows part of the curve y = 1 − . The curve intersects the x-axis at A. The
2x + 12
normal to the curve at A intersects the y-axis at B.

dy
(i) Obtain expressions for and Ó y dx. [4]
dx

........................................................................................................................................................

© UCLES 2019 9709/12/O/N/19

(ii) Find the coordinates of B. [4]

........................................................................................................................................................

(iii) Find, showing all necessary working, the area of the shaded region. [4]

........................................................................................................................................................

© UCLES 2019 9709/12/O/N/19

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.

........................................................................................................................................................................

© UCLES 2019 9709/12/O/N/19

Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*8348568421*

NUMBER NUMBER

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) February/March 2018
1 hour 45 minutes
Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name in the spaces at the top of this page.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
DO NOT WRITE IN ANY BARCODES.

This document consists of 20 printed pages.

JC18 03_9709_12/2R
© UCLES 2018 [Turn over
2

dy −1
1 A curve passes through the point 4, −6 and has an equation for which = x 2 − 3. Find the
dx
equation of the curve. [4]

................................................................................................................................................................

© UCLES 2018 9709/12/F/M/18

2 (i) Find the coefficients of x2 and x3 in the expansion of 1 − 2x7 . [3]

........................................................................................................................................................

(ii) Hence find the coefficient of x3 in the expansion of 2 + 5x 1 − 2x7 . [2]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18 [Turn over

3 On a certain day, the height of a young bamboo plant was found to be 40 cm. After exactly one day its
height was found to be 41.2 cm. Two different models are used to predict its height exactly 60 days
after it was first measured.
³ Model A assumes that the daily amount of growth continues to be constant at the amount found
for the first day.
³ Model B assumes that the daily percentage rate of growth continues to be constant at the
percentage rate of growth found for the first day.

(i) Using model A, find the predicted height in cm of the bamboo plant exactly 60 days after it was
first measured. [2]

........................................................................................................................................................

(ii) Using model B, find the predicted height in cm of the bamboo plant exactly 60 days after it was
first measured. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18

4 A straight line cuts the positive x-axis at A and the positive y-axis at B 0, 2. Angle BAO = 16 0 radians,
where O is the origin.

(i) Find the exact value of the x-coordinate of A. [2]

........................................................................................................................................................

(ii) Find the equation of the perpendicular bisector of AB, giving your answer in the form y = mx + c,
where m is given exactly and c is an integer. [4]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18 [Turn over

5 + 2 tan x
5 (a) Express the equation = 1 + tan x as a quadratic equation in tan x and hence solve the
3 + 2 tan x
equation for 0 ≤ x ≤ 0. [4]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18

(b)
y

y = k sin 1 + !
2

1
O 150Å

The diagram shows part of the graph of y = k sin 1 + !, where k and ! are constants and
0Å < ! < 180Å. Find the value of ! and the value of k. [2]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18 [Turn over

6
P 10 cm O R

2.2 rad

The diagram shows a sector POQ of a circle of radius 10 cm and centre O. Angle POQ is 2.2 radians.
QR is an arc of a circle with centre P and POR is a straight line.

(i) Show that the length of PQ is 17.8 cm, correct to 3 significant figures. [2]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18

(ii) Find the perimeter of the shaded region. [4]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18 [Turn over

7
E

4 C 3 B
1
D C B
3 D k
2
j
O 7 A O i A

Fig. 1 Fig. 2

Fig. 1 shows a rectangle with sides of 7 units and 3 units from which a triangular corner has been
removed, leaving a 5-sided polygon OABCD. The sides OA, AB, BC and DO have lengths of 7 units,
3 units, 3 units and 2 units respectively. Fig. 2 shows the polygon OABCD forming the horizontal
base of a pyramid in which the point E is 8 units vertically above D. Unit vectors i, j and k are parallel
to OA, OD and DE respectively.
−−→
(i) Find CE and the length of CE. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18

11
@ A
(ii) Use a scalar product to find angle ECA, giving your answer in the form cos−1 , where m
m
n
and n are integers. [5]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18 [Turn over

A curve has equation y = 12 x2 − 4x 2 + 8x.

3
8

(i) Find the x-coordinates of the stationary points. [5]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18

d2 y
(ii) Find . [1]
dx2

........................................................................................................................................................

(iii) Find, showing all necessary working, the nature of each stationary point. [2]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18 [Turn over

1
9 A curve has equation y = + c and a line has equation y = cx − 3, where c is a constant.
x
(i) Find the set of values of c for which the curve and the line meet. [4]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18

(ii) The line is a tangent to the curve for two particular values of c. For each of these values find the
x-coordinate of the point at which the tangent touches the curve. [4]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18 [Turn over

10 Functions f and g are defined by

8
f x = +2 for x > 2,
x−2
8
g x = +2 for 2 < x < 4.
x−2

(i) (a) State the range of the function f. [1]

................................................................................................................................................

(b) State the range of the function g. [1]

................................................................................................................................................

(c) State the range of the function fg. [1]

................................................................................................................................................

(ii) Explain why the function gf cannot be formed. [1]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18

(iii) Find the set of values of x satisfying the inequality 6f ′ x + 2f −1 x − 5 < 0. [6]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18 [Turn over

11
y

y = 1 − 2x

1
O A 2 , 0
x

y = 1 − 2x − 1 − 2x3

The diagram
shows part of the curve y = 1 − 2x − 1 − 2x3 intersecting the x-axis at the origin O and
at A 12 , 0 . The line AB intersects the y-axis at B and has equation y = 1 − 2x.

(i) Show that AB is the tangent to the curve at A. [4]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18

19
1

(ii) Show that the area of the shaded region can be expressed as Ó 1 − 2x3 dx.
2
[2]
0

........................................................................................................................................................

(iii) Hence, showing all necessary working, find the area of the shaded region. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/F/M/18 [Turn over

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.

........................................................................................................................................................................

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

© UCLES 2018 9709/12/F/M/18

Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*6653000674*

NUMBER NUMBER

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) May/June 2018
1 hour 45 minutes
Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

This document consists of 20 printed pages.

JC18 06_9709_12/2RP R
© UCLES 2018 [Turn over
2
0 x 16
1 The coefficient of x2 in the expansion of 2 + + a + x5 is 330. Find the value of the constant a.
2
[5]

................................................................................................................................................................

© UCLES 2018 9709/12/M/J/18

2 The equation of a curve is y = x2 − 6x + k, where k is a constant.

(i) Find the set of values of k for which the whole of the curve lies above the x-axis. [2]

........................................................................................................................................................

(ii) Find the value of k for which the line y + 2x = 7 is a tangent to the curve. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18 [Turn over

3 A company producing salt from sea water changed to a new process. The amount of salt obtained
each week increased by 2% of the amount obtained in the preceding week. It is given that in the first
week after the change the company obtained 8000 kg of salt.

(i) Find the amount of salt obtained in the 12th week after the change. [3]

........................................................................................................................................................

(ii) Find the total amount of salt obtained in the first 12 weeks after the change. [2]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18

5

4 The function f is such that f x = a + b cos x for 0 ≤ x ≤ 20. It is given that f 13 0 = 5 and f 0 = 11.

(i) Find the values of the constants a and b. [3]

........................................................................................................................................................

