mathematics 9709/11 read these instructions...

*8461197174*

Cambridge International ExaminationsCambridge International Advanced Subsidiary and Advanced Level

MATHEMATICS 9709/11

Paper 1 Pure Mathematics 1 (P1) May/June 2015

1 hour 45 minutes

Additional Materials: Answer Booklet/Paper

Graph Paper

List of Formulae (MF9)

READ THESE INSTRUCTIONS FIRST

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Write your Centre number, candidate number and name on all the work you hand in.

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Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in

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The use of an electronic calculator is expected, where appropriate.

You are reminded of the need for clear presentation in your answers.

At the end of the examination, fasten all your work securely together.

The number of marks is given in brackets [ ] at the end of each question or part question.

The total number of marks for this paper is 75.

Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger

numbers of marks later in the paper.

This document consists of 4 printed pages.

JC15 06_9709_11/RP

© UCLES 2015 [Turn over

2

1 Given that 1 is an obtuse angle measured in radians and that sin 1 = k, find, in terms of k, an expression

for

(i) cos1, [1]

(ii) tan1, [2]

(iii) sin�1 + 0�. [1]

2

x

y

O

Q

X �−2, 0� P �p, 0�

y = 2x2

The diagram shows the curve y = 2x2 and the points X �−2, 0� and P �p, 0�. The point Q lies on the

curve and PQ is parallel to the y-axis.

(i) Express the area, A, of triangle XPQ in terms of p. [2]

The point P moves along the x-axis at a constant rate of 0.02 units per second and Q moves along the

curve so that PQ remains parallel to the y-axis.

(ii) Find the rate at which A is increasing when p = 2. [3]

3 (i) Find the first three terms, in ascending powers of x, in the expansion of

(a) �1 − x�6, [2]

(b) �1 + 2x�6. [2]

(ii) Hence find the coefficient of x2 in the expansion of ��1 − x��1 + 2x��6. [3]

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

−−→OA =

`3

0

−4

aand

−−→OB =

`6

−3

2

a.

(i) Find the cosine of angle AOB. [3]

The position vector of C is given by−−→OC =

`k

−2k

2k − 3

a.

(ii) Given that AB and OC have the same length, find the possible values of k. [4]

© UCLES 2015 9709/11/M/J/15

3

5 A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius r cm.

(i) Show that the area of the sector, A cm2, is given by A = 12r − r2. [3]

(ii) Express A in the form a − �r − b�2, where a and b are constants. [2]

(iii) Given that r can vary, state the greatest value of A and find the corresponding angle of the sector.

[2]

6 The line with gradient −2 passing through the point P �3t, 2t� intersects the x-axis at A and the y-axis

at B.

(i) Find the area of triangle AOB in terms of t. [3]

The line through P perpendicular to AB intersects the x-axis at C.

(ii) Show that the mid-point of PC lies on the line y = x. [4]

7 (a) The third and fourth terms of a geometric progression are 13

and 29

respectively. Find the sum to

infinity of the progression. [4]

(b) A circle is divided into 5 sectors in such a way that the angles of the sectors are in arithmetic

progression. Given that the angle of the largest sector is 4 times the angle of the smallest sector,

find the angle of the largest sector. [4]

8 The function f : x → 5 + 3 cos�

12x�

is defined for 0 ≤ x ≤ 20.

(i) Solve the equation f�x� = 7, giving your answer correct to 2 decimal places. [3]

(ii) Sketch the graph of y = f�x�. [2]

(iii) Explain why f has an inverse. [1]

(iv) Obtain an expression for f −1�x�. [3]

9 The equation of a curve is y = x3+ px2, where p is a positive constant.

(i) Show that the origin is a stationary point on the curve and find the coordinates of the other

stationary point in terms of p. [4]

(ii) Find the nature of each of the stationary points. [3]

Another curve has equation y = x3+ px2

+ px.

(iii) Find the set of values of p for which this curve has no stationary points. [3]

[Question 10 is printed on the next page.]

© UCLES 2015 9709/11/M/J/15 [Turn over

4

10

x

y

O

B

x = 4

y =8��3x + 4�

P

QA�0, 4�

The diagram shows part of the curve y =8��3x + 4� . The curve intersects the y-axis at A �0, 4�. The

normal to the curve at A intersects the line x = 4 at the point B.

(i) Find the coordinates of B. [5]

(ii) Show, with all necessary working, that the areas of the regions marked P and Q are equal. [6]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable

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be pleased to make amends at the earliest possible opportunity.

To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International

Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after

the live examination series.

Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local

Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.

© UCLES 2015 9709/11/M/J/15

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MATHEMATICS 9709/12


1 hour 45 minutes


Graph Paper



















This document consists of 3 printed pages and 1 blank page.

JC15 06_9709_12/FP


2

1 The function f is such that f ′�x� = 5 − 2x2 and �3, 5� is a point on the curve y = f�x�. Find f�x�. [3]

2

21 radO

A

B

X Y

r

In the diagram, AYB is a semicircle with AB as diameter and OAXB is a sector of a circle with centre

O and radius r. Angle AOB = 21 radians. Find an expression, in terms of r and 1, for the area of the

shaded region. [4]

3 (i) Find the coefficients of x2 and x3 in the expansion of �2 − x�6. [3]

(ii) Find the coefficient of x3 in the expansion of �3x + 1��2 − x�6. [2]

4 Variables u, x and y are such that u = 2x�y − x� and x + 3y = 12. Express u in terms of x and hence

find the stationary value of u. [5]

5 (i) Prove the identitysin 1 − cos1

sin 1 + cos1�

tan1 − 1

tan1 + 1. [1]

(ii) Hence solve the equationsin 1 − cos1

sin 1 + cos1=

tan 1

6, for 0Å ≤ 1 ≤ 180Å. [4]

6 A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel

turns in such a way that the height, h m, of a passenger above the ground is given by the formula

h = 60�1 − cos kt�. In this formula, k is a constant, t is the time in minutes that has elapsed since the

passenger started the ride at ground level and kt is measured in radians.

(i) Find the greatest height of the passenger above the ground. [1]

One complete revolution of the wheel takes 30 minutes.

(ii) Show that k = 1150. [2]

(iii) Find the time for which the passenger is above a height of 90 m. [3]

© UCLES 2015 9709/12/M/J/15

3

7 The point C lies on the perpendicular bisector of the line joining the points A �4, 6� and B �10, 2�.C also lies on the line parallel to AB through �3, 11�.

(i) Find the equation of the perpendicular bisector of AB. [4]

(ii) Calculate the coordinates of C. [3]

8 (a) The first, second and last terms in an arithmetic progression are 56, 53 and −22 respectively.

Find the sum of all the terms in the progression. [4]

(b) The first, second and third terms of a geometric progression are 2k + 6, 2k and k + 2 respectively,

where k is a positive constant.

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

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

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

−−→OA = 2i + 4j + 4k and

−−→OB = 3i + j + 4k.

(i) Use a vector method to find angle AOB. [4]

The point C is such that−−→AB =

−−→BC.

(ii) Find the unit vector in the direction of−−→OC. [4]

(iii) Show that triangle OAC is isosceles. [1]

10 The equation of a curve is y =4

2x − 1.

(i) Find, showing all necessary working, the volume obtained when the region bounded by the

curve, the x-axis and the lines x = 1 and x = 2 is rotated through 360Å about the x-axis. [4]

(ii) Given that the line 2y = x + c is a normal to the curve, find the possible values of the constant c.

[6]

11 The function f is defined by f : x → 2x2− 6x + 5 for x ∈ >.

(i) Find the set of values of p for which the equation f�x� = p has no real roots. [3]

The function g is defined by g : x → 2x2− 6x + 5 for 0 ≤ x ≤ 4.

(ii) Express g�x� in the form a�x + b�2+ c, where a, b and c are constants. [3]

(iii) Find the range of g. [2]

The function h is defined by h : x → 2x2− 6x + 5 for k ≤ x ≤ 4, where k is a constant.

