PAPA CAMBRIDGE
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
International General Certificate of Secondary Education
*1024033730*
CAMBRIDGE INTERNATIONAL MATHEMATICS
0607/21
May/June 2013
Paper 2 (Extended)
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, highlighters, glue or correction fluid.
You may use a 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.
IB13 06_0607_21/3RP
© UCLES 2013
[Turn over
2
Formula List
2
ax + bx + c = 0
For the equation
x=
_ b ± b2 _ 4ac
2a
Curved surface area, A, of cylinder of radius r, height h.
A = 2πrh
Curved surface area, A, of cone of radius r, sloping edge l.
A = πrl
Curved surface area, A, of sphere of radius r.
A = 4πr2
Volume, V, of pyramid, base area A, height h.
V=
Volume, V, of cylinder of radius r, height h.
V = πr2h
Volume, V, of cone of radius r, height h.
V=
Volume, V, of sphere of radius r.
V=
A
1
3
Ah
1
3
4
3
πr2h
πr3
a
b
c
=
=
sin A sin B sin C
a2 = b2 + c2 – 2bc cos A
b
c
Area =
B
© UCLES 2013
a
C
0607/21/M/J/13
1
2
bc sin A
3
Answer all the questions.
1
For
Examiner's
Use
The population of India in 2011 was 1.21 × 109 .
The population of Pakistan in 2011 was 1.77 × 108 .
Calculate the total population of India and Pakistan in 2011.
Give your answer in standard form.
Answer
2
[2]
P is the point (–2, 5) and Q is the point (4, 1).
(a) Find the co-ordinates of the midpoint of PQ.
Answer(a) (
,
)
[1]
(b) Find the gradient of PQ.
Answer(b)
[2]
(c) (i) Find the equation of the line perpendicular to PQ which passes through the point (0, 4).
Answer(c)(i)
[2]
(ii) Find the x co-ordinate of the point where this line cuts the x-axis.
Answer(c)(ii) x =
© UCLES 2013
0607/21/M/J/13
[1]
[Turn over
4
3
Solve these simultaneous equations.
For
Examiner's
Use
y = 2x – 8
3x + 2y = 5
Answer x =
Answer y =
4
[3]
One morning, Ashad carries out a survey on the colours of 200 cars in his town.
These are his results.
Colour
Silver
Black
Red
Blue
Other
Frequency
78
40
36
30
16
Red
Blue
Other
(a) Complete this table of relative frequencies.
Colour
Relative
Frequency
Silver
Black
0.2
[2]
(b) There is a total of 18 000 cars in the town.
Work out an estimate of the number of black cars in the town.
Answer(b)
© UCLES 2013
0607/21/M/J/13
[2]
5
5
For
Examiner's
Use
NOT TO
SCALE
O
130°
D
A
x°
B
y°
C
E
A, B, C and D are points on the circle centre O.
DCE is a straight line.
Angle AOD = 130°.
Find the value of
(a) x,
Answer(a) x =
[2]
Answer(b) y =
[2]
(b) y.
© UCLES 2013
0607/21/M/J/13
[Turn over
6
6
For
Examiner's
Use
U
P
Q
R
On the Venn diagram write the elements a, b and c in the correct subsets using the following
information.
a ∈ (P ∪ Q ∪ R)'
b ∈ P ' ∩ (Q ∩ R)
c ∈ (Q ∪ R)' ∩ P
[3]
7
(a) Write down the value of
(i) log 1000,
Answer(a)(i)
[1]
Answer(a)(ii)
[1]
Answer(b) p =
[2]
(ii) log 0.01 .
(b) Find p when
2log 5 – log 2 = log p .
© UCLES 2013
0607/21/M/J/13
7
8
For
Examiner's
Use
r cm
160°
NOT TO
SCALE
5 cm
The diagrams show a circle with radius 5 cm and the sector of another circle with angle 160° and
radius r cm.
The circle and the sector have the same area.
Calculate the value of r.
9
Answer r =
[4]
Answer(a)
[2]
Answer(b)
[2]
Simplify.
50 +
(a)
8
2
(b)
(5+ 3 )
Questions 10 and 11 are printed on the next page.
© UCLES 2013
0607/21/M/J/13
[Turn over
8
10 Rearrange this equation to make x the subject.
For
Examiner's
Use
ax – 3y = b(x + 2y)
Answer x =
[3]
11
b
a
p
q
r
Write the vectors p, q and r in terms of a and b.
Answer p =
q=
r=
[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.
University of 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 2013
0607/21/M/J/13
PAPA CAMBRIDGE
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
International General Certificate of Secondary Education
*4727863751*
CAMBRIDGE INTERNATIONAL MATHEMATICS
0607/22
May/June 2013
Paper 2 (Extended)
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, highlighters, glue or correction fluid.
You may use a 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.
IB13 06_0607_22/RP
© UCLES 2013
[Turn over
2
Formula List
2
ax + bx + c = 0
For the equation
x=
_ b ± b2 _ 4ac
2a
Curved surface area, A, of cylinder of radius r, height h.
A = 2πrh
Curved surface area, A, of cone of radius r, sloping edge l.
A = πrl
Curved surface area, A, of sphere of radius r.
A = 4πr2
Volume, V, of pyramid, base area A, height h.
V=
Volume, V, of cylinder of radius r, height h.
V = πr2h
Volume, V, of cone of radius r, height h.
V=
Volume, V, of sphere of radius r.
V=
A
1
3
Ah
1
3
4
3
πr2h
πr3
a
b
c
=
=
sin A sin B sin C
a2 = b2 + c2 – 2bc cos A
b
c
Area =
B
© UCLES 2013
a
C
0607/22/M/J/13
1
2
bc sin A
3
Answer all the questions.
For
Examiner's
Use
1
U
A
B
3
7
4
3
1
1
6
5
C
The Venn diagram shows the number of elements in each of the sets A, B and C, and n(U) = 30.
(a) Find
(i) n(A),
Answer(a)(i)
[1]
Answer(a)(ii)
[1]
(ii) n (C ∪ B′) .
(b) Shade the region ( A ∩ B) ∪ C on the Venn diagram.
2
[1]
P
65°
NOT TO
SCALE
O
Q
A
B
R
A, P, Q, B and R lie on a circle, centre O.
Angle APB = 65°.
Find
(a) angle AQB,
Answer(a) Angle AQB =
[1]
Answer(b) Angle AOB =
[1]
Answer(c) Angle ARB =
[1]
(b) angle AOB,
(c) angle ARB.
© UCLES 2013
0607/22/M/J/13
[Turn over
4
3
For
Examiner's
Use
y
5
4
3
P
2
Q
1
–5 –4 –3 –2 –1 0
–1
1
2
3
4
5
6
7
8
9 10
x
–2
–3
–4
–5
–6
–7
–8
–9
–10
(a) Enlarge shape P using centre (3, 4) and scale factor 3.
[2]
(b) Describe fully the single transformation that maps shape P onto shape Q.
[3]
4
(a) Simplify.
(b) 8n =
16x16 ÷ 2x2
Answer(a)
[2]
Answer(b) n =
[2]
1
2
Find the value of n.
© UCLES 2013
0607/22/M/J/13
5
5
Rationalise the denominator in each of the following.
(a)
2
3
(b)
6
For
Examiner's
Use
Answer(a)
[1]
Answer(b)
[2]
1
3 −1
(a) Find the value of ax3 when a = 1200 and x = 5.
Give your answer in standard form.
Answer(a)
[2]
(b) Make x the subject of the formula y = ax3.
Answer(b) x =
© UCLES 2013
0607/22/M/J/13
[2]
[Turn over
6
7
(a) Write 2log(x + 1) – log(x – 1) as a single logarithm.
For
Examiner's
Use
Answer(a)
[2]
Answer(b) p =
[2]
(b) log3 p = 4 where p is an integer.
Find the value of p.
8
These are the first five terms of a sequence.
2
6
12
20
30
(a) Find the next term.
Answer(a)
[1]
Answer(b)
[3]
(b) Find an expression for the nth term.
© UCLES 2013
0607/22/M/J/13
7
9
f(x) = 3 + 2x
For
Examiner's
Use
Find
(a) f(f(– 4)),
Answer(a)
[2]
Answer(b)
[2]
Answer y =
[2]
(b) f –1(x) .
10 y varies inversely as x2.
When x = 2, y = 24.
Find a formula for y in terms of x.
Question 11 is printed on the next page.
© UCLES 2013
0607/22/M/J/13
[Turn over
8
11
For
Examiner's
Use
NOT TO
SCALE
r
R
The diagram shows a circle of radius r inside a circle of radius R.
(a) Find an expression, in terms of π, r and R, for the shaded area.
Factorise your expression completely.
Answer(a)
[2]
Answer(b) r =
[2]
(b) When R = r + 3, the shaded area is 24π.
Find the value of r.
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.
University of 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 2013
0607/22/M/J/13
PAPA CAMBRIDGE
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
International General Certificate of Secondary Education
*4686246803*
CAMBRIDGE INTERNATIONAL MATHEMATICS
0607/23
May/June 2013
Paper 2 (Extended)
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, highlighters, glue or correction fluid.
You may use a 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.