(ii) Find the set of values of k for which the equation f x = k has no solution. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18 [Turn over

5
C

B
k
j

O i A

The diagram shows a three-dimensional shape. The base OAB is a horizontal triangle in which
angle AOB is 90Å. The side OBCD is a rectangle and the side OAD lies in a vertical plane. Unit
vectors i and j are parallel to OA and OB respectively and the unit vector k is vertical. The position
−−→ −−→ −−→
vectors of A, B and D are given by OA = 8i, OB = 5j and OD = 2i + 4k.
−−→ −−→
(i) Express each of the vectors DA and CA in terms of i, j and k. [2]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18

(ii) Use a scalar product to find angle CAD. [4]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18 [Turn over

6
B
r
O
21 rad

T
A

The diagram shows points A and B on a circle with centre O and radius r. The tangents to the circle
at A and B meet at T . The shaded region is bounded by the minor arc AB and the lines AT and BT .
Angle AOB is 21 radians.

(i) In the case where the area of the sector AOB is the same as the area of the shaded region, show
that tan 1 = 21. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18

(ii) In the case where r = 8 cm and the length of the minor arc AB is 19.2 cm, find the area of the
shaded region. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18 [Turn over

7 The function f is defined by f : x → 7 − 2x2 − 12x for x ∈ >.

(i) Express 7 − 2x2 − 12x in the form a − 2 x + b2 , where a and b are constants. [2]

........................................................................................................................................................

(ii) State the coordinates of the stationary point on the curve y = f x. [1]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18

The function g is defined by g : x → 7 − 2x2 − 12x for x ≥ k.

(iii) State the smallest value of k for which g has an inverse. [1]

........................................................................................................................................................

(iv) For this value of k, find g−1 x. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18 [Turn over

8 Points A and B have coordinates h, h and 4h + 6, 5h respectively. The equation of the perpendicular
bisector of AB is 3x + 2y = k. Find the values of the constants h and k. [7]

................................................................................................................................................................

© UCLES 2018 9709/12/M/J/18

................................................................................................................................................................

© UCLES 2018 9709/12/M/J/18 [Turn over

dy
9 A curve is such that = 4x + 1 and 2, 5 is a point on the curve.
dx

(i) Find the equation of the curve. [4]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18

(ii) A point P moves along the curve in such a way that the y-coordinate is increasing at a constant
rate of 0.06 units per second. Find the rate of change of the x-coordinate when P passes through
2, 5. [2]

........................................................................................................................................................

d2 y dy
(iii) Show that × is constant. [2]
dx2 dx

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18 [Turn over

10 (i) Solve the equation 2 cos x + 3 sin x = 0, for 0Å ≤ x ≤ 360Å. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18

(ii) Sketch, on the same diagram, the graphs of y = 2 cos x and y = −3 sin x for 0Å ≤ x ≤ 360Å. [3]

(iii) Use your answers to parts (i) and (ii) to find the set of values of x for 0Å ≤ x ≤ 360Å for which
2 cos x + 3 sin x > 0. [2]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18 [Turn over

11
y
x 6
y= +
2 x

P Q y=4

x
O

x 6
The diagram shows part of the curve y = + . The line y = 4 intersects the curve at the points P
2 x
and Q.

(i) Show that the tangents to the curve at P and Q meet at a point on the line y = x. [6]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18

(ii) Find, showing all necessary working, the volume obtained when the shaded region is rotated
through 360Å about the x-axis. Give your answer in terms of 0 . [6]

........................................................................................................................................................

© UCLES 2018 9709/12/M/J/18 [Turn over

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.

........................................................................................................................................................................

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

© UCLES 2018 9709/12/M/J/18

Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*9945076812*

NUMBER NUMBER

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) October/November 2018
1 hour 45 minutes
Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

This document consists of 20 printed pages.

JC18 11_9709_12/RP
© UCLES 2018 [Turn over
2
@ A
Find the coefficient of 2 in the expansion of 3x + 2 .
1 2 7
1 [4]
x 3x

................................................................................................................................................................

© UCLES 2018 9709/12/O/N/18

3
4@ A
Showing all necessary working, find Ô x + dx.
2
2 [4]
x
1

................................................................................................................................................................

© UCLES 2018 9709/12/O/N/18 [Turn over

3
y
y = 5x

Q R

P
y = x 9 − x2

x
O

The diagram shows part of the curve y = x 9 − x2 and the line y = 5x, intersecting at the origin O and
the point R. Point P lies on the line y = 5x between O and R and the x-coordinate of P is t. Point Q
lies on the curve and PQ is parallel to the y-axis.

(i) Express the length of PQ in terms of t, simplifying your answer. [2]

........................................................................................................................................................

(ii) Given that t can vary, find the maximum value of the length of PQ. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/O/N/18

4 Functions f and g are defined by

f : x → 2 − 3 cos x for 0 ≤ x ≤ 20,

g : x → 12 x for 0 ≤ x ≤ 20.

(i) Solve the equation fg x = 1. [3]

........................................................................................................................................................

(ii) Sketch the graph of y = f x. [3]

© UCLES 2018 9709/12/O/N/18 [Turn over

5 The first three terms of an arithmetic progression are 4, x and y respectively. The first three terms of
a geometric progression are x, y and 18 respectively. It is given that both x and y are positive.

(i) Find the value of x and the value of y. [4]

........................................................................................................................................................

© UCLES 2018 9709/12/O/N/18

(ii) Find the fourth term of each progression. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/O/N/18 [Turn over

6
B

20 cm

1Å
A C
9 cm D

The diagram shows a triangle ABC in which BC = 20 cm and angle ABC = 90Å. The perpendicular
from B to AC meets AC at D and AD = 9 cm. Angle BCA = 1Å.

(i) By expressing the length of BD in terms of 1 in each of the triangles ABD and DBC, show that
20 sin2 1 = 9 cos 1. [4]

........................................................................................................................................................

© UCLES 2018 9709/12/O/N/18

(ii) Hence, showing all necessary working, calculate 1. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/O/N/18 [Turn over

7
R

P Q

C 4
k j
A B
4 O i

The diagram shows a solid cylinder standing on a horizontal circular base with centre O and radius
4 units. Points A, B and C lie on the circumference of the base such that AB is a diameter and
angle BOC = 90Å. Points P, Q and R lie on the upper surface of the cylinder vertically above A, B
and C respectively. The height of the cylinder is 12 units. The mid-point of CR is M and N lies on
BQ with BN = 4 units.

Unit vectors i and j are parallel to OB and OC respectively and the unit vector k is vertically upwards.
−−→ −−→
Evaluate PN . PM and hence find angle MPN . [7]

................................................................................................................................................................

© UCLES 2018 9709/12/O/N/18

................................................................................................................................................................

© UCLES 2018 9709/12/O/N/18 [Turn over

8
Y

8 cm
8 cm

A 12 cm C

The diagram shows an isosceles triangle ACB in which AB = BC = 8 cm and AC = 12 cm. The arc
XC is part of a circle with centre A and radius 12 cm, and the arc YC is part of a circle with centre B
and radius 8 cm. The points A, B, X and Y lie on a straight line.

(i) Show that angle CBY = 1.445 radians, correct to 4 significant figures. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/O/N/18

(ii) Find the perimeter of the shaded region. [4]

........................................................................................................................................................

© UCLES 2018 9709/12/O/N/18 [Turn over

9 The function f is defined by f : x → 2x2 − 12x + 7 for x ∈ >.

(i) Express 2x2 − 12x + 7 in the form 2 x + a2 + b, where a and b are constants. [2]

........................................................................................................................................................

