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Maths Paper 3 Pack

This document is an examination paper for Cambridge International AS & A Level Mathematics, specifically Paper 3 Pure Mathematics 3 from February/March 2021. It includes instructions for candidates, information about the total marks, and a series of mathematical questions covering topics such as logarithms, polynomials, differential equations, and complex numbers. The paper is structured to require candidates to show their workings and provide answers to specified accuracy.

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0% found this document useful (0 votes)

6 views496 pages

Maths Paper 3 Pack

Uploaded by

shaynemakurumure

We take content rights seriously. If you suspect this is your content, claim it here.

Available Formats

Download as PDF, TXT or read online on Scribd

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com

Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*8483881097*

MATHEMATICS 9709/32
Paper 3 Pure Mathematics 3 February/March 2021

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.

JC21 03_9709_32/2R
© UCLES 2021 [Turn over
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2

BLANK PAGE

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3

1 Solve the equation ln x3 − 3 = 3 ln x − ln 3. Give your answer correct to 3 significant figures. [3]

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2 The polynomial ax3 + 5x2 − 4x + b, where a and b are constants, is denoted by p x. It is given that
x + 2 is a factor of p x and that when p x is divided by x + 1 the remainder is 2.

Find the values of a and b. [5]

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3 By first expressing the equation tan x + 45Å = 2 cot x + 1 as a quadratic equation in tan x, solve the
equation for 0Å < x < 180Å. [6]

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4 The variables x and y satisfy the differential equation

dy
1 − cos x = y sin x.
dx
It is given that y = 4 when x = π.

(a) Solve the differential equation, obtaining an expression for y in terms of x. [6]

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

(b) Sketch the graph of y against x for 0 < x < 2π. [1]

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5 (a) Express 7 sin x + 2 cos x in the form R sin x + !, where R > 0 and 0Å < ! < 90Å. State the exact
value of R and give ! correct to 2 decimal places. [3]

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(b) Hence solve the equation 7 sin 21 + 2 cos 21 = 1, for 0Å < 1 < 180Å. [5]

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

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5a
6 Let f x = , where a is a positive constant.
2x − a 3a − x

(a) Express f x in partial fractions. [3]

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11
2a
(b) Hence show that Ó f x dx = ln 6. [4]
a

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` a ` a ` a ` a
1 2 2 1
7 Two lines have equations r = 3 + s −1 and r = 1 + t −1 .
2 3 4 4

(a) Show that the lines are skew. [5]

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(b) Find the acute angle between the directions of the two lines. [3]

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8 The complex numbers u and v are defined by u = −4 + 2i and v = 3 + i.

u
(a) Find in the form x + iy, where x and y are real. [3]
v

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

u
(b) Hence express in the form r ei1 , where r and 1 are exact. [2]
v

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In an Argand diagram, with origin O, the points A, B and C represent the complex numbers u, v and
2u + v respectively.

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

(d) Prove that angle AOB = 34 π. [2]

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e2x + 1
9 Let f x = , for x > 0.
e2x − 1

(a) The equation x = f x has one root, denoted by a.

Verify by calculation that a lies between 1 and 1.5. [2]

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

(b) Use an iterative formula based on the equation in part (a) to determine a correct to 2 decimal
places. Give the result of each iteration to 4 decimal places. [3]

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10
y
M

x
O 1
2π

The diagram shows the curve y = sin 2x cos2 x for 0 ≤ x ≤ 12 π, and its maximum point M .

(a) Using the substitution u = sin x, find the exact area of the region bounded by the curve and the
x-axis. [5]

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(b) Find the exact x-coordinate of M . [6]

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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.

© UCLES 2021 9709/32/F/M/21

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Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*3201784670*

MATHEMATICS 9709/32
Paper 3 Pure Mathematics 3 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_32/RP
© UCLES 2022 [Turn over
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2

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3

1 Solve the inequality 2x + 3 > 3 x + 2. [4]

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2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers z
satisfying the inequalities z + 2 − 3i ≤ 2 and arg z ≤ 34 π. [4]

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3
ln y

(0.31, 1.21)

(1.06, 0.91)

ln x
O

The variables x and y satisfy the equation xn y2 = C, where n and C are constants. The graph of ln y
against ln x is a straight line passing through the points 0.31, 1.21 and 1.06, 0.91, as shown in the
diagram.

Find the value of n and find the value of C correct to 2 decimal places. [5]

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4 The parametric equations of a curve are

x = 1 − cos 1, y = cos 1 − 14 cos 21.

= −2 sin2 12 1 .
dy
Show that [5]
dx

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5 The angles ! and " lie between 0Å and 180Å and are such that
tan ! + " = 2 and tan ! = 3 tan ".
Find the possible values of ! and ". [6]

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6 Find the complex numbers w which satisfy the equation w2 + 2iw* = 1 and are such that Re w ≤ 0.
Give your answers in the form x + iy, where x and y are real. [6]

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7 (a) By sketching a suitable pair of graphs, show that the equation 4 − x2 = sec 12 x has exactly one
root in the interval 0 ≤ x < π. [2]

(b) Verify by calculation that this root lies between 1 and 2. [2]

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

?
(c) Use the iterative formula xn+1 = 4 − sec 21 xn to determine the root correct to 2 decimal places.
Give the result of each iteration to 4 decimal places. [3]

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8 (a) Find the quotient and remainder when 8x3 + 4x2 + 2x + 7 is divided by 4x2 + 1. [3]

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

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8x3 + 4x2 + 2x + 7
1

(b) Hence find the exact value of Ô

4x2 + 1
dx. [5]
0

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9 The variables x and y satisfy the differential equation

x + 1 3x + 1 = y,
dy
dx
and it is given that y = 1 when x = 1.

