Cambridge International Examinations

Cambridge International General Certificate of Secondary Education

* 1 6 4 3 4 4 6 5 9 5 *

CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/21

Paper 2 (Extended) May/June 2015
45 minutes
Candidates answer on the Question Paper.
Additional Materials: Geometrical Instruments

READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
Do not use staples, paper clips, glue or correction fluid.
You may use an HB pencil for any diagrams or graphs.
DO NOT WRITE IN ANY BARCODES.

Answer all the questions.

CALCULATORS MUST NOT BE USED IN THIS PAPER.
All answers should be given in their simplest form.
You must show all the relevant working to gain full marks and you will be given marks for correct methods
even if your answer is incorrect.
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 40.

This document consists of 8 printed pages.

DC (LEG/SW) 99284/5
© UCLES 2015 [Turn over
 2

Formula List

- b ! b 2 - 4ac
For the equation ax 2 + bx + c = 0 x=
2a

Curved surface area, A, of cylinder of radius r, height h. A = 2rrh

Curved surface area, A, of cone of radius r, sloping edge l. A = rrl

Curved surface area, A, of sphere of radius r. A = 4r r 2

1
Volume, V, of pyramid, base area A, height h. V = Ah
3

Volume, V, of cylinder of radius r, height h. V = rr 2 h

1
Volume, V, of cone of radius r, height h. V = rr 2 h
3

4
Volume, V, of sphere of radius r. V = rr 3
3

A a b c
= =
sin A sin B sin C

b a 2 = b 2 + c 2 - 2bc cos A
c

1
Area = bc sin A
2

B a C

© UCLES 2015 0607/21/M/J/15

 3

Answer all the questions.

1 (a) Write 4725.6 correct to two significant figures.

Answer(a) .................................................................. [1]

(b) Write 0.01026 correct to three decimal places.

Answer(b) .................................................................. [1]

2 Expand and simplify.

(a) - 3x ^2 - xh - ^3x 2 - 7h

Answer(a) ................................................................. [2]

(b) ^5x - 3yh^2y - 5xh

Answer(b) ................................................................. [3]

© UCLES 2015 0607/21/M/J/15 [Turn over

 4

1
3 Find the exact value of 27 - 3 .

Answer ................................................................. [2]

1
4 Simplify ^16x 8 y 2h2 .

Answer ................................................................. [2]

5 (a) Simplify.

27 + 147

Answer(a) ................................................................. [2]

(b) Rationalise the denominator.

3- 5
3+ 5

Answer(b) ................................................................. [3]

© UCLES 2015 0607/21/M/J/15

 5

6 Solve.
log x + log 5 - log 25 = log 10

Answer x= ................................................................. [3]

7 There are 400 students at a school.

2
5 of the students are boys.

70% of the girls can swim.

The ratio of boys that cannot swim to girls that cannot swim is 2 : 3.

Complete the table.

Boys Girls Total

Can swim
Cannot swim
Total 400

[4]

© UCLES 2015 0607/21/M/J/15 [Turn over

 6

8
1 cm
y°
NOT TO
2 cm SCALE
x cm

30°

(a) Write down the value of x.

Answer(a) x = ............................................................[1]

(b) Find the value of y.

Answer(b) y = ................................................................. [2]

1
9 f ^xh =
3x - 2
(a) Find f ^4h .

Answer(a) ................................................................. [1]

1
(b) Solve f ^xh = .
4

Answer(b) ................................................................. [2]

(c) Find f -1 ^xh .

Answer(c) ................................................................. [3]

© UCLES 2015 0607/21/M/J/15

 7

10
D M C

NOT TO
SCALE

A N B

ABCD is a trapezium.
AB = 2DC, DM = 2MC and AN = 3NB.
AB = p and AD = q .

(a) Write MC in terms of p.

Answer(a) ................................................................. [2]

(b) Find MN in terms of p and q.

Answer(b) ................................................................. [2]

Question 11 is printed on the next page.

© UCLES 2015 0607/21/M/J/15 [Turn over

 8

11 The point A has co-ordinates (2, 8) and the point B has co-ordinates (6, 6).

Find the equation of the perpendicular bisector of the line AB.

Answer ................................................................. [4]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.

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

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

© UCLES 2015 0607/21/M/J/15

 Cambridge International Examinations
Cambridge International General Certificate of Secondary Education
* 1 0 4 9 5 4 7 5 2 6 *

CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/22

Paper 2 (Extended) May/June 2015
45 minutes
Candidates answer on the Question Paper.
Additional Materials: Geometrical Instruments

READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
Do not use staples, paper clips, glue or correction fluid.
You may use an HB pencil for any diagrams or graphs.
DO NOT WRITE IN ANY BARCODES.

Answer all the questions.

CALCULATORS MUST NOT BE USED IN THIS PAPER.
All answers should be given in their simplest form.
You must show all the relevant working to gain full marks and you will be given marks for correct methods
even if your answer is incorrect.
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 40.

This document consists of 8 printed pages.

DC (AC/SG) 99283/6
© UCLES 2015 [Turn over
 2

Formula List

- b ! b 2 - 4ac
For the equation ax 2 + bx + c = 0 x=
2a

Curved surface area, A, of cylinder of radius r, height h. A = 2rrh

Curved surface area, A, of cone of radius r, sloping edge l. A = rrl

Curved surface area, A, of sphere of radius r. A = 4rr 2

1
Volume, V, of pyramid, base area A, height h. V = Ah
3

Volume, V, of cylinder of radius r, height h. V = rr 2 h

1
Volume, V, of cone of radius r, height h. V = rr 2 h
3

4
Volume, V, of sphere of radius r. V = rr 3
3

A a b c
= =
sin A sin B sin C

b a 2 = b 2 + c 2 - 2bc cos A
c

1
Area = bc sin A
2

B a C

© UCLES 2015 0607/22/M/J/15

 3

Answer all the questions.

1 (a) Work out (0.3) 2 .

Answer(a) ................................................................ [1]

n
= .
5
(b) Find n when
6 24

Answer(b) n = ................................................................ [1]

2 (a) Find the value of

(i) 25 0 ,

Answer(a)(i) ................................................................ [1]

3
(ii) 100 2 .

Answer(a)(ii) ................................................................ [1]

(b) Write as a single power of 5.

5 12
5 # 52
3

Answer(b) ................................................................ [1]

J- 6N
3 Find the magnitude of KK OO .
L 4P
Write your answer in surd form as simply as possible.

Answer ................................................................ [3]

© UCLES 2015 0607/22/M/J/15 [Turn over

 4

4 Anneke, Babar, Céline, and Dieter each throw the same biased die.
They want to find the probability of throwing a six with this die.
They each throw the die a different number of times.

These are their results.

Anneke Babar Céline Dieter

Number of throws 200 40 100 500

Number of sixes 46 12 15 100

(a) Complete the table below to show the relative frequencies of their results.
Write your answers as decimals.

Anneke Babar Céline Dieter

Relative frequency
of throwing a six
[2]

(b) Whose result gives the best estimate of the probability of throwing a six with the biased die?
Give a reason for your answer.

Answer(b) .................................................... because ...............................................................................

............................................................................................................................................................. [1]

(c) The probability of throwing a six with a different biased die is 0.41.
Find the expected number of sixes when this die is thrown 600 times.

Answer(c) ................................................................ [1]

© UCLES 2015 0607/22/M/J/15

 5

5 A is the point (2, 8) and B is the point (6, 0).

(a) Find the co-ordinates of the midpoint of AB.

Answer(a) (........................... , ...........................) [1]

(b) Find the gradient of AB.

Answer(b) ................................................................ [2]

6 Simplify (5 + 3) 2 .

Answer ................................................................ [2]

7 Solve.

2x + 3 G 4 (x - 2)

Answer ................................................................ [3]

© UCLES 2015 0607/22/M/J/15 [Turn over

 6

3m

6m

NOT TO
SCALE

10 m

The diagram shows a shape made from a cylinder and a cone.