(ii) State the range of f. [1]

........................................................................................................................................................

© UCLES 2018 9709/12/O/N/18

The function g is defined by g : x → 2x2 − 12x + 7 for x ≤ k.

(iii) State the largest value of k for which g has an inverse. [1]

........................................................................................................................................................

(iv) Given that g has an inverse, find an expression for g−1 x. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/O/N/18 [Turn over

The equation of a curve is y = 2x + and the equation of a line is y + x = k, where k is a constant.

12
10
x
(i) Find the set of values of k for which the line does not meet the curve. [3]

........................................................................................................................................................

In the case where k = 15, the curve intersects the line at points A and B.

(ii) Find the coordinates of A and B. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/O/N/18

........................................................................................................................................................

(iii) Find the equation of the perpendicular bisector of the line joining A and B. [3]

........................................................................................................................................................

© UCLES 2018 9709/12/O/N/18 [Turn over

11
y

M
y = 3 4x + 1 − 2x

x
O

The diagram shows part of the curve y = 3 4x + 1 − 2x. The curve crosses the y-axis at A and the

stationary point on the curve is M .

and Ó y dx.
dy
(i) Obtain expressions for [5]
dx

........................................................................................................................................................

(ii) Find the coordinates of M . [3]

........................................................................................................................................................

(iii) Find, showing all necessary working, the area of the shaded region. [4]

........................................................................................................................................................

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.

........................................................................................................................................................................

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*8253559189*

NUMBER NUMBER

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) February/March 2017
1 hour 45 minutes
Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

Answer all the questions.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in
degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.

This document consists of 20 printed pages.

1 Find the set of values of k for which the equation 2x2 + 3kx + k = 0 has distinct real roots. [4]

................................................................................................................................................................

3
@
A5
1 2
2 In the expansion of + 2ax , the coefficient of x is 5. Find the value of the constant a. [4]
ax

................................................................................................................................................................

1
2h

The diagram shows a water container in the form of an inverted pyramid, which is such that when the
height of the water level is h cm the surface of the water is a square of side 12 h cm.

(i) Express the volume of water in the container in terms of h. [1]

[The volume of a pyramid having a base area A and vertical height h is 31 Ah.]

........................................................................................................................................................

Water is steadily dripping into the container at a constant rate of 20 cm3 per minute.

(ii) Find the rate, in cm per minute, at which the water level is rising when the height of the water
level is 10 cm. [4]

........................................................................................................................................................

4
C

8 cm

2
7
0 rad
D B 8 cm A

In the diagram, AB = AC = 8 cm and angle CAB = 27 0 radians. The circular arc BC has centre A, the
circular arc CD has centre B and ABD is a straight line.

9 0 radians.
(i) Show that angle CBD = 14 [1]

........................................................................................................................................................

(ii) Find the perimeter of the shaded region. [5]

........................................................................................................................................................

5
y
y = tan x

x
O 0
B
y = cos x

The diagram shows the graphs of y = tan x and y = cos x for 0 ≤ x ≤ 0. The graphs intersect at points
A and B.

(i) Find by calculation the x-coordinate of A. [4]

........................................................................................................................................................

(ii) Find by calculation the coordinates of B. [3]

........................................................................................................................................................

6 Relative to an origin O, the position vectors of the points A and B are given by
−−→ −−→
OA = 2i + 3j + 5k and OB = 7i + 4j + 3k.

(i) Use a scalar product to find angle OAB. [5]

........................................................................................................................................................

(ii) Find the area of triangle OAB. [2]

........................................................................................................................................................

12
3
7 The function f is defined for x ≥ 0 by f x = 4x + 1 2 .

(i) Find f ′ x and f ′′ x. [3]

........................................................................................................................................................

The first, second and third terms of a geometric progression are respectively f 2, f ′ 2 and kf ′′ 2.

(ii) Find the value of the constant k. [5]

........................................................................................................................................................

8 The functions f and g are defined for x ≥ 0 by

f : x → 2x2 + 3,
g : x → 3x + 2.

(i) Show that gf x = 6x2 + 11 and obtain an unsimplified expression for fg x. [2]

........................................................................................................................................................

(ii) Find an expression for fg−1 x and determine the domain of fg−1 . [5]

........................................................................................................................................................

(iii) Solve the equation gf 2x = fg x. [3]

........................................................................................................................................................

9 The point A 2, 2 lies on the curve y = x2 − 2x + 2.

(i) Find the equation of the tangent to the curve at A. [3]

........................................................................................................................................................

The normal to the curve at A intersects the curve again at B.

(ii) Find the coordinates of B. [4]

........................................................................................................................................................

The tangents at A and B intersect each other at C.

(iii) Find the coordinates of C. [4]

........................................................................................................................................................

10
y

y = f x

x
O B C

The diagram shows the curve y = f x defined for x > 0. The curve has a minimum point at A and
dy 2
crosses the x-axis at B and C. It is given that = 2x − 3 and that the curve passes through the point
189 dx x
4, 16 .

(i) Find the x-coordinate of A. [2]

........................................................................................................................................................

(ii) Find f x. [3]

........................................................................................................................................................

(iii) Find the x-coordinates of B and C. [4]

........................................................................................................................................................

[Question 10 (iv) is printed on the next page.]

(iv) Find, showing all necessary working, the area of the shaded region. [4]

........................................................................................................................................................

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*9022343494*

NUMBER NUMBER

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) May/June 2017
1 hour 45 minutes
Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

Answer all the questions.

This document consists of 19 printed pages and 1 blank page.

........................................................................................................................................................

@ A
1 5
(ii) Hence find the coefficient of x in the expansion of 1 + 3x2 2x − . [4]
x

........................................................................................................................................................

2 The point A has coordinates −2, 6. The equation of the perpendicular bisector of the line AB is
2y = 3x + 5.

(i) Find the equation of AB. [3]

........................................................................................................................................................

(ii) Find the coordinates of B. [3]

........................................................................................................................................................

4
@ A2
1 1 − sin 1
3 (i) Prove the identity − tan 1 . [3]
cos 1 1 + sin 1

........................................................................................................................................................

5
@ A2
1
(ii) Hence solve the equation − tan 1 = 12 , for 0Å ≤ 1 ≤ 360Å. [3]
cos 1

........................................................................................................................................................

O
r cm
21 rad

A B
r cm

D C

The diagram shows a circle with radius r cm and centre O. Points A and B lie on the circle and ABCD
is a rectangle. Angle AOB = 21 radians and AD = r cm.

(i) Express the perimeter of the shaded region in terms of r and 1. [3]

........................................................................................................................................................

(ii) In the case where r = 5 and 1 = 16 0 , find the area of the shaded region. [4]

........................................................................................................................................................

12
5 A curve has equation y = 3 + .
2−x

(i) Find the equation of the tangent to the curve at the point where the curve crosses the x-axis. [5]

........................................................................................................................................................

(ii) A point moves along the curve in such a way that the x-coordinate is increasing at a constant rate
of 0.04 units per second. Find the rate of change of the y-coordinate when x = 4. [2]

........................................................................................................................................................