Solve the differential equation and find the exact value of y when x = 3, giving your answer in a
simplified form. [9]

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10 The points A and B have position vectors 2i + j + k and i − 2j + 2k respectively. The line l has vector
equation r = i + 2j − 3k + - i − 3j − 2k.

(a) Find a vector equation for the line through A and B. [3]

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

(b) Find the acute angle between the directions of AB and l, giving your answer in degrees. [3]

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11
y

x
O 1
2π

The diagram shows the curve y = sin x cos 2x for 0 ≤ x ≤ 12 π, and its maximum point M .

(a) Find the x-coordinate of M , giving your answer correct to 3 significant figures. [6]

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(b) Using the substitution u = cos x, find the area of the shaded region enclosed by the curve and the
x-axis in the first quadrant, giving your answer in a simplified exact form. [5]

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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 International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*9337495756*

MATHEMATICS 9709/32
Paper 3 Pure Mathematics 3 February/March 2023

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.

JC23 03_9709_32/2R
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2

1 It is given that x = ln 2y − 3 − ln y + 4.

Express y in terms of x. [3]

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2 (a) On an Argand diagram, shade the region whose points represent complex numbers z satisfying
the inequalities − 13 π ≤ arg z − 1 − 2i ≤ 13 π and Re z ≤ 3. [3]

(b) Calculate the least value of arg z for points in the region from (a). Give your answer in radians
correct to 3 decimal places. [2]

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3 The polynomial 2x4 + ax3 + bx − 1, where a and b are constants, is denoted by p x. When p x is
divided by x2 − x + 1 the remainder is 3x + 2.

Find the values of a and b. [5]

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4 Solve the equation

5z
− zz* + 30 + 10i = 0,
1 + 2i
giving your answers in the form x + iy, where x and y are real. [5]

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5 The parametric equations of a curve are

x = t e 2t , y = t2 + t + 3.

dy
(a) Show that = e−2t . [3]
dx

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@ A
1
(b) Hence show that the normal to the curve, where t = −1, passes through the point 0, 3 − 4 .
e
[3]

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6 (a) Express 5 sin 1 + 12 cos 1 in the form R cos 1 − !, where R > 0 and 0 < ! < 12 π. [3]

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

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(b) Hence solve the equation 5 sin 2x + 12 cos 2x = 6 for 0 ≤ x ≤ π. [4]

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7
B

O x rad

The diagram shows a circle with centre O and radius r . The angle of the minor sector AOB of the
circle is x radians. The area of the major sector of the circle is 3 times the area of the shaded region.

(a) Show that x = 34 sin x + 12 π. [4]

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(b) Show by calculation that the root of the equation in (a) lies between 2 and 2.5. [2]

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

(c) Use an iterative formula based on the equation in (a) to calculate this root correct to 2 decimal
places. Give the result of each iteration to 4 decimal places. [3]

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8
y

1
2
x
O

The diagram shows the curve y = x3 ln x, for x > 0, and its minimum point M .

(a) Find the exact coordinates of M . [4]

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(b) Find the exact area of the shaded region bounded by the curve, the x-axis and the line x = 12 . [5]

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9 The variables x and y satisfy the differential equation

dy
= e3y sin2 2x.
dx
It is given that y = 0 when x = 0.

Solve the differential equation and find the value of y when x = 12 . [7]

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10 With respect to the origin O, the points A, B, C and D have position vectors given by
` a ` a ` a ` a
−−¿ 3 −−¿ 1 −−¿ 1 −−¿ 5
OA = −1 , OB = 2 , OC = −2 and OD = −6 .
2 −3 5 11

−−¿ −−¿
(a) Find the obtuse angle between the vectors OA and OB. [3]

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

The line l passes through the points A and B.

(b) Find a vector equation for the line l. [2]

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5x2 + x + 11
11 Let f x = .
4 + x2 1 + x

(a) Express f x in partial fractions. [5]

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

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2
(b) Hence show that Ó f x dx = ln 54 − 18 π. [5]
0

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

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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.

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

© UCLES 2023 9709/32/F/M/23

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Cambridge International AS & A Level

* 5 5 5 2 0 0 4 5 5 2 *

MATHEMATICS 9709/32
Paper 3 Pure Mathematics 3 February/March 2024

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

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

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

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

DC (LK/SW) 329847/3
© UCLES 2024 [Turn over
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2

BLANK PAGE

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3

1 Find the quotient and remainder when x 4 - 3x 3 + 9x 2 - 12x + 27 is divided by x 2 + 5. [3]

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

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2 (a) Find the coefficient of x 2 in the expansion of (2x - 5) 4 - x . [4]

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

(b) State the set of values of x for which the expansion in part (a) is valid. [1]

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

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3 It is given that z =- 3 + i .

(a) Express z 2 in the form r e ii , where r 2 0 and - r 1 i G r . [3]

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

............................................................................................................................................................
z2
(b) The complex number ~ is such that z 2 ~ is real and = 12 .
~
Find the two possible values of ~, giving your answers in the form Re ia , where R 2 0 and
- r 1 a G r. [3]

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

4 The positive numbers p and q are such that

p
ln e o = a and ln `q 2 pj = b .
q

Express ln ` p 7 qj in terms of a and b. [4]

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

5 (a) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z
satisfying the inequalities z - 4 - 2i G 3 and z H 10 - z . [4]

(b) Find the greatest value of arg z for points in this region. [2]

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

6 The equation of a curve is 2y 2 + 3xy + x = x 2 .

d y 2x - 3y - 1
(a) Show that = . [4]
dx 4y + 3x

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

(b) Hence show that the curve does not have a tangent that is parallel to the x-axis. [3]

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

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7
y

O a x

The diagram shows the curve y = xe 2x - 5x and its minimum point M, where x = a .

ln b l.
1 5
(a) Show that a satisfies the equation a = [3]
2 1 + 2a
............................................................................................................................................................