(iv) State the smallest value of k for which h has an inverse. [1]

(v) For this value of k, find an expression for h−1�x�. [3]

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MATHEMATICS 9709/13


1 hour 45 minutes


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JC15 06_9709_13/RP


2

1 Express 2x2− 12x + 7 in the form a�x + b�2

+ c, where a, b and c are constants. [3]

2 A curve is such thatdy

dx= �2x + 1�

12 and the point �4, 7� lies on the curve. Find the equation of the

curve. [4]

3 (i) Write down the first 4 terms, in ascending powers of x, of the expansion of �a − x�5. [2]

(ii) The coefficient of x3 in the expansion of �1 − ax��a − x�5 is −200. Find the possible values of the

constant a. [4]

4 (i) Express the equation 3 sin 1 = cos1 in the form tan 1 = k and solve the equation for 0Å < 1 < 180Å.

[2]

(ii) Solve the equation 3 sin2 2x = cos2 2x for 0Å < x < 180Å. [4]

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

−−→OA =

`3

2

−3

a,

−−→OB =

`5

−1

−2

aand

−−→OC =

`6

1

2

a.

(i) Show that angle ABC is 90Å. [4]

(ii) Find the area of triangle ABC, giving your answer correct to 1 decimal place. [3]

6

x

y

O

y =1 − 5x

2x

The diagram shows the graph of y = f −1�x�, where f −1 is defined by f −1�x� = 1 − 5x

2xfor 0 < x ≤ 2.

(i) Find an expression for f�x� and state the domain of f. [5]

(ii) The function g is defined by g�x� = 1

xfor x ≥ 1. Find an expression for f −1g�x�, giving your

answer in the form ax + b, where a and b are constants to be found. [2]

© UCLES 2015 9709/13/M/J/15

3

7 The point A has coordinates �p, 1� and the point B has coordinates �9, 3p + 1�, where p is a constant.

(i) For the case where the distance AB is 13 units, find the possible values of p. [3]

(ii) For the case in which the line with equation 2x + 3y = 9 is perpendicular to AB, find the value

of p. [4]

8 The function f is defined by f�x� = 1

x + 1+

1

�x + 1�2for x > −1.

(i) Find f ′�x�. [3]

(ii) State, with a reason, whether f is an increasing function, a decreasing function or neither. [1]

The function g is defined by g�x� = 1

x + 1+

1

�x + 1�2for x < −1.

(iii) Find the coordinates of the stationary point on the curve y = g�x�. [4]

9 (a) The first term of an arithmetic progression is −2222 and the common difference is 17. Find the

value of the first positive term. [3]

(b) The first term of a geometric progression is ï3 and the second term is 2 cos1, where 0 < 1 < 0.

Find the set of values of 1 for which the progression is convergent. [5]

10

x

y

O CB�3, 0�

A �2, 9�

y = 9 + 6x − 3x2

Points A �2, 9� and B �3, 0� lie on the curve y = 9 + 6x − 3x2, as shown in the diagram. The tangent at

A intersects the x-axis at C. Showing all necessary working,

(i) find the equation of the tangent AC and hence find the x-coordinate of C, [4]

(ii) find the area of the shaded region ABC. [5]



4

11

! radO

A

B

C

r

In the diagram, OAB is a sector of a circle with centre O and radius r. The point C on OB is such

that angle ACO is a right angle. Angle AOB is ! radians and is such that AC divides the sector into

two regions of equal area.

(i) Show that sin ! cos! = 12!. [4]

It is given that the solution of the equation in part (i) is ! = 0.9477, correct to 4 decimal places.

(ii) Find the ratio

perimeter of region OAC : perimeter of region ACB,

giving your answer in the form k : 1, where k is given correct to 1 decimal place. [5]

(iii) Find angle AOB in degrees. [1]









© UCLES 2015 9709/13/M/J/15

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Cambridge International ExaminationsCambridge International Advanced Subsidiary Level

MATHEMATICS 9709/21


1 hour 15 minutes


Graph Paper




















JC15 06_9709_21/RP


2

1 (i) Solve the equation �3x + 4 � = �3x − 11�. [3]

(ii) Hence, using logarithms, solve the equation �3 × 2y+ 4 � = �3 × 2y

− 11�, giving the answer correct

to 3 significant figures. [2]

2

x

ln y

O

(2, 1.60)

(5, 2.92)

The variables x and y satisfy the equation

y = Aep�x−1�,

where A and p are constants. The graph of ln y against x is a straight line passing through the points

�2, 1.60� and �5, 2.92�, as shown in the diagram. Find the values of A and p correct to 2 significant

figures. [5]

3 The equation of a curve is

y = 6 sin x − 2 cos 2x.

Find the equation of the tangent to the curve at the point�

160, 2

�. Give the answer in the form

y = mx + c, where the values of m and c are correct to 3 significant figures. [5]

4 The polynomials f�x� and g�x� are defined by

f�x� = x3+ ax2

+ b and g�x� = x3+ bx2

− a,

where a and b are constants. It is given that �x + 2� is a factor of f�x�. It is also given that, when g�x�is divided by �x + 1�, the remainder is −18.

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

(ii) When a and b have these values, find the greatest possible value of g�x� − f�x� as x varies. [2]

5 (i) Given that Óa

0

�3e12x+ 1�dx = 10, show that the positive constant a satisfies the equation

a = 2 ln

@16 − a

6

A. �5�

(ii) Use the iterative formula an+1 = 2 ln

@16 − an

6

Awith a1 = 2 to find the value of a correct to

3 decimal places. Give the result of each iteration to 5 decimal places. [3]

© UCLES 2015 9709/21/M/J/15

3

6 (i) Prove that 2 cosec 21 tan1 � sec21. [3]

(ii) Hence

(a) solve the equation 2 cosec 21 tan1 = 5 for 0 < 1 < 0, [3]

(b) find the exact value of Ó

160

0

2 cosec 4x tan 2x dx. [4]


y3+ 4xy = 16.

(i) Show thatdy

dx= −

4y

3y2+ 4x

. [4]

(ii) Show that the curve has no stationary points. [2]

(iii) Find the coordinates of the point on the curve where the tangent is parallel to the y-axis. [4]

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MATHEMATICS 9709/22


1 hour 15 minutes


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JC15 06_9709_22/RP


2

1 (i) Use logarithms to solve the equation 2x= 205, giving the answer correct to 3 significant figures.

[2]

(ii) Hence determine the number of integers n satisfying

20−5< 2n

< 205. �2�

2 (i) Given that �x + 2� is a factor of

4x3+ ax2

− �a + 1�x − 18,

find the value of the constant a. [3]

(ii) When a has this value, factorise 4x3+ ax2

− �a + 1�x − 18 completely. [3]

3 It is given that 1 is an acute angle measured in degrees such that

2 sec21 + 3 tan 1 = 22.

(i) Find the value of tan1. [3]

(ii) Use an appropriate formula to find the exact value of tan�1 + 135Å�. [3]

4

x

y

O

M

The diagram shows the curve y = ex+ 4e−2x and its minimum point M.

(i) Show that the x-coordinate of M is ln 2. [3]

(ii) The region shaded in the diagram is enclosed by the curve and the lines x = 0, x = ln 2 and y = 0.

Use integration to show that the area of the shaded region is 52. [4]

© UCLES 2015 9709/22/M/J/15

3

5 (i) By sketching a suitable pair of graphs, show that the equation

�3x � = 16 − x4

has two real roots. [3]

(ii) Use the iterative formula xn+1 =4��16 − 3xn� to find one of the real roots correct to 3 decimal

places. Give the result of each iteration to 5 decimal places. [3]

(iii) Hence find the coordinates of each of the points of intersection of the graphs y = �3x � and

y = 16 − x4, giving your answers correct to 3 decimal places. [2]

6

x

y

O

P

The diagram shows part of the curve with equation

y = 4 sin2 x + 8 sin x + 3

and its point of intersection P with the x-axis.

(i) Find the exact x-coordinate of P. [3]

(ii) Show that the equation of the curve can be written

y = 5 + 8 sin x − 2 cos 2x,

and use integration to find the exact area of the shaded region enclosed by the curve and the axes.

[6]

7 (a) Find the gradient of the curve

3 ln x + 4 ln y + 6xy = 6

at the point �1, 1�. [4]

(b) The parametric equations of a curve are

x =10

t− t, y =

��2t − 1�.

