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Cambridge International AS & A Level

CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*5712505207*

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

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

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_12/RP
© UCLES 2021 [Turn over
2

BLANK PAGE

© UCLES 2021 9709/12/F/M/21

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

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

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

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

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

2 By using a suitable substitution, solve the equation

4
2x − 32 − − 3 = 0. [4]
2x − 32

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

© UCLES 2021 9709/12/F/M/21

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

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

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

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

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

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

© UCLES 2021 9709/12/F/M/21

5
y

2
y = f x
1

x
O 1 2 3 4 5 6

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

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

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

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

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

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

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

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

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

© UCLES 2021 9709/12/F/M/21

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

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

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

7 Functions f and g are defined as follows:

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

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

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

© UCLES 2021 9709/12/F/M/21

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

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

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

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

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

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

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

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

© UCLES 2021 9709/12/F/M/21

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

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

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

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

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

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

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

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

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

© UCLES 2021 9709/12/F/M/21

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

Find the 85th term when 1 = 13 π. [4]

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

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

10
A

ka ka
a

E
D

B C

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

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

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

© UCLES 2021 9709/12/F/M/21

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

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

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

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

11
y

A
x
O

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

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

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

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

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

© UCLES 2021 9709/12/F/M/21

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

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

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

© UCLES 2021 9709/12/F/M/21

Additional Page

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

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

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 9709/12/F/M/21

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*2702778238*

MATHEMATICS 9709/22
Paper 2 Pure Mathematics 2 February/March 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 16 pages. Any blank pages are indicated.

JC21 03_9709_22/RP
© UCLES 2021 [Turn over
2

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© UCLES 2021 9709/22/F/M/21

1 (a) Sketch, on the same diagram, the graphs of y = 3x − 5 and y = x + 2. [2]

(b) Solve the equation 3x − 5 = x + 2. [3]

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

© UCLES 2021 9709/22/F/M/21 [Turn over

2 Solve the equation sec2 1 cot 1 = 8 for 0 < 1 < π. [5]

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

© UCLES 2021 9709/22/F/M/21

3 The parametric equations of a curve are

x = e2t cos 4t, y = 3 sin 2t.

Find the gradient of the curve at the point for which t = 0. [5]

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

© UCLES 2021 9709/22/F/M/21 [Turn over

4
y

x
O 1 3

The diagram shows part of the curve with equation y =

5x
4x3 + 1
. The shaded region is bounded by the
curve and the lines x = 1, x = 3 and y = 0.

dy
(a) Find and hence find the x-coordinate of the maximum point. [4]
dx

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

© UCLES 2021 9709/22/F/M/21

(b) Use the trapezium rule with two intervals to find an approximation to the area of the shaded
region. Give your answer correct to 2 significant figures. [3]

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

(c) State, with a reason, whether your answer to part (b) is an over-estimate or under-estimate of the
exact area of the shaded region. [1]

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

© UCLES 2021 9709/22/F/M/21 [Turn over

8
_
(a) Given that 2 ln x + 1 + ln x = ln x + 9, show that x =
9
x+2
5 . [3]

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

© UCLES 2021 9709/22/F/M/21

9
_
(b) It is given that the equation x =
9
x+2
has a single root.

Show by calculation that this root lies between 1.5 and 2.0. [2]

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

(c) Use an iterative formula, based on the equation in part (b), to find the root correct to 3 significant
figures. Give the result of each iteration to 5 significant figures. [3]

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

© UCLES 2021 9709/22/F/M/21 [Turn over

6 The polynomial p x is defined by

p x = x3 + ax + b,

where a and b are constants. It is given that x + 2 is a factor of p x and that the remainder is 5 when
p x is divided by x − 3.

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

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

© UCLES 2021 9709/22/F/M/21

(b) Hence find the exact root of the equation p e2y = 0.

[5]

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

© UCLES 2021 9709/22/F/M/21 [Turn over

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

7 [3]

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

(b) As x varies, find the least possible value of

4 + 5 3 cos x + 5 sin x,

and determine the corresponding value of x where −π < x < π. [3]

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

© UCLES 2021 9709/22/F/M/21

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

(c) Find Ô d1.

1
5 3 cos 31 + 5 sin 312
[3]

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

© UCLES 2021 9709/22/F/M/21

Additional Page

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

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

© UCLES 2021 9709/22/F/M/21

BLANK PAGE

© UCLES 2021 9709/22/F/M/21

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© UCLES 2021 9709/22/F/M/21

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)

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
2

BLANK PAGE

© UCLES 2021 9709/32/F/M/21

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

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

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

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]

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

© UCLES 2021 9709/32/F/M/21

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]

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

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

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]

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

© UCLES 2021 9709/32/F/M/21

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

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

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

8

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]

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

© UCLES 2021 9709/32/F/M/21

9

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

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

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

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

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

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

© UCLES 2021 9709/32/F/M/21

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

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

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

12
` 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]

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

© UCLES 2021 9709/32/F/M/21

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

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

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

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

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

© UCLES 2021 9709/32/F/M/21

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]

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

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

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]

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

© UCLES 2021 9709/32/F/M/21

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

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

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]

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

© UCLES 2021 9709/32/F/M/21

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

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

© UCLES 2021 9709/32/F/M/21

Additional Page

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

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

© UCLES 2021 9709/32/F/M/21

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*9303616557*

MATHEMATICS 9709/42
Paper 4 Mechanics February/March 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 16 pages. Any blank pages are indicated.

JC21 03_9709_42/2R
© UCLES 2021 [Turn over
2

BLANK PAGE

© UCLES 2021 9709/42/F/M/21

1 Two particles P and Q of masses 0.2 kg and 0.3 kg respectively are free to move in a horizontal straight
line on a smooth horizontal plane. P is projected towards Q with speed 0.5 m s−1 . At the same instant
Q is projected towards P with speed 1 m s−1 . Q comes to rest in the resulting collision.

Find the speed of P after the collision. [3]

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

© UCLES 2021 9709/42/F/M/21 [Turn over

2 A car of mass 1400 kg is travelling at constant speed up a straight hill inclined at ! to the horizontal,
where sin ! = 0.1. There is a constant resistance force of magnitude 600 N. The power of the car’s
engine is 22 500 W.

(a) Show that the speed of the car is 11.25 m s−1 . [3]

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

The car, moving with speed 11.25 m s−1 , comes to a section of the hill which is inclined at 2Å to the
horizontal.

(b) Given that the power and resistance force do not change, find the initial acceleration of the car
up this section of the hill. [3]

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

© UCLES 2021 9709/42/F/M/21

3
P

Q 60Å
30Å

A particle Q of mass 0.2 kg is held in equilibrium by two light inextensible strings PQ and QR. P is
a fixed point on a vertical wall and R is a fixed point on a horizontal floor. The angles which strings
PQ and QR make with the horizontal are 60Å and 30Å respectively (see diagram).

Find the tensions in the two strings. [5]

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

© UCLES 2021 9709/42/F/M/21 [Turn over

4
v (m s−1 )

0 t (s)
0 1.5 6 7 13 15 20 21.5

−V

An elevator moves vertically, supported by a cable. The diagram shows a velocity-time graph which
models the motion of the elevator. The graph consists of 7 straight line segments.

The elevator accelerates upwards from rest to a speed of 2 m s−1 over a period of 1.5 s and then travels
at this speed for 4.5 s, before decelerating to rest over a period of 1 s.

The elevator then remains at rest for 6 s, before accelerating to a speed of V m s−1 downwards over a
period of 2 s. The elevator travels at this speed for a period of 5 s, before decelerating to rest over a
period of 1.5 s.

(a) Find the acceleration of the elevator during the first 1.5 s. [1]

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

(b) Given that the elevator starts and finishes its journey on the ground floor, find V . [2]

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

© UCLES 2021 9709/42/F/M/21

(c) The combined weight of the elevator and passengers on its upward journey is 1500 kg. Assuming
that there is no resistance to motion, find the tension in the elevator cable on its upward journey
when the elevator is decelerating. [3]

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

© UCLES 2021 9709/42/F/M/21 [Turn over

5
XN

30Å
5 kg

A block of mass 5 kg is being pulled along a rough horizontal floor by a force of magnitude X N acting
at 30Å above the horizontal (see diagram). The block starts from rest and travels 2 m in the first 5 s of
its motion.

