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com

Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*6552648013*

MATHEMATICS 9709/13
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)

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 11_9709_13/2R
© UCLES 2022 [Turn over
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2

1 Solve the equation 8 sin2 1 + 6 cos 1 + 1 = 0 for 0Å < 1 < 180Å. [3]

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

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3

2 The function f is defined by f x = −2x2 − 8x − 13 for x < −3.

(a) Express f x in the form −2 x + a2 + b, where a and b are integers. [2]

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

(b) Find the range of f. [1]

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

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

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

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4

3 (a) Find the first three terms in ascending powers of x of the expansion of 1 + 2x5 . [2]

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

(b) Find the first three terms in ascending powers of x of the expansion of 1 − 3x4 . [2]

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

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5

4 A large industrial water tank is such that, when the depth of the water in the tank is x metres, the
volume V m3 of water in the tank is given by V = 243 − 13 9 − x3 . Water is being pumped into the
tank at a constant rate of 3.6 m3 per hour.

Find the rate of increase of the depth of the water when the depth is 4 m, giving your answer in
cm per minute. [5]

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

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6

5
y
8, 12
12

x
O 2 4 6 8 10 12

The diagram shows a curve which has a maximum point at 8, 12 and a minimum point at 8, 0. The
curve is the result of applying
@ a Acombination of two transformations to a circle. The first transformation
7
applied is a translation of . The second transformation applied is a stretch in the y-direction.
−3

(a) State the scale factor of the stretch. [1]

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

(b) State the radius of the original circle. [1]

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

(c) State the coordinates of the centre of the circle after the translation has been completed but before
the stretch is applied. [2]

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

(d) State the coordinates of the centre of the original circle. [2]

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

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7
8
6 It is given that ! = cos−1 17 .

1 1
Find, without using the trigonometric functions on your calculator, the exact value of + .
sin ! tan !
[5]

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

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8

−3
7 The curve y = f x is such that f ′ x = .
x + 24

(a) The tangent at a point on the curve where x = a has gradient − 16

27
.

Find the possible values of a. [4]

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

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9

(b) Find f x given that the curve passes through the point −1, 5. [3]

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

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10

r
5
3r
P Q

The diagram shows two identical circles intersecting at points A and B and with centres at P and Q.
The radius of each circle is r and the distance PQ is 53 r.

(a) Find the perimeter of the shaded region in terms of r . [4]

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

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11

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

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

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12

9 The first term of a geometric progression is 216 and the fourth term is 64.

(a) Find the sum to infinity of the progression. [3]

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

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13

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

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

(b) Find the sum of the first 21 terms of the arithmetic progression. [6]

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

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14

10
y

y = 2x − 1

x
O D

B x2 + y2 = 2

The diagram shows the circle x2 + y2 = 2 and the straight line y = 2x − 1 intersecting at the points A
and B. The point D on the x-axis is such that AD is perpendicular to the x-axis.

(a) Find the coordinates of A. [4]

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

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15

(b) Find the volume of revolution when the shaded region is rotated through 360Å about the x-axis.
π
Give your answer in the form b c − d , where a, b, c and d are integers. [4]
a

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

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16

11 The coordinates of points A, B and C are A 5, −2, B 10, 3 and C 2p, p, where p is a constant.

(a) Given that AC and BC are equal in length, find the value of the fraction p. [3]

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

(b) It is now given instead that AC is perpendicular to BC and that p is an integer.

(i) Find the value of p. [4]

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

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17

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

(ii) Find the equation of the circle which passes through A, B and C, giving your answer in the
form x2 + y2 + ax + by + c = 0, where a, b and c are constants. [4]

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

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18

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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19

BLANK PAGE

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20

BLANK PAGE

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 9709/13/O/N/22

www.dynamicpapers.com

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
www.dynamicpapers.com
2

BLANK PAGE

© UCLES 2022 9709/12/O/N/22

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3

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]

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

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4

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]

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

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5

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]

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

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6

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]

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

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7

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]

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

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8

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]

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

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9

(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

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10

sin 1 cos 1 tan2 1 + 1

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

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

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11

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]

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

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12

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]

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

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13

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

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

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14

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]

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

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15

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]

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

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16

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]

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

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17

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

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

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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]

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

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19

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

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]

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

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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 2022 9709/12/O/N/22

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Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*6859654903*

MATHEMATICS 9709/11
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.

JC22 11_9709_11/RP
© UCLES 2022 [Turn over
www.dynamicpapers.com
2

2
1 Solve the equation 3x + 2 = . [3]
x−1

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

© UCLES 2022 9709/11/O/N/22

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3

dy −4
2 The equation of a curve is such that = 12 12 x − 1 . It is given that the curve passes through the
dx
point P 6, 4.

(a) Find the equation of the tangent to the curve at P. [2]

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

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

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

© UCLES 2022 9709/11/O/N/22 [Turn over

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4
1
3 A curve has equation y = ax 2 − 2x, where x > 0 and a is a constant. The curve has a stationary point
at the point P, which has x-coordinate 9.

Find the y-coordinate of P. [5]

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

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5
@ A5
2
4 2
The coefficient of x in the expansion of 1 + x + 1 + px6 is 70.
p

Find the possible values of the constant p. [6]

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

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6

5
8 cm

A B

The diagram shows a sector OAB of a circle with centre O. The length of the arc AB is 8 cm. It is
given that the perimeter of the sector is 20 cm.

(a) Find the perimeter of the shaded segment. [4]

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

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7

(b) Find the area of the shaded segment. [2]

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

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8

6 (a) Show that the equation

1 1
+ =1
sin 1 + cos 1 sin 1 − cos 1
may be expressed in the form a sin2 1 + b sin 1 + c = 0, where a, b and c are constants to be
found. [3]

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

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9

1 1
(b) Hence solve the equation + = 1 for 0Å ≤ 1 ≤ 360Å. [3]
sin 1 + cos 1 sin 1 − cos 1

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

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10

7 A tool for putting fence posts into the ground is called a ‘post-rammer’. The distances in millimetres
that the post sinks into the ground on each impact of the post-rammer follow a geometric progression.
The first three impacts cause the post to sink into the ground by 50 mm, 40 mm and 32 mm respectively.

(a) Verify that the 9th impact is the first in which the post sinks less than 10 mm into the ground.
[3]

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

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11

(b) Find, to the nearest millimetre, the total depth of the post in the ground after 20 impacts. [2]

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

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12

3 p
8 The function f is defined by f x = 2 − for x > , where p is a constant.
4x − p 4

(a) Find f ′ x and hence determine whether f is an increasing function, a decreasing function or
neither. [3]

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

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13

p b
(b) Express f −1 x in the form − , where a, b, c and d are integers. [4]
a cx − d

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

(c) Hence state the value of p for which f −1 x f x. [1]

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

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14

9 Functions f and g are both defined for x ∈ > and are given by

f x = x2 − 4x + 9,
g x = 2x2 + 4x + 12.

(a) Express f x in the form x − a2 + b. [1]

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

(b) Express g x in the form 2 x + c2 + d . [2]

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

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15

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

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

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

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16

10
y

!Å

B 4, 5

1
y = 2x 2 + 1

A 0, 1
y = 12 x2 − x + 1

x
O

1
Curves with equations y = 2x 2 + 1 and y = 12 x2 − x + 1 intersect at A 0, 1 and B 4, 5, as shown in
the diagram.

(a) Find the area of the region between the two curves. [5]

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

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17

The acute angle between the two tangents at B is denoted by !Å, and the scales on the axes are the
same.