Find the gradient of the curve at the point �−3, 3�. [6]

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© UCLES 2015 9709/22/M/J/15

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MATHEMATICS 9709/23


1 hour 15 minutes


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JC15 06_9709_23/FP


2

1 (i) Use logarithms to solve the equation 2x= 205, giving the answer correct to 3 significant figures.

[2]

(ii) Hence determine the number of integers n satisfying

20−5< 2n

< 205. �2�

2 (i) Given that �x + 2� is a factor of

4x3+ ax2

− �a + 1�x − 18,

find the value of the constant a. [3]

(ii) When a has this value, factorise 4x3+ ax2

− �a + 1�x − 18 completely. [3]

3 It is given that 1 is an acute angle measured in degrees such that

2 sec21 + 3 tan 1 = 22.

(i) Find the value of tan1. [3]

(ii) Use an appropriate formula to find the exact value of tan�1 + 135Å�. [3]

4

x

y

O

M

The diagram shows the curve y = ex+ 4e−2x and its minimum point M.

(i) Show that the x-coordinate of M is ln 2. [3]

(ii) The region shaded in the diagram is enclosed by the curve and the lines x = 0, x = ln 2 and y = 0.

Use integration to show that the area of the shaded region is 52. [4]

© UCLES 2015 9709/23/M/J/15

3

5 (i) By sketching a suitable pair of graphs, show that the equation

�3x � = 16 − x4

has two real roots. [3]

(ii) Use the iterative formula xn+1 =4��16 − 3xn� to find one of the real roots correct to 3 decimal

places. Give the result of each iteration to 5 decimal places. [3]

(iii) Hence find the coordinates of each of the points of intersection of the graphs y = �3x � and

y = 16 − x4, giving your answers correct to 3 decimal places. [2]

6

x

y

O

P

The diagram shows part of the curve with equation

y = 4 sin2 x + 8 sin x + 3

and its point of intersection P with the x-axis.

(i) Find the exact x-coordinate of P. [3]

(ii) Show that the equation of the curve can be written

y = 5 + 8 sin x − 2 cos 2x,

and use integration to find the exact area of the shaded region enclosed by the curve and the axes.

[6]

7 (a) Find the gradient of the curve

3 ln x + 4 ln y + 6xy = 6

at the point �1, 1�. [4]

(b) The parametric equations of a curve are

x =10

t− t, y =

��2t − 1�.

Find the gradient of the curve at the point �−3, 3�. [6]

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© UCLES 2015 9709/23/M/J/15

*8254149865*

Cambridge International ExaminationsCambridge International Advanced Level

MATHEMATICS 9709/31


1 hour 45 minutes


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JC15 06_9709_31/2R


2

1 Use logarithms to solve the equation 25x= 32x+1, giving the answer correct to 3 significant figures.

[4]

2 Use the trapezium rule with three intervals to find an approximation to

Ó

3

0

�3x− 10 � dx. �4�

3 Show that, for small values of x2,�1 − 2x2

�−2−�1 + 6x2

�23≈ kx4,

where the value of the constant k is to be determined. [6]


y = 3 cos 2x + 7 sin x + 2.

Find the x-coordinates of the stationary points in the interval 0 ≤ x ≤ 0. Give each answer correct to

3 significant figures. [7]

5 (a) Find Ó �4 + tan2 2x�dx. [3]

(b) Find the exact value of Ô

120

140

sin�x + 1

60�

sin xdx. [5]

6 The straight line l1

passes through the points �0, 1, 5� and �2, −2, 1�. The straight line l2

has equation

r = 7i + j + k + -�i + 2j + 5k�.

(i) Show that the lines l1

and l2

are skew. [6]

(ii) Find the acute angle between the direction of the line l2 and the direction of the x-axis. [3]

7 Given that y = 1 when x = 0, solve the differential equation

dy

dx= 4x�3y2

+ 10y + 3�,

obtaining an expression for y in terms of x. [9]

8 The complex number w is defined by w =22 + 4i

�2 − i�2.

(i) Without using a calculator, show that w = 2 + 4i. [3]

(ii) It is given that p is a real number such that 140 ≤ arg�w + p� ≤ 3

40. Find the set of possible values

of p. [3]

(iii) The complex conjugate of w is denoted by w*. The complex numbers w and w* are represented

in an Argand diagram by the points S and T respectively. Find, in the form �Ï − a � = k, the

equation of the circle passing through S, T and the origin. [3]

© UCLES 2015 9709/31/M/J/15

3

9

x

y

O

M

The diagram shows the curve y = x2e2−x and its maximum point M.

(i) Show that the x-coordinate of M is 2. [3]

(ii) Find the exact value of Ó2

0

x2e2−x dx. [6]

10

x

y

O

P

The diagram shows part of the curve with parametric equations

x = 2 ln�t + 2�, y = t3+ 2t + 3.

(i) Find the gradient of the curve at the origin. [5]

(ii) At the point P on the curve, the value of the parameter is p. It is given that the gradient of the

curve at P is 12.

(a) Show that p =1

3p2+ 2

− 2. [1]

(b) By first using an iterative formula based on the equation in part (a), determine the coordinates

of the point P. Give the result of each iteration to 5 decimal places and each coordinate of

P correct to 2 decimal places. [4]

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MATHEMATICS 9709/32


1 hour 45 minutes


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JC15 06_9709_32/RP


2

1 Use the trapezium rule with three intervals to estimate the value of

Ó

120

0

ln�1 + sin x�dx,

giving your answer correct to 2 decimal places. [3]

2 Using the substitution u = 4x, solve the equation 4x+ 42

= 4x+2, giving your answer correct to

3 significant figures. [4]

3 A curve has equation y = cos x cos 2x. Find the x-coordinate of the stationary point on the curve in

the interval 0 < x < 120, giving your answer correct to 3 significant figures. [6]

4 (i) Express 3 sin 1 + 2 cos1 in the form R sin�1 + !�, where R > 0 and 0Å < ! < 90Å, stating the exact

value of R and giving the value of ! correct to 2 decimal places. [3]

(ii) Hence solve the equation

3 sin 1 + 2 cos1 = 1,

for 0Å < 1 < 180Å. [3]

5

2x rad

A

T

B

O

r

The diagram shows a circle with centre O and radius r. The tangents to the circle at the points A and

B meet at T, and the angle AOB is 2x radians. The shaded region is bounded by the tangents AT and

BT, and by the minor arc AB. The perimeter of the shaded region is equal to the circumference of the

circle.

(i) Show that x satisfies the equation

tan x = 0 − x. �3�

(ii) This equation has one root in the interval 0 < x < 120. Verify by calculation that this root lies

between 1 and 1.3. [2]

(iii) Use the iterative formula

xn+1 = tan−1�0 − xn�

to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal

places. [3]

© UCLES 2015 9709/32/M/J/15

3

6 Let I = Ô

1

0

�x

2 −�

xdx.

(i) Using the substitution u = 2 −�

x, show that I = Ô

2

1

2�2 − u�2

udu. [4]

(ii) Hence show that I = 8 ln 2 − 5. [4]

7 The complex number u is given by u = −1 + �4ï3�i.

(i) Without using a calculator and showing all your working, find the two square roots of u. Give

your answers in the form a + ib, where the real numbers a and b are exact. [5]

(ii) On an Argand diagram, sketch the locus of points representing complex numbers Ï satisfying

the relation �Ï − u � = 1. Determine the greatest value of arg Ï for points on this locus. [4]

8 Let f�x� = 5x2+ x + 6

�3 − 2x��x2+ 4�

.

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

(ii) Hence obtain the expansion of f�x� in ascending powers of x, up to and including the term in x2.

[5]

9 The number of organisms in a population at time t is denoted by x. Treating x as a continuous variable,

the differential equation satisfied by x and t is

dx

dt=

xe−t

k + e−t ,

where k is a positive constant.