(a) Find the acceleration of the block. [2]

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

(b) Given that the coefficient of friction between the block and the floor is 0.4, find X . [4]

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

© UCLES 2021 9709/42/F/M/21

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

The block is now placed on a part of the floor where the coefficient of friction between the block and
the floor has a different value. The value of X is changed to 25, and the block is now in limiting
equilibrium.

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

© UCLES 2021 9709/42/F/M/21 [Turn over

6 A particle moves in a straight line. It starts from rest from a fixed point O on the line. Its velocity at
3
time t s after leaving O is v m s−1 , where v = t2 − 8t 2 + 10t.

(a) Find the displacement of the particle from O when t = 1. [4]

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

© UCLES 2021 9709/42/F/M/21

(b) Show that the minimum velocity of the particle is −125 m s−1 . [7]

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

© UCLES 2021 9709/42/F/M/21 [Turn over

0.5 kg

0.8 N P

m kg
Q
30Å 45Å

Two particles P and Q of masses 0.5 kg and m kg respectively are attached to the ends of a light
inextensible string. The string passes over a fixed smooth pulley which is attached to the top of two
inclined planes. The particles are initially at rest with P on a smooth plane inclined at 30Å to the
horizontal and Q on a plane inclined at 45Å to the horizontal. The string is taut and the particles can
move on lines of greatest slope of the two planes. A force of magnitude 0.8 N is applied to P acting
down the plane, causing P to move down the plane (see diagram).

(a) It is given that m = 0.3, and that the plane on which Q rests is smooth.

Find the tension in the string. [5]

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

© UCLES 2021 9709/42/F/M/21

(b) It is given instead that the plane on which Q rests is rough, and that after each particle has moved
a distance of 1 m, their speed is 0.6 m s−1 . The work done against friction in this part of the
motion is 0.5 J.

Use an energy method to find the value of m. [5]

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

© UCLES 2021 9709/42/F/M/21

Additional Page

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

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

© UCLES 2021 9709/42/F/M/21

BLANK PAGE

© UCLES 2021 9709/42/F/M/21

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© UCLES 2021 9709/42/F/M/21

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*6258618359*

MATHEMATICS 9709/52
Paper 5 Probability & Statistics 1 February/March 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 12 pages.

JC21 03_9709_52/RP
© UCLES 2021 [Turn over
2

1 A fair spinner with 5 sides numbered 1, 2, 3, 4, 5 is spun repeatedly. The score on each spin is the
number on the side on which the spinner lands.

(a) Find the probability that a score of 3 is obtained for the first time on the 8th spin. [1]

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

(b) Find the probability that fewer than 6 spins are required to obtain a score of 3 for the first time.
[2]

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

© UCLES 2021 9709/52/F/M/21

2 Georgie has a red scarf, a blue scarf and a yellow scarf. Each day she wears exactly one of these
scarves. The probabilities for the three colours are 0.2, 0.45 and 0.35 respectively. When she wears a
red scarf, she always wears a hat. When she wears a blue scarf, she wears a hat with probability 0.4.
When she wears a yellow scarf, she wears a hat with probability 0.3.

(a) Find the probability that on a randomly chosen day Georgie wears a hat. [2]

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

(b) Find the probability that on a randomly chosen day Georgie wears a yellow scarf given that she
does not wear a hat. [3]

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

© UCLES 2021 9709/52/F/M/21 [Turn over

3 The time spent by shoppers in a large shopping centre has a normal distribution with mean 96 minutes
and standard deviation 18 minutes.

(a) Find the probability that a shopper chosen at random spends between 85 and 100 minutes in the
shopping centre. [3]

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

88% of shoppers spend more than t minutes in the shopping centre.

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

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

© UCLES 2021 9709/52/F/M/21

4 The random variable X takes the values 1, 2, 3, 4 only. The probability that X takes the value x is
kx 5 − x, where k is a constant.

(a) Draw up the probability distribution table for X , in terms of k. [2]

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

(b) Show that Var X = 1.05. [4]

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

© UCLES 2021 9709/52/F/M/21 [Turn over

5 A driver records the distance travelled in each of 150 journeys. These distances, correct to the nearest
km, are summarised in the following table.

Distance (km) 0−4 5 − 10 11 − 20 21 − 30 31 − 40 41 − 60

Frequency 12 16 32 66 20 4

(a) Draw a cumulative frequency graph to illustrate the data. [4]

© UCLES 2021 9709/52/F/M/21

(b) For 30% of these journeys the distance travelled is d km or more.

Use your graph to estimate the value of d . [2]

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

© UCLES 2021 9709/52/F/M/21 [Turn over

6 (a) Find the total number of different arrangements of the 11 letters in the word CATERPILLAR.
[2]

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

(b) Find the total number of different arrangements of the 11 letters in the word CATERPILLAR in
which there is an R at the beginning and an R at the end, and the two As are not together. [4]

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

© UCLES 2021 9709/52/F/M/21

(c) Find the total number of different selections of 6 letters from the 11 letters of the word
CATERPILLAR that contain both Rs and at least one A and at least one L. [4]

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

© UCLES 2021 9709/52/F/M/21 [Turn over

7 There are 400 students at a school in a certain country. Each student was asked whether they preferred
swimming, cycling or running and the results are given in the following table.

Swimming Cycling Running

Female 104 50 66
Male 31 57 92

A student is chosen at random.

(a) (i) Find the probability that the student prefers swimming. [1]

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

(ii) Determine whether the events ‘the student is male’ and ‘the student prefers swimming’ are
independent, justifying your answer. [2]

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

© UCLES 2021 9709/52/F/M/21

On average at all the schools in this country 30% of the students do not like any sports.

(b) (i) 10 of the students from this country are chosen at random.
Find the probability that at least 3 of these students do not like any sports. [3]

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

(ii) 90 students from this country are now chosen at random.

Use an approximation to find the probability that fewer than 32 of them do not like any
sports. [5]

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

© UCLES 2021 9709/52/F/M/21

Additional Page

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

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

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Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*9228178848*

MATHEMATICS 9709/62
Paper 6 Probability & Statistics 2 February/March 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 12 pages. Any blank pages are indicated.

JC21 03_9709_62/2R
© UCLES 2021 [Turn over
2

BLANK PAGE

© UCLES 2021 9709/62/F/M/21

1 A construction company notes the time, t days, that it takes to build each house of a certain design.
The results for a random sample of 60 such houses are summarised as follows.

Σ t = 4820 Σ t2 = 392 050

(a) Calculate a 98% confidence interval for the population mean time. [6]

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

(b) Explain why it was necessary to use the Central Limit theorem in part (a). [1]

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

© UCLES 2021 9709/62/F/M/21 [Turn over

2
f x

1
2
k

x
O k

The diagram shows the graph of the probability density function, f, of a random variable X .

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

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

(b) Using this value of k, find f x for 0 ≤ x ≤ k and hence find E X . [3]

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

© UCLES 2021 9709/62/F/M/21

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

© UCLES 2021 9709/62/F/M/21 [Turn over

3 An architect wishes to investigate whether the buildings in a certain city are higher, on average, than
buildings in other cities. He takes a large random sample of buildings from the city and finds the
mean height of the buildings in the sample. He calculates the value of the test statistic, z, and finds
that z = 2.41.

(a) Explain briefly whether he should use a one-tail test or a two-tail test. [1]

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

(b) Carry out the test at the 1% significance level. [3]

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

4 On average, 1 in 400 microchips made at a certain factory are faulty. The number of faulty microchips
in a random sample of 1000 is denoted by X .

(a) State the distribution of X , giving the values of any parameters. [1]

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

(b) State an approximating distribution for X , giving the values of any parameters. [2]

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

© UCLES 2021 9709/62/F/M/21

(c) Use this approximating distribution to find each of the following.

(i) P X = 4. [2]

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

(ii) P 2 ≤ X ≤ 4. [2]

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

(d) Use a suitable approximating distribution to find the probability that, in a random sample of 700
microchips, there will be at least 1 faulty one. [3]

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

© UCLES 2021 9709/62/F/M/21 [Turn over

5 The volumes, in litres, of juice in large and small bottles have the distributions N 5.10, 0.0102 and
N 2.51, 0.0036 respectively.