6
y

A 1, 4

x+y=5

4
y=
x B 4, 1

x
O

4
The diagram shows the straight line x + y = 5 intersecting the curve y = at the points A 1, 4 and
x
B 4, 1. Find, showing all necessary working, the volume obtained when the shaded region is rotated
through 360Å about the x-axis. [7]

................................................................................................................................................................

7 (a) The first two terms of an arithmetic progression are 16 and 24. Find the least number of terms of
the progression which must be taken for their sum to exceed 20 000. [4]

........................................................................................................................................................

(b) A geometric progression has a first term of 6 and a sum to infinity of 18. A new geometric
progression is formed by squaring each of the terms of the original progression. Find the sum to
infinity of the new progression. [4]

........................................................................................................................................................

8 Relative to an origin O, the position vectors of three points A, B and C are given by
−−→ −−→ −−→
OA = 3i + pj − 2pk, OB = 6i + p + 4j + 3k and OC = p − 1i + 2j + qk,
where p and q are constants.

(i) In the case where p = 2, use a scalar product to find angle AOB. [4]

........................................................................................................................................................

15
−−→ −−→
(ii) In the case where AB is parallel to OC, find the values of p and q. [4]

........................................................................................................................................................

16

9 The equation of a curve is y = 8 x − 2x.

(i) Find the coordinates of the stationary point of the curve. [3]

........................................................................................................................................................

d2 y
(ii) Find an expression for and hence, or otherwise, determine the nature of the stationary point.
dx2
[2]

........................................................................................................................................................

(iii) Find the values of x at which the line y = 6 meets the curve. [3]

........................................................................................................................................................

(iv) State the set of values of k for which the line y = k does not meet the curve. [1]

........................................................................................................................................................

18

10 The function f is defined by f x = 3 tan 12 x − 2, for − 12 0 ≤ x ≤ 12 0.

(i) Solve the equation f x + 4 = 0, giving your answer correct to 1 decimal place. [3]

........................................................................................................................................................

(ii) Find an expression for f −1 x and find the domain of f −1 . [5]

........................................................................................................................................................

(iii) Sketch, on the same diagram, the graphs of y = f x and y = f −1 x. [3]

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be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*3053484001*

NUMBER NUMBER

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) October/November 2017
1 hour 45 minutes
Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

Answer all the questions.

This document consists of 19 printed pages and 1 blank page.

................................................................................................................................................................

2 A function f is defined by f : x → 4 − 5x for x ∈ >.

(i) Find an expression for f −1 x and find the point of intersection of the graphs of y = f x and
y = f −1 x. [3]

........................................................................................................................................................

(ii) Sketch, on the same diagram, the graphs of y = f x and y = f −1 x, making clear the relationship
between the graphs. [3]

3 (a) Each year, the value of a certain rare stamp increases by 5% of its value at the beginning of the
year. A collector bought the stamp for $10 000 at the beginning of 2005. Find its value at the
beginning of 2015 correct to the nearest $100. [2]

........................................................................................................................................................

(b) The sum of the first n terms of an arithmetic progression is 12 n 3n + 7. Find the 1st term and the
common difference of the progression. [4]

........................................................................................................................................................

4
C

P Q

A B
D O 6 cm

The diagram shows a semicircle with centre O and radius 6 cm. The radius OC is perpendicular to
the diameter AB. The point D lies on AB, and DC is an arc of a circle with centre B.

(i) Calculate the length of the arc DC. [3]

........................................................................................................................................................

(ii) Find the value of

area of region P
,
area of region Q
giving your answer correct to 3 significant figures. [4]

........................................................................................................................................................

5 (i) Show that the equation cos 2x tan2 2x + 3 + 3 = 0 can be expressed as

2 cos2 2x + 3 cos 2x + 1 = 0. [3]

........................................................................................................................................................

(ii) Hence solve the equation cos 2x tan2 2x + 3 + 3 = 0 for 0Å ≤ x ≤ 180Å. [4]

........................................................................................................................................................

10

6 (a) The function f, defined by f : x → a + b sin x for x ∈ >, is such that f 16 0 = 4 and f 12 0 = 3.

(i) Find the values of the constants a and b. [3]

................................................................................................................................................

(ii) Evaluate ff 0. [2]

................................................................................................................................................

(b) The function g is defined by g : x → c + d sin x for x ∈ >. The range of g is given by −4 ≤ g x ≤ 10.
Find the values of the constants c and d . [3]

........................................................................................................................................................

7 Points A and B lie on the curve y = x2 − 4x + 7. Point A has coordinates 4, 7 and B is the stationary
point of the curve. The equation of a line L is y = mx − 2, where m is a constant.

(i) In the case where L passes through the mid-point of AB, find the value of m. [4]

........................................................................................................................................................

(ii) Find the set of values of m for which L does not meet the curve. [4]

........................................................................................................................................................

dy
8 A curve is such that = −x2 + 5x − 4.
dx

(i) Find the x-coordinate of each of the stationary points of the curve. [2]

........................................................................................................................................................

d2 y
(ii) Obtain an expression for and hence or otherwise find the nature of each of the stationary
dx2
points. [3]

........................................................................................................................................................

(iii) Given that the curve passes through the point 6, 2, find the equation of the curve. [4]

........................................................................................................................................................

9
A
O

The diagram shows a trapezium OABC in which ` OAa is parallel` toaCB. The position vectors of A and
−−→ 2 −−→ 6
B relative to the origin O are given by OA = −2 and OB = 1 .
−1 1

(i) Show that angle OAB is 90Å. [3]

........................................................................................................................................................

−−→ −−→
The magnitude of CB is three times the magnitude of OA.

(ii) Find the position vector of C. [3]

........................................................................................................................................................

(iii) Find the exact area of the trapezium OABC, giving your answer in the form a b, where a and b
are integers. [3]

........................................................................................................................................................

10
y

y= 5x − 1

P 2, 3

Q
x
O

The diagram shows part of the curve y = 5x − 1 and the normal to the curve at the point P 2, 3.
This normal meets the x-axis at Q.

(i) Find the equation of the normal at P. [4]

........................................................................................................................................................

(ii) Find, showing all necessary working, the area of the shaded region. [7]

........................................................................................................................................................

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effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) February/March 2016
1 hour 45 minutes
Additional Materials: Answer Booklet/Paper
*5026516164*

Graph Paper
List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
DO NOT WRITE IN ANY BARCODES.

Answer all the questions.

At the end of the examination, fasten all your work securely together.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 75.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger
numbers of marks later in the paper.

This document consists of 5 printed pages and 3 blank pages.

1 (i) Find the coefficients of x4 and x5 in the expansion of 1 − 2x5 . [2]

(ii) It is given that, when 1 + px 1 − 2x5 is expanded, there is no term in x5 . Find the value of the
constant p. [2]

dy 2
2 A curve for which = 3x2 − 3 passes through −1, 3. Find the equation of the curve. [4]
dx x

3 The 12th term of an arithmetic progression is 17 and the sum of the first 31 terms is 1023. Find the
31st term. [5]

4 (a) Solve the equation sin−1 3x = − 13 0, giving the solution in an exact form. [2]

(b) Solve, by factorising, the equation 2 cos 1 sin 1 − 2 cos 1 − sin 1 + 1 = 0 for 0 ≤ 1 ≤ 0. [4]

5 Two points have coordinates A 5, 7 and B 9, −1.

(i) Find the equation of the perpendicular bisector of AB. [3]

The line through C 1, 2 parallel to AB meets the perpendicular bisector of AB at the point X .