(b) Find !. [5]

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

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18

11
y

A 0, 10

B C

D x
O

x2 + y2 = 20

The diagram shows the circle with equation x2 + y2 = 20. Tangents touching the circle at points B and
C pass through the point A 0, 10.

(a) By letting the equation of a tangent be y = mx + 10, find the two possible values of m. [4]

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

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19

(b) Find the coordinates of B and C. [3]

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

The point D is where the circle crosses the positive x-axis.

(c) Find angle BDC in degrees. [3]

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

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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 2022 9709/11/O/N/22

www.dynamicpapers.com

Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*7036013722*

MATHEMATICS 9709/13
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. Any blank pages are indicated.

JC22 06_9709_13/RP
© UCLES 2022 [Turn over
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2

BLANK PAGE

© UCLES 2022 9709/13/M/J/22

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3
@ A4
1
1 The coefficient of x in the expansion of p + x
3
is 144.
p

Find the possible values of the constant p. [4]

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

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4

2
y

0 1
0 π 2π 3π 4π
−1

−2

−3
y = p sin q1 + r
−4

−5

The diagram shows part of the curve with equation y = p sin q1 + r, where p, q and r are constants.

(a) State the value of p. [1]

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

(b) State the value of q. [1]

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

(c) State the value of r. [1]

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

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5

3 An arithmetic progression has first term 4 and common difference d . The sum of the first n terms of
the progression is 5863.

11726
(a) Show that n − 1d = − 8. [1]
n

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

(b) Given that the nth term is 139, find the values of n and d , giving the value of d as a fraction. [4]

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

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6
@ A
−1
4 (a) The curve with equation y = x2 + 2x − 5 is translated by .
3

Find the equation of the translated curve, giving your answer in the form y = ax2 + bx + c. [3]

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

(b) The curve with equation y = x2 + 2x − 5 is transformed to a curve with equation y = 4x2 + 4x − 5.

Describe fully the single transformation that has been applied. [2]

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

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7
2
5 (a) Solve the equation 6 y + − 7 = 0. [4]
y

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

2
(b) Hence solve the equation 6 tan x + − 7 = 0 for 0Å ≤ x ≤ 360Å. [3]
tan x

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

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8

6 The function f is defined by f x = 2x2 − 16x + 23 for x < 3.

(a) Express f x in the form 2 x + a2 + b. [2]

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

(b) Find the range of f. [1]

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

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9

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

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

The function g is defined by g x = 2x + 4 for x < −1.

(d) Find and simplify an expression for fg x. [2]

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

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10

7
y

B 0, 2

O P
x

x − 22 + y + 42 = 20

The diagram shows the circle with equation x − 22 + y + 42 = 20 and with centre C. The point B
has coordinates 0, 2 and the line segment BC intersects the circle at P.

(a) Find the equation of BC. [2]

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

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11

(b) Hence find the coordinates of P, giving your answer in exact form. [5]

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

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12

8
y
1 1
y = x 2 + 4x− 2

A 1, 5
B 16, 5

x
O

−1
The diagram shows the curve with equation y = x 2 + 4x 2 . The line y = 5 intersects the curve at the
1

points A 1, 5 and B 16, 5.

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

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

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13

(b) Calculate the area of the shaded region. [4]

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

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14

9
D

1.8 rad
6 cm
6 cm

C A

The diagram shows triangle ABC with AB = BC = 6 cm and angle ABC = 1.8 radians. The arc CD is
part of a circle with centre A and ABD is a straight line.

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

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

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15

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

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

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16

10 The function f is defined by f x = 4x + 2−2 for x > − 12 .

∞
(a) Find Ó f x dx. [4]
1

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

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17

A point is moving along the curve y = f x in such a way that, as it passes through the point A, its
y-coordinate is decreasing at the rate of k units per second and its x-coordinate is increasing at the
rate of k units per second.

(b) Find the coordinates of A. [6]

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

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18

11 The point P lies on the line with equation y = mx + c, where m and c are positive constants. A curve
m
has equation y = − . There is a single point P on the curve such that the straight line is a tangent to
x
the curve at P.

(a) Find the coordinates of P, giving the y-coordinate in terms of m. [6]

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

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19

The normal to the curve at P intersects the curve again at the point Q.

(b) Find the coordinates of Q in terms of m. [4]

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

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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 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
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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]

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

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3

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

Find the sum to infinity. [4]

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

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4

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, 52 .

Find the equation of the curve. [4]

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

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5

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]

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

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6

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]

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

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7

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

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

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8

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]

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

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9

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

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10

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]

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

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11

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

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

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12

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]

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

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13

(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]

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

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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

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

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15

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

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

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16

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]

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

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17

(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]

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

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18

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]

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

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19

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

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

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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 International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*8494117371*

MATHEMATICS 9709/11
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_11/3R
© UCLES 2022 [Turn over
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2

1 (a) Express x2 − 8x + 11 in the form x + p2 + q where p and q are constants. [2]

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

(b) Hence find the exact solutions of the equation x2 − 8x + 11 = 1. [2]

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

© UCLES 2022 9709/11/M/J/22

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3

2 The thirteenth term of an arithmetic progression is 12 and the sum of the first 30 terms is −15.

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

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

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4
P Q5
4 k2 2
3 The coefficient of x in the expansion of 2x + is a. The coefficient of x2 in the expansion of
x
4
2kx − 1 is b.

(a) Find a and b in terms of the constant k. [3]

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

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5

(b) Given that a + b = 216, find the possible values of k. [3]

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

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6

sin3 1 sin2 1
4 (a) Prove the identity − − tan2 1 1 + sin2 1. [4]
sin 1 − 1 1 + sin 1

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

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7

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

(b) Hence solve the equation

sin3 1 sin2 1
− = tan2 1 1 − sin2 1
sin 1 − 1 1 + sin 1
for 0 < 1 < 2π. [2]

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

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8

5
C
D

A 1

The diagram shows a sector ABC of a circle with centre A and radius r. The line BD is perpendicular
to AC. Angle CAB is 1 radians.

(a) Given that 1 = 16 π, find the exact area of BCD in terms of r. [3]

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

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9

3
(b) Given instead that the length of BD is r, find the exact perimeter of BCD in terms of r. [4]
2

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

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10

6 The function f is defined as follows:

x2 − 4
f x = for x > 2.
x2 + 4

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

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

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11

8 x2 − 4
(b) Show that 1 − can be expressed as and hence state the range of f. [4]
x2 + 4 x2 + 4

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

(c) Explain why the composite function ff cannot be formed. [1]

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

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12

7
y
y = 12 x + 1

B 1
y = 3x − 2 2

x
O

1
The diagram shows the curve with equation y = 3x − 2 2 and the line y = 12 x + 1. The curve and the
line intersect at points A and B.

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

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

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13

(b) Hence find the area of the region enclosed between the curve and the line. [5]

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

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14

8 (a) The curve y = sin x is transformed to the curve y = 4 sin 12 x − 30Å .

Describe fully a sequence of transformations that have been combined, making clear the order
in which the transformations are applied. [5]

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

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15

(b) Find the exact solutions of the equation 4 sin 12 x − 30Å = 2 2 for 0Å ≤ x ≤ 360Å. [3]

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

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16

9 The equation of a circle is x2 + y2 + 6x − 2y − 26 = 0.

(a) Find the coordinates of the centre of the circle and the radius. Hence find the coordinates of the
lowest point on the circle. [4]

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

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17

(b) Find the set of values of the constant k for which the line with equation y = kx − 5 intersects the
circle at two distinct points. [6]

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

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18

d2 y 4
10 The equation of a curve is such that 2
= 6x2 − 3 . The curve has a stationary point at −1, 92 .
dx x

(a) Determine the nature of the stationary point at −1, 29 . [1]

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

(b) Find the equation of the curve. [5]

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

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19

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

(d) A point A is moving along the curve and the y-coordinate of A is increasing at a rate of 5 units
per second.