(a) Find the probability that the total volume of juice in 3 randomly chosen large bottles and
4 randomly chosen small bottles is less than 25.5 litres. [5]

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

© UCLES 2021 9709/62/F/M/21

(b) Find the probability that the volume of juice in a randomly chosen large bottle is at least twice
the volume of juice in a randomly chosen small bottle. [5]

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

© UCLES 2021 9709/62/F/M/21 [Turn over

6 It is known that 8% of adults in a certain town own a Chantor car. After an advertising campaign, a
car dealer wishes to investigate whether this proportion has increased. He chooses a random sample
of 25 adults from the town and notes how many of them own a Chantor car.

(a) He finds that 4 of the 25 adults own a Chantor car.

Carry out a hypothesis test at the 5% significance level. [5]

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

© UCLES 2021 9709/62/F/M/21

(b) Explain which of the errors, Type I or Type II, might have been made in carrying out the test in
part (a). [2]

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

Later, the car dealer takes another random sample of 25 adults from the town and carries out a similar
hypothesis test at the 5% significance level.

(c) Find the probability of a Type I error. [3]

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

© UCLES 2021 9709/62/F/M/21

Additional Page

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

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

© UCLES 2021 9709/62/F/M/21

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*7942312992*

MATHEMATICS 9709/11
Paper 1 Pure Mathematics 1 May/June 2021

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

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

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

JC21 06_9709_11/RP
© UCLES 2021 [Turn over
2

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

Find the equation of the curve. [4]

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

© UCLES 2021 9709/11/M/J/21

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

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

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

© UCLES 2021 9709/11/M/J/21 [Turn over

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

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

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

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

© UCLES 2021 9709/11/M/J/21

4
y

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

−4

−6

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

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

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

© UCLES 2021 9709/11/M/J/21 [Turn over

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

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

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

© UCLES 2021 9709/11/M/J/21

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

tangent to the curve.

Find the value of k. [5]

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

© UCLES 2021 9709/11/M/J/21 [Turn over

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

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

© UCLES 2021 9709/11/M/J/21

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

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

© UCLES 2021 9709/11/M/J/21 [Turn over

8
P Q

S R

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

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

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

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

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

© UCLES 2021 9709/11/M/J/21

11

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

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

© UCLES 2021 9709/11/M/J/21 [Turn over

9 Functions f and g are defined as follows:

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

where a is a constant.

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

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

(b) Find f −1 x. [2]

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

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

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

© UCLES 2021 9709/11/M/J/21

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

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

© UCLES 2021 9709/11/M/J/21 [Turn over

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

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

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

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

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

© UCLES 2021 9709/11/M/J/21

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

© UCLES 2021 9709/11/M/J/21 [Turn over

16

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

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

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

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

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

© UCLES 2021 9709/11/M/J/21

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

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

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

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

© UCLES 2021 9709/11/M/J/21

Additional Page

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

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BLANK PAGE

© UCLES 2021 9709/11/M/J/21

BLANK PAGE

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Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*2355526103*

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

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

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

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

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

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

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

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

Find the value of this root. [2]

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

© UCLES 2021 9709/12/M/J/21

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

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

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

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

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

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

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

4

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

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

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

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

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

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

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

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

© UCLES 2021 9709/12/M/J/21

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

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

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

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

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

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

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

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

© UCLES 2021 9709/12/M/J/21

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

Find the values of p and q. [4]

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

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

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

centre 5, 11 and radius 52.

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

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

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

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

© UCLES 2021 9709/12/M/J/21

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

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

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

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

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

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

9
y

y2 = x − 2

x
0 5

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

Find the volume obtained. [6]

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

© UCLES 2021 9709/12/M/J/21

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

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

1 + sin x 1 − sin x 4 tan x

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

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

© UCLES 2021 9709/12/M/J/21

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

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

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

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

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

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

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

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

© UCLES 2021 9709/12/M/J/21

d2 y
(c) Find . [2]
dx2

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

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

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

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

12
Q
P
A B

F C

E D

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

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

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

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

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

© UCLES 2021 9709/12/M/J/21

17

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

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

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

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

© UCLES 2021 9709/12/M/J/21

Additional Page

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

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Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*6412245051*

MATHEMATICS 9709/13
Paper 1 Pure Mathematics 1 May/June 2021

1 hour 50 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

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

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

JC21 06_9709_13/RP
© UCLES 2021 [Turn over
2

BLANK PAGE

© UCLES 2021 9709/13/M/J/21

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

Find f x. [3]

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

© UCLES 2021 9709/13/M/J/21 [Turn over

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

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

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

© UCLES 2021 9709/13/M/J/21

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

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

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

© UCLES 2021 9709/13/M/J/21 [Turn over

4 (a) Show that the equation

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

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

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

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

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

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

© UCLES 2021 9709/13/M/J/21

5
A

D
4 cm

B C

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

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

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

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

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

© UCLES 2021 9709/13/M/J/21 [Turn over

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

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

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

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

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

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

© UCLES 2021 9709/13/M/J/21

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

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

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

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

© UCLES 2021 9709/13/M/J/21 [Turn over

8 Functions f and g are defined as follows:

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

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

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

© UCLES 2021 9709/13/M/J/21

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

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

© UCLES 2021 9709/13/M/J/21 [Turn over

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

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

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

© UCLES 2021 9709/13/M/J/21

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

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

© UCLES 2021 9709/13/M/J/21 [Turn over

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

(a) Show that angle ABC = 90Å. [2]

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

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

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

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

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

© UCLES 2021 9709/13/M/J/21

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

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

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

© UCLES 2021 9709/13/M/J/21 [Turn over

11
y

1 1
y = x 2 + k2 x− 2

x
O 9 2
4k2
4k

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

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

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

© UCLES 2021 9709/13/M/J/21

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

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

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

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

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

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

© UCLES 2021 9709/13/M/J/21

Additional Page

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

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

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© UCLES 2021 9709/13/M/J/21

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© UCLES 2021 9709/13/M/J/21

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*0484787472*

MATHEMATICS 9709/21
Paper 2 Pure Mathematics 2 May/June 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 16 pages. Any blank pages are indicated.

JC21 06_9709_21/RP
© UCLES 2021 [Turn over
2

BLANK PAGE

© UCLES 2021 9709/21/M/J/21

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

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

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

2 By first expanding sin 1 + 30Å, solve the equation sin 1 + 30Å cosec 1 = 2 for 0Å < 1 < 360Å. [6]

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

© UCLES 2021 9709/21/M/J/21

3 (a) Show that sec x + cos x2 can be expressed as sec2 x + a + b cos 2x, where a and b are constants
to be determined. [2]

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

1π

(b) Hence find the exact value of Ó sec x + cos x2 dx.
4
[4]
0

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

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

4 A curve has parametric equations

x = ln 2t + 6 − ln t, y = t ln t.

(a) Find the value of t at the point P on the curve for which x = ln 4. [3]

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

© UCLES 2021 9709/21/M/J/21

(b) Find the exact gradient of the curve at P. [5]

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

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

5
y

x
O

3x + 2
The diagram shows the curve with equation y = . The curve has a minimum point M .
ln x

3x + 2
and show that the x-coordinate of M satisfies the equation x =
dy
(a) Find an expression for .
dx 3 ln x
[3]

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

© UCLES 2021 9709/21/M/J/21

(b) Use the equation in part (a) to show by calculation that the x-coordinate of M lies between
3 and 4. [2]

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

(c) Use an iterative formula, based on the equation in part (a), to find the x-coordinate of M correct
to 5 significant figures. Give the result of each iteration to 7 significant figures. [3]

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

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

10
4
(a) Use the trapezium rule with three intervals to find an approximation to Ô
6
1+ x
6 dx. Give your
1
answer correct to 5 significant figures. [3]

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

1 x−2
(b) Find the exact value of Ó 2e 2
4
dx. [3]
1

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

© UCLES 2021 9709/21/M/J/21

1
y = 2e 2 x−2

6
y=
1+ x

x
O 1 4

1 x−2
The diagram shows the curves y = and y = 2e 2 which meet at a point with x-coordinate 4.
6
1+ x
The shaded region is bounded by the two curves and the line x = 1.

Use your answers to parts (a) and (b) to find an approximation to the area of the shaded region.
Give your answer correct to 3 significant figures. [2]

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

(d) State, with a reason, whether your answer to part (c) is an over-estimate or under-estimate of the
exact area of the shaded region. [1]

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

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

7 The polynomial p x is defined by

p x = ax3 − 11x2 − 19x − a,

where a is a constant. It is given that x − 3 is a factor of p x.