(ii) Find, by calculation, the distance BX . [5]

6 A vacuum flask (for keeping drinks hot) is modelled as a closed cylinder in which the internal radius
is r cm and the internal height is h cm. The volume of the flask is 1000 cm3. A flask is most efficient
when the total internal surface area, A cm2, is a minimum.

2000
(i) Show that A = 20r2 + . [3]
r
(ii) Given that r can vary, find the value of r, correct to 1 decimal place, for which A has a stationary
value and verify that the flask is most efficient when r takes this value. [5]

7
C

3
P
k B
2.4
j
i
O A
4

The diagram shows a pyramid OABC with a horizontal triangular base OAB and vertical height OC.
Angles AOB, BOC and AOC are each right angles. Unit vectors i, j and k are parallel to OA, OB and
OC respectively, with OA = 4 units, OB = 2.4 units and OC = 3 units. The point P on CA is such
that CP = 3 units.
−−→
(i) Show that CP = 2.4i − 1.8k. [2]
−−→ −−→
(ii) Express OP and BP in terms of i, j and k. [2]

(iii) Use a scalar product to find angle BPC. [4]

1
8 The function f is such that f x = a2 x2 − ax + 3b for x ≤ , where a and b are constants.
2a

(i) For the case where f −2 = 4a2 − b + 8 and f −3 = 7a2 − b + 14, find the possible values of a
and b. [5]

(ii) For the case where a = 1 and b = −1, find an expression for f −1 x and give the domain of f −1 .
[5]

9 (a)
X

!
A B

r r

O
Fig. 1

In Fig. 1, OAB is a sector of a circle with centre O and radius r. AX is the tangent at A to the arc
AB and angle BAX = !.
(i) Show that angle AOB = 2!. [2]
(ii) Find the area of the shaded segment in terms of r and !. [2]

(b)
C

4 cm 4 cm
X

A B
4 cm

Fig. 2

In Fig. 2, ABC is an equilateral triangle of side 4 cm. The lines AX , BX and CX are tangents to
the equal circular arcs AB, BC and CA. Use the results in part (a) to find the area of the shaded
region, giving your answer in terms of 0 and ï3. [6]

10
y

Q 3, 4

1
y= 16
3x − 12

x
O P R

The diagram shows part of the curve y = 16 1 3x − 12 , which touches the x-axis at the point P. The

point Q 3, 4 lies on the curve and the tangent to the curve at Q crosses the x-axis at R.

(i) State the x-coordinate of P. [1]

Showing all necessary working, find by calculation

(ii) the x-coordinate of R, [5]
(iii) the area of the shaded region PQR. [6]

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effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) May/June 2016
1 hour 45 minutes
Additional Materials: List of Formulae (MF9)
*9566317764*

READ THESE INSTRUCTIONS FIRST

An answer booklet is provided inside this question paper. You should follow the instructions on the front cover
of the answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet.

Answer all the questions.

This document consists of 4 printed pages and 1 insert.

1 Functions f and g are defined by

f : x → 10 − 3x, x ∈ >,
10
g : x → , x ∈ >, x ≠ 32 .
3 − 2x
Solve the equation ff x = gf 2. [3]

dy 8
2 A curve is such that = . Given that the curve passes through 2, 7, find the equation of
dx 5 − 2x2
the curve. [4]

3 Relative to an origin O, the position vectors of points A and B are given by

−−→ −−→
OA = 2i − 5j − 2k and OB = 4i − 4j + 2k.
−−→ −−→ −−→
The point C is such that AB = BC. Find the unit vector in the direction of OC. [4]

4 Find the term that is independent of x in the expansion of

@ A
2 6
(i) x − , [2]
x
@ A@ A
3 2 6
(ii) 2 + 2 x − . [4]
x x

5
A

1
30
B x x C
M

In the diagram, triangle ABC is right-angled at C and M is the mid-point of BC. It is given that
angle ABC = 13 0 radians and angle BAM = 1 radians. Denoting the lengths of BM and MC by x,

(i) find AM in terms of x, [3]

@ A
(ii) show that 1 = 16 0 − tan−1
1
. [2]
2ï3

! rad
r cm Q

T
P

The diagram shows a circle with radius r cm and centre O. The line PT is the tangent to the circle
at P and angle POT = ! radians. The line OT meets the circle at Q.

(i) Express the perimeter of the shaded region PQT in terms of r and !. [3]

(ii) In the case where ! = 13 0 and r = 10, find the area of the shaded region correct to 2 significant
figures. [3]

1 + cos 1 1 − cos 1 4
7 (i) Prove the identity − . [4]
1 − cos 1 1 + cos 1 sin 1 tan 1

(ii) Hence solve, for 0Å < 1 < 360Å, the equation

1 + cos 1 1 − cos 1
@ A
sin 1 − = 3. [3]
1 − cos 1 1 + cos 1

8 Three points have coordinates A 0, 7, B 8, 3 and C 3k, k. Find the value of the constant k for
which
(i) C lies on the line that passes through A and B, [4]
(ii) C lies on the perpendicular bisector of AB. [4]

9 A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that
each day a greater amount of water is lost.

(i) On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day.
(a) How many litres will be lost on the 30th day after filling? [2]
(b) The tank becomes empty during the nth day after filling. Find the value of n. [3]

(ii) Assume instead that 10 litres of water are lost on the first day and that the amount of water lost
increases by 10% on each succeeding day. Find what percentage of the original 2000 litres is
left in the tank at the end of the 30th day after filling. [4]

[Questions 10 and 11 are printed on the next page.]

10
y

8
y= + 2x
x

x
O

8
The diagram shows the part of the curve y = + 2x for x > 0, and the minimum point M .
x

dy d2 y
(i) Find expressions for , and Ó y2 dx. [5]
dx dx2

(ii) Find the coordinates of M and determine the coordinates and nature of the stationary point on
the part of the curve for which x < 0. [5]

(iii) Find the volume obtained when the region bounded by the curve, the x-axis and the lines x = 1
and x = 2 is rotated through 360Å about the x-axis. [2]

11 The function f is defined by f : x → 6x − x2 − 5 for x ∈ >.

(i) Find the set of values of x for which f x ≤ 3. [3]

(ii) Given that the line y = mx + c is a tangent to the curve y = f x, show that 4c = m2 − 12m + 16.
[3]

The function g is defined by g : x → 6x − x2 − 5 for x ≥ k, where k is a constant.

(iii) Express 6x − x2 − 5 in the form a − x − b2 , where a and b are constants. [2]

(iv) State the smallest value of k for which g has an inverse. [1]

(v) For this value of k, find an expression for g−1 x. [2]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) October/November 2016
1 hour 45 minutes
Additional Materials: List of Formulae (MF9)
*5199919634*

READ THESE INSTRUCTIONS FIRST

Answer all the questions.