IB13 06_0607_23/3RP
© UCLES 2013
[Turn over
2
Formula List
2
ax + bx + c = 0
For the equation
x=
_ b ± b2 _ 4ac
2a
Curved surface area, A, of cylinder of radius r, height h.
A = 2πrh
Curved surface area, A, of cone of radius r, sloping edge l.
A = πrl
Curved surface area, A, of sphere of radius r.
A = 4πr2
Volume, V, of pyramid, base area A, height h.
V=
Volume, V, of cylinder of radius r, height h.
V = πr2h
Volume, V, of cone of radius r, height h.
V=
Volume, V, of sphere of radius r.
V=
A
1
3
Ah
1
3
4
3
πr2h
πr3
a
b
c
=
=
sin A sin B sin C
a2 = b2 + c2 – 2bc cos A
b
c
Area =
B
© UCLES 2013
a
C
0607/23/M/J/13
1
2
bc sin A
3
Answer all the questions.
1
2
For
Examiner's
Use
Work out (1.6 × 103) ÷ (4 × 105).
Give your answer in standard form.
Answer
[2]
Answer(a) x =
[3]
Answer(b) x =
[3]
Solve the equations.
(a) 2 – 3(1 – 2x) = 4(2 – x)
(b) sinx = ±
© UCLES 2013
3
for 0° Y x Y 360°
2
0607/23/M/J/13
[Turn over
4
3
Find the value of the following.
For
Examiner's
Use
(a) 40
−
Answer(a)
[1]
Answer(b)
[2]
Answer(a)
[2]
Answer(b)
[3]
2
(b) 27 3
4
(a) Simplify.
200 − 98
(b) Rationalise the denominator.
11
5− 3
© UCLES 2013
0607/23/M/J/13
5
5
The diagram shows the graph of y = f(x) for – 4 Y x Y 3.
For
Examiner's
Use
y
3
2
1
–4
–3
–2
–1
0
1
2
3
4
2
3
4
x
–1
–2
–3
(a) On the diagram below, sketch the graph of y = |f(x)|.
y
3
2
1
–4
–3
–2
–1
0
1
x
–1
–2
–3
[3]
(b) On the diagram below, sketch the graph of y = f(x – 1).
y
3
2
1
–4
–3
–2
–1
0
1
2
3
4
x
–1
–2
–3
© UCLES 2013
0607/23/M/J/13
[2]
[Turn over
6
6
Make x the subject of the equation.
For
Examiner's
Use
a
b
=
x+3 x
Answer x =
7
[3]
D
NOT TO
SCALE
E
z°
O
C
70°
x°
y°
30°
B
A
B, C, D and E lie on a circle, centre O.
CE is a diameter, angle DAC = 30° and angle BOE = 70°.
Find the values of x, y and z.
Answer x =
y=
z=
© UCLES 2013
0607/23/M/J/13
[3]
7
8
The points A (1, 9) and B (7, 1) are shown on the diagram below.
For
Examiner's
Use
y
10
A
8
6
4
2
0
B
2
4
6
8
10
x
(a) Calculate the length AB.
Answer(a)
[2]
(b) (i) Find the co-ordinates of the midpoint of the line AB.
Answer(b)(i) (
,
)
[1]
(ii) Find the equation of the perpendicular bisector of the line AB.
Answer(b)(ii)
[3]
Questions 9 and 10 are printed on the next page.
© UCLES 2013
0607/23/M/J/13
[Turn over
8
9
1
hours.
2
She then runs 9 km in 45 minutes.
For
Examiner's
Use
Wendy walks 9 km in 1
Find her average speed in km/h for the whole journey.
Answer
km/h
[3]
10 Paulo goes to a supermarket.
The probability that he buys orange juice is 0.65 .
The probability that he does not buy milk is 0.30 .
The probability that he buys milk but does not buy orange juice is 0.15 .
(a) Complete the table of probabilities.
Buys milk
Does not buy milk
Buys orange juice
Does not buy orange juice
Total
0.65
0.15
Total
0.30
1.00
[2]
(b) Find the probability that Paulo buys either orange juice or milk but not both.
Answer(b)
[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.
University of 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 2013
0607/23/M/J/13
PAPA CAMBRIDGE
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
International General Certificate of Secondary Education
*1170642332*
CAMBRIDGE INTERNATIONAL MATHEMATICS
0607/41
May/June 2013
Paper 4 (Extended)
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, highlighters, glue or correction fluid.
You may use a 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.
For Examiner's Use
This document consists of 16 printed pages.
IB13 06_0607_41/4RP
© UCLES 2013
[Turn over
2
Formula List
2
ax + bx + c = 0
For the equation
x=
_ b ± b2 _ 4ac
2a
Curved surface area, A, of cylinder of radius r, height h.
A = 2πrh
Curved surface area, A, of cone of radius r, sloping edge l.
A = πrl
Curved surface area, A, of sphere of radius r.
A = 4πr2
Volume, V, of pyramid, base area A, height h.
V=
Volume, V, of cylinder of radius r, height h.
V = πr2h
Volume, V, of cone of radius r, height h.
V=
Volume, V, of sphere of radius r.
V=
A
1
3
Ah
1
3
4
3
πr2h
πr3
a
b
c
=
=
sin A sin B sin C
a2 = b2 + c2 – 2bc cos A
b
c
Area =
B
© UCLES 2013
a
C
0607/41/M/J/13
1
2
bc sin A
3
Answer all the questions.
1
For
Examiner's
Use
Each year the value of a car decreases by 12%.
On 1st April 2011 Sami bought a car for $15 840.
(a) Work out
(i) the value of the car on 1st April 2010,
Answer(a)(i) $
[3]
(ii) the value of the car on 1st April 2014,
Answer(a)(ii) $
[3]
(iii) the year in which the value of the car will first be below $5000.
Answer(a)(iii)
[2]
(b) Each year Sami drives 20 000 km in his car.
His yearly motoring costs are
• fuel at $0.68 per litre,
• service and other repairs $950,
• tax and insurance $1020.
The car travels 15 km on each litre of fuel.
Find the total yearly motoring costs as a percentage of the value of the car in 2011.
Answer(b)
© UCLES 2013
0607/41/M/J/13
% [4]
[Turn over
4
2
For
Examiner's
Use
y
3
2
A
1
–3
–1 0
–2
1
2
3
4
5
6
7
x
–1
B
–2
–3
–4
–5
–6
(a) Describe fully the single transformation that maps triangle A onto triangle B.
Answer(a)
[2]
(b) (i) Rotate triangle A through 180° about the point (3, 0). Label the image C.
(ii) Enlarge triangle C with scale factor 2 and centre (6, 0). Label the image D.
[2]
[2]
(iii) Describe fully the single transformation that maps triangle A onto triangle D.
Answer(b)(iii)
[3]
© UCLES 2013
0607/41/M/J/13
5
3
For
Examiner's
Use
y
10
–2
x
0
4
–10
(a) On the diagram, sketch the graph of
(b) Solve the equation
y = x3 – 3x2 + 2 for –2 Y x Y 4.
[2]
x3 – 3x2 + 2 = 0.
Answer(b) x =
or x =
or x =
[2]
(c) (i) Find the co-ordinates of the local maximum and local minimum points.
Answer(c)(i)
(
,
)
(
,
)
[2]
(ii) The equation x3 – 3x2 + 2 = k has 3 solutions.
Write down the range of values for k.
Answer(c)(ii)
[2]
(d) By drawing a suitable line on your diagram show that
solution.
x3 – 3x2 + 2 = 6 – 3x has only one
[2]
© UCLES 2013
0607/41/M/J/13
[Turn over
6
4
For
Examiner's
Use
R
NOT TO
SCALE
Q
C
A
P
B
The diagram shows a vertical radio mast PQR supported by 6 straight wires.
A, B, C and P are on level horizontal ground.
RA = RB = RC and QA = QB = QC.
PQ = 30 m, QR = 20 m and angle AQP = angle BQP = angle CQP = 65°.
R
NOT TO
SCALE
20 m
Q
65°
30 m
P
C
(a) Show that QC = 70.99 m, correct to 2 decimal places.
[2]
© UCLES 2013
0607/41/M/J/13
7
(b) Using the cosine rule, calculate the length RC.
For
Examiner's
Use
Answer(b)
m
[3]
Answer(c)
m
[1]
Answer(d)
m
[2]
m2
[2]
(c) Find the total length of the 6 wires.
(d) Calculate the length PC.
(e) This is a view from above showing A, B, P and C on horizontal ground.
A
NOT TO
SCALE
P
120°
B
C
Calculate the area of triangle BPC.
Answer(e)
© UCLES 2013
0607/41/M/J/13
[Turn over
8
5
For
Examiner's
Use
y
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
x
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
(a) On the grid, find the region satisfied by the following inequalities.
Label the region R.
xY4
x + y Y 12
5x + 2y [ 30
[5]
(b) (h, k) is a point in the region R and h and k are integers.
(i) Find the number of possible points (h, k).
Answer(b)(i)
[1]
Answer(b)(ii)
[1]
(ii) Find the minimum value of h + k.