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

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

(b) When a has this value, factorise p x completely. [3]

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

© UCLES 2021 9709/21/M/J/21

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

© UCLES 2021 9709/21/M/J/21

Additional Page

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

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Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*4966108877*

MATHEMATICS 9709/22
Paper 2 Pure Mathematics 2 May/June 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 12 pages.

JC21 06_9709_22/RP
© UCLES 2021 [Turn over
2

1 (a) Solve the equation ln 2 + x − ln x = 2 ln 3. [3]

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

(b) Hence solve the equation ln 2 + cot y − ln cot y = 2 ln 3 for 0 < y < 12 π. Give your answer correct
to 4 significant figures. [2]

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

© UCLES 2021 9709/22/M/J/21

2 The solutions of the equation 5 x = 5 − 2x are x = a and x = b, where a < b.

Find the value of 3a − 1 + 7b − 1. [5]

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

© UCLES 2021 9709/22/M/J/21 [Turn over

3 Solve the equation sin 21 + 30Å = 5 cos 21 + 60Å for 0Å < 1 < 180Å. [6]

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

© UCLES 2021 9709/22/M/J/21

5
2
4 (a) Find the exact value of Ó 6e2x+1 dx. [3]
0

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

(b) Find Ó tan2 x + 4 sin2 2x dx. [5]

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

© UCLES 2021 9709/22/M/J/21 [Turn over

5 (a) Find the quotient when x4 − 32x + 55 is divided by x − 22 and show that the remainder is 7.
[3]

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

© UCLES 2021 9709/22/M/J/21

(b) Factorise x4 − 32x + 48. [2]

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

© UCLES 2021 9709/22/M/J/21 [Turn over

6
y

A B x
O

The diagram shows the curve with equation

y = ln x2 − 2 ln x.
The curve crosses the x-axis at the points A and B, and has a minimum point M .

(a) Find the exact value of the gradient of the curve at each of the points A and B. [6]

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

© UCLES 2021 9709/22/M/J/21

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

(b) Find the exact x-coordinate of M . [2]

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

© UCLES 2021 9709/22/M/J/21 [Turn over

7
y

x
O

The diagram shows the curve with parametric equations

x = 4t + e2t , y = 6t sin 2t,

for 0 ≤ t ≤ 1. The point P on the curve has parameter p and y-coordinate 3.

1
(a) Show that p = . [1]
2 sin 2p

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

(b) Show by calculation that the value of p lies between 0.5 and 0.6. [2]

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

(c) Use an iterative formula, based on the equation in part (a), to find the value of p correct
to 3 significant figures. Use an initial value of 0.55 and give the result of each iteration to
5 significant figures. [3]

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

© UCLES 2021 9709/22/M/J/21

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

(d) Find the gradient of the curve at P. [5]

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

© UCLES 2021 9709/22/M/J/21

Additional Page

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

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

© UCLES 2021 9709/22/M/J/21

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*0329251367*

MATHEMATICS 9709/23
Paper 2 Pure Mathematics 2 May/June 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 12 pages.

JC21 06_9709_23/FP
© UCLES 2021 [Turn over
2

1 (a) Solve the equation ln 2 + x − ln x = 2 ln 3. [3]

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

(b) Hence solve the equation ln 2 + cot y − ln cot y = 2 ln 3 for 0 < y < 12 π. Give your answer correct
to 4 significant figures. [2]

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

© UCLES 2021 9709/23/M/J/21

2 The solutions of the equation 5 x = 5 − 2x are x = a and x = b, where a < b.

Find the value of 3a − 1 + 7b − 1. [5]

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

© UCLES 2021 9709/23/M/J/21 [Turn over

3 Solve the equation sin 21 + 30Å = 5 cos 21 + 60Å for 0Å < 1 < 180Å. [6]

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

© UCLES 2021 9709/23/M/J/21

5
2
4 (a) Find the exact value of Ó 6e2x+1 dx. [3]
0

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

(b) Find Ó tan2 x + 4 sin2 2x dx. [5]

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

© UCLES 2021 9709/23/M/J/21 [Turn over

5 (a) Find the quotient when x4 − 32x + 55 is divided by x − 22 and show that the remainder is 7.
[3]

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

© UCLES 2021 9709/23/M/J/21

(b) Factorise x4 − 32x + 48. [2]

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

© UCLES 2021 9709/23/M/J/21 [Turn over

6
y

A B x
O

The diagram shows the curve with equation

y = ln x2 − 2 ln x.
The curve crosses the x-axis at the points A and B, and has a minimum point M .

(a) Find the exact value of the gradient of the curve at each of the points A and B. [6]

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

© UCLES 2021 9709/23/M/J/21

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

(b) Find the exact x-coordinate of M . [2]

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

© UCLES 2021 9709/23/M/J/21 [Turn over

7
y

x
O

The diagram shows the curve with parametric equations

x = 4t + e2t , y = 6t sin 2t,

for 0 ≤ t ≤ 1. The point P on the curve has parameter p and y-coordinate 3.

1
(a) Show that p = . [1]
2 sin 2p

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

(b) Show by calculation that the value of p lies between 0.5 and 0.6. [2]

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

© UCLES 2021 9709/23/M/J/21

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

(d) Find the gradient of the curve at P. [5]

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

© UCLES 2021 9709/23/M/J/21

Additional Page

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

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

© UCLES 2021 9709/23/M/J/21

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
© UCLES 2021 [Turn over
2

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

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

© UCLES 2021 9709/31/M/J/21

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]

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

© UCLES 2021 9709/31/M/J/21 [Turn over

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]

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

© UCLES 2021 9709/31/M/J/21

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

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

© UCLES 2021 9709/31/M/J/21 [Turn over

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]

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

© UCLES 2021 9709/31/M/J/21

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

© UCLES 2021 9709/31/M/J/21 [Turn over

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]

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

© UCLES 2021 9709/31/M/J/21

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

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

© UCLES 2021 9709/31/M/J/21 [Turn over

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]

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

© UCLES 2021 9709/31/M/J/21

(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]

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

© UCLES 2021 9709/31/M/J/21 [Turn over

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]

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

© UCLES 2021 9709/31/M/J/21

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

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

© UCLES 2021 9709/31/M/J/21 [Turn over

− 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]

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

© UCLES 2021 9709/31/M/J/21

(b) Show that Ó y dx = 18 ln 2 − 9.

8
[5]
1

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

© UCLES 2021 9709/31/M/J/21 [Turn over

= 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]

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

© UCLES 2021 9709/31/M/J/21

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

© UCLES 2021 9709/31/M/J/21

Additional Page

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

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

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BLANK PAGE

© UCLES 2021 9709/31/M/J/21

BLANK PAGE

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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
2

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

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

© UCLES 2021 9709/32/M/J/21

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

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]

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

© UCLES 2021 9709/32/M/J/21

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

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

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

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]

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

© UCLES 2021 9709/32/M/J/21

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

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

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

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]

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

© UCLES 2021 9709/32/M/J/21

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

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

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]

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

© UCLES 2021 9709/32/M/J/21

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

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

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

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

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

© UCLES 2021 9709/32/M/J/21

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

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

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

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]

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

© UCLES 2021 9709/32/M/J/21

(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]

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

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

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]

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

© UCLES 2021 9709/32/M/J/21

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]

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

© UCLES 2021 9709/32/M/J/21

Additional Page

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

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

© UCLES 2021 9709/32/M/J/21

BLANK PAGE

© UCLES 2021 9709/32/M/J/21

BLANK PAGE

© UCLES 2021 9709/32/M/J/21

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
2

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

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

© UCLES 2021 9709/33/M/J/21

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

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

© UCLES 2021 9709/33/M/J/21 [Turn over

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]

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

© UCLES 2021 9709/33/M/J/21

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

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

© UCLES 2021 9709/33/M/J/21 [Turn over

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]

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

© UCLES 2021 9709/33/M/J/21

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

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

© UCLES 2021 9709/33/M/J/21 [Turn over

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]

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

© UCLES 2021 9709/33/M/J/21

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]

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

© UCLES 2021 9709/33/M/J/21 [Turn over

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

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

© UCLES 2021 9709/33/M/J/21

(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]

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

© UCLES 2021 9709/33/M/J/21 [Turn over

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]

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

© UCLES 2021 9709/33/M/J/21

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

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

© UCLES 2021 9709/33/M/J/21 [Turn over

−−¿ −−¿
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]

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

© UCLES 2021 9709/33/M/J/21

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

© UCLES 2021 9709/33/M/J/21 [Turn over

(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]

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

© UCLES 2021 9709/33/M/J/21

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

© UCLES 2021 9709/33/M/J/21

Additional Page

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

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

© UCLES 2021 9709/33/M/J/21

BLANK PAGE

© UCLES 2021 9709/33/M/J/21

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Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*2776439369*

MATHEMATICS 9709/41
Paper 4 Mechanics May/June 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 12 pages.