Find the rate of increase of the x-coordinate of A at the point where x = 1. [3]

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

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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 2022 9709/11/M/J/22

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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)

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
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2

BLANK PAGE

© UCLES 2022 9709/12/F/M/22

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3

− 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]

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

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4

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]

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

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5

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

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

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6

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]

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

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7

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]

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

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8

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]

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

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9

(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]

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

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10

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

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

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11

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

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

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12

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]

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

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13

(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]

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

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14

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 → x2 − 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]

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

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15

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

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

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16

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]

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

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17

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

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

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

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18

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]

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

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19

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]

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

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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 2022 9709/12/F/M/22

www.dynamicpapers.com

Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*5896421517*

MATHEMATICS 9709/13
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. Any blank pages are indicated.

JC21 11_9709_13/RP
© UCLES 2021 [Turn over
www.dynamicpapers.com
2

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

Describe fully, in the correct order, the two transformations that have been combined. [4]

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

© UCLES 2021 9709/13/O/N/21

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3

2 (a) Find the first three terms, in ascending powers of x, in the expansion of 1 + ax6 . [1]

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

(b) Given that the coefficient of x2 in the expansion of 1 − 3x 1 + ax6 is −3, find the possible values
of the constant a. [4]

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

© UCLES 2021 9709/13/O/N/21 [Turn over

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4

3 (a) Express 5y2 − 30y + 50 in the form 5 y + a2 + b, where a and b are constants. [2]

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

(b) The function f is defined by f x = x5 − 10x3 + 50x for x ∈ >.

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

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

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5

4 The first term of an arithmetic progression is 84 and the common difference is −3.

(a) Find the smallest value of n for which the nth term is negative. [2]

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

It is given that the sum of the first 2k terms of this progression is equal to the sum of the first k terms.

(b) Find the value of k. [3]

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

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6

5
B

9 cm

11 cm
X C

In the diagram, X and Y are points on the line AB such that BX = 9 cm and AY = 11 cm. Arc BC is
part of a circle with centre X and radius 9 cm, where CX is perpendicular to AB. Arc AC is part of a
circle with centre Y and radius 11 cm.

(a) Show that angle XYC = 0.9582 radians, correct to 4 significant figures. [1]

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

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7

(b) Find the perimeter of ABC. [6]

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

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8

6
y

x
O

y = f x

The diagram shows the graph of y = f x.

(a) On this diagram sketch the graph of y = f −1 x. [1]

x
It is now given that f x = − where −2 < x < 2.
4 − x2

(b) Find an expression for f −1 x. [4]

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

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9

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

The function g is defined by g x = 2x for −a < x < a, where a is a constant.

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

(d) Assuming that fg can be formed, find and simplify an expression for fg x. [2]

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

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10
tan x + cos x
7 (a) Show that the equation = k, where k is a constant, can be expressed as
tan x − cos x
k + 1 sin2 x + k − 1 sin x − k + 1 = 0. [4]

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

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11
tan x + cos x
(b) Hence solve the equation = 4 for 0Å ≤ x ≤ 360Å. [4]
tan x − cos x

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

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12

8
y

1
A 4, 2

5 1
y= 2 − x2

1 B 4, 12
y = x− 2
x
O

− 12 1
The diagram shows the curves with equations y = x and y = 52 − x 2 . The curves intersect at the points

A 14 , 2 and B 4, 12 .

(a) Find the area of the region between the two curves. [6]

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

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13

− 12
(b) The normal to the curve y = x at the point 1, 1 intersects the y-axis at the point 0, p.

Find the value of p. [4]

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

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14

9 The line y = 2x + 5 intersects the circle with equation x2 + y2 = 20 at A and B.

(a) Find the coordinates of A and B in surd form and hence find the exact length of the chord AB.
[7]

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

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15

A straight line through the point 10, 0 with gradient m is a tangent to the circle.

(b) Find the two possible values of m. [5]

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

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16

10 A curve has equation y = f x and it is given that

−2
f ′ x = 12 x + k − 1 + k−2 ,

where k is a constant. The curve has a minimum point at x = 2.

(a) Find f ′′ x in terms of k and x, and hence find the set of possible values of k. [3]

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

It is now given that k = −3 and the minimum point is at 2, 3 21 .

(b) Find f x. [4]

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

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

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18

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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.

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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
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2

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

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

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3

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]

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

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4

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]

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

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5

dy 8
4 A curve is such that = 2
. The curve passes through the point 2, 5 32 .
dx 3x + 2

Find the equation of the curve. [4]

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

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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 12 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]

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

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7

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 12 .

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

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

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8

15 cm 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]

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

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9

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

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

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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]

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

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11

(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]

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

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12

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]

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

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13

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

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

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14

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]

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

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15

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

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

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16

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, 56 .

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

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

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17

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

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

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18

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]

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

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19

(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]

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

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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 International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*2554281371*

MATHEMATICS 9709/11
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. Any blank pages are indicated.

JC21 11_9709_11/RP
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2

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@ A2
(a) Expand 1 −
1
1 . [1]
2x

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

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

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

@ A
(c) Hence find the coefficient of x in the expansion of 1 − 1 + 2x6 .
1 2
[2]
2x

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

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4

2 A curve has equation y = kx2 + 2x − k and a line has equation y = kx − 2, where k is a constant.

Find the set of values of k for which the curve and line do not intersect. [5]

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

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5

3 Solve, by factorising, the equation

6 cos 1 tan 1 − 3 cos 1 + 4 tan 1 − 2 = 0,
for 0Å ≤ 1 ≤ 180Å. [4]

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

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6

4 The first term of an arithmetic progression is a and the common difference is −4. The first term
of a geometric progression is 5a and the common ratio is − 14 . The sum to infinity of the geometric
progression is equal to the sum of the first eight terms of the arithmetic progression.

(a) Find the value of a. [4]

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

The kth term of the arithmetic progression is zero.

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

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

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7

5
y

8 y = a cos bx + c

x
0 π 2π
−2

−4

The diagram shows part of the graph of y = a cos bx + c.

(a) Find the values of the positive integers a, b and c. [3]

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

(b) For these values of a, b and c, use the given diagram to determine the number of solutions in the
interval 0 ≤ x ≤ 2π for each of the following equations.

(i) a cos bx + c =

6
π
x [1]

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

(ii) a cos bx + c = 6 −

6
π
x [1]

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

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8

6
C

A 6 cm B

The diagram shows a metal plate ABC in which the sides are the straight line AB and the arcs AC
and BC. The line AB has length 6 cm. The arc AC is part of a circle with centre B and radius 6 cm,
and the arc BC is part of a circle with centre A and radius 6 cm.

(a) Find the perimeter of the plate, giving your answer in terms of π. [3]

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

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9

(b) Find the area of the plate, giving your answer in terms of π and 3.

[4]

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

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10

7 A circle with centre 5, 2 passes through the point 7, 5.

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

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

The line y = 5x − 10 intersects the circle at A and B.

(b) Find the exact length of the chord AB. [7]

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

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11

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

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12

8 (a) Express −3x2 + 12x + 2 in the form −3 x − a2 + b, where a and b are constants. [2]

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

The one-one function f is defined by f : x → −3x2 + 12x + 2 for x ≤ k.