This document consists of 3 printed pages, 1 blank page and 1 insert.

dy 8
1 A curve is such that = . The point 2, 5 lies on the curve. Find the equation of the
dx 4x + 1
curve. [4]

2 (i) Express the equation sin 2x + 3 cos 2x = 3 sin 2x − cos 2x in the form tan 2x = k, where k is a
constant. [2]

(ii) Hence solve the equation for −90Å ≤ x ≤ 90Å. [3]

3 A curve has equation y = 2x2 − 6x + 5.

(i) Find the set of values of x for which y > 13. [3]

(ii) Find the value of the constant k for which the line y = 2x + k is a tangent to the curve. [3]

0 x 1n
4 In the expansion of 3 − 2x 1 + , the coefficient of x is 7. Find the value of the constant n and
2
hence find the coefficient of x2 . [6]

x y
5 The line + = 1, where a and b are positive constants, intersects the x- and y-axes at the points A
a b
and B respectively. The mid-point of AB lies on the line 2x + y = 10 and the distance AB = 10. Find
the values of a and b. [6]

cm
10

A O 1.2 rad C

The diagram shows a metal plate ABCD made from two parts. The part BCD is a semicircle. The
part DAB is a segment of a circle with centre O and radius 10 cm. Angle BOD is 1.2 radians.

(i) Show that the radius of the semicircle is 5.646 cm, correct to 3 decimal places. [2]

(ii) Find the perimeter of the metal plate. [3]

(iii) Find the area of the metal plate. [3]

3
7 The equation of a curve is y = 2 + .
2x − 1
dy
(i) Obtain an expression for . [2]
dx
(ii) Explain why the curve has no stationary points. [1]

At the point P on the curve, x = 2.

(iii) Show that the normal to the curve at P passes through the origin. [4]

(iv) A point moves along the curve in such a way that its x-coordinate is decreasing at a constant
rate of 0.06 units per second. Find the rate of change of the y-coordinate as the point passes
through P. [2]

8 (a) A cyclist completes a long-distance charity event across Africa. The total distance is 3050 km.
He starts the event on May 1st and cycles 200 km on that day. On each subsequent day he
reduces the distance cycled by 5 km.
(i) How far will he travel on May 15th? [2]
(ii) On what date will he finish the event? [3]

(b) A geometric progression is such that the third term is 8 times the sixth term, and the sum of the
first six terms is 31 12 . Find
(i) the first term of the progression, [4]
(ii) the sum to infinity of the progression. [1]

9 Relative to an origin O, the position vectors of the points A, B and C are given by
` a ` a ` a
−−→ 2 −−→ −2 −−→ 2
OA = −2 , OB = 3 and OC = 6 .
−1 6 5

(i) Use a scalar product to find angle AOB. [4]

−−→
(ii) Find the vector which is in the same direction as AC and of magnitude 15 units. [3]
−−→ −−→ −−→
(iii) Find the value of the constant p for which p OA + OC is perpendicular to OB. [3]

10 A function f is defined by f : x → 5 − 2 sin 2x for 0 ≤ x ≤ 0.

(i) Find the range of f. [2]

(ii) Sketch the graph of y = f x. [2]

(iii) Solve the equation f x = 6, giving answers in terms of 0. [3]

The function g is defined by g : x → 5 − 2 sin 2x for 0 ≤ x ≤ k, where k is a constant.

(iv) State the largest value of k for which g has an inverse. [1]

(v) For this value of k, find an expression for g−1 x. [3]

BLANK PAGE

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) May/June 2015
1 hour 45 minutes
Additional Materials: Answer Booklet/Paper
*2057450266*

Graph Paper
List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

Answer all the questions.

This document consists of 3 printed pages and 1 blank page.

1 The function f is such that f ′ x = 5 − 2x2 and 3, 5 is a point on the curve y = f x. Find f x. [3]

2
A

O 21 rad X Y

In the diagram, AYB is a semicircle with AB as diameter and OAXB is a sector of a circle with centre
O and radius r. Angle AOB = 21 radians. Find an expression, in terms of r and 1, for the area of the
shaded region. [4]

3 (i) Find the coefficients of x2 and x3 in the expansion of 2 − x6 . [3]

(ii) Find the coefficient of x3 in the expansion of 3x + 1 2 − x6 . [2]

4 Variables u, x and y are such that u = 2x y − x and x + 3y = 12. Express u in terms of x and hence
find the stationary value of u. [5]

sin 1 − cos 1 tan 1 − 1

5 (i) Prove the identity . [1]
sin 1 + cos 1 tan 1 + 1

sin 1 − cos 1 tan 1

(ii) Hence solve the equation = , for 0Å ≤ 1 ≤ 180Å. [4]
sin 1 + cos 1 6

6 A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel
turns in such a way that the height, h m, of a passenger above the ground is given by the formula
h = 60 1 − cos kt. In this formula, k is a constant, t is the time in minutes that has elapsed since the
passenger started the ride at ground level and kt is measured in radians.

(i) Find the greatest height of the passenger above the ground. [1]

One complete revolution of the wheel takes 30 minutes.

(ii) Show that k = 1

15 0 . [2]

(iii) Find the time for which the passenger is above a height of 90 m. [3]

7 The point C lies on the perpendicular bisector of the line joining the points A 4, 6 and B 10, 2.
C also lies on the line parallel to AB through 3, 11.

(i) Find the equation of the perpendicular bisector of AB. [4]

(ii) Calculate the coordinates of C. [3]

8 (a) The first, second and last terms in an arithmetic progression are 56, 53 and −22 respectively.
Find the sum of all the terms in the progression. [4]

(b) The first, second and third terms of a geometric progression are 2k + 6, 2k and k + 2 respectively,
where k is a positive constant.
(i) Find the value of k. [3]
(ii) Find the sum to infinity of the progression. [2]

9 Relative to an origin O, the position vectors of points A and B are given by

−−→ −−→
OA = 2i + 4j + 4k and OB = 3i + j + 4k.

(i) Use a vector method to find angle AOB. [4]

−−→ −−→
The point C is such that AB = BC.
−−→
(ii) Find the unit vector in the direction of OC. [4]

(iii) Show that triangle OAC is isosceles. [1]

4
10 The equation of a curve is y = .
2x − 1
(i) Find, showing all necessary working, the volume obtained when the region bounded by the
curve, the x-axis and the lines x = 1 and x = 2 is rotated through 360Å about the x-axis. [4]

(ii) Given that the line 2y = x + c is a normal to the curve, find the possible values of the constant c.
[6]

11 The function f is defined by f : x → 2x2 − 6x + 5 for x ∈ >.

(i) Find the set of values of p for which the equation f x = p has no real roots. [3]

The function g is defined by g : x → 2x2 − 6x + 5 for 0 ≤ x ≤ 4.

(ii) Express g x in the form a x + b2 + c, where a, b and c are constants. [3]

(iii) Find the range of g. [2]

The function h is defined by h : x → 2x2 − 6x + 5 for k ≤ x ≤ 4, where k is a constant.

(iv) State the smallest value of k for which h has an inverse. [1]

(v) For this value of k, find an expression for h−1 x. [3]

BLANK PAGE

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) October/November 2015
1 hour 45 minutes
Additional Materials: Answer Booklet/Paper
*0446695683*

Graph Paper
List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

Answer all the questions.

This document consists of 4 printed pages.

1 Functions f and g are defined by

f : x → 3x + 2, x ∈ >,
g : x → 4x − 12, x ∈ >.