JC21 06_9709_41/RP
© UCLES 2021 [Turn over
2

1 A winch operates by means of a force applied by a rope. The winch is used to pull a load of mass
50 kg up a line of greatest slope of a plane inclined at 60Å to the horizontal. The winch pulls the load
a distance of 5 m up the plane at constant speed. There is a constant resistance to motion of 100 N.

Find the work done by the winch. [3]

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

© UCLES 2021 9709/41/M/J/21

A B
m kg 0.1 kg

0.9 m

Two particles A and B have masses m kg and 0.1 kg respectively, where m > 0.1. The particles are
attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and
the particles hang vertically below it. Both particles are at a height of 0.9 m above horizontal ground
(see diagram). The system is released from rest, and while both particles are in motion the tension in
the string is 1.5 N. Particle B does not reach the pulley.

(a) Find m. [4]

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

(b) Find the speed at which A reaches the ground. [2]

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

© UCLES 2021 9709/41/M/J/21 [Turn over

3 Three particles P, Q and R, of masses 0.1 kg, 0.2 kg and 0.5 kg respectively, are at rest in a straight
line on a smooth horizontal plane. Particle P is projected towards Q at a speed of 5 m s−1 . After P
and Q collide, P rebounds with speed 1 m s−1 .

(a) Find the speed of Q immediately after the collision with P. [3]

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

Q now collides with R. Immediately after the collision with Q, R begins to move with speed V m s−1 .

(b) Given that there is no subsequent collision between P and Q, find the greatest possible value
of V . [3]

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

© UCLES 2021 9709/41/M/J/21

4 Two cyclists, Isabella and Maria, are having a race. They both travel along a straight road with
constant acceleration, starting from rest at point A.

Isabella accelerates for 5 s at a constant rate a m s−2 . She then travels at the constant speed she has
reached for 10 s, before decelerating to rest at a constant rate over a period of 5 s.

Maria accelerates at a constant rate, reaching a speed of 5 m s−1 in a distance of 27.5 m. She then
maintains this speed for a period of 10 s, before decelerating to rest at a constant rate over a period
of 5 s.

(a) Given that a = 1.1, find which cyclist travels further. [5]

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

(b) Find the value of a for which the two cyclists travel the same distance. [2]

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

© UCLES 2021 9709/41/M/J/21 [Turn over

5 A particle moving in a straight line starts from rest at a point A and comes instantaneously to rest at a
point B. The acceleration of the particle at time t s after leaving A is a m s−2 , where
1
a = 6t 2 − 2t.

(a) Find the value of t at point B. [3]

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

© UCLES 2021 9709/41/M/J/21

(b) Find the distance travelled from A to the point at which the acceleration of the particle is again
zero. [5]

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

© UCLES 2021 9709/41/M/J/21 [Turn over

6
y

20 N 10 N

30Å !Å
x
O 60Å

25 N

Three coplanar forces of magnitudes 10 N, 25 N and 20 N act at a point O in the directions shown in
the diagram.

(a) Given that the component of the resultant force in the x-direction is zero, find !, and hence find
the magnitude of the resultant force. [4]

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

© UCLES 2021 9709/41/M/J/21

(b) Given instead that ! = 45, find the magnitude and direction of the resultant of the three forces.
[5]

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

© UCLES 2021 9709/41/M/J/21 [Turn over

7
P
35 kg

2.5 m

30Å

A slide in a playground descends at a constant angle of 30Å for 2.5 m. It then has a horizontal section
in the same vertical plane as the sloping section. A child of mass 35 kg, modelled as a particle P,
starts from rest at the top of the slide and slides straight down the sloping section. She then continues
along the horizontal section until she comes to rest (see diagram). There is no instantaneous change
in speed when the child goes from the sloping section to the horizontal section.

The child experiences a resistance force on the horizontal section of the slide, and the work done
against the resistance force on the horizontal section of the slide is 250 J per metre.

(a) It is given that the sloping section of the slide is smooth.

(i) Find the speed of the child when she reaches the bottom of the sloping section. [3]

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

(ii) Find the distance that the child travels along the horizontal section of the slide before she
comes to rest. [2]

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

© UCLES 2021 9709/41/M/J/21

(b) It is given instead that the sloping section of the slide is rough and that the child comes to rest on
the slide 1.05 m after she reaches the horizontal section.

Find the coefficient of friction between the child and the sloping section of the slide. [6]

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

© UCLES 2021 9709/41/M/J/21

Additional Page

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must be clearly shown.

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© UCLES 2021 9709/41/M/J/21

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*4405618753*

MATHEMATICS 9709/42
Paper 4 Mechanics May/June 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 12 pages.

1 A particle of mass 0.6 kg is projected with a speed of 4 m s−1 down a line of greatest slope of a smooth
plane inclined at 10Å to the horizontal.

Use an energy method to find the speed of the particle after it has moved 15 m down the plane. [3]

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

2
34 N

30 N

26 N

Coplanar forces of magnitudes 34 N, 30 N and 26 N act at a point in the directions shown in the
diagram.

5 and sin 1 = 8 , find the magnitude and direction of the resultant of the three
Given that sin ! = 13 17
forces. [6]

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

3 A ring of mass 0.3 kg is threaded on a horizontal rough rod. The coefficient of friction between the
ring and the rod is 0.8. A force of magnitude 8 N acts on the ring. This force acts at an angle of 10Å
above the horizontal in the vertical plane containing the rod.

Find the time taken for the ring to move, from rest, 0.6 m along the rod. [6]

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

4 A particle of mass 12 kg is stationary on a rough plane inclined at an angle of 25Å to the horizontal. A
pulling force of magnitude P N acts at an angle of 8Å above a line of greatest slope of the plane. This
force is used to keep the particle in equilibrium. The coefficient of friction between the particle and
the plane is 0.3.

Find the greatest possible value of P. [6]

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

5 A car of mass 1250 kg is pulling a caravan of mass 800 kg along a straight road. The resistances to the
motion of the car and caravan are 440 N and 280 N respectively. The car and caravan are connected
by a light rigid tow-bar.

(a) The car and caravan move along a horizontal part of the road at a constant speed of 30 m s−1 .
(i) Calculate, in kW, the power developed by the engine of the car. [2]

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

(ii) Given that this power is suddenly decreased by 8 kW, find the instantaneous deceleration of
the car and caravan and the tension in the tow-bar. [4]

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

(b) The car and caravan now travel along a part of the road inclined at sin−1 0.06 to the horizontal.
The car and caravan travel up the incline at constant speed with the engine of the car working at
28 kW.
(i) Find this constant speed. [3]

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

(ii) Find the increase in the potential energy of the caravan in one minute. [2]

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

6 A particle A is projected vertically upwards from level ground with an initial speed of 30 m s−1 . At
the same instant a particle B is released from rest 15 m vertically above A. The mass of one of the
particles is twice the mass of the other particle. During the subsequent motion A and B collide and
coalesce to form particle C.

Find the difference between the two possible times at which C hits the ground. [8]

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

7 A particle P moving in a straight line starts from rest at a point O and comes to rest 16 s later. At time
t s after leaving O, the acceleration a m s−2 of P is given by
a = 6 + 4t 0 ≤ t < 2,
a = 14 2 ≤ t < 4,
a = 16 − 2t 4 ≤ t ≤ 16.
There is no sudden change in velocity at any instant.

(a) Find the values of t when the velocity of P is 55 m s−1 . [5]

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

(b) Complete the sketch of the velocity-time diagram. [2]

v (m s−1 )

t (s)
0 2 4 16

(c) Find the distance travelled by P when it is decelerating. [3]

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

Additional Page

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must be clearly shown.

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

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*9782984789*

MATHEMATICS 9709/43
Paper 4 Mechanics May/June 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 12 pages.

1 Particles P of mass 0.4 kg and Q of mass 0.5 kg are free to move on a smooth horizontal plane. P and
Q are moving directly towards each other with speeds 2.5 m s−1 and 1.5 m s−1 respectively. After P
and Q collide, the speed of Q is twice the speed of P.