The cylinder and cone have a common radius of 6 m.
The height of the cylinder is 10 m and the height of the cone is 3 m.

Calculate the total volume of the shape.

Leave your answer as a multiple of π.

Answer ........................................................... m3 [3]

9 Solve these simultaneous equations.

5x + 2y = 11
4x - 3y = 18

Answer x = ................................................................

y = ................................................................ [4]

© UCLES 2015 0607/22/M/J/15

 7

10 Solve the following equations.

(a) log x + log 3 = log 12

Answer(a) x = ................................................................ [1]

(b) log x = 3

Answer(b) x = ................................................................ [1]

(c) 2log x – log 5 = log 20

Answer(c) x = ................................................................ [3]

11 A, B, C and D are points on the circle, centre O. Q

B
PQ is a tangent to the circle at the point C.
Angle PCD = 55° and angle ADO = 40°.
A
NOT TO
SCALE
O C
55°

40°

D P
Find

(a) angle COD,

Answer(a) ................................................................ [2]

(b) angle DAC,

Answer(b) ................................................................ [1]

(c) angle ABC.

Answer(c) ................................................................ [1]

Question 12 is printed on the next page.

© UCLES 2015 0607/22/M/J/15 [Turn over
 8

12 These are sketches of the graphs of six functions.

A y B y

0 x 0 x

C y D y

0 x 0 x

y y
E F

0 x 0 x

In the table below are four functions.

Write the correct letter in the table to match each function with its graph.

Function Graph

f (x) = 2x - 3

f (x) = (x - 2) 2

f (x) = 4x - x 3
f (x) = 5 - 2x
[4]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.

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

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

© UCLES 2015 0607/22/M/J/15

 Cambridge International Examinations
Cambridge International General Certificate of Secondary Education
* 4 2 1 2 2 9 3 3 0 7 *

CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/23

Paper 2 (Extended) May/June 2015
45 minutes
Candidates answer on the Question Paper.
Additional Materials: Geometrical Instruments

READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
Do not use staples, paper clips, glue or correction fluid.
You may use an HB pencil for any diagrams or graphs.
DO NOT WRITE IN ANY BARCODES.

Answer all the questions.

CALCULATORS MUST NOT BE USED IN THIS PAPER.
All answers should be given in their simplest form.
You must show all the relevant working to gain full marks and you will be given marks for correct methods
even if your answer is incorrect.
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 40.

This document consists of 12 printed pages.

DC (RW/SW) 99290/3
© UCLES 2015 [Turn over
 2

Formula List

- b ! b 2 - 4ac
For the equation ax 2 + bx + c = 0 x=
2a

Curved surface area, A, of cylinder of radius r, height h. A = 2rrh

Curved surface area, A, of cone of radius r, sloping edge l. A = rrl

Curved surface area, A, of sphere of radius r. A = 4rr 2

1
Volume, V, of pyramid, base area A, height h. V = Ah
3

Volume, V, of cylinder of radius r, height h. V = rr 2 h

1
Volume, V, of cone of radius r, height h. V = rr 2 h
3

4
Volume, V, of sphere of radius r. V = rr 3
3

A a b c
= =
sin A sin B sin C

b a 2 = b 2 + c 2 - 2bc cos A
c

1
Area = bc sin A
2

B a C

© UCLES 2015 0607/23/M/J/15

 3

Answer all the questions.

1 Round these numbers to 3 significant figures.

(a) 0.000 604 83

Answer(a) .................................................................. [1]

(b) 6 998 800

Answer(b) .................................................................. [1]

2 By rounding each number to 1 significant figure, estimate the value of

0.583 # 311.6 .
1.82 + 10.43
Show your working.

Answer .................................................................. [2]

© UCLES 2015 0607/23/M/J/15 [Turn over

 4

3 a = 23 # 3 # 52 b = 22 # 32 # 76

(a) Find, giving each answer as the product of prime factors,

(i) the highest common factor (HCF) of a and b,

Answer(a)(i) .................................................................. [1]

(ii) b.

Answer(a)(ii) .................................................................. [1]

(b) ap is a cube number.

Find the smallest integer value of p.

Answer(b) .................................................................. [1]

© UCLES 2015 0607/23/M/J/15

 5

3 cm
NOT TO
8 cm SCALE

5 cm

3 cm

The diagram shows a rectangle, two semicircles and two right-angled triangles.

(a) Find the total area of the shape.

Give your answer in the form a + br .

Answer(a) ...........................................................cm2 [3]

(b) Describe fully the symmetry of the shape.

Answer(b) .................................................................................................................................................

.............................................................................................................................................................. [2]

5 Solve.
5 ^x + 2h 1 2 ^4x - 7h

Answer .................................................................. [3]

© UCLES 2015 0607/23/M/J/15 [Turn over

 6

6 François and George each ask a sample of students at their college how they travel to college.

These are their results.

Total number
Walk Cycle Bus Train Car
of students
François 7 3 4 1 5 20
George 46 24 44 11 25 150

(a) Explain why George’s results will give the better estimates of the probabilities of the different types of
travel.

Answer(a) ............................................................................................................................................. [1]

(b) A student is selected at random.

(i) Use George’s results to estimate the probability that the student cycles to college.

Answer(b)(i) .................................................................. [1]

(ii) There are 3000 students at the college.

Use George’s results to estimate the number of students who cycle to college.

Answer(b)(ii) .................................................................. [1]

© UCLES 2015 0607/23/M/J/15

 7

7
y

NOT TO
SCALE

0 x

The diagram shows the lines x =- 2 , y = 12 x + 1 and 3x + 4y = 20 .

(a) Use simultaneous equations to find the co-ordinates of the point A.

Answer(a) ( .................. , .................. ) [3]

(b) (i) P is a point in the region such that

x 1- 2 , y 2 12 x + 1 and 3x + 4y 1 20 .

On the diagram, mark and label a possible position of P. [1]

(ii) Q is a point in the region such that

x 2- 2 , y = 12 x + 1 and 3x + 4y 1 20 .

On the diagram, mark and label a possible position of Q. [1]

© UCLES 2015 0607/23/M/J/15 [Turn over

 8

8
C

B
NOT TO
SCALE

35°
A D

In the diagram, A, B, C, D and E are points on the circle.

AD is a diameter and angle CAD = 35° .

Find

(a) angle ACD,

Answer(a) .................................................................. [1]

(b) angle CBD,

Answer(b) .................................................................. [1]

(c) angle AEC.

Answer(c) .................................................................. [2]

© UCLES 2015 0607/23/M/J/15

 9

9 The sets P, Q and R are subsets of the universal set U.

• PkR ! Q
• Q is a subset of R
• QkP = Q

Complete the Venn diagram to show the sets P, Q, and R.

[3]

© UCLES 2015 0607/23/M/J/15 [Turn over

 10

10 (a) Factorise x 2 - 3x - 10 .

Answer(a) .................................................................. [2]

3
x
(b) Make x the subject of y = .
a

Answer(b) x = ........................................................... [2]

© UCLES 2015 0607/23/M/J/15

 11

1
11 (a) Find log 5 .
25

Answer(a) .................................................................. [1]

(b) Find x when

(i) log x - log 2 = log 6 ,

Answer(b)(i) .................................................................. [1]

1
(ii) log x 4 = .
2

Answer(b)(ii) .................................................................. [1]

Question 12 is printed on the next page.

© UCLES 2015 0607/23/M/J/15 [Turn over

 12

12
y

NOT TO
SCALE

–3 0 2 x

(0, –12)

The diagram shows a sketch of the graph of y = ax 2 + bx + c .

The graph goes through the points ^- 3, 0h , ^0, -12h and ^2, 0h .

Find the values a, b and c.