Solve the equation f −1 x = gf x. [4]

2 In the expansion of x + 2k7 , where k is a non-zero constant, the coefficients of x4 and x5 are equal.
Find the value of k. [4]

3
D C
m
40 c
F A B
B 30Å 30Å
A
30Å 30Å
h cm

E E
Fig. 1 Fig. 2

Fig. 1 shows an open tank in the shape of a triangular prism. The vertical ends ABE and DCF are
identical isosceles triangles. Angle ABE = angle BAE = 30Å. The length of AD is 40 cm. The tank is
fixed in position with the open top ABCD horizontal. Water is poured into the tank at a constant rate
of 200 cm3 s−1 . The depth of water, t seconds after filling starts, is h cm (see Fig. 2).

(i) Show that, when the depth of water in the tank is h cm, the volume, V cm3 , of water in the tank
is given by V = 40ï3h2 . [3]

(ii) Find the rate at which h is increasing when h = 5. [3]

@ A2
1 1 1 − cos x
4 (i) Prove the identity − . [4]
sin x tan x 1 + cos x
@ A2
1 1
(ii) Hence solve the equation − = 25 for 0 ≤ x ≤ 20. [3]
sin x tan x

5
C

0.6 rad
O 6 cm A

The diagram shows a metal plate OABC, consisting of a right-angled triangle OAB and a sector OBC
of a circle with centre O. Angle AOB = 0.6 radians, OA = 6 cm and OA is perpendicular to OC.

(i) Show that the length of OB is 7.270 cm, correct to 3 decimal places. [1]

(ii) Find the perimeter of the metal plate. [3]

(iii) Find the area of the metal plate. [3]

6 Points A, B and C have coordinates A −3, 7, B 5, 1 and C −1, k, where k is a constant.

(i) Given that AB = BC, calculate the possible values of k. [3]

The perpendicular bisector of AB intersects the x-axis at D.

(ii) Calculate the coordinates of D. [5]

7 Relative to an origin O, the position vectors of points A, B and C are given by

` a ` a ` a
−−→ 0 −−→ 2 −−→ 3
OA = 2 , OB = 5 and OC = p .
−3 −2 q

(i) In the case where ABC is a straight line, find the values of p and q. [4]

(ii) In the case where angle BAC is 90Å, express q in terms of p. [2]

(iii) In the case where p = 3 and the lengths of AB and AC are equal, find the possible values of q.
[3]

8 The function f is defined, for x ∈ >, by f : x → x2 + ax + b, where a and b are constants.

(i) In the case where a = 6 and b = −8, find the range of f. [3]

(ii) In the case where a = 5, the roots of the equation f x = 0 are k and −2k, where k is a constant.
Find the values of b and k. [3]

(iii) Show that if the equation f x + a = a has no real roots, then a2 < 4 b − a. [3]

12
9 The curve y = f x has a stationary point at 2, 10 and it is given that f ′′ x = .
x3
(i) Find f x. [6]

(ii) Find the coordinates of the other stationary point. [2]

(iii) Find the nature of each of the stationary points. [2]

10
y

y= 9 − 2x2

P 2, 1

B
x
A O

The diagram shows part of the curve y = 9 − 2x2 . The point P 2, 1 lies on the curve and the
normal to the curve at P intersects the x-axis at A and the y-axis at B.

(i) Show that B is the mid-point of AP. [6]

The shaded region is bounded by the curve, the y-axis and the line y = 1.

(ii) Find, showing all necessary working, the exact volume obtained when the shaded region is
rotated through 360Å about the y-axis. [5]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.
To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International
Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after
the live examination series.
Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local
Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) May/June 2014
1 hour 45 minutes
Additional Materials: Answer Booklet/Paper
*6140629685*

Graph Paper
List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

Answer all the questions.

This document consists of 4 printed pages.

1 Find the coordinates of the point at which the perpendicular bisector of the line joining 2, 7 to
10, 3 meets the x-axis. [5]

@ A6
x 4
2 Find the coefficient of x2 in the expansion of 1 + x2 − . [5]
2 x

3 The reflex angle 1 is such that cos 1 = k, where 0 < k < 1.

(i) Find an expression, in terms of k, for

(a) sin 1, [2]
(b) tan 1. [1]

(ii) Explain why sin 21 is negative for 0 < k < 1. [2]

4
O
1 rad r cm

A B

The diagram shows a sector of a circle with radius r cm and centre O. The chord AB divides the
sector into a triangle AOB and a segment AXB. Angle AOB is 1 radians.

(i) In the case where the areas of the triangle AOB and the segment AXB are equal, find the value
of the constant p for which 1 = p sin 1. [2]

(ii) In the case where r = 8 and 1 = 2.4, find the perimeter of the segment AXB. [3]

1 cos 1
5 (i) Prove the identity − tan 1. [4]
cos 1 1 + sin 1

1 cos 1
(ii) Solve the equation − + 2 = 0 for 0Å ≤ 1 ≤ 360Å. [3]
cos 1 1 + sin 1

6 The 1st, 2nd and 3rd terms of a geometric progression are the 1st, 9th and 21st terms respectively
of an arithmetic progression. The 1st term of each progression is 8 and the common ratio of the
geometric progression is r, where r ≠ 1. Find
(i) the value of r, [4]
(ii) the 4th term of each progression. [3]

7
C

D
B
A

The diagram shows a trapezium ABCD in which BA is parallel to CD. The position vectors of A, B
and C relative to an origin O are given by
` a ` a ` a
−−→ 3 −−→ 1 −−→ 4
OA = 4 , OB = 3 and OC = 5 .
0 2 6

(i) Use a scalar product to show that AB is perpendicular to BC. [3]

(ii) Given that the length of CD is 12 units, find the position vector of D. [4]

d2 y
8 The equation of a curve is such that = 2x − 1. Given that the curve has a minimum point at
dx2
3, −10, find the coordinates of the maximum point. [8]

9
y

y = 8 − ï 4 − x
P 3, 7

x
O

The diagram shows part of the curve y = 8 − 4 − x and the tangent to the curve at P 3, 7.

dy
(i) Find expressions for and Ó y dx. [5]
dx

(ii) Find the equation of the tangent to the curve at P in the form y = mx + c. [2]

(iii) Find, showing all necessary working, the area of the shaded region. [4]

10 Functions f and g are defined by

f : x → 2x − 3, x ∈ >,
g : x → x + 4x,
2
x ∈ >.

(i) Solve the equation ff x = 11. [2]

(ii) Find the range of g. [2]

(iii) Find the set of values of x for which g x > 12. [3]

(iv) Find the value of the constant p for which the equation gf x = p has two equal roots. [3]

Function h is defined by h : x → x2 + 4x for x ≥ k, and it is given that h has an inverse.

(v) State the smallest possible value of k. [1]

(vi) Find an expression for h−1 x. [4]

Cambridge International Examinations
Cambridge International Advanced Subsidiary and Advanced Level

MATHEMATICS 9709/12
Paper 1 Pure Mathematics 1 (P1) October/November 2014
1 hour 45 minutes
Additional Materials: Answer Booklet/Paper
*1603328313*

Graph Paper
List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

Answer all the questions.

This document consists of 4 printed pages.

1
y

2, 5

y = x2 + 1

0, 1

x
O

The diagram shows part of the curve y = x2 + 1. Find the volume obtained when the shaded region is
rotated through 360 about the y-axis. [4]

2
P B
Q
5 cm

A O
12 cm

The diagram shows a triangle AOB in which OA is 12 cm, OB is 5 cm and angle AOB is a right angle.
Point P lies on AB and OP is an arc of a circle with centre A. Point Q lies on AB and OQ is an arc of
a circle with centre B.