Find the two possible values of the speed of P after the collision. [4]

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

2 A cyclist is travelling along a straight horizontal road. She is working at a constant rate of 150 W. At
an instant when her speed is 4 m s−1 , her acceleration is 0.25 m s−2 . The resistance to motion is 20 N.

(a) Find the total mass of the cyclist and her bicycle. [3]

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

The cyclist comes to a straight hill inclined at an angle 1 above the horizontal. She ascends the hill at
constant speed 3 m s−1 . She continues to work at the same rate as before and the resistance force is
unchanged.

(b) Find the value of 1. [2]

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

3
20 N
FN

1Å 60Å
"Å !Å
30 N

40 N

Four coplanar forces act at a point. The magnitudes of the forces are 20 N, 30 N, 40 N and F N. The
directions of the forces are as shown in the diagram, where sin !Å = 0.28 and sin "Å = 0.6.

Given that the forces are in equilibrium, find F and 1. [6]

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

4 A particle is projected vertically upwards with speed u m s−1 from a point on horizontal ground. After
2 seconds, the height of the particle above the ground is 24 m.

(a) Show that u = 22. [2]

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

(b) The height of the particle above the ground is more than h m for a period of 3.6 s.

Find h. [4]

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

5 A car of mass 1400 kg is towing a trailer of mass 500 kg down a straight hill inclined at an angle of 5Å
to the horizontal. The car and trailer are connected by a light rigid tow-bar. At the top of the hill the
speed of the car and trailer is 20 m s−1 and at the bottom of the hill their speed is 30 m s−1 .

(a) It is given that as the car and trailer descend the hill, the engine of the car does 150 000 J of work,
and there are no resistance forces.

Find the length of the hill. [5]

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

(b) It is given instead that there is a resistance force of 100 N on the trailer, the length of the hill is
200 m, and the acceleration of the car and trailer is constant.

Find the tension in the tow-bar between the car and trailer. [4]

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

6 A particle moves in a straight line and passes through the point A at time t = 0. The velocity of the
particle at time t s after leaving A is v m s−1 , where

v = 2t2 − 5t + 3.

(a) Find the times at which the particle is instantaneously at rest. Hence or otherwise find the
minimum velocity of the particle. [4]

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

(b) Sketch the velocity-time graph for the first 3 seconds of motion. [3]

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

7
4N P
0.3 kg
1

A particle P of mass 0.3 kg rests on a rough plane inclined at an angle 1 to the horizontal, where
7 . A horizontal force of magnitude 4 N, acting in the vertical plane containing a line of
sin 1 = 25
greatest slope of the plane, is applied to P (see diagram). The particle is on the point of sliding up the
plane.

(a) Show that the coefficient of friction between the particle and the plane is 43 . [4]

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

The force acting horizontally is replaced by a force of magnitude 4 N acting up the plane parallel to a
line of greatest slope.

(b) Find the acceleration of P. [3]

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

Find the total distance travelled until P comes to instantaneous rest. [3]

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

Additional Page

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

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

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*2153885552*

MATHEMATICS 9709/51
Paper 5 Probability & Statistics 1 May/June 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 16 pages. Any blank pages are indicated.

1 A bag contains 12 marbles, each of a different size. 8 of the marbles are red and 4 of the marbles are
blue.

How many different selections of 5 marbles contain at least 4 marbles of the same colour? [4]

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

2 A company produces a particular type of metal rod. The lengths of these rods are normally distributed
with mean 25.2 cm and standard deviation 0.4 cm. A random sample of 500 of these rods is chosen.

How many rods in this sample would you expect to have a length that is within 0.5 cm of the mean
length? [5]

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

3 (a) How many different arrangements are there of the 8 letters in the word RELEASED? [1]

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

(b) How many different arrangements are there of the 8 letters in the word RELEASED in which the
letters LED appear together in that order? [3]

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

(c) An arrangement of the 8 letters in the word RELEASED is chosen at random.

Find the probability that the letters A and D are not together. [4]

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

4 To gain a place at a science college, students first have to pass a written test and then a practical test.

Each student is allowed a maximum of two attempts at the written test. A student is only allowed
a second attempt if they fail the first attempt. No student is allowed more than one attempt at the
practical test. If a student fails both attempts at the written test, then they cannot attempt the practical
test.

The probability that a student will pass the written test at the first attempt is 0.8. If a student fails the
first attempt at the written test, the probability that they will pass at the second attempt is 0.6. The
probability that a student will pass the practical test is always 0.3.

(a) Draw a tree diagram to represent this information, showing the probabilities on the branches.
[3]

(b) Find the probability that a randomly chosen student will succeed in gaining a place at the college.
[2]

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

(c) Find the probability that a randomly chosen student passes the written test at the first attempt
given that the student succeeds in gaining a place at the college. [2]

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

5 The times taken by 200 players to solve a computer puzzle are summarised in the following table.

Time (t seconds) 0 ≤ t < 10 10 ≤ t < 20 20 ≤ t < 40 40 ≤ t < 60 60 ≤ t < 100

Number of players 16 54 78 32 20

(a) Draw a histogram to represent this information. [4]

(b) Calculate an estimate of the mean time taken by these 200 players. [2]

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

6 In Questa, 60% of the adults travel to work by car.

(a) A random sample of 12 adults from Questa is taken.

Find the probability that the number who travel to work by car is less than 10. [3]

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

(b) A random sample of 150 adults from Questa is taken.

Use an approximation to find the probability that the number who travel to work by car is less
than 81. [5]

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

(c) Justify the use of your approximation in part (b). [1]

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

7 Sharma knows that she has 3 tins of carrots, 2 tins of peas and 2 tins of sweetcorn in her cupboard.
All the tins are the same shape and size, but the labels have all been removed, so Sharma does not
know what each tin contains.

Sharma wants carrots for her meal, and she starts opening the tins one at a time, chosen randomly,
until she opens a tin of carrots. The random variable X is the number of tins that she needs to open.

6.
(a) Show that P X = 3 = 35 [2]

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

(b) Draw up the probability distribution table for X . [4]

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

(c) Find Var X . [3]

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

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

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Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*0009306270*

MATHEMATICS 9709/52
Paper 5 Probability & Statistics 1 May/June 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 12 pages.

1 An ordinary fair die is thrown repeatedly until a 5 is obtained. The number of throws taken is denoted
by the random variable X .

(a) Write down the mean of X . [1]

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

(b) Find the probability that a 5 is first obtained after the 3rd throw but before the 8th throw. [2]

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

2 The weights of bags of sugar are normally distributed with mean 1.04 kg and standard deviation 3 kg.
In a random sample of 2000 bags of sugar, 72 weighed more than 1.10 kg.

Find the value of 3. [4]

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

3 On each day that Alexa goes to work, the probabilities that she travels by bus, by train or by car are
0.4, 0.35 and 0.25 respectively. When she travels by bus, the probability that she arrives late is 0.55.
When she travels by train, the probability that she arrives late is 0.7. When she travels by car, the
probability that she arrives late is x.

On a randomly chosen day when Alexa goes to work, the probability that she does not arrive late
is 0.48.

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

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

(b) Find the probability that Alexa travels to work by train given that she arrives late. [3]

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

4 A fair spinner has sides numbered 1, 2, 2. Another fair spinner has sides numbered −2, 0, 1. Each
spinner is spun. The number on the side on which a spinner comes to rest is noted. The random
variable X is the sum of the numbers for the two spinners.

(a) Draw up the probability distribution table for X . [3]

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

(b) Find E X and Var X . [3]

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

5 Every day Richard takes a flight between Astan and Bejin. On any day, the probability that the flight
arrives early is 0.15, the probability that it arrives on time is 0.55 and the probability that it arrives
late is 0.3.

(a) Find the probability that on each of 3 randomly chosen days, Richard’s flight does not arrive late.
[1]

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

(b) Find the probability that for 9 randomly chosen days, Richard’s flight arrives early at least 3
times. [3]

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

(c) 60 days are chosen at random.

Use an approximation to find the probability that Richard’s flight arrives early at least 12 times.
[5]

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

6 (a) Find the total number of different arrangements of the 8 letters in the word TOMORROW. [2]

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

(b) Find the total number of different arrangements of the 8 letters in the word TOMORROW that
have an R at the beginning and an R at the end, and in which the three Os are not all together.
[3]

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

Four letters are selected at random from the 8 letters of the word TOMORROW.