(i) Given that x = 10 when t = 0, solve the differential equation, obtaining a relation between x, k

and t. [6]

(ii) Given also that x = 20 when t = 1, show that k = 1 −2

e. [2]

(iii) Show that the number of organisms never reaches 48, however large t becomes. [2]

10 The points A and B have position vectors given by−−→OA = 2i − j + 3k and

−−→OB = i + j + 5k. The line l

has equation r = i + j + 2k + -�3i + j − k�.

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

(ii) Find the equation of the plane containing the line l and the point A. Give your answer in the

form ax + by + cÏ = d. [6]

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4

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MATHEMATICS 9709/33


1 hour 45 minutes


Graph Paper




















JC15 06_9709_33/RP


2

1 Solve the equation ln�x + 4� = 2 ln x + ln 4, giving your answer correct to 3 significant figures. [4]

2 Solve the inequality �x − 2 � > 2x − 3. [4]

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

4 The curve with equation y =e2x

4 + e3xhas one stationary point. Find the exact values of the coordinates

of this point. [6]

5 The parametric equations of a curve are

x = a cos4 t, y = a sin4 t,

where a is a positive constant.

(i) Expressdy

dxin terms of t. [3]

(ii) Show that the equation of the tangent to the curve at the point with parameter t is

x sin2 t + y cos2 t = a sin2 t cos2 t. �3�

(iii) Hence show that if the tangent meets the x-axis at P and the y-axis at Q, then

OP + OQ = a,

where O is the origin. [2]

6 It is given that Óa

0

x cos x dx = 0.5, where 0 < a < 120.

(i) Show that a satisfies the equation sin a =1.5 − cos a

a. [4]

(ii) Verify by calculation that a is greater than 1. [2]

(iii) Use the iterative formula

an+1= sin−1

P1.5 − cos an

an

Q

to determine the value of a correct to 4 decimal places, giving the result of each iteration to

6 decimal places. [3]

© UCLES 2015 9709/33/M/J/15

3

7 The number of micro-organisms in a population at time t is denoted by M. At any time the variation

in M is assumed to satisfy the differential equation

dM

dt= k�ïM� cos�0.02t�,

where k is a constant and M is taken to be a continuous variable. It is given that when t = 0, M = 100.

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

(ii) Given also that M = 196 when t = 50, find the value of k. [2]

(iii) Obtain an expression for M in terms of t and find the least possible number of micro-organisms.

[2]

8 The complex number 1 − i is denoted by u.

(i) Showing your working and without using a calculator, express

i

u

in the form x + iy, where x and y are real. [2]

(ii) On an Argand diagram, sketch the loci representing complex numbers Ï satisfying the equations

�Ï − u � = �Ï � and �Ï − i � = 2. [4]

(iii) Find the argument of each of the complex numbers represented by the points of intersection of

the two loci in part (ii). [3]

9 Two planes have equations x + 3y − 2Ï = 4 and 2x + y + 3Ï = 5. The planes intersect in the straight

line l.

(i) Calculate the acute angle between the two planes. [4]

(ii) Find a vector equation for the line l. [6]

10 Let f�x� = 11x + 7

�2x − 1��x + 2�2.

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

(ii) Show that Ó2

1

f�x�dx = 14+ ln

�94

�. [5]

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4

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MATHEMATICS 9709/41

Paper 4 Mechanics 1 (M1) May/June 2015

1 hour 15 minutes


Graph Paper












Where a numerical value for the acceleration due to gravity is needed, use 10 m s−2.









JC15 06_9709_41/RP


2

1 A block B of mass 2.7 kg is pulled at constant speed along a straight line on a rough horizontal floor.

The pulling force has magnitude 25 N and acts at an angle of 1 above the horizontal. The normal

component of the contact force acting on B has magnitude 20 N.

(i) Show that sin 1 = 0.28. [2]

(ii) Find the work done by the pulling force in moving the block a distance of 5 m. [2]

2

x

y

O

25 N

F N

63 N

1tan−1 0.75

Three horizontal forces of magnitudes F N, 63 N and 25 N act at O, the origin of the x-axis and y-axis.

The forces are in equilibrium. The force of magnitude F N makes an angle 1 anticlockwise with the

positive x-axis. The force of magnitude 63 N acts along the negative y-axis. The force of magnitude

25 N acts at tan−1 0.75 clockwise from the negative x-axis (see diagram). Find the value of F and the

value of tan1. [5]

3 A block of weight 6.1 N slides down a slope inclined at tan−1�

1160

�to the horizontal. The coefficient of

friction between the block and the slope is 14. The block passes through a point A with speed 2 m s−1.

Find how far the block moves from A before it comes to rest. [5]

4 A lorry of mass 14 000 kg moves along a road starting from rest at a point O. It reaches a point A,

and then continues to a point B which it reaches with a speed of 24 m s−1. The part OA of the road

is straight and horizontal and has length 400 m. The part AB of the road is straight and is inclined

downwards at an angle of 1Å to the horizontal and has length 300 m.

(i) For the motion from O to B, find the gain in kinetic energy of the lorry and express its loss in

potential energy in terms of 1. [3]

The resistance to the motion of the lorry is 4800 N and the work done by the driving force of the lorry

from O to B is 5000 kJ.

(ii) Find the value of 1. [3]

© UCLES 2015 9709/41/M/J/15

3

5 A cyclist and her bicycle have a total mass of 84 kg. She works at a constant rate of P W while

moving on a straight road which is inclined to the horizontal at an angle 1, where sin 1 = 0.1. When

moving uphill, the cyclist’s acceleration is 1.25 m s−2 at an instant when her speed is 3 m s−1. When

moving downhill, the cyclist’s acceleration is 1.25 m s−2 at an instant when her speed is 10 m s−1. The

resistance to the cyclist’s motion, whether the cyclist is moving uphill or downhill, is R N. Find the

values of P and R. [8]

6 Two particles A and B start to move at the same instant from a point O. The particles move in the

same direction along the same straight line. The acceleration of A at time t s after starting to move is

a m s−2, where a = 0.05 − 0.0002t.

(i) Find A’s velocity when t = 200 and when t = 500. [4]

B moves with constant acceleration for the first 200 s and has the same velocity as A when t = 200. B

moves with constant retardation from t = 200 to t = 500 and has the same velocity as A when t = 500.

(ii) Find the distance between A and B when t = 500. [6]

7A

B

0.5 m

Particles A and B, of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light

inextensible string. Particle A is held at rest on a rough horizontal table with the string passing over

a smooth pulley fixed at the edge of the table. The coefficient of friction between A and the table

is 0.2. Particle B hangs vertically below the pulley at a height of 0.5 m above the floor (see diagram).

The system is released from rest and 0.25 s later the string breaks. A does not reach the pulley in the

subsequent motion. Find

(i) the speed of B immediately before it hits the floor, [9]

(ii) the total distance travelled by A. [3]

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MATHEMATICS 9709/43


1 hour 15 minutes


Graph Paper





















JC15 06_9709_43/RP


2

1 A block is pulled along a horizontal floor by a horizontal rope. The tension in the rope is 500 N and

the block moves at a constant speed of 2.75 m s−1. Find the work done by the tension in 40 s and find

the power applied by the tension. [4]

2

A

B

0.35 kg

0.15 kg

h m

Particles A and B, of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light

inextensible string. A is held at rest on a smooth horizontal surface with the string passing over a

small smooth pulley fixed at the edge of the surface. B hangs vertically below the pulley at a distance

h m above the floor (see diagram). A is released and the particles move. B reaches the floor and A

subsequently reaches the pulley with a speed of 3 m s−1.

(i) Explain briefly why the speed with which B reaches the floor is 3 m s−1. [1]

(ii) Find the value of h. [4]

3 A car of mass 860 kg travels along a straight horizontal road. The power provided by the car’s engine

is P W and the resistance to the car’s motion is R N. The car passes through one point with speed

4.5 m s−1 and acceleration 4 m s−2. The car passes through another point with speed 22.5 m s−1 and

acceleration 0.3 m s−2. Find the values of P and R. [6]

4 A lorry of mass 12 000 kg moves up a straight hill of length 500 m, starting at the bottom with a speed

of 24 m s−1 and reaching the top with a speed of 16 m s−1. The top of the hill is 25 m above the level

of the bottom of the hill. The resistance to motion of the lorry is 7500 N. Find the driving force of the

lorry. [6]

© UCLES 2015 9709/43/M/J/15

3

516 N

12 N

8 N

4 N30Å

30Å30Å

Fig. 1

Four coplanar forces of magnitudes 4 N, 8 N, 12 N and 16 N act at a point. The directions in which

the forces act are shown in Fig. 1.