Answer a = .....................................

b = .....................................

c = ...................................... [3]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.

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

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

© UCLES 2015 0607/23/M/J/15

 Cambridge International Examinations
Cambridge International General Certificate of Secondary Education
* 7 2 6 0 8 7 7 3 2 5 *

CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/41

Paper 4 (Extended) May/June 2015
2 hours 15 minutes
Candidates answer on the Question Paper.
Additional Materials: Geometrical Instruments
Graphics Calculator

READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
Do not use staples, paper clips, glue or correction fluid.
You may use an HB pencil for any diagrams or graphs.
DO NOT WRITE IN ANY BARCODES.

Answer all the questions.

Unless instructed otherwise, give your answers exactly or correct to three significant figures as appropriate.
Answers in degrees should be given to one decimal place.
For π, use your calculator value.
You must show all the relevant working to gain full marks and you will be given marks for correct methods,
including sketches, even if your answer is incorrect.
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 120.

This document consists of 20 printed pages.

DC (KN/SW) 99287/7
© UCLES 2015 [Turn over
 2

Formula List

- b ! b 2 - 4ac
For the equation ax 2 + bx + c = 0 x=
2a

Curved surface area, A, of cylinder of radius r, height h. A = 2rrh

Curved surface area, A, of cone of radius r, sloping edge l. A = rrl

Curved surface area, A, of sphere of radius r. A = 4rr 2

1
Volume, V, of pyramid, base area A, height h. V = Ah
3

Volume, V, of cylinder of radius r, height h. V = rr 2 h

1
Volume, V, of cone of radius r, height h. V = rr 2 h
3

4
Volume, V, of sphere of radius r. V = rr 3
3

A a b c
= =
sin A sin B sin C

b a 2 = b 2 + c 2 - 2bc cos A
c

1
Area = bc sin A
2

B a C

© UCLES 2015 0607/41/M/J/15

 3

Answer all the questions.

1 The table shows the marks that 80 students scored in an examination.

Mark 0 1 2 3 4 5 6 7 8 9 10

Number of
1 5 6 8 9 10 12 8 16 3 2
students

(a) Write down the mode.

Answer(a) ................................................................ [1]

(b) Write down the range.

Answer(b) ................................................................ [1]

(c) Find the median.

Answer(c) ................................................................ [1]

(d) Find the interquartile range.

Answer(d) ................................................................ [2]

(e) Calculate the mean.

Answer(e) ................................................................ [1]

© UCLES 2015 0607/41/M/J/15 [Turn over

 4

2 Solve the simultaneous equations.

You must show all your working.

5x – 2y = 11.5
4x + 3y = 0

Answer x = ................................................................

y = ................................................................ [4]

3 A car of length 4.5 metres is travelling at 72 km/h.

The car approaches a tunnel of length 260 metres.

(a) Change 72 km/h into m/s.

Answer(a) ......................................................... m/s [1]

(b) Find the time it will take for the car to pass completely through the tunnel.
Give your answer in seconds.

Answer(b) .............................................................. s [2]

© UCLES 2015 0607/41/M/J/15

 5

4
y
10

7
A B
6

–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 x

(a) Describe fully the single transformation that maps triangle A onto triangle B.

Answer(a) .................................................................................................................................................

.............................................................................................................................................................. [2]

(b) Rotate triangle B through 90° clockwise, centre (–1, 6). Draw this triangle and label it C. [3]

(c) Describe fully the single transformation that maps triangle C onto triangle A.

Answer(c) ..................................................................................................................................................

.............................................................................................................................................................. [2]

© UCLES 2015 0607/41/M/J/15 [Turn over

 6

5 (a) y varies inversely as the square root of x.

y = 5 when x = 9.

(i) Find the value of y when x = 25.

Answer(a)(i) y = ................................................................ [2]

(ii) Find the value of x when y = 25.

Answer(a)(ii) x = ................................................................ [2]

(iii) Find x in terms of y.

Answer(a)(iii) x = ................................................................ [2]

(b)
y

24

–4 0 2 x

Find the equation of this quadratic curve.

Answer(b) ................................................................ [3]

© UCLES 2015 0607/41/M/J/15

 7

6 The Venn diagram shows the sets A, B and C.

U
A C

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}

A = {factors of 12}
B = {factors of 6}
C = {11, 12, 13, 14}

(a) List the elements of sets A and B.

Answer(a) A = { ...............................................................}

B = { ............................................................... } [2]

(b) Write all the elements of U in the correct regions of the Venn diagram above. [3]

(c) List the elements of

(i) A + B,

Answer(c)(i) { ............................................................... } [1]

(ii) Al + C ,

Answer(c)(ii) { ............................................................... } [1]

(iii) B , C l.

Answer(c)(iii) { ............................................................... } [1]

(d) Find

(i) n (A , B , C)l ,

Answer(d)(i) ................................................................ [1]

(ii) n (A + B + C)l .

Answer(d)(ii) ................................................................ [1]

© UCLES 2015 0607/41/M/J/15 [Turn over

 8

7 Squash balls have radius 1.5 cm.

They are sold in boxes. Each box is a cuboid.
Each box has length 15 cm, width 12 cm and height 3 cm.

(a) Show that the maximum number of balls in a box is 20.

[1]

(b) Calculate the volume of one ball.

Answer(b) ......................................................... cm3 [2]

(c) Calculate the total volume of 20 balls.

Answer(c) ......................................................... cm3 [1]

(d) Write your answer to part (c) in standard form.

Answer(d) ......................................................... cm3 [1]

(e) Calculate the percentage of the volume of the box that the 20 balls fill.

Answer(e) ............................................................ % [2]

© UCLES 2015 0607/41/M/J/15

 9

8
A
NOT TO
SCALE
C

X
O P
26°

A, B and C lie on a circle, centre O.

AP and BP are tangents to the circle.
AB intersects OP at the point X and angle OPB = 26°.

(a) Find the size of

(i) angle ABP,

Answer(a)(i) ................................................................ [1]

(ii) angle OBA,

Answer(a)(ii) ................................................................ [1]

(iii) angle ACB.

Answer(a)(iii) ................................................................ [1]

(b) Write down the mathematical name of quadrilateral AOBP.

Answer(b) ................................................................ [1]

(c) Complete these statements.

(i) Triangle OBP is congruent to triangle ...................................................................................... . [1]

(ii) Triangle OBP is similar, but not congruent to, triangle ........................................................... . [1]

© UCLES 2015 0607/41/M/J/15 [Turn over

 10

9 The table shows the amount in dollars, y, that 10 families of different size, x, spend in one week.

Number in family, (x) 2 2 3 3 5 5 6 6 6 6

Amount in dollars, (y). 60 65 80 75 100 105 120 135 125 115

(a) (i) Complete the scatter diagram.

The first four points have been plotted for you.

y
150

140

130

120

110
Amount
in dollars 100

90

80

70

60

50
0 1 2 3 4 5 6 7 8 x
Number in family
[2]

(ii) What type of correlation is shown by the scatter diagram?

Answer(a)(ii) ................................................................ [1]

© UCLES 2015 0607/41/M/J/15

 11

(b) Find

(i) the mean family size,

Answer(b)(i) ................................................................ [1]

(ii) the mean amount spent in one week.

Answer(b)(ii) $ ................................................................ [1]

(c) (i) Find the equation of the regression line in the form y = mx + c .

Answer(c)(i) y = ................................................................ [2]

(ii) Use your answer to part (c)(i) to estimate the amount spent in one week by a family of 4.

Answer(c)(ii) $ ................................................................ [1]

© UCLES 2015 0607/41/M/J/15 [Turn over

 12

10
B

72° NOT TO
SCALE

65 m 80 m

A C

64 m

58°

(a) Find AC.

Answer(a) ............................................................ m [2]

© UCLES 2015 0607/41/M/J/15

 13

(b) Calculate angle CAD.