© UCLES 2013
0607/41/M/J/13
9
6
For
Examiner's
Use
NOT TO
SCALE
5 cm
6 cm
2 cm
2.5 cm
2 cm
The diagram shows a child’s wooden brick.
The brick is a cuboid with a semicircular hole cut in the bottom.
(a) Find the volume of the brick.
Answer(a)
cm3 [3]
(b) Each cubic centimetre of wood has a mass of 0.8 g.
Find the mass of the brick.
Answer(b)
g [1]
Answer(c)
cm2 [5]
(c) Find the total surface area of the brick.
© UCLES 2013
0607/41/M/J/13
[Turn over
10
7
(a) The speeds, v km/h, of 140 cars were measured on road A.
For
Examiner's
Use
The cumulative frequency graph shows the speeds of these cars.
140
130
120
110
100
90
80
Cumulative
frequency 70
60
50
40
30
20
10
0
v
10
20
30
40
50
60
70
80
90
100
Speed (km/h)
(i) Find the median speed.
Answer(a)(i)
km/h [1]
(ii) Find the inter-quartile range of the speeds.
Answer(a)(ii)
© UCLES 2013
0607/41/M/J/13
km/h [2]
11
(b) The speeds of another 140 cars were measured on road B.
The results are shown in this table.
For
Examiner's
Use
Speed
20 < v Y 30 30 < v Y 40 40 < v Y 45 45 < v Y 50 50 < v Y 60 60 < v Y 80 80 < v Y 100
(v km/h)
Frequency
6
12
15
20
32
30
v Y 60
v Y 80
25
(i) Complete this table of cumulative frequencies for road B.
Speed
(v km/h)
Cumulative
frequency
v Y 20
v Y 30
v Y 40
0
6
18
v Y 45
v Y 50
v Y 100
140
[2]
(ii) On the grid in part (a), draw the cumulative frequency curve for road B.
[3]
(iii) Make a comparison between the distributions of speeds on roads A and B.
Answer(b)(iii)
[2]
(iv) Calculate an estimate for the mean speed of the 140 cars on road B.
Answer(b)(iv)
km/h [2]
(v) On the grid below, complete the histogram to show the speeds of the cars on road B.
4
3
Frequency
density
2
1
0
v
10
20
30
40
50
60
70
80
90
100
Speed (km/h)
[4]
© UCLES 2013
0607/41/M/J/13
[Turn over
12
8
The table shows the number of left-handed and right-handed girls and boys in a class.
Left-handed
Right-handed
Total
Girls
4
14
18
Boys
3
11
14
Total
7
25
32
For
Examiner's
Use
(a) Two students are chosen at random from the whole class.
Find the probability that they are both left-handed.
Answer(a)
[2]
(b) Two of the girls are chosen at random.
Find the probability that exactly one of these girls is left-handed.
Answer(b)
[3]
(c) Two of the right-handed students are chosen at random.
Find the probability that at least one is a girl.
Answer(c)
© UCLES 2013
0607/41/M/J/13
[3]
13
9
The resistance, R ohms, of a standard length of wire varies inversely as the square of its
diameter, d mm.
For
Examiner's
Use
(a) The resistance of a standard length of wire of diameter 0.5 mm is 0.8 ohms.
(i) Find a formula for R in terms of d.
Answer(a)(i) R =
[3]
(ii) Find the resistance of a standard length of the same type of wire with diameter 2 mm.
Answer(a)(ii)
ohms [1]
(iii) The resistance of a standard length of the same type of wire is 4 ohms.
Find the diameter of this wire.
Answer(a)(iii)
mm
[2]
(b) For a different type of wire the resistance of a standard length is 2 ohms.
Find the resistance of a standard length of this wire when the diameter is doubled.
Answer(b)
© UCLES 2013
0607/41/M/J/13
ohms [2]
[Turn over
14
10
For
Examiner's
Use
y
10
x
0
–6
3
–10
(a) On the diagram, sketch the graph of
f(x) =
( x − 1)
( x + 3)
y = f(x), where
between x = –6 and x = 3.
[3]
(b) Find the co-ordinates of the point where the graph crosses the x-axis.
Answer(b)
(c) Find the equations of the asymptotes of
(
,
)
[1]
y = f(x).
Answer(c)
and
[2]
(d) Find the range of f(x) for x [ 0.
Answer(d)
(e) Find the solutions to the equation
( x − 1)
= –5 – 2x .
( x + 3)
Answer(e) x =
(f) On the diagram, sketch the graph of
© UCLES 2013
[2]
y = f(x – 3).
0607/41/M/J/13
or x =
[3]
[2]
15
11
A
6 cm
3 cm
For
Examiner's
Use
B
NOT TO
SCALE
O
5 cm
D
C
The diagram shows a trapezium ABCD with diagonals intersecting at O.
AB is parallel to DC.
(a) Explain why triangle AOB is similar to triangle COD.
Answer(a)
[2]
(b) Calculate the length of CD.
Answer(b)
cm [2]
(c) Find the value of these fractions.
(i) Area of triangle ABO
Area of triangle CBO
Answer(c)(i)
[1]
Answer(c)(ii)
[1]
Answer(c)(iii)
[1]
(ii) Area of triangle ABO
Area of triangle CDO
(iii)
Area of triangle ABO
Area of trapezium ABCD
Question 12 is printed on the next page.
© UCLES 2013
0607/41/M/J/13
[Turn over
16
12 An aircraft travels 5500 km from Dubai to London.
The average speed is x km/h.
For
Examiner's
Use
(a) Write down an expression, in terms of x, for the time taken for this journey.
Answer(a)
hours [1]
(b) The return journey from London to Dubai is
• 60 km/h faster
• half-an-hour shorter
than the journey from Dubai to London.
Write down an equation in x and show that it simplifies to
x2 + 60x – 660 000 = 0 .
[4]
(c) Solve the equation x2 + 60x – 660 000 = 0 .
Give your answers correct to the nearest whole number.
Answer(c) x =
or x =
[3]
(d) The time that the aircraft leaves Dubai is 09 40 local time.
The time in London is 4 hours behind the time in Dubai.
Use your answer to part (c) to find the arrival time in London.
Answer(d)
[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.
University of 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 2013
0607/41/M/J/13
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
International General Certificate of Secondary Education
*1604463664*
CAMBRIDGE INTERNATIONAL MATHEMATICS
0607/42
May/June 2013
Paper 4 (Extended)
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, highlighters, glue or correction fluid.
You may use a 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.
For Examiner's Use
This document consists of 19 printed pages and 1 blank page.
IB13 06_0607_42/2RP
© UCLES 2013
[Turn over
2
Formula List
2
ax + bx + c = 0
For the equation
x=
_ b ± b2 _ 4ac
2a
Curved surface area, A, of cylinder of radius r, height h.
A = 2πrh
Curved surface area, A, of cone of radius r, sloping edge l.
A = πrl
Curved surface area, A, of sphere of radius r.
A = 4πr2
Volume, V, of pyramid, base area A, height h.
V=
Volume, V, of cylinder of radius r, height h.
V = πr2h
Volume, V, of cone of radius r, height h.
V=
Volume, V, of sphere of radius r.
V=
A
1
3
Ah
1
3
4
3
πr2h
πr3
a
b
c
=
=
sin A sin B sin C
a2 = b2 + c2 – 2bc cos A
b
c
Area =
B
© UCLES 2013
a
C
0607/42/M/J/13
1
2
bc sin A
3
Answer all the questions.
1
For
Examiner's
Use
(a) (i) Kim’s wage is $720 each month.
She spends $196 each month on food.
Calculate $196 as a percentage of $720.
Answer(a)(i)
%
[1]
(ii) She pays 25% of the $720 in taxes.
Find the ratio money spent on food : money paid in taxes.
Give your answer in its simplest form.
Answer(a)(ii)
:
[2]
(iii) The $720 is an increase of 44% on Kim’s previous wage.
Calculate her previous wage.
Answer(a)(iii) $
[3]
(iv) Next year the $720 will increase by 4%.
Calculate next year’s monthly wage.
Answer(a)(iv) $
[2]
(b) Jay’s monthly wage is $650.
Each year Jay’s monthly wage increases by 5%.
Calculate the number of years it will take for Jay’s monthly wage to exceed $1000.
© UCLES 2013
Answer(b)
[3]
0607/42/M/J/13
[Turn over
4
2
(a)
x+3
2x
For
Examiner's
Use
NOT TO
SCALE
3x
2x + 1
The areas of the rectangles are equal.
Find the value of x.
Show all your working.
Answer(a) x =
[4]
(b)
2y
θ
NOT TO
SCALE
y+1
Find the value of y when tan θ =
1
.
3
Show all your working.
Answer(b) y =
© UCLES 2013
0607/42/M/J/13
[3]
5
(c) Jo walks 10 km at w kilometres per hour.
Sam cycles 10 km at (w + 9) kilometres per hour.
The difference between the times taken by Jo and Sam is 2
(i) Show that
For
Examiner's
Use
1
hours.
2
w2 + 9w – 36 = 0.
[4]
(ii) Find the time, in hours and minutes, taken by Jo to walk the 10 km.