(i) Show that angle BAO is 0.3948 radians, correct to 4 decimal places. [1]

(ii) Calculate the area of the shaded region. [5]

3 (i) Find the first 3 terms, in ascending powers of x, in the expansion of 1 + x5 . [2]
5
The coefficient of x2 in the expansion of 1 + px + x2 is 95.

(ii) Use the answer to part (i) to find the value of the positive constant p. [3]

12
4 A curve has equation y = .
3 − 2x
dy
(i) Find . [2]
dx
A point moves along this curve. As the point passes through A, the x-coordinate is increasing at a
rate of 0.15 units per second and the y-coordinate is increasing at a rate of 0.4 units per second.

(ii) Find the possible x-coordinates of A. [4]

5 (i) Show that the equation 1 + sin x tan x = 5 cos x can be expressed as

6 cos2 x − cos x − 1 = 0. 3

(ii) Hence solve the equation 1 + sin x tan x = 5 cos x for 0 ≤ x ≤ 180. [3]

6 The equation of a curve is y = x3 + ax2 + bx, where a and b are constants.

(i) In the case where the curve has no stationary point, show that a2 < 3b. [3]

(ii) In the case where a = −6 and b = 9, find the set of values of x for which y is a decreasing function
of x. [3]

7
X

M
10
C
B
k

j D
8
O
i
A

The diagram shows a pyramid OABCX . The horizontal square base OABC has side 8 units and the
centre of the base is D. The top of the pyramid, X , is vertically above D and XD = 10 units. The
−−→ −−→
mid-point of OX is M . The unit vectors i and j are parallel to OA and OC respectively and the unit
vector k is vertically upwards.
−−→ −−→
(i) Express the vectors AM and AC in terms of i, j and k. [3]

(ii) Use a scalar product to find angle MAC. [4]

8 (a) The sum, Sn , of the first n terms of an arithmetic progression is given by Sn = 32n − n2 . Find the
first term and the common difference. [3]

(b) A geometric progression in which all the terms are positive has sum to infinity 20. The sum of
the first two terms is 12.8. Find the first term of the progression. [5]

[Questions 9, 10 and 11 are printed on the next page.]

9
y

A
2, 6

C 8, 3

x
O

B 5, −3

The diagram shows a trapezium ABCD in which AB is parallel to DC and angle BAD is 90. The
coordinates of A, B and C are 2, 6, 5, −3 and 8, 3 respectively.

(i) Find the equation of AD. [3]

(ii) Find, by calculation, the coordinates of D. [3]

The point E is such that ABCE is a parallelogram.

(iii) Find the length of BE. [2]

d2 y 24
10 A curve is such that = 3 − 4. The curve has a stationary point at P where x = 2.
dx2 x
(i) State, with a reason, the nature of this stationary point. [1]

dy
(ii) Find an expression for . [4]
dx
(iii) Given that the curve passes through the point 1, 13, find the coordinates of the stationary
point P. [4]

11 The function f : x → 6 − 4 cos 12 x is defined for 0 ≤ x ≤ 2.

(i) Find the exact value of x for which fx = 4. [3]

(ii) State the range of f. [2]

(iii) Sketch the graph of y = fx. [2]

(iv) Find an expression for f −1 x. [3]

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Cambridge International AS & A Level

You must answer on the question paper.

You will need: List of formulae (MF19)

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

© UCLES 2022 9709/12/F/M/22

Find f x. [4]

© UCLES 2022 9709/12/F/M/22 [Turn over

© UCLES 2022 9709/12/F/M/22

3 Find the term independent of x in each of the following expansions.

© UCLES 2022 9709/12/F/M/22 [Turn over

© UCLES 2022 9709/12/F/M/22

5 (a) Express 2x2 − 8x + 14 in the form 2 x − a2 + b. [2]

The functions f and g are defined by

© UCLES 2022 9709/12/F/M/22 [Turn over

(a) Find, by calculation, the coordinates of A and B. [4]

© UCLES 2022 9709/12/F/M/22

© UCLES 2022 9709/12/F/M/22 [Turn over

sin 1 + 2 cos 1 sin 1 − 2 cos 1 4

© UCLES 2022 9709/12/F/M/22

sin 1 + 2 cos 1 sin 1 − 2 cos 1

© UCLES 2022 9709/12/F/M/22 [Turn over

(a) Find, by calculation, the coordinates of A and B. [3]

© UCLES 2022 9709/12/F/M/22

© UCLES 2022 9709/12/F/M/22 [Turn over

9 Functions f, g and h are defined as follows:

© UCLES 2022 9709/12/F/M/22

(b) Given that f x  gh x, find the values of m and n. [4]

© UCLES 2022 9709/12/F/M/22 [Turn over

(a) Find the perimeter of the shaded region. [5]

© UCLES 2022 9709/12/F/M/22

(b) Find the area of the shaded region. [3]

© UCLES 2022 9709/12/F/M/22 [Turn over

11 It is given that a curve has equation y = k 3x − k−1 + 3x, where k is a constant.

© UCLES 2022 9709/12/F/M/22

The function f has a stationary value at x = a and is defined by

(c) The function g is defined by g x = − 3x + 1−1 + 3x for x ≥ 0.

Determine, making your reasoning clear, whether g is an increasing function, a decreasing

© UCLES 2022 9709/12/F/M/22

© UCLES 2022 9709/12/F/M/22

You must answer on the question paper.

You will need: List of formulae (MF19)

This document has 20 pages.

Find the value of the positive constant a. [4]

© UCLES 2022 9709/12/M/J/22

Find the sum to infinity. [4]

© UCLES 2022 9709/12/M/J/22 [Turn over

Find the equation of the curve. [4]

© UCLES 2022 9709/12/M/J/22

(a) Find the value of the constant k. [2]

© UCLES 2022 9709/12/M/J/22 [Turn over

© UCLES 2022 9709/12/M/J/22

© UCLES 2022 9709/12/M/J/22 [Turn over

© UCLES 2022 9709/12/M/J/22

© UCLES 2022 9709/12/M/J/22 [Turn over

(a) Find the angle BOC. [2]

© UCLES 2022 9709/12/M/J/22

© UCLES 2022 9709/12/M/J/22 [Turn over

8 The equation of a circle is x2 + y2 + ax + by − 12 = 0. The points A 1, 1 and B 2, −6 lie on the

© UCLES 2022 9709/12/M/J/22

© UCLES 2022 9709/12/M/J/22 [Turn over

© UCLES 2022 9709/12/M/J/22

© UCLES 2022 9709/12/M/J/22 [Turn over

10 Functions f and g are defined as follows:

The diagram shows part of the graph of y = f x.

State the domain of f −1 . [1]

(b) Find an expression for f −1 x. [3]

(c) Find gf −1 3. [2]

© UCLES 2022 9709/12/M/J/22

(d) Explain why g−1 x cannot be found. [1]

© UCLES 2022 9709/12/M/J/22 [Turn over

11 The function f is given by f x = 4 cos4 x + cos2 x − k for 0 ≤ x ≤ 2π, where k is a constant.

© UCLES 2022 9709/12/M/J/22

© UCLES 2022 9709/12/M/J/22

© UCLES 2022 9709/12/M/J/22

You must answer on the question paper.

You will need: List of formulae (MF19)

(b) Given that f x gh x, find the values of m and n. [4]