Answer(b) ................................................................ [3]

(c) Calculate the area of the quadrilateral ABCD.

Answer(c) ........................................................... m2 [4]

© UCLES 2015 0607/41/M/J/15 [Turn over

 14

11 Paula invests $3000 in Bank A and $3000 in Bank B.

(a) Bank A pays compound interest at a rate of 4% each year.

(i) Find the total amount that Paula has in Bank A at the end of 3 years.

Answer(a)(i) $ ................................................................. [2]

(ii) After how many complete years is the total amount that Paula has in Bank A greater than $4000?

Answer(a)(ii) ................................................................ [3]

(b) Bank B pays simple interest at a rate of 5% each year.

(i) Find the total amount that Paula has in Bank B at the end of 3 years.

Answer(b)(i) $ ................................................................. [1]

(ii) After how many complete years is the total amount that Paula has in Bank B greater than $4000?

Answer(b)(ii) .................................................................. [1]

(c) After how many complete years will the total amount that Paula has in Bank A be greater than the total
amount that Paula has in Bank B?

Answer(c) .................................................................. [3]

© UCLES 2015 0607/41/M/J/15

 15

12 Bag 1 only contains 6 blue balls and 4 red balls.

Bag 2 only contains 8 blue balls and 2 red balls.
Marco chooses a ball at random from Bag 1 and puts it in Bag 2.
He then chooses a ball at random from Bag 2 and puts it in Bag 1.

(a) Complete the tree diagram.

Bag 1 Bag 2
blue
........

blue
6
10

........ red

blue
........

........
red

........ red
[2]
(b) Find the probability that the two balls chosen are

(i) both blue,

Answer(b)(i) ................................................................ [2]

(ii) one red and one blue.

Answer(b)(ii) ................................................................ [3]

(c) Find the probability that, after Marco chooses the two balls, there are exactly 6 blue balls in Bag 1.

Answer(c) ................................................................ [3]

© UCLES 2015 0607/41/M/J/15 [Turn over

 16

13 In this question all lengths are in centimetres.

6x + 1 NOT TO
SCALE
2x – 1

5x + 4

(a) Write down a quadratic equation, in terms of x, and show that it simplifies to

7x 2 - 24x - 16 = 0.

[3]

(b) Factorise 7x 2 - 24x - 16.

Answer(b) ................................................................ [2]

© UCLES 2015 0607/41/M/J/15

 17

(c) Show that the area of the triangle is 84 cm2.

[2]

(d) The area of this rectangle is equal to the area of the triangle.
y+2
Find the value of y.
NOT TO
SCALE
y

Answer(d) y = ................................................................ [4]

© UCLES 2015 0607/41/M/J/15 [Turn over

 18

14 In this question all measurements are in metres.

A rectangular garden has length p and width q.

The garden is divided into 3 sections as shown in the diagram.

2p
3

1q
Vegetables
4
NOT TO
x SCALE
Flowers

Grass

(a) Write down an expression, in terms of p and q, for the area for flowers.

Answer(a) ............................................................m2 [1]

3
(b) Show that x = p .
4

[2]

© UCLES 2015 0607/41/M/J/15

 19

(c) Find an expression, in terms of p and q, for the area for grass.
Give your answer in its simplest form.

Answer(c) ............................................................m2 [2]

(d) Find the ratio area for vegetables : area for grass .

Answer(d) .............................. : .............................. [2]

Question 15 is printed on the next page.

© UCLES 2015 0607/41/M/J/15 [Turn over

 20

x 2 + 4x + 3
15 The diagram shows a sketch of the graph of y = f ^xh where f ^xh = .
x 2 - 4x + 3
y

NOT TO
SCALE

O x

(a) (i) Find the equations of the three asymptotes.

Answer(a)(i) .............................. , .............................. , .............................. [3]

(ii) Find the co-ordinates of the local maximum point.

Answer(a)(ii) ( .............................. , ..............................) [2]

(iii) Find the co-ordinates of the local minimum point.

Answer(a)(iii) ( .............................. , ..............................) [2]

(b) Find the values of k, when

(i) f ^xh = k has no solutions,

Answer(b)(i) ................................................................ [2]

(ii) f ^xh = k has one solution.

Answer(b)(ii) ................................................................ [1]

(c) Solve the inequality f ^xh 2 0 .

Answer(c) ................................................................ [3]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.

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

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

© UCLES 2015 0607/41/M/J/15

 Cambridge International Examinations
Cambridge International General Certificate of Secondary Education
* 5 4 8 3 7 0 5 8 8 5 *

CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/42

Paper 4 (Extended) May/June 2015
2 hours 15 minutes
Candidates answer on the Question Paper.
Additional Materials: Geometrical Instruments
Graphics Calculator

READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
Do not use staples, paper clips, glue or correction fluid.
You may use an HB pencil for any diagrams or graphs.
DO NOT WRITE IN ANY BARCODES.

Answer all the questions.

Unless instructed otherwise, give your answers exactly or correct to three significant figures as appropriate.
Answers in degrees should be given to one decimal place.
For π, use your calculator value.
You must show all the relevant working to gain full marks and you will be given marks for correct methods,
including sketches, even if your answer is incorrect.
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 120.

This document consists of 20 printed pages.

DC (NH/SW) 99286/6
© UCLES 2015 [Turn over
 2

Formula List

- b ! b 2 - 4ac
For the equation ax 2 + bx + c = 0 x=
2a

Curved surface area, A, of cylinder of radius r, height h. A = 2rrh

Curved surface area, A, of cone of radius r, sloping edge l. A = rrl

Curved surface area, A, of sphere of radius r. A = 4rr 2

1
Volume, V, of pyramid, base area A, height h. V = Ah
3

Volume, V, of cylinder of radius r, height h. V = rr 2 h

1
Volume, V, of cone of radius r, height h. V = rr 2 h
3

4
Volume, V, of sphere of radius r. V = rr 3
3

A a b c
= =
sin A sin B sin C

b a 2 = b 2 + c 2 - 2bc cos A
c

1
Area = bc sin A
2

B a C

© UCLES 2015 0607/42/M/J/15

 3

Answer all the questions.

1 An art gallery values its paintings every five years.

The value of one painting increased by 90% every five years from 1990.
The value in 1995 was $76 000.

(a) Calculate the exact value of the painting in

(i) 1990,

Answer(a)(i) $ .................................................................. [3]

(ii) 2010.

Answer(a)(ii) $ .................................................................. [3]

(b) The value of the painting continues to increase by 90% every five years.

In which year’s valuation will the value of the painting first be over $10 million?

Answer(b) .................................................................. [2]

© UCLES 2015 0607/42/M/J/15 [Turn over

 4

y
7

5
C
4

2
B A
1

–4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 x

–1

–2

–3

(a) Describe fully the single transformation that maps triangle A onto triangle B.

Answer(a) .................................................................................................................................................

.............................................................................................................................................................. [3]

(b) Complete the statement.

J N
K O
Triangle A can be mapped onto triangle C by a translation with vector K OO followed by
K
L P
a reflection in the line ............................................... . [2]

(c) Stretch triangle A with the x-axis invariant and stretch factor 2. [2]

© UCLES 2015 0607/42/M/J/15

 5

3 Jean-Paul goes on holiday and drives 780 km.

He leaves at 06 45 and arrives at 16 10.

(a) Find the average speed for the whole journey.

Answer(a) .........................................................km/h [3]

(b) He travels partly on autoroutes and partly on other roads.

He travels for 520 km on autoroutes at an average speed of 105 km/h.

Find the average speed for the part of the journey on other roads.

Answer(b) .........................................................km/h [3]

(c) For every 100 km travelled on autoroutes, Jean-Paul’s car uses 6 litres of fuel.
For every 100 km travelled on other roads, it uses 8 litres of fuel.
Fuel costs 1.63 euros per litre.
The total autoroute toll charges are 15.20 euros.