(b) State the largest possible value of the constant k. [1]

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

It is now given that k = −1.

(c) State the range of f. [1]

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

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13

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

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

−3
@ A
The result of translating the graph of y = f x by is the graph of y = g x.
1

(e) Express g x in the form px2 + qx + r , where p, q and r are constants. [3]

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

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14

A curve has equation y = f x, and it is given that f ′ x = 2x2 − 7 −

4
9 .
x2

(a) Given that f 1 = − 13 , find f x. [4]

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

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15

(b) Find the coordinates of the stationary points on the curve. [5]

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

(c) Find f ′′ x. [1]

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

(d) Hence, or otherwise, determine the nature of each of the stationary points. [2]

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

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16
∞
(a) Find Ô
1
10 dx. [4]
3x − 2 2
3
1

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

y
1
y= 3
3x − 2 2

x
O 1 2

The diagram shows the curve with equation y =

1
. The shaded region is bounded by the
3x − 2 2
3

curve, the x-axis and the lines x = 1 and x = 2. The shaded region is rotated through 360Å about the
x-axis.

(b) Find the volume of revolution. [4]

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

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17

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

The normal to the curve at the point 1, 1 crosses the y-axis at the point A.

(c) Find the y-coordinate of A. [4]

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

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18

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must be clearly shown.

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19

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Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*6412245051*

MATHEMATICS 9709/13
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_13/RP
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2

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3

8
1 A curve with equation y = f x is such that f ′ x = 6x2 − . It is given that the curve passes through
x2
the point 2, 7.

Find f x. [3]

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

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4
3
2 The function f is defined by f x = 13 2x − 1 2 − 2x for 21 < x < a. It is given that f is a decreasing
function.

Find the maximum possible value of the constant a. [4]

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

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5

3 A line with equation y = mx − 6 is a tangent to the curve with equation y = x2 − 4x + 3.

Find the possible values of the constant m, and the corresponding coordinates of the points at which
the line touches the curve. [6]

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

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6

4 (a) Show that the equation

tan x + sin x
= k,
tan x − sin x
where k is a constant, may be expressed as
1 + cos x
= k. [2]
1 − cos x

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

(b) Hence express cos x in terms of k. [2]

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

tan x + sin x
(c) Hence solve the equation = 4 for −π < x < π. [2]
tan x − sin x

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

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7

5
A

D
4 cm

B C

The diagram shows a triangle ABC, in which angle ABC = 90Å and AB = 4 cm. The sector ABD is
part of a circle with centre A. The area of the sector is 10 cm2.

(a) Find angle BAD in radians. [2]

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

(b) Find the perimeter of the shaded region. [4]

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

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8

6 Functions f and g are both defined for x ∈ > and are given by

f x = x2 − 2x + 5,
g x = x2 + 4x + 13.

(a) By first expressing each of f x and g x in completed square form, express g x in the form
f x + p + q, where p and q are constants. [4]

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

(b) Describe fully the transformation which transforms the graph of y = f x to the graph of y = g x.
[2]

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

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9

7 (a) Write down the first four terms of the expansion, in ascending powers of x, of a − x6 . [2]

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

@ A
2 2
(b) Given that the coefficient of x in the expansion of 1 + a − x6 is −20, find in exact form
ax
the possible values of the constant a. [5]

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

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10

8 Functions f and g are defined as follows:

f : x → x2 − 1 for x < 0,
1
g:x→ for x < − 12 .
2x + 1

(a) Solve the equation fg x = 3. [4]

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

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11

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

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

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12

9 (a) A geometric progression is such that the second term is equal to 24% of the sum to infinity.

Find the possible values of the common ratio. [3]

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

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13

(b) An arithmetic progression P has first term a and common difference d . An arithmetic progression
Q has first term 2 a + 1 and common difference d + 1. It is given that
5th term of P 1 Sum of first 5 terms of P 2
= and = .
12th term of Q 3 Sum of first 5 terms of Q 3
Find the value of a and the value of d . [6]

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

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14

10 Points A −2, 3, B 3, 0 and C 6, 5 lie on the circumference of a circle with centre D.

(a) Show that angle ABC = 90Å. [2]

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

(b) Hence state the coordinates of D. [1]

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

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

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

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15

The point E lies on the circumference of the circle such that BE is a diameter.

(d) Find an equation of the tangent to the circle at E. [5]

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

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16

11
y

1 1
y = x 2 + k2 x− 2

x
O 9 2
4k2
4k

1 −1
The diagram shows part of the curve with equation y = x 2 + k2 x 2 , where k is a positive constant.

(a) Find the coordinates of the minimum point of the curve, giving your answer in terms of k. [4]

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

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17

The tangent at the point on the curve where x = 4k2 intersects the y-axis at P.

(b) Find the y-coordinate of P in terms of k. [4]

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

The shaded region is bounded by the curve, the x-axis and the lines x = 94 k2 and x = 4k2 .

(c) Find the area of the shaded region in terms of k. [3]

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

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18

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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19

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20

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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
www.dynamicpapers.com
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]

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

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3

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 31 in the
x-direction.

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

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

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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]

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

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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]

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

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6

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]

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

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7

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]

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

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8

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]

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

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9

8 The first, second and third terms of an arithmetic progression are a, 23 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]

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

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10

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]

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

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11

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

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12

1 + sin x 1 − sin x 4 tan x

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

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

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13

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

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

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14

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]

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

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15

d2 y
(c) Find . [2]
dx2

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

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

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

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16

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]

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

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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]

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

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18

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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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19

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Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*7942312992*

MATHEMATICS 9709/11
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_11/RP
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2

dy 3
1 The equation of a curve is such that = 4 + 32x3 . It is given that the curve passes through the point
1 dx x
2
, 4 .

Find the equation of the curve. [4]

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

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3

2 The sum of the first 20 terms of an arithmetic progression is 405 and the sum of the first 40 terms
is 1410.

Find the 60th term of the progression. [5]

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

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4

3 (a) Find the first three terms in the expansion of 3 − 2x5 in ascending powers of x. [3]

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

(b) Hence find the coefficient of x2 in the expansion of 4 + x2 3 − 2x5 . [3]

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

© UCLES 2021 9709/11/M/J/21

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5

4
y

x
− 14 π 0 1
4π
1
2π
3
4π
π 5
4π
3
2π
7
4π
−2

−4

−6

The diagram shows part of the graph of y = a tan x − b + c.

Given that 0 < b < π, state the values of the constants a, b and c. [3]

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

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6

5 The fifth, sixth and seventh terms of a geometric progression are 8k, −12 and 2k respectively.

Given that k is negative, find the sum to infinity of the progression. [4]

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

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7

6 The equation of a curve is y = 2k − 3x2 − kx − k − 2, where k is a constant. The line y = 3x − 4 is a

tangent to the curve.

Find the value of k. [5]

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

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8

1 − 2 sin2 1
7 (a) Prove the identity 1 − tan2 1. [2]
1 − sin2 1

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

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9

1 − 2 sin2 1
(b) Hence solve the equation = 2 tan4 1 for 0Å ≤ 1 ≤ 180Å. [3]
1 − sin2 1

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

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10

8
P Q

S R

The diagram shows a symmetrical metal plate. The plate is made by removing two identical pieces
from a circular disc with centre C. The boundary of the plate consists of two arcs PS and QR of the
original circle and two semicircles with PQ and RS as diameters. The radius of the circle with centre
C is 4 cm, and PQ = RS = 4 cm also.