(i) Find the magnitude and direction of the resultant of the four forces. [5]

16 N

12 N

8 N

4 N

30Å

30Å30Å

Fig. 2

The forces of magnitudes 4 N and 16 N exchange their directions and the forces of magnitudes 8 N

and 12 N also exchange their directions (see Fig. 2).

(ii) State the magnitude and direction of the resultant of the four forces in Fig. 2. [2]

6 A small box of mass 5 kg is pulled at a constant speed of 2.5 m s−1 down a line of greatest slope

of a rough plane inclined at 10Å to the horizontal. The pulling force has magnitude 20 N and acts

downwards parallel to a line of greatest slope of the plane.

(i) Find the coefficient of friction between the box and the plane. [5]

The pulling force is removed while the box is moving at 2.5 m s−1.

(ii) Find the distance moved by the box after the instant at which the pulling force is removed. [4]



4

7 A particle P moves on a straight line. It starts at a point O on the line and returns to O 100 s later.

The velocity of P is v m s−1 at time t s after leaving O, where

v = 0.0001t3− 0.015t2

+ 0.5t.

(i) Show that P is instantaneously at rest when t = 0, t = 50 and t = 100. [2]

(ii) Find the values of v at the times for which the acceleration of P is zero, and sketch the velocity-

time graph for P’s motion for 0 ≤ t ≤ 100. [7]

(iii) Find the greatest distance of P from O for 0 ≤ t ≤ 100. [4]









© UCLES 2015 9709/43/M/J/15

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MATHEMATICS 9709/51


1 hour 15 minutes


Graph Paper





















JC15 06_9709_51/RP


2

1 One end of a light elastic string of natural length 0.7 m is attached to a fixed point A on a smooth

horizontal surface. The other end of the string is attached to a particle P of mass 0.3 kg which is held

at a point B on the horizontal surface, where AB = 1.2 m. It is given that P is released from rest at B

and that when AP = 0.9 m, the particle has speed 4 m s−1. Calculate the modulus of elasticity of the

string. [3]

2 A stone is projected from a point O on horizontal ground. The equation of the trajectory of the stone

is

y = 1.2x − 0.15x2,

where x m and y m are respectively the horizontal and vertically upwards displacements of the stone

from O. Find

(i) the greatest height of the stone, [2]

(ii) the distance from O of the point where the stone strikes the ground. [2]

3

P

A

O5 rad s−1

1

One end of a light inextensible string is attached to a fixed point A and the other end of the string is

attached to a particle P. The particle P moves with constant angular speed 5 rad s−1 in a horizontal

circle which has its centre O vertically below A. The string makes an angle 1 with the vertical (see

diagram). The tension in the string is three times the weight of P.

(i) Show that the length of the string is 1.2 m. [3]

(ii) Find the speed of P. [4]

© UCLES 2015 9709/51/M/J/15

3

4

B

15 m s−1

O30Å

A small ball B is projected from a point O above horizontal ground, with initial speed 15 m s−1 at an

angle of projection of 30Å above the horizontal (see diagram). The ball strikes the ground 3 s after

projection.

(i) Calculate the speed and direction of motion of the ball immediately before it strikes the ground.

[5]

(ii) Find the height of O above the ground. [2]

5 A particle P of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.9 m and

modulus of elasticity 18 N. The other end of the string is attached to a fixed point O which is 3 m

above the ground.

(i) Find the extension of the string when P is in the equilibrium position. [2]

P is projected vertically downwards from the equilibrium position with initial speed 6 m s−1. At the

instant when the tension in the string is 12 N the string breaks. P continues to descend vertically.

(ii) (a) Calculate the height of P above the ground at the instant when the string breaks. [2]

(b) Find the speed of P immediately before it strikes the ground. [4]

6 A particle P of mass 0.1 kg moves with decreasing speed in a straight line on a smooth horizontal

surface. A horizontal resisting force of magnitude 0.2e−x N acts on P, where x m is the displacement

of P from a fixed point O on the line. The velocity of P is v m s−1 when its displacement from O

is x m.

(i) Show that

vdv

dx= ke−x,

where k is a constant to be found. [2]

P passes through O with velocity 2.2 m s−1.

(ii) Calculate the value of x at the instant when the velocity of P is 2 m s−1. [4]

(iii) Show that the speed of P does not fall below 0.917 m s−1, correct to 3 significant figures. [2]



4

7

45Å

B

A O

ED

C

0.8 m

0.6 m

The diagram shows the cross-section OABCDE through the centre of mass of a uniform prism

on a rough inclined plane. The portion ADEO is a rectangle in which AD = OE = 0.6 m and

DE = AO = 0.8 m; the portion BCD is an isosceles triangle in which angle BCD is a right angle,

and A is the mid-point of BD. The plane is inclined at 45Å to the horizontal, BC lies along a line of

greatest slope of the plane and DE is horizontal.

(i) Calculate the distance of the centre of mass of the prism from BD. [3]

The weight of the prism is 21 N, and it is held in equilibrium by a horizontal force of magnitude P N

acting along ED.

(ii) (a) Find the smallest value of P for which the prism does not topple. [2]

(b) It is given that the prism is about to slip for this smallest value of P. Calculate the coefficient

of friction between the prism and the plane. [3]

The value of P is gradually increased until the prism ceases to be in equilibrium.

(iii) Show that the prism topples before it begins to slide, stating the value of P at which equilibrium

is broken. [5]









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*4179854811*


MATHEMATICS 9709/52


1 hour 15 minutes


Graph Paper





















JC15 06_9709_52/RP


2

1 A particle P of mass 0.6 kg is on the rough surface of a horizontal disc with centre O. The distance

OP is 0.4 m. The disc and P rotate with angular speed 3 rad s−1 about a vertical axis which passes

through O. Find the magnitude of the frictional force which the disc exerts on the particle, and state

the direction of this force. [3]

2 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 30 N is attached to

a fixed point O. The other end of the string is attached to a particle P which hangs in equilibrium

vertically below O, with OP = 0.8 m.

(i) Show that the mass of P is 1.8 kg. [2]

The particle is pulled vertically downwards and released from rest from the point where OP = 1.2 m.

(ii) Find the speed of P at the instant when the string first becomes slack. [3]

3 A triangular frame ABC consists of two uniform rigid rods each of length 0.8 m and weight

3 N, and a longer uniform rod of weight 4 N. The triangular frame has AB = BC, and

angle BAC = angle BCA = 30Å.

(i) Calculate the distance of the centre of mass of the frame from AC. [3]

F N

30Å

30Å

A

C

B

0.8

m

0.8

m

The vertex A of the frame is attached to a smooth hinge at a fixed point. The frame is held in

equilibrium with AC vertical by a vertical force of magnitude F N applied to the frame at B (see

diagram).

(ii) Calculate F, and state the magnitude and direction of the force acting on the frame at the hinge.

[3]

© UCLES 2015 9709/52/M/J/15

3

4 One end of a light inextensible string of length 0.5 m is attached to a fixed point A. The other end of

the string is attached to a particle P of weight 6 N. Another light inextensible string of length 0.5 m

connects P to a fixed point B which is 0.8 m vertically below A. The particle P moves with constant

speed in a horizontal circle with centre at the mid-point of AB. Both strings are taut.

(i) Calculate the speed of P when the tension in the string BP is 2 N. [5]

(ii) Show that the angular speed of P must exceed 5 rad s−1. [3]

5

A

B

C

D

0.4m

0.4m

30Å

30Å

A uniform solid cube with edges of length 0.4 m rests in equilibrium on a rough plane inclined at an

angle of 30Å to the horizontal. ABCD is a cross-section through the centre of mass of the cube, with

AB along a line of greatest slope. B lies below the level of A. One end of a light elastic string with

modulus of elasticity 12 N and natural length 0.4 m is attached to C. The other end of the string is

attached to a point below the level of B on the same line of greatest slope, such that the string makes

an angle of 30Å with the plane (see diagram). The cube is on the point of toppling. Find

(i) the tension in the string, [3]

(ii) the weight of the cube. [4]

[Questions 6 and 7 are printed on the next page.]