Answer(c)(ii)
© UCLES 2013
0607/42/M/J/13
h
min
[4]
[Turn over
6
3
For
Examiner's
Use
y
5
4
3
2
1
–5
–4
–3
–2
–1
0
1
2
3
4
5
x
–1
–2
–3
–4
–5
L
(a) Find the equation of the line L.
Answer(a)
(b) (i) On the grid, draw the line y = 2x + 4.
[2]
[2]
(ii) On the grid, shade the region where y [ 0 and y [ 2x + 4.
[2]
(c) P is the point (1, – 4) and Q is the point (3, 2).
Find the equation of the line passing through P and Q.
Answer(c)
© UCLES 2013
0607/42/M/J/13
[3]
7
4
The masses of 100 apples are measured.
The results are shown in the table.
For
Examiner's
Use
Mass (m grams)
20 < m Y 100
100 < m Y 150
150 < m Y 240
Frequency
28
45
27
(a) Calculate an estimate of the mean mass.
Answer(a)
g [2]
(b) Use the information in the table to complete the histogram.
1.0
0.9
0.8
0.7
0.6
Frequency
0.5
density
0.4
0.3
0.2
0.1
0
m
20
40
60
80
100
120
140
160
180
200
220
240
Mass (grams)
[3]
© UCLES 2013
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[Turn over
8
5
For
Examiner's
Use
H
NOT TO
SCALE
30°
988 km
1060 km
B
P
1185 km
998 km
K
The diagram shows some straight line distances between Bangkok (B), Hanoi (H), Phnom Penh (P)
and Kuala Lumpur (K). Angle BHP = 30°.
(a) Calculate BP and show that it rounds to 535 km, correct to the nearest kilometre.
[3]
© UCLES 2013
0607/42/M/J/13
9
(b) Calculate angle BKP.
For
Examiner's
Use
Answer(b)
[3]
Answer(c)
[1]
(c) The bearing of P from K is 020°.
Find the bearing of B from K.
© UCLES 2013
0607/42/M/J/13
[Turn over
10
6
R
For
Examiner's
Use
C
6 cm
NOT TO
SCALE
A
P
12 cm
Q
B
20 cm
The diagram shows a triangular prism of length 20 cm.
The cross-section of the prism is triangle ABC with angle BAC = 90°, AC = 6 cm and AB = 12 cm.
(a) Calculate the volume of the prism.
Answer(a)
© UCLES 2013
0607/42/M/J/13
cm3 [2]
11
(b) (i) Calculate the total surface area of the prism.
Answer(b)(i)
For
Examiner's
Use
cm2 [4]
(ii) The surface of the prism is painted at a cost of $0.005 per square centimetre.
Calculate the cost of painting the surface of the prism.
Answer(b)(ii) $
[1]
(c) Calculate the angle between the diagonal line CQ and the base ABQP.
Answer(c)
© UCLES 2013
0607/42/M/J/13
[3]
[Turn over
12
7
A flight from London, England to Auckland, New Zealand departs at 14 00 on February 7th.
1
The journey takes 27 hours and the distance is 18 400 km.
2
The time in New Zealand is 13 hours ahead of the time in England.
For
Examiner's
Use
(a) Find the time and the date that the flight arrives in Auckland.
Answer(a) Time
Date
[3]
(b) Calculate the average speed of the journey.
Answer(b)
km/h [1]
(c) The cost of a ticket for the flight is 3600 pounds (£).
£1 = 2.09 New Zealand dollars (NZD).
(i) Calculate the cost of the ticket in NZD.
Answer(c)(i)
NZD [1]
(ii) Calculate the cost of the journey, in NZD per kilometre.
Give your answer correct to 2 decimal places.
Answer(c)(ii)
© UCLES 2013
0607/42/M/J/13
NZD/km [2]
13
8
(a) Solve the equation
2
= x3 + 2 .
x
For
Examiner's
Use
Answer(a) x =
or x =
(b) Solve the inequality
2
[ x3 + 2 .
x
Answer(b)
© UCLES 2013
[4]
0607/42/M/J/13
[3]
[Turn over
14
9
For
Examiner's
Use
B
8 cm
O
NOT TO
SCALE
70°
A
AB is a chord of the circle centre O.
Calculate
(a) the length of the chord AB,
Answer(a)
cm [3]
Answer(b)
cm [2]
Answer(c)
cm2 [4]
(b) the length of the arc AB,
(c) the area of the shaded region.
© UCLES 2013
0607/42/M/J/13
15
10
For
Examiner's
Use
y
2
0
x
180°
360°
–2
f(x) = cos x
x
g(x) = 2sin  
2
 
(a) On the diagram, sketch the following graphs.
(i) y = f(x)
[2]
(ii) y = g(x)
[2]
(b) Write down the equation of the line of symmetry of the graphs.
Answer(b)
[1]
(c) Write down the co-ordinates of the local minimum point on the graph of
y = f(x) for 0° Y x Y 360°.
Answer(c)
(
,
)
[2]
(d) Write down the period and amplitude of g(x).
Answer(d) period =
[2]
amplitude =
(e) Write down the range of g(x) for the following domains.
(i) 0° Y x Y 360°
Answer(e)(i)
[1]
Answer(e)(ii)
[1]
(ii) o
(f) Solve the equation f(x) = g(x) for 0° Y x Y 360°.
Answer(f) x =
(g) Shade the regions on the diagram where y Y f(x) and y [ g(x).
© UCLES 2013
0607/42/M/J/13
or x =
[2]
[1]
[Turn over
16
11
For
Examiner's
Use
1
2
10
1
1
5
The diagram shows a disc, with six equal sectors, and an arrow.
When the disc is spun, each sector is equally likely to stop next to the arrow.
(a) The disc is spun.
Write down the probability that the sector next to the arrow is labelled with
(i) 1 or 2,
Answer(a)(i)
[1]
Answer(a)(ii)
[1]
Answer(a)(iii)
[1]
(ii) an even number,
(iii) a number which is a factor of 10.
(b) The disc is spun twice.
(i) Complete the tree diagram by writing the missing probabilities on each branch.
second number
first number
10
........
10
1
6
........
not 10
10
........
........
not 10
........
not 10
[2]
© UCLES 2013
0607/42/M/J/13
17
(ii) Find the probability that the arrow is next to the number 10 twice.
Answer(b)(ii)
For
Examiner's
Use
[2]
(iii) Find the probability that the arrow is next to the number 10 at least once.
Answer(b)(iii)
[2]
(c) The disc is spun n times until it stops with the number 10 next to the arrow.
625
.
Find n when the probability that this happens is
7776
Answer(c) n =
© UCLES 2013
0607/42/M/J/13
[2]
[Turn over
18
12
Month
Jan
Feb Mar Apr May Jun
Jul
Aug Sep
Oct Nov Dec
Temperature (t ˚C)
13
13
15
16
19
23
25
26
24
20
18
13
Rainfall (r mm)
59
49
62
46
25
6
1
3
28
62
63
66
The table shows the average monthly temperature, t, and rainfall, r, in Malaga, Spain.
(a) Find the mean, median, upper quartile and range of the average monthly temperatures.
Answer(a)
mean =
°C
median =
°C
upper quartile =
°C
range =
°C [4]
(b) (i) Find the equation of the line of regression for this data, giving r in terms of t.
Answer(b)(i) r =
[2]
(ii) Describe the type of correlation between r and t.
Answer(b)(ii)
[1]
(iii) Calculate an estimate of the rainfall when the temperature is 22°C.
Answer(b)(iii)
© UCLES 2013
0607/42/M/J/13
[1]
For
Examiner's
Use
19
13
For
Examiner's
Use
P
p
O
NOT TO
SCALE
X
q
Q
Y
The diagram shows a triangle OPQ.
The point X is on PQ so that PX : XQ = 1 : 2.
= p and
= q.
in terms of p and q.
(a) Find
Give your answer in its simplest form.
Answer(a)
[2]
Answer(b)
[3]
(b) OQY is a straight line and OY = 2OQ.
in terms of p and q.
Find
Give your answer in its simplest form.
 3
(c) p =   and |p| = 5.
k 
Find the two possible values of k.
Answer(c) k =
© UCLES 2013
0607/42/M/J/13
or k =
[2]
20
BLANK PAGE
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.
University of 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 2013
0607/42/M/J/13
PAPA CAMBRIDGE
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
International General Certificate of Secondary Education
*4434452004*
CAMBRIDGE INTERNATIONAL MATHEMATICS
0607/43
May/June 2013
Paper 4 (Extended)
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, highlighters, glue or correction fluid.
You may use a 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.
For Examiner's Use
This document consists of 20 printed pages.
IB13 06_0607_43/6RP
© UCLES 2013
[Turn over
2
Formula List
2
ax + bx + c = 0
For the equation
x=
_ b ± b2 _ 4ac
2a
Curved surface area, A, of cylinder of radius r, height h.
A = 2πrh
Curved surface area, A, of cone of radius r, sloping edge l.
A = πrl
Curved surface area, A, of sphere of radius r.
A = 4πr2
Volume, V, of pyramid, base area A, height h.
V=
Volume, V, of cylinder of radius r, height h.
V = πr2h
Volume, V, of cone of radius r, height h.
V=
Volume, V, of sphere of radius r.