(a) Show that angle PCS = 23 π radians. [2]

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

(b) Find the exact perimeter of the plate. [3]

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

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11

(c) Show that the area of the plate is 20
3
π + 8 3 cm2. [5]

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

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12

9 Functions f and g are defined as follows:

f x = x − 22 − 4 for x ≥ 2,
g x = ax + 2 for x ∈ >,

where a is a constant.

(a) State the range of f. [1]

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

(b) Find f −1 x. [2]

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

(c) Given that a = − 53 , solve the equation f x = g x. [3]

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

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13

(d) Given instead that ggf −1 12 = 62, find the possible values of a. [5]

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

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14

10 The equation of a circle is x2 + y2 − 4x + 6y − 77 = 0.

(a) Find the x-coordinates of the points A and B where the circle intersects the x-axis. [2]

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

(b) Find the point of intersection of the tangents to the circle at A and B. [6]

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

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15

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

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16

11 The equation of a curve is y = 2 3x + 4 − x.

(a) Find the equation of the normal to the curve at the point 4, 4, giving your answer in the form
y = mx + c. [5]

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

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

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

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17

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

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

(d) Find the exact area of the region bounded by the curve, the x-axis and the lines x = 0 and x = 4.
[4]

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

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18

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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19

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20

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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
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2

BLANK PAGE

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3

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]

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

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4

2 By using a suitable substitution, solve the equation

4
2x − 32 − − 3 = 0. [4]
2x − 32

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

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5

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

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

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6

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

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7

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]

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

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8

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]

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

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9

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

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

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10

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

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11

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

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

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

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

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12

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

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13

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

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

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

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14

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]

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

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15

(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]

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

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16

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]

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

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17

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]

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

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18

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]

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

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19

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

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

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

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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 International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*8677467398*

MATHEMATICS 9709/13
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_13/RP
© UCLES 2020 [Turn over
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2

1 (a) Express x2 + 6x + 5 in the form x + a2 + b, where a and b are constants. [2]

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

(b) The curve with equation y = x2 is transformed to the curve with equation y = x2 + 6x + 5.

Describe fully the transformation(s) involved. [2]

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

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3

The function f is defined by f x = for x > −2.

2
x + 22
2

∞
(a) Find Ó f x dx. [3]
1

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

= f x. It is given that the point −1, −1 lies on the

dy
(b) The equation of a curve is such that
dx
curve.

Find the equation of the curve. [2]

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

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4

Solve the equation 3 tan2 1 + 1 = for 0Å < 1 < 180Å.

2
tan2 1
3 [5]

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

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5

4 A curve has equation y = 3x2 − 4x + 4 and a straight line has equation y = mx + m − 1, where m is a
constant.

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

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

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6

5 In the expansion of a + bx7 , where a and b are non-zero constants, the coefficients of x, x2 and x4
are the first, second and third terms respectively of a geometric progression.

a
Find the value of . [5]
b

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

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7

The function f is defined by f x = for x > 13 .

2x
3x − 1
6

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

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

+
2 2 2x
3 3 3x − 1 3x − 1
(b) Show that can be expressed as . [2]

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

(c) State the range of f. [1]

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

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8

tan2 1
−
1
cos2 1 cos2 1
7 The first and second terms of an arithmetic progression are and , respectively, where
0 < 1 < 12 π.

(a) Show that the common difference is −

1
cos4 1
. [4]

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

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9

(b) Find the exact value of the 13th term when 1 = 16 π. [3]

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

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10

The equation of a curve is y = 2x + 1 + for x > − 12 .

1
2x + 1
8

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

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

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11

(b) Find the coordinates of the stationary point and determine the nature of the stationary point. [5]

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

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12

9
B

12 cm
8 cm

A 8 cm O C

In the diagram, arc AB is part of a circle with centre O and radius 8 cm. Arc BC is part of a circle
with centre A and radius 12 cm, where AOC is a straight line.

(a) Find angle BAO in radians. [2]

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

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13

(b) Find the area of the shaded region. [4]

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

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

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

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14

−1
A curve has equation y = x + x 2 + 2 where x > 0 and k is a positive constant.
1 12 1
10
k k

(a) It is given that when x = 14 , the gradient of the curve is 3.

Find the value of k. [4]

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

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15

k2 @ A
−1
(b) It is given instead that Ô x + x 2 + 2 dx = 13
1 12 1
12
.
1 k2 k k
4

Find the value of k. [5]

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

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16

11 A circle with centre C has equation x − 82 + y − 42 = 100.

(a) Show that the point T −6, 6 is outside the circle. [3]

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

Two tangents from T to the circle are drawn.

(b) Show that the angle between one of the tangents and CT is exactly 45Å. [2]

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

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17

The two tangents touch the circle at A and B.

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

(d) Find the x-coordinates of A and B. [3]

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

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18

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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19

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20

BLANK PAGE

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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
www.dynamicpapers.com
2

BLANK PAGE

© UCLES 2020 9709/12/O/N/20

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3

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

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4

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

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5

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]

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

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6

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

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7

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

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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

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

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9

− 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

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10

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]

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

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11

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

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

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12

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]

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

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13

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

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

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

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14

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

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

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15

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

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

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16

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]

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

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17

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]

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

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18

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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19

BLANK PAGE

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20

BLANK PAGE

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Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*2028675937*

MATHEMATICS 9709/11
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_11/RP
© UCLES 2020 [Turn over
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2

1 Find the set of values of m for which the line with equation y = mx − 3 and the curve with equation
y = 2x2 + 5 do not meet. [3]

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

© UCLES 2020 9709/11/O/N/20

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3

dy 1
2 The equation of a curve is such that = + x. It is given that the curve passes through the
dx x − 32
point 2, 7.

Find the equation of the curve. [4]

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

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4

3 Air is being pumped into a balloon in the shape of a sphere so that its volume is increasing at a constant
rate of 50 cm3 s−1 .

Find the rate at which the radius of the balloon is increasing when the radius is 10 cm. [3]

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

© UCLES 2020 9709/11/O/N/20

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5

4
y

1 y = cos 1
1
0 π 2π 3π 4π
−1

In the diagram, the lower curve has equation y = cos 1. The upper curve shows the result of applying
a combination of transformations to y = cos 1.

Find, in terms of a cosine function, the equation of the upper curve. [3]

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

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6
0 a 16
5 In the expansion of 2x2 + , the coefficients of x6 and x3 are equal.
x
(a) Find the value of the non-zero constant a. [4]

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

0 a 16
(b) Find the coefficient of x6 in the expansion of 1 − x3 2x2 + . [1]
x

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

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7

6 The equation of a curve is y = 2 + 25 − x2 .

Find the coordinates of the point on the curve at which the gradient is 34 . [5]

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

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8

sin 1 sin 1
7 (a) Show that − 2 tan2 1. [3]
1 − sin 1 1 + sin 1

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

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9

sin 1 sin 1
(b) Hence solve the equation − = 8, for 0Å < 1 < 180Å. [3]
1 − sin 1 1 + sin 1

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

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10

8 A geometric progression has first term a, common ratio r and sum to infinity S. A second geometric
progression has first term a, common ratio R and sum to infinity 2S.

(a) Show that r = 2R − 1. [3]

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

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11

It is now given that the 3rd term of the first progression is equal to the 2nd term of the second
progression.

(b) Express S in terms of a. [4]

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

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12

The diagram shows a circle with centre A passing through the point B. A second circle has centre B
and passes through A. The tangent at B to the first circle intersects the second circle at C and D.

The coordinates of A are −1, 4 and the coordinates of B are 3, 2.