4

6

U m s−1

1ÅO

60Å

B

18 m s−1

Fig. 1

60Å

V m s−1

B

A

Fig. 2

A small ball B is projected with speed U m s−1 at an angle of 1Å above the horizontal from a point O.

At time 2 s after the instant of projection, B strikes a smooth wall which slopes at 60Å to the horizontal.

The speed of B is 18 m s−1 and its direction of motion is perpendicular to the wall at the instant of

impact (see Fig. 1). B bounces off the wall with speed V m s−1 in a direction perpendicular to the wall.

At time 0.8 s after B bounces off the wall, B strikes the wall again at a lower point A (see Fig. 2).

(i) Find U and 1. [5]

(ii) By considering the motion of B after it bounces off the wall, calculate V. [4]

7 A force of magnitude 0.4t N, applied at an angle of 30Å above the horizontal, acts on a particle P,

where t s is the time since the force starts to act. P is at rest on rough horizontal ground when t = 0.

The mass of P is 0.2 kg and the coefficient of friction between P and the ground is -.

(i) Given that P is about to slip when t = 2, find - and the value of t for the instant when P loses

contact with the ground. [5]

(ii) While P is moving on the ground, it has velocity v m s−1 at time t s. Show that

dv

dt= 2.165t − 4.330,

where the coefficients are correct to 4 significant figures. [3]

(iii) Calculate the speed of P when it loses contact with the ground. [4]









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*4439522840*


MATHEMATICS 9709/53


1 hour 15 minutes


Graph Paper





















JC15 06_9709_53/RP


2

1 A uniform semicircular lamina has diameter AB of length 0.8 m.

(i) Find the distance of the centre of mass of the lamina from AB. [2]

The lamina rests in a vertical plane, with the point B of the lamina in contact with a rough horizontal

surface and with A vertically above B. Equilibrium is maintained by a force of magnitude 6 N in the

plane of the lamina, applied to the lamina at A and acting at an angle of 20Å below the horizontal.

(ii) Calculate the mass of the lamina. [3]

2 A particle P is projected with speed V m s−1 at an angle of 60Å above the horizontal from a point O

on horizontal ground. P is moving at an angle of 45Å above the horizontal at the instant 1.5 s after

projection.

(i) Find V. [3]

(ii) Hence calculate the horizontal and vertical displacements of P from O at the instant 1.5 s after

projection. [2]

3 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to

a fixed point A on a smooth plane inclined at 30Å to the horizontal. The other end of the string is

attached to a particle P of mass 0.5 kg which rests in equilibrium on the plane.

(i) Calculate the extension of the string. [2]

P is projected down the plane from the equilibrium position with speed 5 m s−1. The extension of the

string is e m when the speed of the particle is 2 m s−1 for the first time.

(ii) Find e. [4]

4 A small ball B is projected from a point 1.5 m above horizontal ground with initial speed 29 m s−1 at

an angle of 30Å above the horizontal.

(i) Show that B strikes the ground 3 s after projection. [2]

(ii) Find the speed and direction of motion of B immediately before it strikes the ground. [4]

© UCLES 2015 9709/53/M/J/15

3

5

A

B

CD0.25 m

0.4 m 0.5 m

0.3 m

A uniform triangular prism of weight 20 N rests on a horizontal table. ABC is the cross-section through

the centre of mass of the prism, where BC = 0.5 m, AB = 0.4 m, AC = 0.3 m and angle BAC = 90Å.

The vertical plane ABC is perpendicular to the edge of the table. The point D on AC is at the edge of

the table, and AD = 0.25 m. One end of a light elastic string of natural length 0.6 m and modulus of

elasticity 48 N is attached to C and a particle of mass 2.5 kg is attached to the other end of the string.

The particle is released from rest at C and falls vertically (see diagram).

(i) Show that the tension in the string is 60 N at the instant when the prism topples. [3]

(ii) Calculate the speed of the particle at the instant when the prism topples. [5]

6 A cyclist and her bicycle have a total mass of 60 kg. The cyclist rides in a horizontal straight line, and

exerts a constant force in the direction of motion of 150 N. The motion is opposed by a resistance of

magnitude 12v N, where v m s−1 is the cyclist’s speed at time t s after passing through a fixed point A.

(i) Show that 5dv

dt= 12.5 − v. [2]

(ii) Given that the cyclist passes through A with speed 11.5 m s−1, solve this differential equation to

show that v = 12.5 − e−0.2t. [4]

(iii) Express the displacement of the cyclist from A in terms of t. [3]

7 A particle P of mass 0.7 kg is attached to one end of a light inextensible string of length 0.5 m. The

other end of the string is attached to a fixed point A which is h m above a smooth horizontal surface. P

moves in contact with the surface with uniform circular motion about the point on the surface which

is vertically below A.

(i) Given that h = 0.14, find an inequality for the angular speed of P. [4]

(ii) Given instead that the magnitude of the force exerted by the surface on P is 1.4 N and that the

speed of P is 2.5 m s−1, calculate the tension in the string and the value of h. [7]

© UCLES 2015 9709/53/M/J/15

*3202615932*


MATHEMATICS 9709/61

Paper 6 Probability & Statistics 1 (S1) May/June 2015

1 hour 15 minutes


Graph Paper




















JC15 06_9709_61/2R


2

1 The lengths, in metres, of cars in a city are normally distributed with mean - and standard deviation

0.714. The probability that a randomly chosen car has a length more than 3.2 metres and less than

- metres is 0.475. Find -. [4]

2 The table summarises the lengths in centimetres of 104 dragonflies.

Length (cm) 2.0 − 3.5 3.5 − 4.5 4.5 − 5.5 5.5 − 7.0 7.0 − 9.0

Frequency 8 25 28 31 12

(i) State which class contains the upper quartile. [1]

(ii) Draw a histogram, on graph paper, to represent the data. [4]

3 Jason throws two fair dice, each with faces numbered 1 to 6. Event A is ‘one of the numbers obtained

is divisible by 3 and the other number is not divisible by 3’. Event B is ‘the product of the two

numbers obtained is even’.

(i) Determine whether events A and B are independent, showing your working. [5]

(ii) Are events A and B mutually exclusive? Justify your answer. [1]

4

xTake fewer than 100 photos

Take at least 100 photos

0.76

0.90

View fewer than 3 times

View at least 3 times

View fewer than 3 times

View at least 3 times

A survey is undertaken to investigate how many photos people take on a one-week holiday and also

how many times they view past photos. For a randomly chosen person, the probability of taking

fewer than 100 photos is x. The probability that these people view past photos at least 3 times is 0.76.

For those who take at least 100 photos, the probability that they view past photos fewer than 3 times

is 0.90. This information is shown in the tree diagram. The probability that a randomly chosen person

views past photos fewer than 3 times is 0.801.

(i) Find x. [3]

(ii) Given that a person views past photos at least 3 times, find the probability that this person takes

at least 100 photos. [4]

© UCLES 2015 9709/61/M/J/15

3

5 The table shows the mean and standard deviation of the weights of some turkeys and geese.

Number of birds Mean (kg) Standard deviation �kg�Turkeys 9 7.1 1.45

Geese 18 5.2 0.96

(i) Find the mean weight of the 27 birds. [2]

(ii) The weights of individual turkeys are denoted by xt kg and the weights of individual geese by

xg kg. By first finding Σ x2t and Σ x2

g, find the standard deviation of the weights of all 27 birds.