Find the total cost of the journey.

Answer(c) ....................................................... euros [4]

© UCLES 2015 0607/42/M/J/15 [Turn over

 6

4
y
25

–2 0 4 x

–15

f ^xh = x 3 - 3x 2 + 6

(a) On the diagram, sketch the graph of y = f ^xh for - 2 G x G 4 . [2]

(b) Find the co-ordinates of the local maximum point and the local minimum point.

Answer(b) Maximum ( ................. , ................. )

Minimum ( ................. , ................. ) [2]

(c) Find the range of values of k for which the equation f ^xh = k has 3 different solutions.

Answer(c) .................................................................. [2]

© UCLES 2015 0607/42/M/J/15

 7

(d) Describe fully the symmetry of the graph of y = f ^xh .

Answer(d) ..................................................................................................................................................

.............................................................................................................................................................. [3]
J 0N
(e) The graph of y = g ^xh is the translation of the graph of y = f ^xh with vector K O .
L- 2P
Write down and simplify g ^xh .

Answer(e) g(x) = ................................................................. [1]

© UCLES 2015 0607/42/M/J/15 [Turn over

 8

5 The table shows the number of goals scored in a season, x, and the average attendance at matches in
thousands, y, for ten teams in a league.

Team A B C D E F G H I J
Number of
goals scored in 86 66 75 72 66 55 71 53 47 45
a season (x)
Average
attendance in 76 46 41 60 36 36 45 25 20 35
thousands (y)

(a) Complete the scatter diagram.

The first five points have been plotted for you.

y
80

75

70

65

60

55
Average
attendance
in 50
thousands
45

40

35

30

25

20

15
30 40 50 60 70 80 90 x
Number of goals scored in a season
[2]

© UCLES 2015 0607/42/M/J/15

 9

(b) What type of correlation is shown by the scatter diagram?

Answer(b) .................................................................. [1]

(c) Find the mean

(i) number of goals scored,

Answer(c)(i) .................................................................. [1]

(ii) average attendance.

Answer(c)(ii) ..................................................thousand [1]

(d) Find the equation of the line of regression in the form y = mx + c .

Answer(d) y = ................................................................. [2]

(e) Use your answer to part (d) to estimate the average attendance for a team that scored 80 goals in a
season.

Answer(e) .................................................................. [1]

© UCLES 2015 0607/42/M/J/15 [Turn over

 10

A B NOT TO
SCALE

120 cm

E D C
180 cm

The diagram shows a fence panel ABCDE.

The vertical edges AE and BC are of length 120 cm and the horizontal base EC is of length 180 cm.
D is the midpoint of EC.

(a) Calculate AD.

Answer(a) ............................................................ cm [2]

(b) Show that angle ADB = 73.74° correct to 2 decimal places.

[3]

(c) AB is an arc of a circle centre D.

Find the area of the fence panel.

Answer(c) ...........................................................cm2 [3]

© UCLES 2015 0607/42/M/J/15

 11

(d) Stefan’s fence has 8 panels, each identical to ABCDE.

He wishes to paint both sides of all the panels.
Each litre of paint covers an area of 6 square metres.

Calculate the number of litres Stefan needs to paint both sides of the whole fence.

Answer(d) ......................................................... litres [3]

© UCLES 2015 0607/42/M/J/15 [Turn over

 12

7 y
9

–3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 11 x
–1

–2

–3

–4

–5

–6

–7

–8

–9

(a) On the grid, show clearly the region defined by these inequalities.

x H-1 yH2 y H 2x - 3 3x + 5y G 30 [7]

(b) Use your diagram to estimate

(i) the greatest value of y in the region,

Answer(b)(i) .................................................................. [1]

(ii) the greatest value of x + y in the region.

Answer(b)(ii) .................................................................. [1]

© UCLES 2015 0607/42/M/J/15

 13

8 (a) Give an example of

(i) discrete data,

Answer(a)(i) ................................................................................................................................. [1]

(ii) continuous data.

Answer(a)(ii) ................................................................................................................................ [1]

(b) The table shows the heights, h cm, of 30 students in a class.

Height
150 < h  155 155 < h  160 160 < h  165 165 < h  170 170 < h  175 175 < h  180
(h cm)

Frequency 2 4 8 7 5 4

(i) Write down the modal interval.

Answer(b)(i) ........................... < h  ........................... [1]

(ii) Write down the interval that contains the median.

Answer(b)(ii) .......................... < h  .......................... [1]

(iii) Calculate an estimate of the mean.

Answer(b)(iii) ............................................................ cm [2]

(iv) Explain why the answer to part (b)(iii) is an estimate and not an exact answer.

Answer(b)(iv) ....................................................................................................................................

...................................................................................................................................................... [1]

© UCLES 2015 0607/42/M/J/15 [Turn over

 14

9 Gitte has a bag containing coloured wristbands.

There are 5 blue wristbands, 2 yellow wristbands and 4 pink wristbands.

Gitte takes a wristband at random from the bag.

If it is yellow, she puts it back in the bag.
If it is blue or pink she puts it on her wrist.
She then takes another wristband at random from the bag.

(a) Complete the tree diagram.

1st wristband 2nd wristband

blue

........

........
blue yellow

5 ........ pink
11
blue

........
2
11 ........
yellow yellow

........ pink
4
blue
11
........

........
pink yellow

........ pink
[3]

© UCLES 2015 0607/42/M/J/15

 15

(b) If the second wristband is yellow, Gitte puts it back in the bag.
If it is blue or pink she puts it on her other wrist.

After choosing the second wristband, find the probability that she is wearing

(i) no wristbands,

Answer(b)(i) .................................................................. [2]

(ii) a matching pair of wristbands,

Answer(b)(ii) .................................................................. [3]

(iii) only one wristband.

Answer(b)(iii) .................................................................. [3]

© UCLES 2015 0607/42/M/J/15 [Turn over

 16

10
y
20

–90 0 360 x

–20

f(x) = 2tan (x + 30)°

(a) On the diagram, sketch the graph of y = f(x) for values of x between –90 and 360. [3]

(b) Solve the equation f(x) = 5 for values of x between –90 and 360.

Answer(b) x = ...................... or x = ....................... [2]

(c) Write down the equations of the two asymptotes to this graph for values of x between –90 and 360.

Answer(c) .................................................................

.................................................................. [2]

© UCLES 2015 0607/42/M/J/15

 17

(d) On the diagram below, sketch the graph of y = 2 tan ^x + 30h ° for values of x between –90 and 360.

y
20

–90 0 360 x

–20

[2]

© UCLES 2015 0607/42/M/J/15 [Turn over

 18

11
C
NOT TO
45 m SCALE

B
55 m
70 m

35°

A 80 m D

The diagram shows the plan of a field ABCD with a path from A to C.

(a) Calculate

(i) the obtuse angle ABC,

Answer(a)(i) .................................................................. [4]

(ii) angle CAD.

Answer(a)(ii) .................................................................. [4]

(b) Waqar walks along the path AC.

Calculate his shortest distance from B.

Answer(b) .............................................................. m [2]

© UCLES 2015 0607/42/M/J/15

 19

6 1
12 f ^xh = 5x - 2 g ^xh = , x !- h(x) = 5x2 + 3x – 2
4x + 1 4
(a) Find f(g(1)) .

Answer(a) .................................................................. [2]

(b) Find and simplify these expressions.

(i) g(f(x))

Answer(b)(i) .................................................................. [2]

(ii) f –1(x)

Answer(b)(ii) .................................................................. [2]

(c) Simplify.

(i) f(x)
h(x)

Answer(c)(i) .................................................................. [3]

1
(ii) g(x) –
f ^xh

Answer(c)(ii) .................................................................. [3]

Question 13 is printed on the next page.