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(b) Verify by calculation that a lies between 0.4 and 0.5 . [2]

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

(c) Use an iterative formula based on the equation in part (a) to determine a correct to 2 decimal
places. Give the result of each iteration to 4 decimal places. [3]

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

8 (a) Express 3 sin x + 2 2 cos bx + 14 rl in the form R sin (x + a) , where R 2 0 and 0 1 a 1 12 r . State
the exact value of R and give a correct to 3 decimal places. [4]

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(b) Hence solve the equation

6 sin 12 i + 4 2 cos b12 i + 14 rl = 3

for - 4r 1 i 1 4r . [5]

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9 Relative to the origin O, the position vectors of the points A, B and C are given by

OA = 5i - 2 j + k, OB = 8i + 2 j - 6k and OC = 3i + 4 j - 7k .

(a) Show that OABC is a rectangle. [4]

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(b) Use a scalar product to find the acute angle between the diagonals of OABC. [4]

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36a 2
10 Let f (x) = , where a is a positive constant.
(2a + x) (2a - x) (5a - 2x)

(a) Express f (x) in partial fractions. [5]

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a
(b) Hence find the exact value of y-a
f (x) d x , giving your answer in the form p ln q + r ln s where p and

r are integers and q and s are prime numbers. [5]

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18

11 The variables y and i satisfy the differential equation

dy
(1 + y) (1 + cos 2i) = e 3y .
di
It is given that y = 0 when i = 14 r .

Solve the differential equation and find the exact value of tan i when y = 1. [9]

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

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19

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

Additional page

If you use the following page to complete the answer to any question, the question number 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.

© UCLES 2024 9709/32/F/M/24

www.dynamicpapers.com

Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*3602942828*

MATHEMATICS 9709/31
Paper 3 Pure Mathematics 3 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_31/RP
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2

1 Solve the inequality 23x − 1 < x + 1. [4]

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

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3

2ex + e−x
= 3, giving your answer correct to 3 decimal places.
2 + ex
2 Find the real root of the equation
Your working should show clearly that the equation has only one real root. [5]

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

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4

2− 3

(a) Given that cos x − 30Å = 2 sin x + 30Å, show that tan x =
1−2 3
3 . [4]

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

(b) Hence solve the equation

cos x − 30Å = 2 sin x + 30Å,
for 0Å < x < 360Å. [2]

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

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5

1 − cos 21
tan2 1.
1 + cos 21
4 (a) Prove that [2]

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

1π
1 − cos 21
(b) Hence find the exact value of Ô d1.
3

1 + cos 21
[4]
1π
6

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

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6

5 (a) Solve the equation z2 − 2piz − q = 0, where p and q are real constants. [2]

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

In an Argand diagram with origin O, the roots of this equation are represented by the distinct points
A and B.

(b) Given that A and B lie on the imaginary axis, find a relation between p and q. [2]

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

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7

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

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8

6 The parametric equations of a curve are

x = ln 2 + 3t, y=
t
2 + 3t
.

(a) Show that the gradient of the curve is always positive. [5]

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

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9

(b) Find the equation of the tangent to the curve at the point where it intersects the y-axis. [3]

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

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10

7
y

x
O a

tan−1 x
The diagram shows the curve y = and its maximum point M where x = a.
x

(a) Show that a satisfies the equation

@ A
a = tan
2a
1 + a2
. [4]

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

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11

(b) Verify by calculation that a lies between l.3 and 1.5. [2]

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

(c) Use an iterative formula based on the equation in part (a) to determine a correct to 2 decimal
places. Give the result of each iteration to 4 decimal places. [3]

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

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12
` a
−−¿ 1
8 With respect to the origin O, the points A and B have position vectors given by OA = 2 and
` a ` a ` a 1
−−¿ 3 2 1
OB = 1 . The line l has equation r = 3 + , −2 .
−2 1 1

(a) Find the acute angle between the directions of AB and l. [4]

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

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13

(b) Find the position vector of the point P on l such that AP = BP. [5]

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

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14

− 23
9 The equation of a curve is y = x ln x for x > 0. The curve has one stationary point.

(a) Find the exact coordinates of the stationary point. [5]

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

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15

(b) Show that Ó y dx = 18 ln 2 − 9.

8
[5]
1

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

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16

= x2 1 + 2x, and x = 1 when t = 0.

dx
10 The variables x and t satisfy the differential equation
dt

Using partial fractions, solve the differential equation, obtaining an expression for t in terms of x.
[11]

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

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17

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

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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
*4535261400*

MATHEMATICS 9709/32
Paper 3 Pure Mathematics 3 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_32/RP
© UCLES 2021 [Turn over
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2

1 Solve the inequality 2x − 1 < 3 x + 1. [4]

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

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3

2 On a sketch of an Argand diagram, shade the region whose points represent complex numbers z
satisfying the inequalities z + 1 − i ≤ 1 and arg z − 1 ≤ 34 π. [4]

© UCLES 2021 9709/32/M/J/21 [Turn over

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4

3 The variables x and y satisfy the equation x = A 3−y , where A is a constant.

(a) Explain why the graph of y against ln x is a straight line and state the exact value of the gradient
of the line. [3]

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

It is given that the line intersects the y-axis at the point where y = 1.3.

(b) Calculate the value of A, giving your answer correct to 2 decimal places. [2]

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

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5
2
4 Using integration by parts, find the exact value of Ó tan−1 12 x dx. [5]
0

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

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6

5 The complex number u is given by u = 10 − 4 6i.

Find the two square roots of u, giving your answers in the form a + ib, where a and b are real and
exact. [5]

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

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7

6 (a) Prove that cosec 21 − cot 21 tan 1. [3]

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

1π

(b) Hence show that Ó cosec 21 − cot 21 d1 = 12 ln 2.

3
[4]
1π
4

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

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8
y
7 A curve is such that the gradient at a general point with coordinates x, y is proportional to .
x+1
The curve passes through the points with coordinates 0, 1 and 3, e.