[5]

6 (i) In a certain country, 68% of households have a printer. Find the probability that, in a random

sample of 8 households, 5, 6 or 7 households have a printer. [4]

(ii) Use an approximation to find the probability that, in a random sample of 500 households, more

than 337 households have a printer. [5]

(iii) Justify your use of the approximation in part (ii). [1]

7 (a) Find how many different numbers can be made by arranging all nine digits of the number

223 677 888 if

(i) there are no restrictions, [2]

(ii) the number made is an even number. [4]

(b) Sandra wishes to buy some applications (apps) for her smartphone but she only has enough

money for 5 apps in total. There are 3 train apps, 6 social network apps and 14 games apps

available. Sandra wants to have at least 1 of each type of app. Find the number of different

possible selections of 5 apps that Sandra can choose. [5]

© UCLES 2015 9709/61/M/J/15

*3926170419*


MATHEMATICS 9709/62


1 hour 15 minutes


Graph Paper




















JC15 06_9709_62/2R


2

1 A fair die is thrown 10 times. Find the probability that the number of sixes obtained is between 3

and 5 inclusive. [3]

2 120 people were asked to read an article in a newspaper. The times taken, to the nearest second, by

the people to read the article are summarised in the following table.

Time (seconds) 1 − 25 26 − 35 36 − 45 46 − 55 56 − 90

Number of people 4 24 38 34 20

Calculate estimates of the mean and standard deviation of the reading times. [5]

3

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70

80

90

100

Time in seconds

Cu

mu

lati

ve

freq

uen

cy

In an open-plan office there are 88 computers. The times taken by these 88 computers to access a

particular web page are represented in the cumulative frequency diagram.

(i) On graph paper draw a box-and-whisker plot to summarise this information. [4]

An ‘outlier’ is defined as any data value which is more than 1.5 times the interquartile range above

the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.

(ii) Show that there are no outliers. [2]

© UCLES 2015 9709/62/M/J/15

3

4

0.3Nikita buys a scarf

Nikita buys a handbag

0.72

x

Mother likes her present

Mother does not like her present

Mother likes her present

Mother does not like her present

Nikita goes shopping to buy a birthday present for her mother. She buys either a scarf, with

probability 0.3, or a handbag. The probability that her mother will like the choice of scarf is 0.72.

The probability that her mother will like the choice of handbag is x. This information is shown on the

tree diagram. The probability that Nikita’s mother likes the present that Nikita buys is 0.783.

(i) Find x. [3]

(ii) Given that Nikita’s mother does not like her present, find the probability that the present is a

scarf. [4]

5 A box contains 5 discs, numbered 1, 2, 4, 6, 7. William takes 3 discs at random, without replacement,

and notes the numbers on the discs.

(i) Find the probability that the numbers on the 3 discs are two even numbers and one odd number.

[3]

The smallest of the numbers on the 3 discs taken is denoted by the random variable S.

(ii) By listing all possible selections (126, 246 and so on) draw up the probability distribution table

for S. [5]

6 (a) Find the number of different ways the 7 letters of the word BANANAS can be arranged

(i) if the first letter is N and the last letter is B, [3]

(ii) if all the letters A are next to each other. [3]

(b) Find the number of ways of selecting a group of 9 people from 14 if two particular people cannot

both be in the group together. [3]

7 (a) Once a week Zak goes for a run. The time he takes, in minutes, has a normal distribution with

mean 35.2 and standard deviation 4.7.

(i) Find the expected number of days during a year (52 weeks) for which Zak takes less than

30 minutes for his run. [4]

(ii) The probability that Zak’s time is between 35.2 minutes and t minutes, where t > 35.2,

is 0.148. Find the value of t. [3]

(b) The random variable X has the distribution N�-, 32�. It is given that P�X < 7� = 0.2119 and

P�X < 10� = 0.6700. Find the values of - and 3. [5]

© UCLES 2015 9709/62/M/J/15

*6355909940*


MATHEMATICS 9709/63


1 hour 15 minutes


Graph Paper




















JC15 06_9709_63/4R


2

1 The weights, in grams, of onions in a supermarket have a normal distribution with mean - and standard

deviation 22. The probability that a randomly chosen onion weighs more than 195 grams is 0.128.

Find the value of -. [3]

2 When Joanna cooks, the probability that the meal is served on time is 15. The probability that the

kitchen is left in a mess is 35. The probability that the meal is not served on time and the kitchen is not

left in a mess is 310

. Some of this information is shown in the following table.

Kitchen left Kitchen not Totalin a mess left in a mess

Meal served on time 15

Meal not served on time 310

Total 1

(i) Copy and complete the table. [3]

(ii) Given that the kitchen is left in a mess, find the probability that the meal is not served on time.

[2]

3 On a production line making cameras, the probability of a randomly chosen camera being substandard

is 0.072. A random sample of 300 cameras is checked. Find the probability that there are fewer than

18 cameras which are substandard. [5]

4 A pet shop has 9 rabbits for sale, 6 of which are white. A random sample of two rabbits is chosen

without replacement.

(i) Show that the probability that exactly one of the two rabbits in the sample is white is 12. [2]

(ii) Construct the probability distribution table for the number of white rabbits in the sample. [3]

(iii) Find the expected value of the number of white rabbits in the sample. [1]

5 The heights of books in a library, in cm, have a normal distribution with mean 21.7 and standard

deviation 6.5. A book with a height of more than 29 cm is classified as ‘large’.

(i) Find the probability that, of 8 books chosen at random, fewer than 2 books are classified as large.

[6]

(ii) n books are chosen at random. The probability of there being at least 1 large book is more

than 0.98. Find the least possible value of n. [3]

© UCLES 2015 9709/63/M/J/15

3

6 Seventy samples of fertiliser were collected and the nitrogen content was measured for each sample.

The cumulative frequency distribution is shown in the table below.

Nitrogen content ≤ 3.5 ≤ 3.8 ≤ 4.0 ≤ 4.2 ≤ 4.5 ≤ 4.8

Cumulative frequency 0 6 18 41 62 70

(i) On graph paper draw a cumulative frequency graph to represent the data. [3]

(ii) Estimate the percentage of samples with a nitrogen content greater than 4.4. [2]

(iii) Estimate the median. [1]

(iv) Construct the frequency table for these results and draw a histogram on graph paper. [5]

7 Rachel has 3 types of ornament. She has 6 different wooden animals, 4 different sea-shells and

3 different pottery ducks.

(i) She lets her daughter Cherry choose 5 ornaments to play with. Cherry chooses at least 1 of each

type of ornament. How many different selections can Cherry make? [5]

Rachel displays 10 of the 13 ornaments in a row on her window-sill. Find the number of different

arrangements that are possible if

(ii) she has a duck at each end of the row and no ducks anywhere else, [3]

(iii) she has a duck at each end of the row and wooden animals and sea-shells are placed alternately

in the positions in between. [3]

© UCLES 2015 9709/63/M/J/15

*0494557328*


MATHEMATICS 9709/71


1 hour 15 minutes


Graph Paper




















JC15 06_9709_71/2R


2

1

x

f�x�

a

aO

The random variable X has probability density function, f, as shown in the diagram, where a is a

constant. Find the value of a and hence show that E�X� = 0.943 correct to 3 significant figures. [5]

2 Sami claims that he can read minds. He asks each of 50 people to choose one of the 5 letters A, B, C,

D or E. He then tells each person which letter he believes they have chosen. He gets 13 correct. Sami

says “This shows that I can read minds, because 13 is more than I would have got right if I were just

guessing.”

(i) State null and alternative hypotheses for a test of Sami’s claim. [1]

(ii) Test at the 10% significance level whether Sami’s claim is justified. [5]

3 The daily times, in minutes, that Yu Ming takes showering, getting dressed and having breakfast are

independent and have the distributions N�9, 2.22�, N�8, 1.32� and N�17, 2.62� respectively. The total

daily time that Yu Ming takes for all three activities is denoted by T minutes.

(i) Find the mean and variance of T. [2]

(ii) Yu Ming notes the value of T on each day in a random sample of 70 days and calculates the

sample mean. Find the probability that the sample mean is between 33 and 35. [4]

4 In the past, the time taken by vehicles to drive along a particular stretch of road has had mean

12.4 minutes and standard deviation 2.1 minutes. Some new signs are installed and it is expected that

the mean time will increase. In order to test whether this is the case, the mean time for a random

sample of 50 vehicles is found. You may assume that the standard deviation is unchanged.