(a) Find the equation of the tangent CBD. [2]

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

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13

(b) Find an equation of the circle with centre B. [3]

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

(c) Find, by calculation, the x-coordinates of C and D. [3]

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

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14

10
C

1 rad

F E
r
O

A B
D

The diagram shows a sector CAB which is part of a circle with centre C. A circle with centre O and
radius r lies within the sector and touches it at D, E and F, where COD is a straight line and angle
ACD is 1 radians.

(a) Find CD in terms of r and sin 1. [3]

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

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15

It is now given that r = 4 and 1 = 16 π.

(b) Find the perimeter of sector CAB in terms of π. [3]

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

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16

11 The functions f and g are defined by

f x = x2 + 3 for x > 0,
g x = 2x + 1 for x > − 12 .

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

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

(b) Find an expression for fg−1 x and state the domain of fg−1 . [4]

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

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17

(c) Solve the equation fg x − 3 = gf x. [4]

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

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18

12
y

A 4, 0
O x
1
y = 4x 2 − 2x
y=3−x

1
The diagram shows a curve with equation y = 4x 2 − 2x for x ≥ 0, and a straight line with equation
y = 3 − x. The curve crosses the x-axis at A 4, 0 and crosses the straight line at B and C.

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

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

(b) Show that B is a stationary point on the curve. [2]

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

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19

(c) Find the area of the shaded region. [6]

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

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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 9709/11/O/N/20

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Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*1360524731*

MATHEMATICS 9709/13
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 20 pages. Blank pages are indicated.

JC20 06_9709_13/FP
© UCLES 2020 [Turn over
www.dynamicpapers.com
2

1 Find the set of values of m for which the line with equation y = mx + 1 and the curve with equation
y = 3x2 + 2x + 4 intersect at two distinct points. [4]

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

© UCLES 2020 9709/13/M/J/20

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3

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

Find the equation of the curve. [4]

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

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4

3 In each of parts (a), (b) and (c), the graph shown with solid lines has equation y = f x. The graph
shown with broken lines is a transformation of y = f x.
(a)
y

2
y = f x
1

x
−3 −2 −1 0 1 2 3

State, in terms of f, the equation of the graph shown with broken lines. [1]

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

(b)
y

2
y = f x
1

x
0 1 2 3

State, in terms of f, the equation of the graph shown with broken lines. [1]

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

2
y = f x
1

x
−3 −2 −1 0 1 2 3
−1

−2

State, in terms of f, the equation of the graph shown with broken lines. [2]

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

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5

4 (a) Expand 1 + a5 in ascending powers of a up to and including the term in a3 . [1]

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

(b) Hence expand 1 + x + x2 5 in ascending powers of x up to and including the term in x3 ,
simplifying your answer. [3]

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

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6

O
5
A B

The diagram shows a cord going around a pulley and a pin. The pulley is modelled as a circle with
centre O and radius 5 cm. The thickness of the cord and the size of the pin P can be neglected. The
pin is situated 13 cm vertically below O. Points A and B are on the circumference of the circle such
that AP and BP are tangents to the circle. The cord passes over the major arc AB of the circle and
under the pin such that the cord is taut.

Calculate the length of the cord. [6]

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

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7

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

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8

6 A point P is moving along a curve in such a way that the x-coordinate of P is increasing at a constant
1
rate of 2 units per minute. The equation of the curve is y = 5x − 1 2 .

(a) Find the rate at which the y-coordinate is increasing when x = 1. [4]

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

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9

(b) Find the value of x when the y-coordinate is increasing at 58 units per minute. [3]

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

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10

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

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

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11

tan 1 tan 1 6
(b) Hence solve the equation + = for 0Å < 1 < 180Å. [4]
1 + cos 1 1 − cos 1 tan 1

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12

8 The first term of a progression is sin2 1, where 0 < 1 < 12 π. The second term of the progression is
sin2 1 cos2 1.

(a) Given that the progression is geometric, find the sum to infinity. [3]

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

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13

It is now given instead that the progression is arithmetic.

(b) (i) Find the common difference of the progression in terms of sin 1. [3]

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

(ii) Find the sum of the first 16 terms when 1 = 13 π. [3]

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

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9 The functions f and g are defined by

f x = x2 − 4x + 3 for x > c, where c is a constant,

1
g x = for x > −1.
x+1

(a) Express f x in the form x − a2 + b. [2]

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

It is given that f is a one-one function.

(b) State the smallest possible value of c. [1]

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

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15

It is now given that c = 5.

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

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

(d) Find an expression for gf x and state the range of gf. [3]

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

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10 (a) The coordinates of two points A and B are −7, 3 and 5, 11 respectively.

Show that the equation of the perpendicular bisector of AB is 3x + 2y = 11. [4]

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

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17

(b) A circle passes through A and B and its centre lies on the line 12x − 5y = 70.

Find an equation of the circle. [5]

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

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18

11
y

A
y = x3 − 2bx2 + b2 x

x
O a b

The diagram shows part of the curve with equation y = x3 − 2bx2 + b2 x and the line OA, where A is
the maximum point on the curve. The x-coordinate of A is a and the curve has a minimum point at
b, 0, where a and b are positive constants.

(a) Show that b = 3a. [4]

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

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19

(b) Show that the area of the shaded region between the line and the curve is ka4 , where k is a fraction
to be found. [7]

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

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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.

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

www.dynamicpapers.com

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
www.dynamicpapers.com
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

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

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3

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]

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

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4

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]

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

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5

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]

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

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6

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]

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

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7

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]

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

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8

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]

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

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9

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

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

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

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

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10

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]

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

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(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]

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

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12

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.

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

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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]

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

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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]

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

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15

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]

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

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16

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.

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

www.dynamicpapers.com

Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*2851798509*

MATHEMATICS 9709/11
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 20 pages. Blank pages are indicated.

1 The sum of the first nine terms of an arithmetic progression is 117. The sum of the next four terms
is 91.

Find the first term and the common difference of the progression. [4]

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

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3
@ A @ A
1 1 5 2 8
2 The coefficient of in the expansion of kx + + 1− is 74.
x x x

Find the value of the positive constant k. [5]

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

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4

3 Each year the selling price of a diamond necklace increases by 5% of the price the year before. The
selling price of the necklace in the year 2000 was $36 000.

(a) Write down an expression for the selling price of the necklace n years later and hence find the
selling price in 2008. [3]

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

(b) The company that makes the necklace only sells one each year. Find the total amount of money
obtained in the ten-year period starting in the year 2000. [2]

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

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5

4
y

y = f x

x
O π

The diagram shows the graph of y = f x, where f x = 32 cos 2x + 12 for 0 ≤ x ≤ π.

(a) State the range of f. [2]

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

A function g is such that g x = f x + k, where k is a positive constant. The x-axis is a tangent to the
curve y = g x.

(b) State the value of k and hence describe fully the transformation that maps the curve y = f x on
to y = g x. [2]

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

(c) State the equation of the curve which is the reflection of y = f x in the x-axis. Give your answer
in the form y = a cos 2x + b, where a and b are constants. [1]

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

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6

5 The equation of a line is y = mx + c, where m and c are constants, and the equation of a curve is
xy = 16.

(a) Given that the line is a tangent to the curve, express m in terms of c. [3]

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

(b) Given instead that m = −4, find the set of values of c for which the line intersects the curve at
two distinct points. [3]

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

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7

6 Functions f and g are defined for x ∈ > by

f : x → 12 x − a,
g : x → 3x + b,

where a and b are constants.