By setting up and solving a differential equation, find the equation of the curve, expressing y in terms
of x. [7]

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

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9

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

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10

8 The equation of a curve is y = e−5x tan2 x for − 12 π < x < 12 π.

Find the x-coordinates of the stationary points of the curve. Give your answers correct to 3 decimal
places where appropriate. [8]

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

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11

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

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12

14 − 3x + 2x2
9 Let f x = .
2 + x 3 + x2

(a) Express f x in partial fractions. [5]

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

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13

(b) Hence obtain the expansion of f x in ascending powers of x, up to and including the term in x2 .
[5]

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

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14

10
D C

r r

x rad x rad
A r M r B

The diagram shows a trapezium ABCD in which AD = BC = r and AB = 2r . The acute angles BAD
and ABC are both equal to x radians. Circular arcs of radius r with centres A and B meet at M , the
midpoint of AB.

(a) Given that the sum of the areas of the shaded sectors is 90% of the area of the trapezium, show
that x satisfies the equation x = 0.9 2 − cos x sin x. [3]

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

(b) Verify by calculation that x lies between 0.5 and 0.7. [2]

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

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15

(c) Show that if a sequence of values in the interval 0 < x < 12 π given by the iterative formula
P Q
−1 xn
xn+1 = cos 2−
0.9 sin xn

converges, then it converges to the root of the equation in part (a). [2]

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

(d) Use this iterative formula to determine x correct to 2 decimal places. Give the result of each
iteration to 4 decimal places. [3]

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

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16
−−¿
11 With respect to the origin O, the points A and B have position vectors given by OA = 2i − j and
−−¿
OB = j − 2k.

(a) Show that OA = OB and use a scalar product to calculate angle AOB in degrees. [4]

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

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17

The midpoint of AB is M . The point P on the line through O and M is such that PA : OA = 7 : 1.

(b) Find the possible position vectors of P. [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.

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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
*2912457036*

MATHEMATICS 9709/33
Paper 3 Pure Mathematics 3 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_33/RP
© UCLES 2021 [Turn over
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2

Expand 1 + 3x 3 in ascending powers of x, up to and including the term in x3 , simplifying the
2
1
coefficients. [4]

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

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3

2 Solve the equation 4x = 3 + 4−x . Give your answer correct to 3 decimal places. [5]

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

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4

3 The parametric equations of a curve are

x = t + ln t + 2, y = t − 1e−2t ,
where t > −2.

dy
(a) Express in terms of t, simplifying your answer. [5]
dx

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

(b) Find the exact y-coordinate of the stationary point of the curve. [2]

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

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5

15 − 6x
Let f x =
1 + 2x 4 − x
4 .

(a) Express f x in partial fractions. [3]

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

0a1
(b) Hence find Ó f x dx, giving your answer in the form ln
2
, where a and b are integers. [4]
1 b

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

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6

5 (a) By first expanding tan 21 + 21, show that the equation tan 41 = 12 tan 1 may be expressed as
tan4 1 + 2 tan2 1 − 7 = 0. [4]

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

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7

(b) Hence solve the equation tan 41 = 12 tan 1, for 0Å < 1 < 180Å. [3]

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

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8

6 (a) By sketching a suitable pair of graphs, show that the equation cot 21 x = 1 + e−x has exactly one
root in the interval 0 < x ≤ π. [2]

(b) Verify by calculation that this root lies between 1 and 1.5. [2]

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

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9
A @
(c) Use the iterative formula xn+1 = 2 tan−1
1
1 + e−xn
to determine the root correct to 2 decimal
places. Give the result of each iteration to 4 decimal places. [3]

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

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10

7
y

x
O M N

For the curve shown in the diagram, the normal to the curve at the point P with coordinates x, y
meets the x-axis at N . The point M is the foot of the perpendicular from P to the x-axis.

The curve is such that for all values of x in the interval 0 ≤ x < 12 π, the area of triangle PMN is equal
to tan x.

=
MN dy
(a) (i) Show that . [1]
y dx

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

= tan x.
dy
(ii) Hence show that x and y satisfy the differential equation 12 y2 [2]
dx

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

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11

(b) Given that y = 1 when x = 0, solve this differential equation to find the equation of the curve,
expressing y in terms of x. [6]

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

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12

8
y
M

x
O

The diagram shows the curve y =

ln x
and its maximum point M .
x4
(a) Find the exact coordinates of M . [4]

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

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13
a
(b) By using integration by parts, show that for all a > 1, Ô dx < 19 .
ln x
[6]
1
x4

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

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14

−−¿ −−¿
9 The quadrilateral ABCD is a trapezium in which AB and DC are parallel. With respect to the
origin O, the position vectors of A, B and C are given by OA = −i + 2j + 3k, OB = i + 3j + k and
−−¿
OC = 2i + 2j − 3k.
−−¿ −−¿
(a) Given that DC = 3AB, find the position vector of D. [3]

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

(b) State a vector equation for the line through A and B. [1]

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

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15

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

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16

(a) Verify that −1 + 2i is a root of the equation z4 + 3z2 + 2z + 12 = 0.

10 [3]

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

(b) Find the other roots of this equation. [7]

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

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17

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

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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
*5794103083*

MATHEMATICS 9709/31
Paper 3 Pure Mathematics 3 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_31/RP
© UCLES 2022 [Turn over
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2

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3

1 Solve the equation 2 32x−1 = 4x+1 , giving your answer correct to 2 decimal places. [4]

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

© UCLES 2022 9709/31/M/J/22 [Turn over

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4
−2
(a) Expand 2 − x2

2 in ascending powers of x, up to and including the term in x4 , simplifying the
coefficients. [4]

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

(b) State the set of values of x for which the expansion is valid. [1]

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

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5

3 Solve the equation 2 cot 2x + 3 cot x = 5, for 0Å < x < 180Å. [6]

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

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6

4 The variables x and y satisfy the differential equation

=
dy xy
dx 1 + x2
,

and y = 2 when x = 0.