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

7 The heights, in cm, of the 11 basketball players in each of two clubs, the Amazons and the Giants, are
shown below.

Amazons 205 198 181 182 190 215 201 178 202 196 184
Giants 175 182 184 187 189 192 193 195 195 195 204

(a) State an advantage of using a stem-and-leaf diagram compared to a box-and-whisker plot to

illustrate this information. [1]

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

(b) Represent the data by drawing a back-to-back stem-and-leaf diagram with Amazons on the
left-hand side of the diagram. [4]

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

Four new players join the Amazons. The mean height of the 15 players in the Amazons is now
191.2 cm. The heights of three of the new players are 180 cm, 185 cm and 190 cm.

(d) Find the height of the fourth new player. [3]

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

Additional Page

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

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

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*7050283781*

MATHEMATICS 9709/53
Paper 5 Probability & Statistics 1 May/June 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 16 pages. Any blank pages are indicated.

1 The heights in cm of 160 sunflower plants were measured. The results are summarised on the following
cumulative frequency curve.

160

140

120
Cumulative frequency

100

0
0 40 80 120 160 200 240

Height (cm)

(a) Use the graph to estimate the number of plants with heights less than 100 cm. [1]

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

(b) Use the graph to estimate the 65th percentile of the distribution. [2]

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

2 The random variable X can take only the values −2, −1, 0, 1, 2. The probability distribution of X is
given in the following table.

x −2 −1 0 1 2
P X = x p p 0.1 q q

Given that P X ≥ 0 = 3P X < 0, find the values of p and q. [4]

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

3 A sports club has a volleyball team and a hockey team. The heights of the 6 members of the volleyball
team are summarised by Σ x = 1050 and Σ x2 = 193 700, where x is the height of a member in cm.
The heights of the 11 members of the hockey team are summarised by Σ y = 1991 and Σ y2 = 366 400,
where y is the height of a member in cm.

(a) Find the mean height of all 17 members of the club. [2]

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

(b) Find the standard deviation of the heights of all 17 members of the club. [3]

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

4 Three fair six-sided dice, each with faces marked 1, 2, 3, 4, 5, 6, are thrown at the same time,
repeatedly. For a single throw of the three dice, the score is the sum of the numbers on the top faces.

(a) Find the probability that the score is 4 on a single throw of the three dice. [3]

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

(b) Find the probability that a score of 18 is obtained for the first time on the 5th throw of the three
dice. [3]

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

5 The lengths of the leaves of a particular type of tree are modelled by a normal distribution. A scientist
measures the lengths of a random sample of 500 leaves from this type of tree and finds that 42 are less
than 4 cm long and 100 are more than 10 cm long.

(a) Find estimates for the mean and standard deviation of the lengths of leaves from this type of tree.
[5]

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

The lengths, in cm, of the leaves of a different type of tree have the distribution N -, 3 2 . The scientist
takes a random sample of 800 leaves from this type of tree.

(b) Find how many of these leaves the scientist would expect to have lengths, in cm, between - − 23
and - + 23. [4]

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

6 (a) How many different arrangements are there of the 11 letters in the word REQUIREMENT? [2]

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

(b) How many different arrangements are there of the 11 letters in the word REQUIREMENT in
which the two Rs are together and the three Es are together? [1]

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

(c) How many different arrangements are there of the 11 letters in the word REQUIREMENT in
which there are exactly three letters between the two Rs? [3]

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

Five of the 11 letters in the word REQUIREMENT are selected.

(d) How many possible selections contain at least two Es and at least one R? [4]

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

7 In the region of Arka, the total number of households in the three villages Reeta, Shan and Teber is 800.
Each of the households was asked about the quality of their broadband service. Their responses are
summarised in the following table.

Quality of broadband service

Excellent Good Poor
Reeta 75 118 32
Village Shan 223 177 40
Teber 12 60 63

(a) (i) Find the probability that a randomly chosen household is in Shan and has poor broadband
service. [1]

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

(ii) Find the probability that a randomly chosen household has good broadband service given
that the household is in Shan. [2]

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

In the whole of Arka there are a large number of households. A survey showed that 35% of households
in Arka have no broadband service.

(b) (i) 10 households in Arka are chosen at random.

Find the probability that fewer than 3 of these households have no broadband service. [3]

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

(ii) 120 households in Arka are chosen at random.

Use an approximation to find the probability that more than 32 of these households have no
broadband service. [5]

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

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Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*1417278707*

MATHEMATICS 9709/61
Paper 6 Probability & Statistics 2 May/June 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 16 pages. Any blank pages are indicated.

1 Accidents at two factories occur randomly and independently. On average, the numbers of accidents
per month are 3.1 at factory A and 1.7 at factory B.

Find the probability that the total number of accidents in the two factories during a 2-month period is
more than 3. [4]

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

2 The time, in minutes, taken by students to complete a test has the distribution N 125, 36.

(a) Find the probability that the mean time taken to complete the test by a random sample of
40 students is less than 123 minutes. [3]

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

(b) Explain whether it was necessary to use the Central Limit theorem in the solution to part (a).
[1]

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

3 The graph of the probability density function of a random variable X is symmetrical about the line
x = 4.

Given that P X < 5 = 20

27
, find P 3 < X < 5. [2]

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

4 100 randomly chosen adults each throw a ball once. The length, l metres, of each throw is recorded.
The results are summarised below.
n = 100 Σ l = 3820 Σ l2 = 182 200
Calculate a 94% confidence interval for the population mean length of throws by adults. [6]

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

5 On average, 1 in 75 000 adults has a certain genetic disorder.

(a) Use a suitable approximating distribution to find the probability that, in a random sample of
10 000 people, at least 1 has the genetic disorder. [3]

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

(b) In a random sample of n people, where n is large, the probability that no-one has the genetic
disorder is more than 0.9.

Find the largest possible value of n. [4]

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

6 The probability density function, f, of a random variable X is given by

T
k 6x − x2 0 ≤ x ≤ 6,
f x =
0 otherwise,
where k is a constant.

State the value of E X and show that Var X = 95 . [6]

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

7 The masses, in kilograms, of large and small sacks of flour have the distributions N 55, 32 and
N 27, 2.52 respectively.

(a) Some sacks are loaded onto a boat. The maximum load of flour that the boat can carry safely is
340 kg.

Find the probability that the boat can carry safely 3 randomly chosen large sacks of flour and
6 randomly chosen small sacks of flour. [5]

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

(b) Find the probability that the mass of a randomly chosen large sack of flour is greater than the
total mass of two randomly chosen small sacks of flour. [5]

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

8 At a certain large school it was found that the proportion of students not wearing correct uniform
was 0.15. The school sent a letter to parents asking them to ensure that their children wear the correct
uniform. The school now wishes to test whether the proportion not wearing correct uniform has been
reduced.

(a) It is suggested that a random sample of the students in Grade 12 should be used for the test.

Give a reason why this would not be an appropriate sample. [1]

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

A suitable sample of 50 students is selected and the number not wearing correct uniform is noted.
This figure is used to carry out a test at the 5% significance level.

(b) State suitable null and alternative hypotheses. [1]

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

(d) In fact 4 students out of the 50 are not wearing correct uniform.

State the conclusion of the test, explaining your answer. [2]

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

(e) State, with a reason, which of the errors, Type I or Type II, may have been made. [2]

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

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Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*7977464529*

MATHEMATICS 9709/62
Paper 6 Probability & Statistics 2 May/June 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 12 pages. Any blank pages are indicated.

BLANK PAGE

1 In a game, a ball is thrown and lands in one of 4 slots, labelled A, B, C and D. Raju wishes to test
whether the probability that the ball will land in slot A is 41 .

(a) State suitable null and alternative hypotheses for Raju’s test. [1]

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

The ball is thrown 100 times and it lands in slot A 15 times.

(b) Use a suitable approximating distribution to carry out the test at the 2% significance level. [5]

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

2 The random variable X has the distribution B 400, 0.01.

(a) Find Var 4X + 2. [3]

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

(b) (i) State an appropriate approximating distribution for X , giving the values of any parameters.
Justify your choice of approximating distribution. [2]

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

(ii) Use your approximating distribution to find P 2 ≤ X ≤ 5. [2]

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

3
f x

x
O 1 p

The random variable X takes values in the range 1 ≤ x ≤ p, where p is a constant. The graph of the
probability density function of X is shown in the diagram.