V=
A
1
3
Ah
1
3
4
3
πr2h
πr3
a
b
c
=
=
sin A sin B sin C
a2 = b2 + c2 – 2bc cos A
b
c
Area =
B
© UCLES 2013
a
C
0607/43/M/J/13
1
2
bc sin A
3
Answer all the questions.
1
For
Examiner's
Use
y varies inversely as the square root of x.
y = 16 when x = 4.
(a) Find the value of y when x = 16.
Answer(a) y =
[3]
Answer(b) x =
[2]
Answer(c) x =
[3]
(b) Find the value of x when y = 64.
(c) Find x in terms of y.
© UCLES 2013
0607/43/M/J/13
[Turn over
4
2
(a) Solve the equation.
For
Examiner's
Use
2log 6 – log 9 + log x = 3
Answer(a) x =
[3]
(b) Solve the simultaneous equations.
3x – 4y = 10
5x – 3y = 2
Answer(b) x =
y=
© UCLES 2013
0607/43/M/J/13
[4]
5
3
For each Venn diagram, describe the shaded region using set notation.
For
Examiner's
Use
(a)
U
A
B
C
Answer(a)
[1]
(b)
U
A
B
C
Answer(b)
[1]
(c)
U
A
B
C
Answer(c)
[1]
(d)
U
A
B
C
Answer(d)
© UCLES 2013
0607/43/M/J/13
[2]
[Turn over
6
4
(a)
7 cm
C
For
Examiner's
Use
D
8 cm
B
4.5 cm
NOT TO
SCALE
E
x
A
In the diagram, BE is parallel to CD.
The perpendicular height between the lines BE and CD is 8 cm.
The perpendicular height from the point A to the line BE is x.
Show that x = 14.4 cm.
[2]
© UCLES 2013
0607/43/M/J/13
7
(b)
For
Examiner's
Use
7 cm
NOT TO
SCALE
8 cm
4.5 cm
The diagram shows a plastic cup.
The diameter of the circular base is 4.5 cm and the diameter of the circular top is 7 cm.
The height of the cup is 8 cm.
Using part (a), calculate the volume of the cup.
Give your answer correct to the nearest cubic centimetre.
Answer(b)
© UCLES 2013
0607/43/M/J/13
cm3 [3]
[Turn over
8
5
(a) Solve the equation 10x2 = 5 – x .
Give your answers correct to 2 decimal places.
Answer(a) x =
For
Examiner's
Use
or x =
[4]
(b) Solve the inequality 10x2 > 5 – x .
Answer(b)
6
[2]
The transformation P is a rotation of 180° about the origin.
The transformation Q is a reflection in the line y = x.
(a) Find the image of the point (6, 2) under the transformation P.
Answer(a) (
,
)
[1]
,
)
[1]
(b) Find the image of the point (6, 2) under the transformation Q.
Answer(b) (
(c) Describe fully the single transformation equivalent to P followed by Q.
Answer (c)
[2]
© UCLES 2013
0607/43/M/J/13
9
7
For
Examiner's
Use
y
4
x
0
–4
4
–4
(a) On the diagram, sketch the graph of y = f(x), where
f(x) =
( x − 1)
between x = – 4 and x = 4 .
( x 2 − 4)
[4]
(b) Write down the equations of the three asymptotes.
Answer(b)
[3]
( x − 1)
three times.
( x 2 − 4)
Find the values of the x co-ordinates of the points of intersection.
(c) The line y = x intersects the curve y =
Answer(c) x =
x=
x=
© UCLES 2013
0607/43/M/J/13
[3]
[Turn over
10
8
For
Examiner's
Use
B
NOT TO
SCALE
6 cm
A
7 cm
C
8 cm
The diagram shows a triangle ABC.
(a) Use the cosine rule to find angle ABC.
Answer(a)
[3]
(b) Find the area of triangle ABC, giving your answer correct to 2 decimal places.
Answer(b)
cm2 [3]
(c) Find the length of the perpendicular line from C to the line AB.
Answer(c)
© UCLES 2013
0607/43/M/J/13
cm [2]
11
9
The British Lions squad for the 2009 tour of South Africa originally contained 40 players from
England, Ireland, Scotland and Wales.
The playing positions, either Forward or Back, of these players is shown in the table.
England
Ireland
Scotland
Wales
Forward
6
5
2
6
Back
3
9
2
7
For
Examiner's
Use
(a) A player is selected at random from the squad to visit a local hospital.
Calculate the probability that the player chosen is
(i) a Forward from Ireland,
Answer(a)(i)
[1]
Answer(a)(ii)
[1]
(ii) not from Wales.
(b) A player is chosen at random from the Backs to give a TV interview.
Calculate the probability that he is from England.
Answer(b)
[2]
(c) Three Forwards are chosen at random to take part in a ‘tug-o-war’ competition.
Calculate the probability they are all from Wales.
Answer(c)
© UCLES 2013
0607/43/M/J/13
[3]
[Turn over
12
10 (a)
For
Examiner's
Use
NOT TO
SCALE
1.6 cm
2.4 cm
The diagram shows a brass washer.
The washer is made by removing a circular disc of diameter 1.6 cm from a circular disc of
diameter 2.4 cm.
(i) Find the area of the top surface of the washer in square centimetres.
Answer(a)(i)
cm2 [2]
(ii) The washer is 2 mm thick.
Calculate the volume of the washer in cubic centimetres.
Answer(a)(ii)
© UCLES 2013
0607/43/M/J/13
cm3 [2]
13
(b)
For
Examiner's
Use
NOT TO
SCALE
The diagram shows a globe made from brass.
Globes are hollow spheres.
The outside diameter of this globe is 32 cm and the inside diameter is 30 cm.
(i) Find the volume of brass used to make this globe in cubic centimetres.
Answer(b)(i)
cm3 [2]
(ii) A number of globes are to be made by melting 1 000 000 of the brass washers in part (a).
Find the maximum number of globes that can be made.
Answer(b)(ii)
© UCLES 2013
0607/43/M/J/13
[3]
[Turn over
14
11 Carlos delivers computers from a factory to a town that is 720 km away.
When he drives at an average speed of x km/h the journey takes one hour longer than if he drives at
(x +10) km/h.
(a) Write down an equation in x and show that it simplifies to x2 + 10x – 7200 = 0 .
[4]
(b) (i) Factorise x2 + 10x – 7200.
Answer(b)(i)
[2]
(ii) Solve the equation x2 + 10x – 7200 = 0 .
or x =
Answer(b)(ii) x =
[1]
(iii) Carlos drives the 720 km at x km/h.
Work out the time of his journey.
Answer(b)(iii)
© UCLES 2013
0607/43/M/J/13
hours [1]
For
Examiner's
Use
15
12
For
Examiner's
Use
y
6
x
0
–4
4
–4
(a) (i) On the diagram, sketch the graph of y = f(x), where
f(x) = 2 –
1
(2 x + 3)
between x = – 4 and x = 4 .
[2]
(ii) Write down the co-ordinates of the points where the graph crosses the axes.
Answer(a)(ii) (
,
)
(
,
)
[2]
(iii) Find f(0.25).
(b) Solve the inequality 2 –
Answer(a)(iii)
[1]
Answer(b)
[4]
Answer(c)
[4]
Answer(d) x =
[2]
1
<4.
(2 x + 3)
(c) Find f–1(x).
(d) Solve f–1(x) = 1 .
© UCLES 2013
0607/43/M/J/13
[Turn over
16
13 The masses of 200 tomatoes are given in the table.
Mass (m grams)
Frequency
0 < m Y 20
12
20 < m Y 30
34
30 < m Y 40
40
40 < m Y 45
60
45 < m Y 50
42
50 < m Y 80
12
For
Examiner's
Use
(a) Calculate an estimate of the mean mass of a tomato.
Give your answer correct to the nearest gram.
Answer(a)
g [3]
(b) (i) Complete the frequency density column in this table.
Mass (m grams)
Frequency
0 < m Y 20
12
20 < m Y 30
34
30 < m Y 40
40
40 < m Y 45
60
45 < m Y 50
42
50 < m Y 80
12
Frequency density
[2]
(ii) On the grid opposite, draw an accurate histogram to show this information.
Mark a suitable scale on the frequency density axis.
© UCLES 2013
0607/43/M/J/13
17
Frequency
density
0
m
10
20
30
40
50
60
70
80
Mass (grams)
[4]
© UCLES 2013
0607/43/M/J/13
[Turn over
18
14 Zaira works at an ice-cream shop.
She wants to find out if there is a correlation between the maximum daily temperature, x °C, and the
shop’s daily income, $y.
Zaira recorded the following results.
Temperature (x °C)
23
18
27
19
25
20
22
28
17
24
Income ($y)
430
320
510
380
510
430
450
530
310
490
(a) (i) Complete the scatter diagram.
The first four points have been plotted for you.
y
650
600
550
500
Income
($)
450
400
350
300
250
0
x
10
12
14
16
18
20
22
24
26
28
30
32
Temperature (°C)
[3]
(ii) Describe the type of correlation between the temperature and the income.