Solve the differential equation, obtaining a simplified expression for y in terms of x. [7]

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

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7

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

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8

5 The polynomial ax3 − 10x2 + bx + 8, where a and b are constants, is denoted by p x. It is given that
x − 2 is a factor of both p x and p′ x.

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

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

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9

(b) When a and b have these values, factorise p x completely. [3]

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

© UCLES 2022 9709/31/M/J/22 [Turn over

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10
3
Let I = Ô
27
6 2 dx.
9 + x2

0

1π

(a) Using the substitution x = 3 tan 1, show that I = Ó cos2 1 d1.

4
[4]
0

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

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11

(b) Hence find the exact value of I . [4]

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

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12

2 − a 2i

The complex number u is defined by u =
1 + 2i
7 , where a is a positive integer.

(a) Express u in terms of a, in the form x + iy, where x and y are real and exact. [3]

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

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13

It is now given that a = 3.

(b) Express u in the form rei1 , where r > 0 and −π < 1 ≤ π, giving the exact values of r and 1. [2]

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

(c) Using your answer to part (b), find the two square roots of u. Give your answers in the form rei1 ,
where r > 0 and −π < 1 ≤ π, giving the exact values of r and 1. [3]

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

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14

8 The equation of a curve is x3 + y3 + 2xy + 8 = 0.

dy
(a) Express in terms of x and y. [4]
dx

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

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15

The tangent to the curve at the point where x = 0 and the tangent at the point where y = 0 intersect at
the acute angle !.

(b) Find the exact value of tan !. [5]

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

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16

9
G F

D
k E

O C
j

A N B

In the diagram, OABCDEFG is a cuboid in which OA = 2 units, OC = 4 units and OG = 2 units. Unit
vectors i, j and k are parallel to OA, OC and OG respectively. The point M is the midpoint of DF.
The point N on AB is such that AN = 3NB.
−−−¿ −−−¿
(a) Express the vectors OM and MN in terms of i, j and k. [3]

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

(b) Find a vector equation for the line through M and N . [2]

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

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17
?
(c) Show that the length of the perpendicular from O to the line through M and N is 53 . [4]
6

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

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18

10
y

x
O a π

The curve y = x sin x has one stationary point in the interval 0 < x < π, where x = a (see diagram).

(a) Show that tan a = − 12 a. [4]

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

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19

(b) Verify by calculation that a lies between 2 and 2.5. [2]

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

(c) Show that if a sequence of values in the interval 0 < x < π given by the iterative formula
xn+1 = π − tan−1 12 xn converges, then it converges to a, the root of the equation in part (a). [2]

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

(d) Use the iterative formula given in part (c) to determine a correct to 2 decimal places. Give the
result of each iteration to 4 decimal places. [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 9709/31/M/J/22

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Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*0788354854*

MATHEMATICS 9709/32
Paper 3 Pure Mathematics 3 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_32/RP
© UCLES 2022 [Turn over
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2

BLANK PAGE

© UCLES 2022 9709/32/M/J/22

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3

1 Solve the equation ln e2x + 3 = 2x + ln 3, giving your answer correct to 3 decimal places. [4]

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

© UCLES 2022 9709/32/M/J/22 [Turn over

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4

2 Solve the equation 3 cos 21 = 3 cos 1 + 2, for 0Å ≤ 1 ≤ 360Å. [5]

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

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5

3 The polynomial ax3 + x2 + bx + 3 is denoted by p x. It is given that p x is divisible by 2x − 1 and

that when p x is divided by x + 2 the remainder is 5.

Find the values of a and b. [5]

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

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6

4 The equation of a curve is y = cos3 x sin x. It is given that the curve has one stationary point in the
interval 0 < x < 12 π.

Find the x-coordinate of this stationary point, giving your answer correct to 3 significant figures. [6]

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

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7

5 (a) By sketching a suitable pair of graphs, show that the equation ln x = 3x − x2 has one real root.
[2]

(b) Verify by calculation that the root lies between 2 and 2.8. [2]

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

/
(c) Use the iterative formula xn+1 = 3xn − ln xn to determine the root correct to 2 decimal places.
Give the result of each iteration to 4 decimal places. [3]

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

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8

6 The variables x and y satisfy the differential equation

= xey−x ,
dy
dx
and y = 0 when x = 0.

(a) Solve the differential equation, obtaining an expression for y in terms of x. [7]

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

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9

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

(b) Find the value of y when x = 1, giving your answer in the form a − ln b, where a and b are
integers. [1]

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

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7 The equation of a curve is x3 + 3x2 y − y3 = 3.

dy x2 + 2xy
(a) Show that = 2 . [4]
dx y − x2

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11

(b) Find the coordinates of the points on the curve where the tangent is parallel to the x-axis. [5]

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

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12

x2 + 9x
8 Let f x = .
3x − 1 x2 + 3

(a) Express f x in partial fractions. [5]

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

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13
3
(b) Hence find Ó f x dx, giving your answer in a simplified exact form. [5]
1

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

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14

9 The lines l and m have vector equations

r = −i + 3j + 4k + , 2i − j − k and r = 5i + 4j + 3k + - ai + bj + k
respectively, where a and b are constants.

(a) Given that l and m intersect, show that 2b − a = 4. [4]

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

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15

(b) Given also that l and m are perpendicular, find the values of a and b. [4]

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

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10 The complex number −1 + 7i is denoted by u. It is given that u is a root of the equation

2x3 + 3x2 + 14x + k = 0,

where k is a real constant.