(a) Show that p = 2. [2]

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

(b) Find E X . [5]

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

4 Wendy’s journey to work consists of three parts: walking to the train station, riding on the train and
then walking to the office. The times, in minutes, for the three parts of her journey are independent
and have the distributions N 15.0, 1.12 , N 32.0, 3.52 and N 8.6, 1.22 respectively.

(a) Find the mean and variance of the total time for Wendy’s journey. [2]

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

If Wendy’s journey takes more than 60 minutes, she is late for work.

(b) Find the probability that, on a randomly chosen day, Wendy will be late for work. [3]

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

(c) Find the probability that the mean of Wendy’s journey times over 15 randomly chosen days will
be less than 54.5 minutes. [3]

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

5 The time, in minutes, spent by customers at a particular gym has the distribution N -, 38.2. In the
past the value of - has been 42.4. Following the installation of some new equipment the management
wishes to test whether the value of - has changed.

(a) State what is meant by a Type I error in this context. [1]

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

(b) The mean time for a sample of 20 customers is found to be 45.6 minutes.

Test at the 2.5% significance level whether the value of - has changed. [5]

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

6 The heights, h centimetres, of a random sample of 100 fully grown animals of a certain species were
measured. The results are summarised below.
n = 100 Σ h = 7570 Σ h2 = 588 050

(a) Find unbiased estimates of the population mean and variance. [3]

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

(b) Calculate a 99% confidence interval for the mean height of animals of this species. [3]

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

Four random samples were taken and a 99% confidence interval for the population mean, -, was found
from each sample.

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

7 Customers arrive at a particular shop at random times. It has been found that the mean number of
customers who arrive during a 5-minute interval is 2.1.

(a) Find the probability that exactly 4 customers arrive during a 10-minute interval. [2]

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

(b) Find the probability that at least 4 customers arrive during a 20-minute interval. [2]

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

(c) Use a suitable approximating distribution to find the probability that fewer than 40 customers
arrive during a 2-hour interval. [4]

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

Additional Page

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

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

Cambridge International AS & A Level
CANDIDATE
NAME

CENTRE CANDIDATE
NUMBER NUMBER
*5443685767*

MATHEMATICS 9709/63
Paper 6 Probability & Statistics 2 May/June 2021

1 hour 15 minutes

You must answer on the question paper.

You will need: List of formulae (MF19)

INFORMATION
³ The total mark for this paper is 50.
³ The number of marks for each question or part question is shown in brackets [ ].

This document has 16 pages. Any blank pages are indicated.

BLANK PAGE

1 The number of goals scored by a team in a match is independent of other matches, and is denoted by
the random variable X , which has a Poisson distribution with mean 1.36. A supporter offers to make
a donation of $5 to the team for each goal that they score in the next 10 matches.

Find the expectation and standard deviation of the amount that the supporter will pay. [5]

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

2 In the past, the time, in hours, for a particular train journey has had mean 1.40 and standard deviation
0.12. Following the introduction of some new signals, it is required to test whether the mean journey
time has decreased.

(a) State what is meant by a Type II error in this context. [1]

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

(b) The mean time for a random sample of 50 journeys is found to be 1.36 hours.

Assuming that the standard deviation of journey times is still 0.12 hours, test at the 2.5%
significance level whether the population mean journey time has decreased. [5]

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

3 The local council claims that the average number of accidents per year on a particular road is 0.8.
Jane claims that the true average is greater than 0.8. She looks at the records for a random sample of
3 recent years and finds that the total number of accidents during those 3 years was 5.

(a) Assume that the number of accidents per year follows a Poisson distribution.
(i) State null and alternative hypotheses for a test of Jane’s claim. [1]

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

(ii) Test at the 5% significance level whether Jane’s claim is justified. [4]

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

(b) Jane finds that the number of accidents per year has been gradually increasing over recent years.

State how this might affect the validity of the test carried out in part (a)(ii). [1]

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

4 The masses, m kilograms, of flour in a random sample of 90 sacks of flour are summarised as follows.

n = 90 Σ m = 4509 Σ m2 = 225 950

(a) Find unbiased estimates of the population mean and variance. [3]

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

(b) Calculate a 98% confidence interval for the population mean. [3]

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

(d) Find the probability that the confidence interval found in part (b) is wholly above the true value
of the population mean. [2]

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

5 Most plants of a certain type have three leaves. However, it is known that, on average, 1 in 10 000
of these plants have four leaves, and plants with four leaves are called ‘lucky’. The number of lucky
plants in a random sample of 25 000 plants is denoted by X .

(a) State, with a justification, an approximating distribution for X , giving the values of any parameters.
[2]

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

Use your approximating distribution to answer parts (b) and (c).

(b) Find P X ≤ 3. [2]

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

(c) Given that P X = k = 2P X = k + 1, find k. [2]

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

The number of lucky plants in a random sample of n plants, where n is large, is denoted by Y .

(d) Given that P Y ≥ 1 = 0.963, correct to 3 significant figures, use a suitable approximating
distribution to find the value of n. [3]

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

6 Alethia models the length of time, in minutes, by which her train is late on any day by the random
variable X with probability density function given by
T 3
x − 202 0 ≤ x ≤ 20,
f x = 8000
0 otherwise.

(a) Find the probability that the train is more than 10 minutes late on each of two randomly chosen
days. [4]

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

(b) Find E X . [4]

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

(c) The median of X is denoted by m.

Show that m satisfies the equation m − 203 = −4000, and hence find m correct to 3 significant
figures. [4]

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

(d) State one way in which Alethia’s model may be unrealistic. [1]

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

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

You must answer on the question paper.

You will need: List of formulae (MF19)

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

© UCLES 2021 9709/12/F/M/21

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

2 By using a suitable substitution, solve the equation

© UCLES 2021 9709/12/F/M/21

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

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

© UCLES 2021 9709/12/F/M/21

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

© UCLES 2021 9709/12/F/M/21

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

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

7 Functions f and g are defined as follows:

© UCLES 2021 9709/12/F/M/21

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

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

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

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

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

© UCLES 2021 9709/12/F/M/21

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

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

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

© UCLES 2021 9709/12/F/M/21

Find the 85th term when 1 = 13 π. [4]

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

© UCLES 2021 9709/12/F/M/21

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

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

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

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

© UCLES 2021 9709/12/F/M/21

© UCLES 2021 9709/12/F/M/21

© UCLES 2021 9709/12/F/M/21

You must answer on the question paper.

You will need: List of formulae (MF19)

This document has 16 pages. Any blank pages are indicated.

© UCLES 2021 9709/22/F/M/21

(b) Solve the equation 3x − 5 = x + 2. [3]

© UCLES 2021 9709/22/F/M/21 [Turn over

2 Solve the equation sec2 1 cot 1 = 8 for 0 < 1 < π. [5]

© UCLES 2021 9709/22/F/M/21

3 The parametric equations of a curve are

x = e2t cos 4t, y = 3 sin 2t.

© UCLES 2021 9709/22/F/M/21 [Turn over

The diagram shows part of the curve with equation y =

© UCLES 2021 9709/22/F/M/21

© UCLES 2021 9709/22/F/M/21 [Turn over

© UCLES 2021 9709/22/F/M/21

© UCLES 2021 9709/22/F/M/21 [Turn over

6 The polynomial p x is defined by

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

© UCLES 2021 9709/22/F/M/21

(b) Hence find the exact root of the equation p e2y = 0.

© UCLES 2021 9709/22/F/M/21 [Turn over

(b) As x varies, find the least possible value of

and determine the corresponding value of x where −π < x < π. [3]

© UCLES 2021 9709/22/F/M/21

(c) Find Ô   d1.

© UCLES 2021 9709/22/F/M/21

© UCLES 2021 9709/22/F/M/21

© UCLES 2021 9709/22/F/M/21

© UCLES 2021 9709/22/F/M/21

You must answer on the question paper.

You will need: List of formulae (MF19)

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

© UCLES 2021 9709/32/F/M/21

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

Find the values of a and b. [5]

© UCLES 2021 9709/32/F/M/21

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

4 The variables x and y satisfy the differential equation

© UCLES 2021 9709/32/F/M/21

(b) Solve the equation 3x − 5 = x + 2. [3]

(c) Find Ô d1.