Answer(a)(ii)
© UCLES 2013
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[1]
For
Examiner's
Use
19
(b) Find
For
Examiner's
Use
(i) the mean temperature,
Answer(b)(i)
°C [1]
(ii) the mean income.
Answer(b)(ii) $
[1]
(c) (i) Find the equation of the regression line for y in terms of x.
Answer(c)(i) y =
[2]
(ii) Estimate the income when the temperature is 21°C.
Answer(c)(ii) $
[1]
(iii) Estimate the income when the temperature is 32°C.
Answer(c)(iii) $
[1]
(iv) Explain which of your answers to parts (c)(ii) and (c)(iii) is likely to be the most reliable.
[2]
Question 15 is printed on the next page.
© UCLES 2013
0607/43/M/J/13
[Turn over
20
15 Find the next term and the nth term in each of the following sequences.
For
Examiner's
Use
(a) 6, 18, 54, 162, 486, .....
Answer(a) next term =
nth term =
[3]
(b) –1, 1, 5, 11, 19, .....
Answer(b) next term =
nth term =
[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.
University of 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 2013
0607/43/M/J/13
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
International General Certificate of Secondary Education
*6915781925*
CAMBRIDGE INTERNATIONAL MATHEMATICS
Paper 6 (Extended)
0607/06
May/June 2013
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, highlighters, glue or correction fluid.
You may use a 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 8 printed pages.
IB13 06_0607_06/2RP
© UCLES 2013
[Turn over
2
Answer both parts A and B.
A
INVESTIGATION
DIAGONALS OF RECTANGLES (20 marks)
You are advised to spend 45 minutes on part A.
Rectangles are drawn on a grid.
The sides of each rectangle lie on gridlines and the length is greater than or equal to the width.
This investigation looks for a method for calculating the number of small squares through which a
diagonal passes.
1
The diagram shows a rectangle with length 5 and width 3.
The diagonal crosses 4 vertical gridlines inside the rectangle.
Write down
(a) the number of horizontal gridlines that the diagonal crosses inside the rectangle,
(b) the total number of gridlines that the diagonal crosses inside the rectangle.
2
A rectangle has length x and width y.
x and y do not have a common factor.
(a) Write down an expression for
(i) the number of vertical gridlines that a diagonal crosses inside the rectangle, in terms of x,
(ii) the number of horizontal gridlines that a diagonal crosses inside the rectangle, in terms
of y,
(iii) the total number of gridlines, N, which a diagonal crosses inside the rectangle, in terms of x
and y. Write your answer in its simplest form.
N=
© UCLES 2013
0607/06/M/J/13
For
Examiner's
Use
3
(b) S is the number of squares through which the diagonal passes.
For example, the diagonal in question 1 passes through 7 squares.
For
Examiner's
Use
(i) Write S in terms of N.
S=
(ii) Write S in terms of x and y.
S=
(c) Show that your formula for S in part (b)(ii) gives the correct value for an 8 by 5 rectangle.
Use the grid to show clearly how many squares the diagonal passes through.
© UCLES 2013
0607/06/M/J/13
[Turn over
4
3
In question 2, x and y did not have a common factor.
In this question, x and y do have a common factor.
(a) (i) Show clearly that your formula for S does not give the correct value for a 9 by 6 rectangle.
(ii) 9 and 6 have a common factor of 3.
Show how you use the value of S for a 3 by 2 rectangle to calculate S for a 9 by 6 rectangle.
© UCLES 2013
0607/06/M/J/13
For
Examiner's
Use
5
(b) Use your method in part (a)(ii) to find S for each of these rectangles.
(i) 93 by 90
For
Examiner's
Use
(ii) 60 by 35
4
The diagonal of a rectangle passes through 6 squares.
Use question 2 and question 3 to find the length and the width of each possible rectangle.
© UCLES 2013
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[Turn over
6
B
MODELLING
DRILLING A TUNNEL (20 marks)
For
Examiner's
Use
You are advised to spend 45 minutes on part B.
North
B
300 m
NOT TO
C SCALE
On the plan, A is south of B and C is east of B.
AB = 500 metres and BC = 300 metres.
Engineers want to drill a tunnel from A to C.
The tunnel has one or more straight sections.
500 m
Hard
rock
Normal
rock
Normal
rock
A
1
Calculate the length of the shortest possible tunnel from A to C.
Give your answer correct to the nearest metre.
m
2
Write down the length of the tunnel if the engineers drill through as little hard rock as possible.
m
3
P is a point which is x metres south of B.
North
The engineers decide to drill from A to P to C.
B
300 m
NOT TO
C SCALE
xm
500 m P
Normal
rock
Hard
rock
A
Through normal rock, from A to P, the drill moves forward at 2 metres per hour.
Through the hard rock, from P to C, the drill moves forward at 1 metre per hour.
© UCLES 2013
0607/06/M/J/13
Normal
rock
7
(a) Explain why the time in hours, T, that it takes to drill the tunnel, can be modelled by this
equation.
T=
For
Examiner's
Use
500 − x
+ 90000 + x 2
2
(b) All the measurements are accurate.
Write down a practical reason why the time given by the model may be different from the actual
time.
(c) On the diagram, sketch the graph of T against x.
T
600
Time
500
(hours)
400
0
Distance (metres)
x
500
(d) (i) Find, to the nearest metre, the position of P which gives the minimum time to drill the
tunnel.
metres from B
(ii) Find this minimum time correct to the nearest 10 hours.
hours
© UCLES 2013
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[Turn over
8
4
To drill through normal rock costs 2 thousand dollars per hour.
To drill through the hard rock costs 3 thousand dollars per hour.
For
Examiner's
Use
(a) The total cost of drilling the tunnel is n thousand dollars.
Write down a model for n in terms of x.
n=
(b) (i) Find, to the nearest metre, the position of P which gives the minimum cost.
metres from B
(ii) Write, in full, this minimum cost to the nearest ten thousand dollars.
$
5
The model for the time taken to drill the tunnel is T =
500 − x
+ 90000 + x 2 .
2
(a) The position of B and C are fixed.
Investigate the position of P which gives the minimum time when A is more than 500 m south
of B.
(b) If AB = d metres explain, using part (a), why the minimum time in hours is
d
T = + k , where k = 260 correct to 3 significant figures.
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.
University of 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 2013
0607/06/M/J/13
PAPA CAMBRIDGE
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
International General Certificate of Secondary Education
*3769158093*
0607/04
CAMBRIDGE INTERNATIONAL MATHEMATICS
October/November 2013
Paper 4 (Extended)
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, highlighters, glue or correction fluid.
You may use a 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.
For Examiner's Use
This document consists of 18 printed pages and 2 blank pages.
IB13 11_0607_04/2RP
© UCLES 2013
[Turn over
2
Formula List
2
ax + bx + c = 0
For the equation
x=
_ b ± b2 _ 4ac
2a
Curved surface area, A, of cylinder of radius r, height h.
A = 2πrh
Curved surface area, A, of cone of radius r, sloping edge l.
A = πrl
Curved surface area, A, of sphere of radius r.
A = 4πr2
Volume, V, of pyramid, base area A, height h.
V=
Volume, V, of cylinder of radius r, height h.
V = πr2h
Volume, V, of cone of radius r, height h.
V=
Volume, V, of sphere of radius r.
V=
A
1
3
Ah
1
3
4
3
πr2h
πr3
a
b
c
=
=
sin A sin B sin C
a2 = b2 + c2 – 2bc cos A
b
c
Area =
B
© UCLES 2013
a
C
0607/04/O/N/13
1
2
bc sin A
3
Answer all the questions.
1
For
Examiner's
Use
Manuel buys a car for $8000.
(a) Each year the value of the car decreases by 8% of its value at the start of the year.
(i) Calculate the value of the car after 5 years.
Answer(a)(i) $
[2]
(ii) Calculate how many more years it takes for the value of the car to be less than $4000.
Answer(a)(ii)
[2]
(b) Manuel has a journey of 235 km.
The journey takes 3 h 15 min and the car uses 19.7 litres of fuel.
(i) Calculate the average speed of the journey in kilometres per hour.
Answer(b)(i)
km/h [2]
(ii) Find the rate at which the car uses fuel.
Give your answer in litres per 100 km.
Answer(b)(ii)
© UCLES 2013
0607/04/O/N/13
l/100 km
[1]
[Turn over
4
2
For
Examiner's
Use
y
5
4
3
2
W
T
1
–5
–4
–3
–2
–1
0
1
2
3
4
5
x
–1
–2
–3
–4
–5
(a) (i) Reflect triangle T in the x-axis. Label the image U.
(ii) Rotate triangle U clockwise through 90° about (0, 0). Label the image V.
[2]
[2]
(iii) Describe fully the single transformation that maps triangle T onto triangle V.
[2]
(b) Describe fully the single transformation that maps triangle T onto triangle W.
[3]
© UCLES 2013
0607/04/O/N/13
5
3
For
Examiner's
Use
North
A
346 km
NOT TO
SCALE
493 km
N
271 km
W
The diagram shows the straight line distances between the cities Auckland (A), Napier (N) and
Wellington (W) in New Zealand.
(a) The bearing of W from A is 179°.
Calculate the bearing of N from A.
Answer(a)
[4]
(b) A map shows the three cities.