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

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

(b) Find the other two roots of the equation. [4]

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

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17

(c) On an Argand diagram, sketch the locus of points representing complex numbers z satisfying
the equation z − u = 2. [2]

(d) Determine the greatest value of arg z for points on this locus, giving your answer in radians. [2]

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

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20

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Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*9937725522*

MATHEMATICS 9709/33
Paper 3 Pure Mathematics 3 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_33/2R
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2

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3

1 Find, in terms of a, the set of values of x satisfying the inequality

23x + a < 2x + 3a ,
where a is a positive constant. [4]

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

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4

2 Solve the equation cos 1 − 60Å = 3 sin 1, for 0Å ≤ 1 ≤ 360Å. [5]

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

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5

3 (a) Show that the equation log3 2x + 1 = 1 + 2 log3 x − 1 can be written as a quadratic equation
in x. [3]

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

(b) Hence solve the equation log3 4y + 1 = 1 + 2 log3 2y − 1, giving your answer correct to 2 decimal
places. [2]

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

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6

4 The curve y = e−4x tan x has two stationary points in the interval 0 ≤ x < 12 π.

dy
(a) Obtain an expression for and show it can be written in the form sec2 x a + b sin 2xe−4x , where
dx
a and b are constants. [4]

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

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7

(b) Hence find the exact x-coordinates of the two stationary points. [3]

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

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8

5 The complex number 3 − i is denoted by u.

(a) Show, on an Argand diagram with origin O, the points A, B and C representing the complex
numbers u, u* and u* − u respectively.

State the type of quadrilateral formed by the points O, A, B and C. [3]

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

u*
(b) Express in the form x + iy, where x and y are real. [3]
u

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

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9

u*
(c) By considering the argument of , or otherwise, prove that tan−1 34 = 2 tan−1 13 . [2]
u

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

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10

1
6 The parametric equations of a curve are x = , y = ln tan t, where 0 < t < 12 π.
cos t

dy cos t
(a) Show that = . [5]
dx sin2 t

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

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11

(b) Find the equation of the tangent to the curve at the point where y = 0. [3]

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

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12

5x2 + 8x − 3
7 Let f x = .
x − 2 2x2 + 3

(a) Express f x in partial fractions. [5]

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

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13

(b) Hence obtain the expansion of f x in ascending powers of x, up to and including the term in x2 .
[5]

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

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14

8 At time t days after the start of observations, the number of insects in a population is N . The variation
dN
= kN 2 cos 0.02t, where
3
in the number of insects is modelled by a differential equation of the form
dt
k is a constant and N is a continuous variable. It is given that when t = 0, N = 100.

(a) Solve the differential equation, obtaining a relation between N , k and t. [5]

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

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15

(b) Given also that N = 625 when t = 50, find the value of k. [2]

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

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16
−−¿
9 With respect to the origin O, the point A has position vector given by OA = i + 5j + 6k. The line l has
vector equation r = 4i + k + , −i + 2j + 3k.

(a) Find in degrees the acute angle between the directions of OA and l. [3]

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

(b) Find the position vector of the foot of the perpendicular from A to l. [4]

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

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17

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

(c) Hence find the position vector of the reflection of A in l. [2]

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

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18
a
10 The constant a is such that Ó x2 ln x dx = 4.
1

@ A 13
35
(a) Show that a = . [5]
3 ln a − 1

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

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19

(b) Verify by calculation that a lies between 2.4 and 2.8. [2]

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

(c) Use an iterative formula based on the equation in part (a) to determine a correct to 2 decimal
places. Give the result of each iteration to 4 decimal places. [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.

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

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Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*2968750175*

MATHEMATICS 9709/31
Paper 3 Pure Mathematics 3 May/June 2023

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.

JC23 06_9709_31/2R
© UCLES 2023 [Turn over
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2

1 Solve the equation

3e2x − 4e−2x = 5.
Give the answer correct to 3 decimal places. [3]

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

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3

2 (a) Sketch the graph of y = 2x + 3. [1]

(b) Solve the inequality 3x + 8 > 2x + 3. [3]

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

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4

3 Find the coefficient of x3 in the binomial expansion of 3 + x 1 + 4x. [4]

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

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5

4 (a) Show that the equation sin 21 + cos 21 = 2 sin2 1 can be expressed in the form

cos2 1 + 2 sin 1 cos 1 − 3 sin2 1 = 0. [2]

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

(b) Hence solve the equation sin 21 + cos 21 = 2 sin2 1 for 0Å < 1 < 180Å. [4]

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

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6

5 The equation of a curve is x2 y − ay2 = 4a3 , where a is a non-zero constant.

dy 2xy
(a) Show that = . [4]
dx 2ay − x2

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

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7

(b) Hence find the coordinates of the points where the tangent to the curve is parallel to the y-axis.
[4]

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

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8

6 Relative to the origin O, the points A, B and C have position vectors given by
` a ` a ` a
−−¿ 2 −−¿ 4 −−¿ 3
OA = 1 , OB = 3 and OC = −2 .
3 2 −4
The quadrilateral ABCD is a parallelogram.

(a) Find the position vector of D. [3]

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

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9

(b) The angle between BA and BC is 1.

Find the exact value of cos 1. [3]

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

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10

7 The variables x and y satisfy the differential equation

dy 4 tan 2x
cos 2x = ,
dx sin2 3y

where 0 ≤ x < 14 π. It is given that y = 0 when x = 16 π.

Solve the differential equation to obtain the value of x when y = 16 π. Give your answer correct to
3 decimal places. [8]

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

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11

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

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12

3 − 3x2
8 Let f x = .
2x + 1 x + 22

(a) Express f x in partial fractions. [5]

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

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13
4
(b) Hence find the exact value of Ó f x dx, giving your answer in the form a + b ln c, where a, b
0
and c are integers. [5]

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

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14
a
9 The constant a is such that Ó xe−2x dx = 18 .
0

(a) Show that a = 12 ln 4a + 2. [5]

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

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15

(b) Verify by calculation that a lies between 0.5 and 1. [2]

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

(c) Use an iterative formula based on the equation in (a) to determine a correct to 2 decimal places.
Give the result of each iteration to 4 decimal places. [3]

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

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16

10 The polynomial x3 + 5x2 + 31x + 75 is denoted by p x.

(a) Show that x + 3 is a factor of p x. [2]

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

(b) Show that z = −1 + 2 6i is a root of p z = 0. [3]

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

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17

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

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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
*7943719452*

MATHEMATICS 9709/32
Paper 3 Pure Mathematics 3 May/June 2023

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.