© UCLES 2015 0607/42/M/J/15 [Turn over

 20

13
E
NOT TO
F SCALE
A D

B C

ABCD is a parallelogram.
BFE and CDE are straight lines.

(a) Explain why triangles AFB and DFE are similar.

Answer(a) .................................................................................................................................................

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

.............................................................................................................................................................. [2]

(b) BC = 10 cm, FD = 4 cm and EC = 8 cm.

(i) Calculate the length of AB.

Answer(b)(i) ............................................................ cm [3]

(ii) Find the value of Area of DFE .

Area of AFB

Answer(b)(ii) .................................................................. [1]

(iii) Find the value of Area of DFE .

Area of ABCD

Answer(b)(iii) .................................................................. [2]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.

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

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

© UCLES 2015 0607/42/M/J/15

 Cambridge International Examinations
Cambridge International General Certificate of Secondary Education
* 5 6 5 0 4 7 5 9 6 2 *

CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/43

Paper 4 (Extended) May/June 2015
2 hours 15 minutes
Candidates answer on the Question Paper.
Additional Materials: Geometrical Instruments
Graphics Calculator

READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
Do not use staples, paper clips, glue or correction fluid.
You may use an HB pencil for any diagrams or graphs.
DO NOT WRITE IN ANY BARCODES.

Answer all the questions.

Unless instructed otherwise, give your answers exactly or correct to three significant figures as appropriate.
Answers in degrees should be given to one decimal place.
For p, use your calculator value.
You must show all the relevant working to gain full marks and you will be given marks for correct methods,
including sketches, even if your answer is incorrect.
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 120.

This document consists of 20 printed pages.

DC (ST/FD) 99285/4
© UCLES 2015 [Turn over
 2

Formula List

- b ! b 2 - 4ac
For the equation ax2 + bx + c = 0 x=
2a

Curved surface area, A, of cylinder of radius r, height h. A = 2prh

Curved surface area, A, of cone of radius r, sloping edge l. A = prl

Curved surface area, A, of sphere of radius r. A = 4pr2

1
Volume, V, of pyramid, base area A, height h. V= Ah
3

Volume, V, of cylinder of radius r, height h. V = pr2h

1 2
Volume, V, of cone of radius r, height h. V= rr h
3

4 3
Volume, V, of sphere of radius r. V= rr
3

A
a b c
= =
sin A sin B sin C

c b a2 = b2 + c2 – 2bc cos A

1
Area = bc sin A
2
B a C

© UCLES 2015 0607/43/M/J/15

 3

Answer all the questions.

1 Sancha flew from Santiago to Paris, a distance of 11 585 km.

The average speed of the flight was 852.9 km/h.

(a) Find the length of time for the flight.

Give your answer in hours and minutes.

Answer(a) .............................. h .................... min [3]

(b) The journey back from Paris to Santiago took 14 hours 30 minutes.
The plane left Paris at 23 20.
The local time in Santiago is 6 hours behind the local time in Paris.

Find the local time this plane arrived in Santiago.

Answer(b) ............................................................. [2]

(c) Find the overall average speed for the total journey from Santiago to Paris and back to Santiago.

Answer(c) .................................................... km/h [3]

© UCLES 2015 0607/43/M/J/15 [Turn over

 4

2
y

2
A B
1

x
–5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9
–1

–2

–3

–4

(a) (i) Rotate triangle A through 90° anticlockwise about the origin.
Label the image C. [2]

(ii) Reflect triangle C in the x-axis.

Label the image D. [2]

(iii) Describe fully the single transformation that is equivalent to a rotation through
90° anticlockwise about the origin followed by a reflection in the x-axis.

Answer(a)(iii) ...........................................................................................................................................

.......................................................................................................................................................... [2]

(b) Describe fully the single transformation that maps triangle A onto triangle B.

Answer(b) .................................................................................................................................................

.......................................................................................................................................................... [3]

© UCLES 2015 0607/43/M/J/15

 5

3 Sinitta makes necklaces.

Each necklace costs Sinitta $56 to make.
They are sold through an internet shop at a selling price of $80.

(a) (i) The internet shop charges her 7% of the selling price.

Find the amount that Sinitta receives from the shop for a necklace.

Answer(a)(i) $ ............................................................. [2]

(ii) The shop increases the charge to 12% of the selling price of $80.

Calculate the percentage reduction in Sinitta’s profit.

Answer(a)(ii) ......................................................... % [4]

(b) Sinitta also makes silver rings.

Each ring contains 22 g of silver.
In the last year the cost of silver has increased by 8% to $143.10 per 100 grams.

(i) Find the cost of each 100 g of silver before the increase.

Answer(b)(i) $ ............................................................. [2]

(ii) Find the increase in the cost of the silver in a ring.

Answer(b)(ii) $ ............................................................. [2]

© UCLES 2015 0607/43/M/J/15 [Turn over

 6

4 P is the point (0, 4), Q is the point (6, 0) and R is the point (2, 7).

NOT TO
SCALE

Q
x
0

(a) S is the point such that RS = QP .

Find the co-ordinates of S.

Answer(a) ( .............................., ........................... ) [2]

(b) Calculate QP .

Answer(b) ............................................................. [2]

(c) Find the equation of the line PQ.

Answer(c) ............................................................. [2]

© UCLES 2015 0607/43/M/J/15

 7

(d) Write down the co-ordinates of N, the midpoint of PQ.

Answer(d) ( .............................., ........................... ) [1]

(e) Find the equation of the perpendicular bisector of PQ.

Answer(e) ............................................................. [3]

(f) A and B are points on the perpendicular bisector of PQ such that AN ! BN .

What is the mathematical name given to the quadrilateral PAQB?

Answer(f) ............................................................. [1]

© UCLES 2015 0607/43/M/J/15 [Turn over

 8

5
40 cm

NOT TO
SCALE
A B

30 cm

x cm D C

x cm

The diagram shows a rectangle, with sides 40 cm and 30 cm, made from a metal sheet.

A square of side x cm is cut from each of the four corners of the rectangle.
The remaining shape is folded up to make a rectangular open box with ABCD as the base.
The height of the box is x cm.

(a) Show that the volume of the box is 1200x - 140x 2 + 4x 3 .

[3]

© UCLES 2015 0607/43/M/J/15

 9

(b) On the diagram, sketch the graph of y = 1200x - 140x 2 + 4x 3 for 0 G x G 25.

5000

x
0 25
–1000

[2]

(c) Solve the equation 1200x - 140x 2 + 4x 3 = 2000 .

Answer(c) x = .................................. or x = .................................. or x = ................................. [3]

(d) Which solution to part (c) is not a possible value of x when the volume of the box is 2000 cm3? Give a
reason for your answer.

Answer(d) .................................................................................................................................................

.......................................................................................................................................................... [1]

(e) What is the maximum volume of the box?

For this volume what is the length of the box?

Answer(e) Maximum volume = .............................................................. cm3

Length = ....................................................... cm [2]

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 10

6 (a) (i) Find an expression for the nth term of this sequence.

2, 6, 10, 14, ...

Answer(a)(i) ............................................................. [2]

(ii) Use your answer to part (a)(i) to find an expression for u, the nth term of this sequence.

2 # 10 2 , 6 # 10 3 , 10 # 10 4 , 14 # 10 5 , ...

Answer(a)(ii) u = ............................................................. [1]

(b) The nth term, t, of another sequence, is given by t = 2 # 10 (3 - 2n) .

(i) Write down the first 4 terms in this sequence, giving your answers in standard form.

Answer(b)(i) ............................. , ............................. , ............................. , ............................. [2]

u
(ii) Using your answer to part (a)(ii), find and simplify an expression for .
t

Answer(b)(ii) ............................................................. [3]

© UCLES 2015 0607/43/M/J/15

 11

7
North

B NOT TO
SCALE

70°
A 55°

150 m
120 m

C
235 m
D

The diagram shows a field ABCD with a path from A to C.