The scale of the map is 1 : 10 000 000.
Calculate the area of triangle ANW on the map.
Give your answer in square centimetres.
Answer(b)
© UCLES 2013
0607/04/O/N/13
cm2 [3]
[Turn over
6
4
For
Examiner's
Use
O
NOT TO
SCALE
14 cm
P
12 cm
C
The diagram shows a hollow cone of height 12 cm and sloping edge, OP, 14 cm.
C is the centre of the base of the cone.
(a) Calculate
(i) the radius of the base of the cone,
Answer(a)(i)
cm
[3]
Answer(a)(ii)
cm3
[2]
(ii) the volume of the cone.
© UCLES 2013
0607/04/O/N/13
7
(b) The cone is cut along the sloping edge OP and opened out to make a sector of a circle.
For
Examiner's
Use
B
A
NOT TO
SCALE
O
(i) Calculate the area of the sector and show that it rounds to 317 cm2, correct to 3 significant
figures.
[2]
(ii) Calculate the reflex angle AOB.
Answer(b)(ii)
© UCLES 2013
0607/04/O/N/13
[3]
[Turn over
8
5
200 students each record the number of hours, h, they spend on homework in one week.
The cumulative frequency curve shows the results.
For
Examiner's
Use
200
180
160
Cumulative frequency
140
120
100
80
60
40
20
0
5
10
15
20
25
30
35
h
Time spent on homework (hours)
(a) Find
(i) the median,
Answer(a)(i)
h [1]
Answer(a)(ii)
h [1]
Answer(a)(iii)
h [1]
Answer(a)(iv)
h [1]
(ii) the lower quartile,
(iii) the inter-quartile range,
(iv) the 90th percentile,
(v) the number of students who spend more than 10 hours on homework.
Answer(a)(v)
© UCLES 2013
0607/04/O/N/13
[2]
9
(b) (i) Use the cumulative frequency curve to complete the frequency table.
Time spent on
homework
h hours
0 < h Ğ 10
10 < h Ğ 15
Frequency
20
20
15 < h Ğ 20
For
Examiner's
Use
20 < h Ğ 25
25 < h Ğ 35
50
[2]
(ii) Calculate an estimate of the mean number of hours spent on homework.
Answer(b)(ii)
h [2]
(iii) The data is used to draw a histogram.
Complete the frequency density table.
(Do not draw the histogram.)
Time spent on
homework
h hours
Frequency
density
0 < h Ğ 15
15 < h Ğ 20
20 < h Ğ 25
25 < h Ğ 35
10
[3]
© UCLES 2013
0607/04/O/N/13
[Turn over
10
6
For
Examiner's
Use
y
8
0
–10
10
x
–8
f(x) =
(2 x − 3)
( x + 2)
(a) On the diagram, sketch the graph of y = f(x).
[3]
(b) Write down the value of f(0).
Answer(b)
[1]
Answer(c)
[1]
(c) Solve the equation f(x) = 0.
(d) Write down the equations of the asymptotes.
Answer(d)
[2]
(e) Find the range of f(x) for the domain 0 Ğ x Ğ 8 .
Answer(e)
© UCLES 2013
0607/04/O/N/13
[2]
11
g(x) = 3 – x
(f)
(i) On the diagram, sketch the graph of y = g(x).
[1]
For
Examiner's
Use
(ii) Solve the equation f(x) = g(x).
or x =
Answer(f)(ii) x =
[2]
(iii) Show that the equation f(x) = g(x) can be re-arranged into x2 + x – 9 = 0 .
[3]
(iv) The exact solutions of the equation x2 + x – 9 = 0 are
−1± k
.
2
Find the value of k.
Answer(f)(iv) k =
© UCLES 2013
0607/04/O/N/13
[2]
[Turn over
12
7
For
Examiner's
Use
y
A
NOT TO
SCALE
B
x
0
A (1, 6) is joined to B (5, 2) by the line AB.
(a) Calculate the length of the line AB.
Answer(a)
[3]
(b) Find the equation of the straight line that passes through A and B.
Answer(b)
[3]
(c) (i) Find the equation of the line which is perpendicular to AB and passes through the origin.
Answer(c)(i)
[2]
(ii) Find the co-ordinates of the point of intersection of the line in part (c)(i) and the line AB.
Answer(c)(ii) (
© UCLES 2013
0607/04/O/N/13
,
)
[1]
13
8
Find the nth term of each of the following sequences.
(a)
(b)
(c)
(d)
© UCLES 2013
21,
3,
1
,
4
0,
17,
6,
4
,
5
6,
13,
12,
9
,
6
24,
9,
24,
16
,
7
60,
For
Examiner's
Use
5, ………
Answer(a)
[2]
Answer(b)
[2]
48, ………
25
, ………
8
Answer(c)
[2]
Answer(d)
[4]
120, ………
0607/04/O/N/13
[Turn over
14
9
If the weather is fine, the probability that Alex goes to the beach is
9
.
10
If the weather is not fine, the probability that Alex goes to the beach is
5
.
6
The probability that the weather will be fine is
For
Examiner's
Use
3
.
10
(a) Complete the tree diagram.
Weather is fine
Alex goes to the beach
........
........
Yes
........
........
Yes
........
No
Yes
No
........
No
[3]
(b) Find the probability that Alex goes to the beach.
Answer(b)
(c) Which combination of these events has a probability of
Answer(c)
© UCLES 2013
[3]
1
?
12
[1]
0607/04/O/N/13
15
10
D
x°
For
Examiner's
Use
8.55 cm
3x°
C
6x° 2.78 cm
NOT TO
SCALE
X
B
9.23 cm
A
A, B, C and D lie on the circumference of a circle.
AC and BD intersect at X.
(a) Angle CDX = x°, angle DCX = 3x° and angle CXD = 6x°.
Show that angle ABX = 54°.
[3]
(b) (i) Complete the statement
Triangles CDX and BAX are
[1]
(ii) AB = 9.23 cm, DC = 8.55 cm and XC = 2.78 cm.
Calculate the length of BX.
Answer(b)(ii)
(iii) Find the value of
cm [2]
Area of triangle CDX
.
Area of triangle BAX
Give your answer correct to 2 decimal places.
Answer(b)(iii)
© UCLES 2013
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[2]
[Turn over
16
11 (a)
For
Examiner's
Use
y
NOT TO
SCALE
x
0
The sketch shows the graph of y = logax .
On the same diagram, sketch the graph of y = 2logax .
[2]
3log x = log16 – 2log x
(b)
Find the value of x.
Answer(b) x =
[3]
y
(c) Solve the equation 5 = 100 .
Give your answer correct to 4 significant figures.
Answer(c) y =
© UCLES 2013
0607/04/O/N/13
[3]
17
12
For
Examiner's
Use
5x
NOT TO
SCALE
2x
The diagram shows a rectangle with length 5x and width 2x.
One of the shorter sides is joined to a semicircle with radius x.
(a) Find a formula, in terms of x and π, for the total area, A, of the shape.
Answer(a) A =
[2]
(b) Make x the subject of your formula in part (a).
Answer(b) x =
[3]
Answer(c) x =
[1]
(c) Find the value of x when A = 200.
© UCLES 2013
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[Turn over
18
13 (a) (i) Factorise.
For
Examiner's
Use
2x2 – x – 1
Answer(a)(i)
[2]
(ii) Write as a single fraction in its simplest form.
1
4
+
x −1
2x2 − x −1
Answer(a)(ii)
[3]
(b) Simplify.
p 2 − 25q 2
p + 5q − pt − 5qt
Answer(b)
© UCLES 2013
0607/04/O/N/13
[4]
19
BLANK PAGE
© UCLES 2013
0607/04/O/N/13
20
BLANK PAGE
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.
University of 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 2013
0607/04/O/N/13
PAPA CAMBRIDGE
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
International General Certificate of Secondary Education
*3859409843*
0607/02
CAMBRIDGE INTERNATIONAL MATHEMATICS
October/November 2013
Paper 2 (Extended)
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, highlighters, glue or correction fluid.
You may use a 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.
IB13 11_0607_02/2RP
© UCLES 2013
[Turn over
2
Formula List
2
ax + bx + c = 0
For the equation
x=
_ b ± b2 _ 4ac
2a
Curved surface area, A, of cylinder of radius r, height h.
A = 2πrh
Curved surface area, A, of cone of radius r, sloping edge l.
A = πrl
Curved surface area, A, of sphere of radius r.
A = 4πr2
Volume, V, of pyramid, base area A, height h.
V=
Volume, V, of cylinder of radius r, height h.
V = πr2h
Volume, V, of cone of radius r, height h.
V=
Volume, V, of sphere of radius r.
V=
A
1
3
Ah
1
3
4
3
πr2h
πr3
a
b
c
=
=
sin A sin B sin C
a2 = b2 + c2 – 2bc cos A
b
c
Area =
B
© UCLES 2013
a
C
0607/02/O/N/13
1
2
bc sin A
3
Answer all the questions.
1
For
Examiner's
Use
Solve the simultaneous equations.
3g – 2h = 11
g – 2h = 5
Answer g =
h=
2
[2]
(a)
U
A
B
n(U) = 20, n(A ∪ B)' = 3, n(A) = 11, n(B) = 13.