JC23 06_9709_32/2R
© UCLES 2023 [Turn over
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2

BLANK PAGE

© UCLES 2023 9709/32/M/J/23

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3

1 Solve the inequality 5x − 3 < 23x − 7. [4]

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

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4

2 Solve the equation ln 2x2 − 3 = 2 ln x − ln 2, giving your answer in an exact form. [3]

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

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5

3 (a) On an Argand diagram, sketch the locus of points representing complex numbers z satisfying
z + 3 − 2i = 2. [2]

(b) Find the least value of z for points on this locus, giving your answer in an exact form. [2]

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

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6

4 Solve the equation 2 cos x − cos 21 x = 1 for 0 ≤ x ≤ 2π. [5]

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

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7

5 The complex number 2 + yi is denoted by a, where y is a real number and y < 0. It is given that
f a = a3 − a2 − 2a.

(a) Find a simplified expression for f a in terms of y. [3]

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

(b) Given that Re f a = −20, find arg a. [3]

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

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8

6 The equation cot 21 x = 3x has one root in the interval 0 < x < π, denoted by !.

(a) Show by calculation that ! lies between 0.5 and 1. [2]

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

(b) Show that, if a sequence of positive values given by the iterative formula
P @ AQ
−1 1
xn+1 = 3 xn + 4 tan
1
3xn
converges, then it converges to !. [2]

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

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9

(c) Use this iterative formula to calculate ! correct to 2 decimal places. Give the result of each
iteration to 4 decimal places. [3]

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

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10

7 The equation of a curve is 3x2 + 4xy + 3y2 = 5.

dy 3x + 2y
(a) Show that =− . [4]
dx 2x + 3y

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

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11

(b) Hence find the exact coordinates of the two points on the curve at which the tangent is parallel
to y + 2x = 0. [5]

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

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12

8 (a) The variables x and y satisfy the differential equation

dy 4 + 9y2
= 2x+1 .
dx e
It is given that y = 0 when x = 1.

Solve the differential equation, obtaining an expression for y in terms of x. [7]

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

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13

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

(b) State what happens to the value of y as x tends to infinity. Give your answer in an exact form.
[1]

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

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14

2x2 + 17x − 17
9 Let f x = .
1 + 2x 2 − x2

(a) Express f x in partial fractions. [5]

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

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15
1
(b) Hence show that Ó f x dx = 52 − ln 72. [5]
0

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

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16

10
y
M

x
O

The diagram shows the curve y = x + 5 3 − 2x and its maximum point M .

(a) Find the exact coordinates of M . [5]

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

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17

(b) Using the substitution u = 3 − 2x, find by integration thearea of the shaded region bounded by
the curve and the x-axis. Give your answer in the form a 13, where a is a rational number. [5]

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

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18

11 The points A and B have position vectors i + 2j − 2k and 2i − j + k respectively. The line l has equation
r = i − j + 3k + - 2i − 3j + 4k.

(a) Show that l does not intersect the line passing through A and B. [5]

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

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19

(b) Find the position vector of the foot of the perpendicular from A to l. [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
*5471210840*

MATHEMATICS 9709/33
Paper 3 Pure Mathematics 3 May/June 2023

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.

JC23 06_9709_33/2R
© UCLES 2023 [Turn over
www.dynamicpapers.com
2

1 Solve the equation ln x + 5 = 5 + ln x. Give your answer correct to 3 decimal places. [4]

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

© UCLES 2023 9709/33/M/J/23

www.dynamicpapers.com
3

2 Find the quotient and remainder when 2x4 − 27 is divided by x2 + x + 3. [3]

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

© UCLES 2023 9709/33/M/J/23 [Turn over

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4

3 On a sketch of an Argand diagram, shade the region whose points represent complex numbers z
satisfying the inequalities z − 3 − i ≤ 3 and z ≥ z − 4i. [4]

© UCLES 2023 9709/33/M/J/23

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5

4 The parametric equations of a curve are

cos 1
x= , y = 1 + 2 cos 1.
2 − sin 1
dy
Show that = 2 − sin 12 . [5]
dx

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

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6

5
y

x
O a 1
6π

The diagram shows the part of the curve y = x2 cos 3x for 0 ≤ x ≤ 16 π, and its maximum point M , where
x = a.
@ A
−1 2
(a) Show that a satisfies the equation a = 3 tan
1 . [3]
3a

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7

(b) Use an iterative formula based on the equation in (a) to determine a correct to 2 decimal places.
Give the result of each iteration to 4 decimal places. [3]

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

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8

6 (a) Express 3 cos x + 2 cos x − 60Å in the form R cos x − !, where R > 0 and 0Å < ! < 90Å.
State the exact value of R and give ! correct to 2 decimal places. [4]

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

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9

(b) Hence solve the equation

3 cos 21 + 2 cos 21 − 60Å = 2.5
for 0Å < 1 < 180Å. [4]

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10

7 (a) Use the substitution u = cos x to show that

π 1
Ó sin 2x e2 cos x dx = Ó 2ue2u du. [4]
0 −1

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

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11
π
(b) Hence find the exact value of Ó sin 2x e2 cos x dx. [4]
0

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

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12

8 The variables x and y satisfy the differential equation

dy y2 + 4
=
dx x y + 4

for x > 0. It is given that x = 4 when y = 2 3.