(a) Given that gg 2 = 10 and f −1 2 = 14, find the values of a and b. [4]

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

(b) Using these values of a and b, find an expression for gf x in the form cx + d , where c and d are
constants. [2]

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

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8

1 + sin 1 cos 1 2
7 (a) Prove the identity + . [3]
cos 1 1 + sin 1 cos 1

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

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9

1 + sin 1 cos 1 3
(b) Hence solve the equation + = , for 0 ≤ 1 ≤ 2π. [3]
cos 1 1 + sin 1 sin 1

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

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10

8
15 cm

A C
6 cm O X

In the diagram, ABC is a semicircle with diameter AC, centre O and radius 6 cm. The length of the
arc AB is 15 cm. The point X lies on AC and BX is perpendicular to AX .

Find the perimeter of the shaded region BXC. [6]

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

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11

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

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12

9 The equation of a curve is y = 3 − 2x3 + 24x.

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

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

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13

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

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

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

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

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14

10 The coordinates of the points A and B are −1, −2 and 7, 4 respectively.

(a) Find the equation of the circle, C, for which AB is a diameter. [4]

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

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15

(b) Find the equation of the tangent, T , to circle C at the point B. [4]

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

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16

11
y

2y + x = 8

C
B 8
y=
x+2
x
O

8
The diagram shows part of the curve y = and the line 2y + x = 8, intersecting at points A and B.
x+2
The point C lies on the curve and the tangent to the curve at C is parallel to AB.

(a) Find, by calculation, the coordinates of A, B and C. [6]

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

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17

(b) Find the volume generated when the shaded region, bounded by the curve and the line, is rotated
through 360Å about the x-axis. [6]

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

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18

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.

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

www.dynamicpapers.com
19

BLANK PAGE

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20

BLANK PAGE

www.dynamicpapers.com

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.

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]

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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]

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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]

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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]

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5 Solve the equation

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

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@ 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]

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

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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]

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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]

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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]

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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]

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

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

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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]

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

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

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

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

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(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]

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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]

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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]

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The two circles intersect at points R and S.

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

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Additional Page

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

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BLANK PAGE

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Cambridge Assessment International Education

Cambridge International Advanced Subsidiary and Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*3117579859*

NUMBER NUMBER

MATHEMATICS 9709/13
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

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.

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.

This document consists of 20 printed pages.

1 (i) Expand 1 + y6 in ascending powers of y as far as the term in y2 . [1]

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6
(ii) In the expansion of 1 + px − 2x2 the coefficient of x2 is 48. Find the value of the positive
constant p. [3]

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2 The function g is defined by g x = x2 − 6x + 7 for x > 4. By first completing the square, find an
expression for g−1 x and state the domain of g−1 . [5]

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3 The equation of a curve is y = x3 + x2 − 8x + 7. The curve has no stationary points in the interval
a < x < b. Find the least possible value of a and the greatest possible value of b. [4]

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4
C

A r O r B

The diagram shows a semicircle ACB with centre O and radius r. Arc OC is part of a circle with
centre A.

(i) Express angle CAO in radians in terms of 0. [1]

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

(ii) Find the area of the shaded region in terms of r, 0 and ï3, simplifying your answer. [4]

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x cm 4x cm

2x cm

The dimensions of a cuboid are x cm, 2x cm and 4x cm, as shown in the diagram.

(i) Show that the surface area S cm2 and the volume V cm3 are connected by the relation
2
S = 7V 3 . [3]

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(ii) When the volume of the cuboid is 1000 cm3 the surface area is increasing at 2 cm2 s−1 . Find the
rate of increase of the volume at this instant. [4]

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6 A line has equation y = 3kx − 2k and a curve has equation y = x2 − kx + 2, where k is a constant.

(i) Find the set of values of k for which the line and curve meet at two distinct points. [4]

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(ii) For each of two particular values of k, the line is a tangent to the curve. Show that these two
tangents meet on the x-axis. [3]

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7 (i) Show that the equation 3 cos4 1 + 4 sin2 1 − 3 = 0 can be expressed as 3x2 − 4x + 1 = 0, where
x = cos2 1. [2]

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(ii) Hence solve the equation 3 cos4 1 + 4 sin2 1 − 3 = 0 for 0Å ≤ 1 ≤ 180Å. [5]

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1
8 A function f is defined for x > 12 and is such that f ′ x = 3 2x − 1 2 − 6.

(i) Find the set of values of x for which f is decreasing. [4]

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(ii) It is now given that f 1 = −3. Find f x. [4]

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9 The first, second and third terms of a geometric progression are 3k, 5k − 6 and 6k − 4, respectively.

(i) Show that k satisfies the equation 7k2 − 48k + 36 = 0. [2]

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(ii) Find, showing all necessary working, the exact values of the common ratio corresponding to
each of the possible values of k. [4]

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(iii) One of these ratios gives a progression which is convergent. Find the sum to infinity. [2]

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10 Relative to an origin O, the position vectors of the points A, B and X are given by
` a ` a ` a
−−→ −8 −−→ 10 −−→ −2
OA = −4 , OB = 2 and OX = −2 .
2 11 5

−−→
(i) Find AX and show that AXB is a straight line. [3]

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` a
−−→ 1
The position vector of a point C is given by OC = −8 .
3

(ii) Show that CX is perpendicular to AX . [3]

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(iii) Find the area of triangle ABC. [3]

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11
y

A 2, 3

y = x − 1−2 + 2

x
O 1 3

The diagram shows part of the curve y = x − 1−2 + 2, and the lines x = 1 and x = 3. The point A on
the curve has coordinates 2, 3. The normal to the curve at A crosses the line x = 1 at B.

(i) Show that the normal AB has equation y = 12 x + 2. [3]

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(ii) Find, showing all necessary working, the volume of revolution obtained when the shaded region
is rotated through 360Å about the x-axis. [8]

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Additional Page

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

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

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.

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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.

BLANK PAGE

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@ 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

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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

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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]

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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]

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

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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 13 0r2 h.]

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(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]

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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]

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(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]

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

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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]

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

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

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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]

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

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(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]

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

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

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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]

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(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]

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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

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

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(iii) Find, showing all necessary working, the area of the shaded region. [4]

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Additional Page

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

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

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Cambridge Assessment International Education

Cambridge International Advanced Subsidiary and Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*2590531708*

NUMBER NUMBER

MATHEMATICS 9709/11
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 20 printed pages.

JC19 11_9709_11/RP
© UCLES 2019 [Turn over
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2
@ A6
1
1 Find the term independent of x in the expansion of 2x + 2 . [3]
4x

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2 An increasing function, f, is defined for x > n, where n is an integer. It is given that f ′ x = x2 − 6x + 8.

Find the least possible value of n. [3]

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3 The line y = ax + b is a tangent to the curve y = 2x3 − 5x2 − 3x + c at the point 2, 6. Find the values
of the constants a, b and c. [5]

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4 A runner who is training for a long-distance race plans to run increasing distances each day for 21 days.
She will run x km on day 1, and on each subsequent day she will increase the distance by 10% of the
previous day’s distance. On day 21 she will run 20 km.