(i) The mean time for the sample of 50 vehicles is found to be 12.9 minutes. Test at the 2.5%

significance level whether the population mean time has increased. [4]

(ii) State what is meant by a Type II error in this context. [2]

(iii) State what extra piece of information would be needed in order to find the probability of a Type II

error. [1]

© UCLES 2015 9709/71/M/J/15

3

5 The masses, m grams, of a random sample of 80 strawberries of a certain type were measured and

summarised as follows.

n = 80 Σm = 4200 Σm2= 229 000

(i) Find unbiased estimates of the population mean and variance. [3]

(ii) Calculate a 98% confidence interval for the population mean. [3]

50 random samples of size 80 were taken and a 98% confidence interval for the population mean, -,

was found from each sample.

(iii) Find the number of these 50 confidence intervals that would be expected to include the true value

of -. [1]

6 A publishing firm has found that errors in the first draft of a new book occur at random and that, on

average, there is 1 error in every 3 pages of a first draft. Find the probability that in a particular first

draft there are

(i) exactly 2 errors in 10 pages, [2]

(ii) at least 3 errors in 6 pages, [3]

(iii) fewer than 50 errors in 200 pages. [4]

7 The independent variables X and Y are such that X ∼ B�10, 0.8� and Y ∼ Po�3�. Find

(i) E�7X + 5Y − 2�, [2]

(ii) Var�4X − 3Y + 3�, [4]

(iii) P�2X − Y = 18�. [4]

© UCLES 2015 9709/71/M/J/15

*1776845998*


MATHEMATICS 9709/72


1 hour 15 minutes


Graph Paper




















JC15 06_9709_72/3R


2

1 The independent random variables X and Y have standard deviations 3 and 6 respectively. Calculate

the standard deviation of 4X − 5Y. [3]

2 Cloth made at a certain factory has been found to have an average of 0.1 faults per square metre.

Suki claims that the cloth made by her machine contains, on average, more than 0.1 faults per square

metre. In a random sample of 5 m2 of cloth from Suki’s machine, it was found that there were 2 faults.

Assuming that the number of faults per square metre has a Poisson distribution,

(i) state null and alternative hypotheses for a test of Suki’s claim, [1]

(ii) test at the 10% significance level whether Suki’s claim is justified. [4]

3 In a golf tournament, the number of times in a day that a ‘hole-in-one’ is scored is denoted by the

variable X, which has a Poisson distribution with mean 0.15. Mr Crump offers to pay $200 each time

that a hole-in-one is scored during 5 days of play. Find the expectation and variance of the amount

that Mr Crump pays. [5]

4 In the past, the flight time, in hours, for a particular flight has had mean 6.20 and standard

deviation 0.80. Some new regulations are introduced. In order to test whether these new regulations

have had any effect upon flight times, the mean flight time for a random sample of 40 of these flights

is found.

(i) State what is meant by a Type I error in this context. [2]

(ii) The mean time for the sample of 40 flights is found to be 5.98 hours. Assuming that the standard

deviation of flight times is still 0.80 hours, test at the 5% significance level whether the population

mean flight time has changed. [4]

(iii) State, with a reason, which of the errors, Type I or Type II, might have been made in your answer

to part (ii). [2]

5 The volumes, v millilitres, of juice in a random sample of 50 bottles of Cooljoos are measured and

summarised as follows.

n = 50 Σv = 14 800 Σv2= 4 390 000

(i) Find unbiased estimates of the population mean and variance. [3]

(ii) An !% confidence interval for the population mean, based on this sample, is found to have a

width of 5.45 millilitres. Find !. [4]

Four random samples of size 10 are taken and a 96% confidence interval for the population mean is

found from each sample.

(iii) Find the probability that these 4 confidence intervals all include the true value of the population

mean. [2]

© UCLES 2015 9709/72/M/J/15

3

6 The waiting time, T minutes, for patients at a doctor’s surgery has probability density function given

by

f�t� =T

k�225 − t2� 0 ≤ t ≤ 15,

0 otherwise,

where k is a constant.

(i) Show that k = 12250

. [3]

(ii) Find the probability that a patient has to wait for more than 10 minutes. [3]

(iii) Find the mean waiting time. [4]

7 In a certain lottery, 10 500 tickets have been sold altogether and each ticket has a probability of 0.0002

of winning a prize. The random variable X denotes the number of prize-winning tickets that have

been sold.

(i) State, with a justification, an approximating distribution for X. [3]

(ii) Use your approximating distribution to find P�X < 4�. [3]

(iii) Use your approximating distribution to find the conditional probability that X < 4, given that

X ≥ 1. [4]

© UCLES 2015 9709/72/M/J/15

*8038919694*


MATHEMATICS 9709/73


1 hour 15 minutes


Graph Paper




















JC15 06_9709_73/RP


2

1 Jyothi wishes to choose a representative sample of 5 students from the 82 members of her school year.

(i) She considers going into the canteen and choosing a table with five students from her year

sitting at it, and using these five people as her sample. Give two reasons why this method is

unsatisfactory. [2]

(ii) Jyothi decides to use another method. She numbers all the students in her year from 1 to 82.

Then she uses her calculator and generates the following random numbers.

231492 762305 346280

From these numbers, she obtains the student numbers 23, 14, 76, 5, 34 and 62. Explain how

Jyothi obtained these student numbers from the list of random numbers. [3]

2 Marie claims that she can predict the winning horse at the local races. There are 8 horses in each

race. Nadine thinks that Marie is just guessing, so she proposes a test. She asks Marie to predict

the winners of the next 10 races and, if she is correct in 3 or more races, Nadine will accept Marie’s

claim.

(i) State suitable null and alternative hypotheses. [1]

(ii) Calculate the probability of a Type I error. [3]

(iii) State the significance level of the test. [1]

3 A die is biased so that the probability that it shows a six on any throw is p.

(i) In an experiment, the die shows a six on 22 out of 100 throws. Find an approximate 97%

confidence interval for p. [4]

(ii) The experiment is repeated and another 97% confidence interval is found. Find the probability

that exactly one of the two confidence intervals includes the true value of p. [2]

4 The marks, x, of a random sample of 50 students in a test were summarised as follows.

n = 50 Σ x = 1508 Σ x2= 51 825

(i) Calculate unbiased estimates of the population mean and variance. [3]

(ii) Each student’s mark is scaled using the formula y = 1.5x + 10. Find estimates of the population

mean and variance of the scaled marks, y. [3]

© UCLES 2015 9709/73/M/J/15

3

5 The mean breaking strength of cables made at a certain factory is supposed to be 5 tonnes. The quality

control department wishes to test whether the mean breaking strength of cables made by a particular

machine is actually less than it should be. They take a random sample of 60 cables. For each cable

they find the breaking strength by gradually increasing the tension in the cable and noting the tension

when the cable breaks.

(i) Give a reason why it is necessary to take a sample rather then testing all the cables produced by

the machine. [1]

(ii) The mean breaking strength of the 60 cables in the sample is found to be 4.95 tonnes. Given that

the population standard deviation of breaking strengths is 0.15 tonnes, test at the 1% significance

level whether the population mean breaking strength is less than it should be. [4]

(iii) Explain whether it was necessary to use the Central Limit theorem in the solution to part (ii).

[2]

6 People arrive at a checkout in a store at random, and at a constant mean rate of 0.7 per minute. Find

the probability that

(i) exactly 3 people arrive at the checkout during a 5-minute period, [2]

(ii) at least 30 people arrive at the checkout during a 1-hour period. [4]

People arrive independently at another checkout in the store at random, and at a constant mean rate

of 0.5 per minute.

(iii) Find the probability that a total of more than 3 people arrive at this pair of checkouts during a

2-minute period. [4]

7 The probability density function of the random variable X is given by

f�x� =T 3

4x�c − x� 0 ≤ x ≤ c,

0 otherwise,

where c is a constant.

(i) Show that c = 2. [3]

(ii) Sketch the graph of y = f�x� and state the median of X. [3]

(iii) Find P�X < 1.5�. [4]

(iv) Hence write down the value of P�0.5 < X < 1�. [1]

© UCLES 2015 9709/73/M/J/15

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