Solve the differential equation to obtain the value of x when y = 2. [8]

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

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13

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

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14

9 The lines l and m have equations

l : r = ai + 3j + bk + , ci − 2j + 4k,
m : r = i + 2j + 3k + - 2i − 3j + k.
Relative to the origin O, the position vector of the point P is 4i + 7j − 2k.

(a) Given that l is perpendicular to m and that P lies on l, find the values of the constants a, b and c.
[4]

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

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15

(b) The perpendicular from P meets line m at Q. The point R lies on PQ extended, with
PQ : QR = 2 : 3.

Find the position vector of R. [6]

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

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16

21 − 8x − 2x2
10 Let f x = .
1 + 2x 3 − x2

(a) Express f x in partial fractions. [5]

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

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17

(b) Hence obtain the expansion of f x in ascending powers of x, up to and including the term in x2 .
[5]

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

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18

5a − 2i
11 The complex number z is defined by z = , where a is an integer. It is given that arg z = − 14 π.
3 + ai

(a) Find the value of a and hence express z in the form x + iy, where x and y are real. [6]

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

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19

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

(b) Express z3 in the form rei1 , where r > 0 and −π < 1 ≤ π. Give the simplified exact values of
r and 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.

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

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Cambridge International AS & A Level

* 3 3 5 3 6 0 8 2 6 2 *

MATHEMATICS 9709/31
Paper 3 Pure Mathematics 3 May/June 2024

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

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

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

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Expand (3 + x) (1 - 2x) 2 in ascending powers of x, up to and including the term in x 2 , simplifying the
1
1
coefficients. [4]

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2 Solve the equation ln (x - 5) = 7 - ln x . Give your answer correct to 2 decimal places. [4]

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3
y

(1.31, 1.50)

(0.336, 1.00)

O ln x

The variables x and y satisfy the equation a y = bx , where a and b are constants. The graph of y
against ln x is a straight line passing through the points (0.336, 1.00) and (1.31, 1.50), as shown in the
diagram.

Find the values of a and b. Give each value correct to the nearest integer. [4]

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4 The complex number u is given by u = - 1 - i 3 .

(a) Express u in the form r (cos i + i sin i) , where r 2 0 and - r 1 i G r . Give the exact values of r
and i . [2]

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The complex number v is given by v = 5 bcos 16 r + i sin 16 rl.

v
(b) Express the complex number in the form re ii where r 2 0 and - r 1 i G r . [2]
u
............................................................................................................................................................

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e sinx
5 The equation of a curve is y = for 0 G x G 2r .
cos 2 x
dy
Find and hence find the x-coordinates of the stationary points of the curve. [7]
dx
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6 (a) By sketching a suitable pair of graphs, show that the equation cosec 12 x = e x - 3 has exactly one
root, denoted by a , in the interval 0 1 x 1 r . [2]

(b) Verify by calculation that a lies between 1 and 2. [2]

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xn + 1 = ln (cosec 12 xn + 3)

converges, then it converges to a . [1]

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(d) Use this iterative formula with an initial value of 1.4 to determine a correct to 2 decimal places.
Give the result of each iteration to 4 decimal places. [3]

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(e) State the minimum number of calculated iterations needed with this initial value to determine a
correct to 2 decimal places. [1]

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7 (a) On a single Argand diagram sketch the loci given by the equations z - 3 + 2i = 2 and
w - 3 + 2i = w + 3 - 4i where z and w are complex numbers. [4]

(b) Hence find the least value of z - w for points on these loci. Give your answer in an exact form.
[2]

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8 Use the substitution u = 1 - sin x to find the exact value of

3
2r sin 2x
y r 1 - sin x
dx .

Give your answer in the form a + b 2 where a and b are rational numbers to be determined. [7]

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9 The equations of two straight lines l1 and l2 are

l1 : r = i - 2j + 3k + m (2i - j + ak) and l2 : r =- i - j - k + n (3i - 2j - 2k) ,

where a is a constant.

The lines l1 and l2 are perpendicular.

(a) Show that a = 4 . [1]

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The lines l1 and l2 also intersect.

(b) Find the position vector of the point of intersection. [4]

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The point A has position vector - 5i + j - 9k .

(c) Show that A lies on l1 . [2]

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The point B is the image of A after a reflection in the line l2 .

(d) Find the position vector of B. [2]

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dy 2
10 (a) Given that 2x = tan y , show that = . [3]
dx 1 + 4x 2
............................................................................................................................................................

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3

(b) Hence find the exact value of y 1

2
x tan -1 (2x) dx . [7]
2

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11 In a field there are 300 plants of a certain species, all of which can be infected by a particular disease. At
time t after the first plant is infected there are x infected plants. The rate of change of x is proportional
to the product of the number of plants infected and the number of plants that are not yet infected. The
dx
variables x and t are treated as continuous, and it is given that = 0.2 and x = 1 when t = 0 .
dt
(a) Show that x and t satisfy the differential equation
dx
1495 = x (300 - x) . [2]
dt
............................................................................................................................................................

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

(b) Using partial fractions, solve the differential equation and obtain an expression for t in terms of a
single logarithm involving x. [9]

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Additional page

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

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

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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.

© UCLES 2024 9709/31/M/J/24

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Cambridge International AS & A Level

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MATHEMATICS 9709/31
Paper 3 Pure Mathematics 3 May/June 2025

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

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

This document has 20 pages.

DC (PQ/SW) 342946/3
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1 (a) Sketch the graph of y = 2x - 3 . [1]

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(b) Solve the inequality 3x - 1 1 2x - 3 . [2]

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2 It is given that 2 ln p + ln ( p - 1) - 12 ln (q + 1) = 3.

Find q in terms of p. [3]

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z + 5i
3 Find the complex numbers z for which is real and z = 17 . Give your answers in the form
z-5
z = x + iy , where x and y are real. [6]

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