Find n(A ∩ B' ).
Answer(a)
[2]
(b) On each Venn diagram, shade the region indicated.
U
P
U
Q
(P ∩ Q)'
S
T
S ∪ T'
[2]
3
Tiago buys a concert ticket and then sells it for $15.
He makes a profit of 20%.
Calculate how much Tiago paid for the ticket.
Answer $
© UCLES 2013
0607/02/O/N/13
[3]
[Turn over
4
4
C
D
A
52°
36°
For
Examiner's
Use
E
P
NOT TO
SCALE
B
Q
ABCD is a parallelogram and BQPC is a rhombus.
DCE is a straight line.
Angle DAB = 52° and angle ECP = 36°.
Find the size of angle BPC.
5
(a) Simplify
Answer
[3]
Answer(a)
[1]
72 .
2+2
= p+q 2
2 −1
(b)
Find the values of p and q.
Answer(b) p =
q=
© UCLES 2013
0607/02/O/N/13
[3]
5
6
Simplify the following.
For
Examiner's
Use
(a) 2y2 × 3y3
(b)
7
3
Answer(a)
[2]
Answer(b)
[2]
27 p 27
(a) Find the amplitude and period of the function f(x) = 4cos(4x).
Answer(a) Amplitude =
Period =
[2]
g(x) = 4cos(4x) – 4
(b)
Describe fully the single transformation that maps the graph of y = f(x) onto the graph of
y = g(x).
Answer(b)
[2]
© UCLES 2013
0607/02/O/N/13
[Turn over
6
1
8
For
Examiner's
Use
(a) Write down the value of 8 3 .
Answer(a)
[1]
Answer(b)
[2]
−2
4
(b) Find the exact value of   .
3
9
NOT TO
SCALE
x°
The diagram shows a cube of side length 1.
Find the value of tan x°.
Answer
© UCLES 2013
0607/02/O/N/13
[3]
7
10
For
Examiner's
Use
x cm
30°
NOT TO
SCALE
12 cm
Find the exact value of x.
Answer x =
11
M
D
[3]
C
q
N
A
p
NOT TO
SCALE
B
ABCD is a parallelogram.
DM = MC and CN = 2NB.
= p and
= q.
(a) Write down
(b) Find
in terms of q.
Answer(a)
[1]
Answer(b)
[1]
in terms of p and q.
Question 12 is printed on the next page.
© UCLES 2013
0607/02/O/N/13
[Turn over
8
f(x) = 3x – 1
12
g(x) = 12 – x
For
Examiner's
Use
Find
(a) f(g(8)),
Answer(a)
[2]
Answer(b)
[2]
Answer(c) g– 1(x) =
[1]
(b) f(g(x)), in its simplest form,
(c) g– 1(x).
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.
University of 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 2013
0607/02/O/N/13
PAPA CAMBRIDGE
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
International General Certificate of Secondary Education
*8778931706*
0607/06
CAMBRIDGE INTERNATIONAL MATHEMATICS
Paper 6 (Extended)
October/November 2013
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, highlighters, glue or correction fluid.
You may use a 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 11 printed pages and 1 blank page.
IB13 11_0607_06/2RP
© UCLES 2013
[Turn over
2
Answer both parts A and B.
A
INVESTIGATION
For
Examiner's
Use
SUMS OF SEQUENCES (20 marks)
You are advised to spend no more than 45 minutes on this part.
Here is the method to construct a sequence for this investigation.
Method
Example
Write down any two numbers for the first two terms.
3 and 7
Add these two terms to make the third term.
3 + 7 = 10
Add the second and third terms to make the fourth term.
7 + 10 = 17
Add the third and fourth terms to make the fifth term.
10 + 17 = 27
Continue in this way to construct the sequence.
This example makes the sequence:
1
3, 7, 10, 17, 27, 44,
Show that the sum of the first six terms in this sequence, divided by the fifth term, is 4.
© UCLES 2013
0607/06/O/N/13
3
2
(a) The first two terms of another sequence are 4.52 and 16.9 .
For
Examiner's
Use
(i) Use the method to write down the next four terms in this sequence.
Do not round any of your numbers.
,
4.52, 16.9,
,
,
(ii) Work out the sum of the first six terms in this sequence and divide it by the fifth term.
(b) (i) Choose two negative numbers to be the first two terms of a sequence.
Use the method to work out the next four terms in this sequence.
Write down the first six terms in your sequence.
,
,
,
,
,
(ii) Work out the sum of the first six terms in your sequence and divide it by the fifth term.
© UCLES 2013
0607/06/O/N/13
[Turn over
4
3
The first two terms of a new sequence are p and q.
The table shows the working for the first five terms.
For
Examiner's
Use
Working
Term
p
p
q
q
p
+
q
p+q
q
+
p+q
p + 2q
p+q
+
p + 2q
2p + 3q
(a) Complete the table.
(b) Find an expression for the sum of these first six terms.
Simplify your answer.
(c) Find an equation to connect the fifth term and the sum of the first six terms.
© UCLES 2013
0607/06/O/N/13
5
4
(a) Find the next four terms in the sequence in question 3.
7th term
For
Examiner's
Use
8th term
9th term
10th term
(b) Find an expression for the sum of the first ten terms in this sequence.
Simplify your answer.
(c) Find an equation to connect the seventh term and the sum of the first ten terms.
© UCLES 2013
0607/06/O/N/13
[Turn over
6
5
(a) Find the next four terms in the sequence in question 4.
11th term
12th term
13th term
14th term
(b) Find an expression for the sum of the first fourteen terms.
Simplify your answer.
(c) This sum is a multiple of one of the terms in question 4(a).
Find this multiple.
(d) Prove this connection.
© UCLES 2013
0607/06/O/N/13
For
Examiner's
Use
7
6
In question 2 the connection between the sum of the six terms and the fifth term is
sum of the first 6 terms =
4 times 5th term
For
Examiner's
Use
Complete the following statements.
sum of the first 10 terms =
sum of the first 14 terms =
sum of the first 18 terms =
© UCLES 2013
0607/06/O/N/13
[Turn over
8
B
MODELLING
THE EARTH’S TEMPERATURE (20 marks)
For
Examiner's
Use
You are advised to spend no more than 45 minutes on this part.
Logarithms to base 10 are written as log.
Scientists have been measuring the temperature of the earth since 1860.
The table shows the increase in Earth’s temperature since 1860.
The increases are averages for each decade (10 years) correct to 2 decimal places.
1
Final year of
each decade
Number of years since 1860
(N)
Temperature increase since 1860 in °C
(T)
1890
30
0.02
1900
40
0.03
1910
50
0.04
1920
60
0.06
1930
70
0.08
1940
80
0.10
1950
90
0.13
1960
100
0.18
1970
110
0.24
1980
120
0.32
(a) On the grid, plot the temperature increase (T) against the number of years since 1860 (N),
for 30 Ğ N Ğ 120.
Draw a smooth curve that shows the increase in temperature.
T
0.4
0.3
Temperature
increase (°C) 0.2
0.1
0
N
30
40
50
60
70
80
90
100
Number of years since 1860
© UCLES 2013
0607/06/O/N/13
110
120
130
9
(b) (i) Which of the following models best fits this graph?
T = aN + b
T = asinbN
T = aN b
For
Examiner's
Use
T = | aN + b |
(ii) Use the values of N and T for 1900 and 1940 in your model to write down two equations in
a and b.
(iii) Use your equations to show that 0.3 = 0.5b.
(iv) Find the value of b and show that it rounds to 1.74, correct to 3 significant figures.
(v) Find the value of a. Give your answer correct to 2 significant figures.
a=
(vi) Show that your model gives a suitable value of T for 1920.
© UCLES 2013
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[Turn over
10
(a) (i) Complete the table to give the value of log T for each value of N.
Give each answer correct to 2 decimal places.
2
For
Examiner's
Use
Final year of
each decade
Number of years since
1860 (N)
Temperature increase since
1860 in °C (T)
log T
1890
30
0.02
–1.70
1900
40
0.03
1910
50
0.04
1920
60
0.06
1930
70
0.08
1940
80
0.10
1950
90
0.13
1960
100
0.18
1970
110
0.24
–0.62
1980
120
0.32
–0.49
–1.22
–0.89
(ii) On the grid plot log T against N, for 30 Ğ N Ğ 120.
log T
0.5
0
10
20
30
40
50
60
70
80
90
–0.5
–1
–1.5
–2
–2.5
© UCLES 2013
0607/06/O/N/13
100
110
120
130
140
150
160
N
11
(iii) The mean point is (75, –1.07).
On the grid, draw the line of best fit.
For
Examiner's
Use
(iv) Use your line of best fit to predict the temperature increase by 2020 (160 years since 1860).
(b) (i) A model for the line of best fit is logT = mN + c.
Find the values of m and c.
Give your answers correct to 2 significant figures.
c=
m=
(ii) Use this model to predict the temperature increase by 2020.
(iii) Comment on your two predictions for the temperature increase by 2020.
© UCLES 2013
0607/06/O/N/13
12
BLANK PAGE
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.
University of 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 2013
0607/06/O/N/13