(i) Find the distance she must run on day 1 in order to achieve this. Give your answer in km correct
to 1 decimal place. [3]

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(ii) Find the total distance she runs over the 21 days. [2]

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1
5 (i) Given that 4 tan x + 3 cos x + = 0, show, without using a calculator, that sin x = − 23 . [3]
cos x

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(ii) Hence, showing all necessary working, solve the equation

1
4 tan 2x − 20Å + 3 cos 2x − 20Å + =0
cos 2x − 20Å
for 0Å ≤ x ≤ 180Å. [4]

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6 A straight line has gradient m and passes through the point 0, −2. Find the two values of m for
which the line is a tangent to the curve y = x2 − 2x + 7 and, for each value of m, find the coordinates
of the point where the line touches the curve. [7]

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7 Functions f and g are defined by

3
f:x→ for x > 0,
2x + 1
1
g : x → + 2 for x > 0.
x

(i) Find the range of f and the range of g. [3]

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ax
(ii) Find an expression for fg x, giving your answer in the form , where a, b and c are integers.
bx + c
[2]

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(iii) Find an expression for fg−1 x, giving your answer in the same form as for part (ii). [3]

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8
C

6 cm

3
8 0 rad

O A

The diagram shows a sector OAC of a circle with centre O. Tangents AB and CB to the circle meet
at B. The arc AC is of length 6 cm and angle AOC = 38 0 radians.

(i) Find the length of OA correct to 4 significant figures. [2]

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

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(iii) Find the area of the shaded region. [4]

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dy 1
9 A curve for which = 5x − 1 2 − 2 passes through the point 2, 3.
dx

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

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d2 y
(ii) Find . [2]
dx2

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(iii) Find the coordinates of the stationary point on the curve and, showing all necessary working,
determine the nature of this stationary point. [4]

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10
D C

A B

Relative to an origin O, the position vectors of the points A, B, C and D, shown in the diagram, are
given by
` a ` a ` a ` a
−−→ −1 −−→ 2 −−→ 4 −−→ 2
OA = 3 , OB = −3 , OC = −2 and OD = 2 .
−4 5 5 −1

(i) Show that AB is perpendicular to BC. [3]

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

(ii) Show that ABCD is a trapezium. [3]

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(iii) Find the area of ABCD, giving your answer correct to 2 decimal places. [3]

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11
y

y = x2 + 4x + 3

x
−2 O
y = −1
−1

The diagram shows a shaded region bounded by the y-axis, the line y = −1 and the part of the curve
y = x2 + 4x + 3 for which x ≥ −2.

(i) Express y = x2 + 4x + 3 in the form y = x + a2 + b, where a and b are constants. Hence, for
x ≥ −2, express x in terms of y. [4]

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(ii) Hence, showing all necessary working, find the volume obtained when the shaded region is
rotated through 360Å about the y-axis. [6]

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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.

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

www.dynamicpapers.com

Cambridge Assessment International Education

Cambridge International Advanced Subsidiary and Advanced Level

CANDIDATE
NAME

CENTRE CANDIDATE
*0008885838*

NUMBER NUMBER

MATHEMATICS 9709/13
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 19 printed pages and 1 blank page.

1 The function f is defined by f x = x2 − 4x + 8 for x ∈ >.

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

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(ii) Hence find the set of values of x for which f x < 9, giving your answer in exact form. [3]

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@ A
1 5
2 (i) In the binomial expansion of 2x − , the first three terms are 32x5 − 40x3 + 20x. Find the
2x
remaining three terms of the expansion. [3]

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@ 1 5
A
(ii) Hence find the coefficient of x in the expansion of 1 + 4x2 2x − . [2]
2x

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3
A

8 cm
E

1
5 0 rad
B D C

The diagram shows triangle ABC which is right-angled at A. Angle ABC = 15 0 radians and AC = 8 cm.
The points D and E lie on BC and BA respectively. The sector ADE is part of a circle with centre A
and is such that BDC is the tangent to the arc DE at D.

(i) Find the length of AD. [3]

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

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48
4 The function f is defined by f x = for 3 ≤ x ≤ 7. The function g is defined by g x = 2x − 4 for
x−1
a ≤ x ≤ b, where a and b are constants.

(i) Find the greatest value of a and the least value of b which will permit the formation of the
composite function gf. [2]

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

It is now given that the conditions for the formation of gf are satisfied.

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

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(iii) Find an expression for gf−1 x. [2]

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5 Two heavyweight boxers decide that they would be more successful if they competed in a lower
weight class. For each boxer this would require a total weight loss of 13 kg. At the end of week 1
they have each recorded a weight loss of 1 kg and they both find that in each of the following weeks
their weight loss is slightly less than the week before.

Boxer A’s weight loss in week 2 is 0.98 kg. It is given that his weekly weight loss follows an arithmetic
progression.

(i) Write down an expression for his total weight loss after x weeks. [1]

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

(ii) He reaches his 13 kg target during week n. Use your answer to part (i) to find the value of n. [2]

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7

Boxer B’s weight loss in week 2 is 0.92 kg and it is given that his weekly weight loss follows a
geometric progression.

(iii) Calculate his total weight loss after 20 weeks and show that he can never reach his target. [4]

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

www.dynamicpapers.com
8

6
F
N
2 E
D 4

7
C

4 k
j
M
A i 8 B

The diagram shows a solid figure ABCDEF in which the horizontal base ABC is a triangle right-angled
at A. The lengths of AB and AC are 8 units and 4 units respectively and M is the mid-point of AB.
The point D is 7 units vertically above A. Triangle DEF lies in a horizontal plane with DE, DF and
FE parallel to AB, AC and CB respectively and N is the mid-point of FE . The lengths of DE and DF
−−→ −−→ −−→
are 4 units and 2 units respectively. Unit vectors i, j and k are parallel to AB, AC and AD respectively.
−−−→
(i) Find MF in terms of i, j and k. [1]

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

−−→
(ii) Find FN in terms of i and j. [1]

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

−−−→
(iii) Find MN in terms of i, j and k. [1]

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

www.dynamicpapers.com
9

(iv) Use a scalar product to find angle FMN . [4]

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

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10

7 The coordinates of two points A and B are 1, 3 and 9, −1 respectively and D is the mid-point of
AB. A point C has coordinates x, y, where x and y are variables.

(i) State the coordinates of D. [1]

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

(ii) It is given that CD2 = 20. Write down an equation relating x and y. [1]

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

(iii) It is given that AC and BC are equal in length. Find an equation relating x and y and show that
it can be simplified to y = 2x − 9. [3]

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

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11

(iv) Using the results from parts (ii) and (iii), and showing all necessary working, find the possible
coordinates of C. [4]

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

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12

dy
8 A curve is such that = 3x2 + ax + b. The curve has stationary points at −1, 2 and 3, k. Find
dx
the values of the constants a, b and k. [8]

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

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13

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

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14

9
y

y = f x

x
O 1 0
20

The function f : x → p sin2 2x + q is defined for 0 ≤ x ≤ 0, where p and q are positive constants. The
diagram shows the graph of y = f x.

(i) In terms of p and q, state the range of f. [2]

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

(ii) State the number of solutions of the following equations.

(a) f x = p + q [1]

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

(b) f x = q [1]

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

www.dynamicpapers.com
15

(iii) For the case where p = 3 and q = 2, solve the equation f x = 4, showing all necessary working.
[5]

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

www.dynamicpapers.com
16

10
y

1
y = 3x + 4 2

x
O 4

1
The diagram shows part of the curve with equation y = 3x + 4 2 and the tangent to the curve at the
point A. The x-coordinate of A is 4.

(i) Find the equation of the tangent to the curve at A. [5]

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

www.dynamicpapers.com
17

(ii) Find, showing all necessary working, the area of the shaded region. [5]

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

[Question 10 (iii) is printed on the next page.]

www.dynamicpapers.com
18

(iii) A point is moving along the curve. At the point P the y-coordinate is increasing at half the rate
at which the x-coordinate is increasing. Find the x-coordinate of P. [3]

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

www.dynamicpapers.com
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.

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

www.dynamicpapers.com
20

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