AC = 150 m, AD = 120 m and CD = 235 m.
Angle ABC = 90°, angle BAC = 55° and the bearing of B from A is 070°.

(a) Calculate the length of AB.

Answer(a) ......................................................... m [2]

(b) Calculate the bearing of D from A.

Answer(b) ............................................................. [4]

(c) Calculate the area of the field ABCD.

Answer(c) ........................................................ m2 [3]

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 12

8 100 light bulbs were tested.

The length of life, t, in thousands of hours was recorded.
The results are shown in this table.

Length of life (t) in

41tG5 51tG6 61tG7 71tG8 81tG9 9 1 t G 10 10 1 t G 12
thousands of hours
Frequency 8 21 31 23 10 5 2

(a) Calculate an estimate of the mean value of t.

Answer(a) ............................................................. [2]

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 13

(b) Draw a cumulative frequency curve for the length of life of the light bulbs.

100

90

80

70

60
Cumulative
frequency
50

40

30

20

10

0 t
4 5 6 7 8 9 10 11 12
Length of life / in thousands of hours
[5]
(c) Use your graph to estimate

(i) the number of light bulbs that lasted longer than 8500 hours,

Answer(c)(i) ............................................................. [2]

(ii) the interquartile range.

Answer(c)(ii) ................................................... hours [2]

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 14

9 (a)
A 20 cm B

NOT TO
40 cm SCALE

C 15 cm D

The diagram shows two similar triangles EAB and ECD.

AB = 20 cm, CD = 15 cm, AC = 40 cm and angle CAB = 90°.

(i) Show that EC = 120 cm.

[2]

(ii) Find ED.

Answer(a)(ii) ....................................................... cm [2]

(iii) Find DB.

Answer(a)(iii) ....................................................... cm [2]

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(b)

20 cm NOT TO
SCALE

40 cm

15 cm

The diagram shows an open waste paper bin made from metal.

The radius of the circular top is 20 cm.

The radius of the circular base is 15 cm.
The perpendicular height of the bin is 40 cm.

Using answers from part (a), calculate

(i) the volume of the waste paper bin,

Answer(b)(i) ...................................................... cm3 [3]

(ii) the area of metal needed to make the bin.

Answer(b)(ii) ...................................................... cm2 [4]

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 16

10 Tricia has 2 bags.

In the first bag there are 6 white balls and 4 red balls.
In the second bag there are 4 blue balls, 3 white balls and 2 red balls.

She takes a ball at random out of the first bag.

She then takes a ball at random out of the second bag.

(a) Complete the tree diagram to show the probability of all the possible outcomes for the two balls.

First ball Second ball

Blue

...........

...........
White
White

...........
........... Red

Blue

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

Red ...........
White

...........
Red
[2]

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 17

(b) Calculate the probability that Tricia’s two balls are

(i) both white,

Answer(b)(i) ............................................................. [2]

(ii) one white and one red,

Answer(b)(ii) ............................................................. [3]

(iii) of different colours.

Answer(b)(iii) ............................................................. [3]

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 18

11
y

12

x
0 4
–6

–12

(1 - 2x)
f (x) =
(x + 3)

(a) On the diagram, sketch the graph of y = f (x) for values of x between x =- 6 and x = 4 . [3]

(b) Write down the equations of the asymptotes of the graph of y = f (x) .

Answer(b) ......................................................................

.............................................................. [2]

(c) Find the range of values for y when x H 0 .

Answer(c) ............................................................. [2]

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(d)
y

12

x
–6 0 4

–12

(1 - 2x)
On this diagram, sketch the graph of y = . [2]
(x + 3)

(1 - 2x)
(e) Solve = 6.
(x + 3)

Answer(e) x = ............................ or x = ...................... [2]

Question 12 is printed on the next page.

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 20

12 f ^xh = 3x - 1 g (x) = 4 - 2x

(a) Find

(i) g (3) ,

Answer(a)(i) ............................................................. [1]

(ii) f (g (3)) .

Answer(a)(ii) ............................................................. [1]

(b) Find and simplify expressions for

(i) g (f (x)) ,

Answer(b)(i) ............................................................. [2]

(ii) g -1 (x) ,

Answer(b)(ii) ............................................................. [2]

2 3
(iii) .
f (x) g (x)
-

Answer(b)(iii) ............................................................. [3]

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.

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

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

© UCLES 2015 0607/43/M/J/15

 Cambridge International Examinations
Cambridge International General Certificate of Secondary Education
* 6 7 1 4 1 0 0 3 2 9 *

CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61

Paper 6 (Extended) May/June 2015
1 hour 30 minutes
Candidates answer on the Question Paper.
Additional Materials: Graphics Calculator

READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
Do not use staples, paper clips, glue or correction fluid.
You may use an HB pencil for any diagrams or graphs.
DO NOT WRITE IN ANY BARCODES.

Answer both parts A and B.

You must show all the relevant working to gain full marks for correct methods, including sketches.
In this paper you will also be assessed on your ability to provide full reasons and communicate your
mathematics clearly and precisely.
At the end of the examination, fasten all your work securely together.
The total number of marks for this paper is 40.

This document consists of 12 printed pages.

DC (AC/FD) 101335/3
© UCLES 2015 [Turn over
 2

Answer both parts A and B.

A INVESTIGATION STAIRCASES (20 marks)

You are advised to spend no more than 45 minutes on this part.

This investigation looks at the number of cubes that make different types of staircase.

1 This is an UP staircase of height 3 made using 6 cubes.

It is a 3-step UP staircase because it has a height of 3 cubes.

(a) Write down the number of cubes that make an UP staircase of height 2.

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

(b) On the grid below draw an UP staircase of height 4.

© UCLES 2015 0607/61/M/J/15

 3

(c) Complete the table for the number of cubes that make these UP staircases.

Height 1 2 3 4 5 6
Number
1 6
of cubes

(d) Find an expression, in terms of n, for the number of cubes that make an UP staircase of height n.

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

(e) Find how many cubes make an UP staircase of height 10.

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

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 4

2 This is an UP AND DOWN staircase of height 3 made using 9 cubes.

It is a 3-step UP AND DOWN staircase because it has a height of 3 cubes.

(a) Find how many cubes make an UP AND DOWN staircase of height 4.

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

(b) Complete the table for the number of cubes that make these UP AND DOWN staircases.

Height 1 2 3 4 5 6
Number
1 9
of cubes

(c) Find an expression, in terms of n, for the number of cubes that make an UP AND DOWN staircase of
height n.

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

(d) Find how many cubes make an UP AND DOWN staircase of height 10.

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

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 5

3 This is a DOUBLE staircase of height 3 made using 12 cubes.

It is a 3-step DOUBLE staircase because it has a height of 3 cubes.

(a) Complete the table for the number of cubes that make these DOUBLE staircases.

Height 1 2 3 4 5 6
Number
2 12
of cubes

(b) Find an expression, in terms of n, for the number of cubes that make a DOUBLE staircase of height n.

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

(c) Find how many cubes make a DOUBLE staircase of height 10.

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

(d) Find the height of a DOUBLE staircase made from 240 cubes.

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

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 6

4 This is a sequence of MULTIPLE staircases of heights 1, 2 and 3.

These are MULTIPLE staircases because, for each staircase, the width, the height and the depth are the
same.

(a) Complete the table for the number of cubes that make these MULTIPLE staircases.

Height 1 2 3 4 5 6
Number
1 6 18
of cubes

(b) Find an expression, in terms of n, for the number of cubes that make a MULTIPLE staircase of
height n.

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

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5 There are 1800 cubes available.

Use your expressions in questions 1(d), 3(b) and 4(b) to complete the table.

Type of staircase Maximum height using 1800 cubes Number of cubes left over

UP

UP AND DOWN 42 36

DOUBLE

MULTIPLE

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 8

B MODELLING BOAT TRIPS (20 marks)

You are advised to spend no more than 45 minutes on this part.

A boat travels up and down a river.

The time taken for a journey depends on the speed of the boat and the speed of the water current.

1 In this model the water is still and so the speed of the water current is zero km / h.
The speed of the boat in still water is 15 km / h.

(a) Find how many minutes it will take for the boat to travel 10 km.

................................................................ min

(b) The boat travels for 24 minutes.

Find the distance that it travels.

................................................................. km

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 9

When a boat travels against the current it goes in When a boat travels with the current it goes in
exactly the opposite direction to the current. exactly the same direction as the current.

boat boat
current current

2 In this model the water is not still.

The speed of the water current is 2 km / h.
The speed of the boat in still water is 15 km / h.

(a) Show that it will now take the boat approximately 46 minutes to travel 10 km against the current.

(b) The boat travels for 20 minutes against the current.

Find the distance it travels.

................................................................. km

(c) How far will the boat travel in 46 minutes with the current?

................................................................. km

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 10

3 The boat travels 20 km up the river before returning to where it started.

The speed of the water current is 2 km / h.
The speed of the boat in still water is v km / h.

(a) (i) Find a model for the total travelling time, T hours, for this whole journey.

T = ......................................................................

40v
(ii) Show that your model simplifies to T = .
v2 - 4

40v
(iii) Sketch the graph of T = 2 for 0 G v G 20 .
v -4
T

0
20 v

(iv) The model is only appropriate for v 2 k .

Find the value of k and give a practical reason why k must have this value.

k = .................................. because ....................................................................................................

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

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 11

(b) When the speed of the boat in still water is 18 km / h, find the time taken for the whole journey.

............................................................. hours

(c) The return journey takes the boat 3 hours.

Find the speed of the boat in still water.

............................................................. km / h

4 The boat travels 20 km up the river before returning to where it started.

The speed of the boat in still water is v km / h.
A whole journey takes the boat 3 hours.

(a) (i) Adjust the model in question 3(a)(ii) for a water current of 3 km / h.

T = ......................................................................

(ii) Find the speed of the boat in still water.

............................................................. km / h

(b) The speed of the boat in still water is now 15 km / h.

Adjust the model in question 3(a)(ii) and find the speed of the water current.

............................................................. km / h

Question 5 is printed on the next page.

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 12

5 (a) There is a change in the boat’s journey.

Explain how the journey has changed when the model in question 3(a)(ii) becomes
80v
T= 2 .
v -4

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

...................................................................................................................................................................
40v
(b) Describe fully the single transformation that maps the graph of T = 2 onto the graph
v -4
80v
of T = .
v2 - 4

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

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

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.

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

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

© UCLES 2015 0607/61/M/J/15

 Cambridge International Examinations
Cambridge International General Certificate of Secondary Education
* 4 0 1 6 8 6 6 9 8 7 *

CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62

Paper 6 (Extended) May/June 2015
1 hour 30 minutes
Candidates answer on the Question Paper.
Additional Materials: Graphics calculator

READ THESE INSTRUCTIONS FIRST

Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
Do not use staples, paper clips, glue or correction fluid.
You may use an HB pencil for any diagrams or graphs.
DO NOT WRITE IN ANY BARCODES.

Answer both parts A and B.

You must show all relevant working to gain full marks for correct methods, including sketches.
In this paper you will also be assessed on your ability to provide full reasons and communicate your
mathematics clearly and precisely.
At the end of the examination, fasten all your work securely together.
The total number of marks for this paper is 40.

This document consists of 12 printed pages.

DC (AC/CGW) 100447/1
© UCLES 2015 [Turn over
 2

Answer both parts A and B.

A INVESTIGATION MOLECULES (20 marks)

You are advised to spend no more than 45 minutes on this part.

This investigation looks at the number of spheres and rods that you need to make models of molecules.

1 Chemists use small spheres and rods to make models of molecules.

These diagrams show a sequence of molecules of height 1.

Molecule 1 Molecule 2 Molecule 3

(a) Draw the next two molecules in this sequence.

(b) Complete this table for molecules of height 1.

Molecule Number of spheres Number of rods

m s r
1 1 0

2 2 1

3 3 2

(c) Write down a formula for s in terms of m.

s = ......................................................................

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2 These diagrams show a sequence of molecules of height 2.

Molecule 1 Molecule 2 Molecule 3 Molecule 4

(a) Complete this table for molecules of height 2.

Molecule Number of spheres Number of rods

m s r
1 2 1

2 4 4

3 6 7

(b) Find, in terms of m, a formula for

(i) s,

s = ......................................................................

(ii) r.

r = ......................................................................

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 4

3 These diagrams show a sequence of molecules of height 3.

Molecule 1 Molecule 2 Molecule 3 Molecule 4

(a) Complete this table for molecules of height 3.

Molecule Number of spheres Number of rods

m s r
1 3 2

2 6 7

3 9 12

(b) Find, in terms of m, a formula for

(i) s,

s = ......................................................................

(ii) r.

r = ......................................................................

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4 (a) Use your answers to questions 1(c), 2(b) and 3(b) to help you complete the table for molecules of
height h.

Number of spheres (s) Number of rods (r)

Height (h)
in terms of m in terms of m

1 m−1

5 5m 9m − 5

(b) Find, in terms of m and h, a formula for

(i) s,

s = ......................................................................

(ii) r.

r = ......................................................................

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 6

(c) Use your answer to part (b)(i) to find a formula for m in terms of s and h.

m = ......................................................................

(d) Find a formula for r in terms of s and h.

r = ......................................................................

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5 (a) A molecule has height h and width w.

For example, Molecule 4 in question 3 has h = 3 and w = 4.

Use your answer to question 4(d) to show that r = 2hw – h − w.

(b) Can a square molecule have 544 rods?

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 8

B MODELLING WHERE IS THE HORIZON? (20 marks)

You are advised to spend no more than 45 minutes on this part.

This table shows the distance to the horizon (y kilometres) at different heights above sea level (x metres).

x 1 2 3 4 5 6 7 8 9 10
y 3.6 5.0 6.2 7.1 8.0 8.7 9.4 10.1 10.7 11.3

y
12

11

10

7
Distance to horizon
6
(km)
5

0 1 2 3 4 5 6 7 8 9 10 x
Height above sea level (m)

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 9

1 (a) On the grid, plot the points given in the table.

The points (2, 5) and (5, 8) have been plotted for you.

(b) The simplest way to model the data is with a straight line.

(i) On the grid, draw the straight line passing through (2, 5) and (5, 8).

Find the equation of this line.

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

(ii) Using this model what is the distance to the horizon when at sea level?

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

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 10

2 Another model for the data is y = ax 2 + bx + c .

Assume that, when the height is 0, the distance to the horizon is 0.

(a) Show that c = 0.

(b) When c = 0, the model is y = ax 2 + bx .

(i) Use the point (2, 5) to form an equation in a and b.

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

(ii) Use the point (5, 8) to form another equation in a and b.

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

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 11

(c) Solve the equations in part (b) and write down your model.

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

(d) Sketch the graph of y against x for 0 G x G 10 .

0 x

(e) Comment on the validity of this model.

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

Question 3 is printed on the next page.

© UCLES 2015 0607/62/M/J/15

 12

3 Another model for this data is y = ax b .

(a) Use (2, 5) and (5, 8) to write down two equations in a and b.

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

(b) Show that 1.6 = 2.5 b .

(c) Show that b = 0.5, correct to 1 decimal place.

(d) Find the value of a and write down your model.

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

(e) Compare the model in this question with the data on page 8.

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

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

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable
effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will
be pleased to make amends at the earliest possible opportunity.

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

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

© UCLES 2015 0607/62/M/J/15