Cambridge International Examinations Cambridge ...maxpapers.com/wp-content/uploads/2012/11/0580_s14_qp_all.pdfCandidates answer on the Question Paper. Additional Materials: Electronic

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.You may use an HB pencil for any diagrams or graphs.Do not use staples, paper clips, glue or correction fl uid.DO NOT WRITE IN ANY BARCODES.

Answer all questions.If working is needed for any question it must be shown below that question.Electronic calculators should be used.If the degree of accuracy is not specifi ed in the question, and if the answer is not exact, give the answer to three signifi cant fi gures. Give answers in degrees to one decimal place.For π, use either your calculator value or 3.142.

At the end of the examination, fasten all your work securely together.The number of marks is given in brackets [ ] at the end of each question or part question.The total of the marks for this paper is 56.

MATHEMATICS 0580/17

Paper 1 (Core) May/June 2014

1 hour

Candidates answer on the Question Paper.

Additional Materials: Electronic calculator Geometrical instruments Tracing paper (optional)

Cambridge International ExaminationsCambridge International General Certifi cate of Secondary Education

This document consists of 11 printed pages and 1 blank page.

[Turn overIB14 06_0580_17/FP© UCLES 2014

*9769970262*

The syllabus is approved for use in England, Wales and Northern Ireland as a Cambridge International Level 1/Level 2 Certifi cate.

2

0580/17/M/J/14© UCLES 2014

1 Write down a factor of 21.

Answer ................................................ [1]__________________________________________________________________________________________

2 Write in fi gures the number four hundred and two thousand nine hundred and six.

Answer ................................................ [1]__________________________________________________________________________________________

3 Write down the mathematical name of this shape.

Answer ................................................ [1]__________________________________________________________________________________________

4 In a desert the noon temperature was 28 °C. At midnight the temperature was 33 °C lower than the noon temperature.

Find the temperature at midnight.

Answer ........................................... °C [1]__________________________________________________________________________________________

5 Work out the value of x.

76°x°

NOT TOSCALE

Answer x = ................................................ [1]__________________________________________________________________________________________

3

0580/17/M/J/14© UCLES 2014 [Turn over

6 Choose a symbol from the list below to make each statement correct.

= < >

(a) 118 ................... 72% [1]

(b) 0.004 ................... 4% [1]__________________________________________________________________________________________

7 (a) Write down the order of rotational symmetry of this shape.

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

(b) Draw the lines of symmetry on this shape.

[1]__________________________________________________________________________________________

8 Insert one pair of brackets into each of these calculations to make the answer correct.

(a) 6 + 14 ÷ 2 – 3 = 7 [1]

(b) 9 + 42 × 3 + 2 = 89 [1]__________________________________________________________________________________________

4

0580/17/M/J/14© UCLES 2014

9 A L G E B R A

(a) A letter is chosen at random from the list.

Find the probability that the letter chosen is A.

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

(b) A letter is chosen at random from the list and then replaced. This is done 63 times.

Work out the number of times the letter A is expected to be chosen.

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

10 During a football match a player ran 7.8 km, correct to 1 decimal place.

Complete the statement about the distance, d km, the player ran during the football match.

Answer ....................... Y d < ....................... [2]__________________________________________________________________________________________

11 Sara invests $600 at a rate of 4% per year compound interest.

Calculate the total amount Sara has after 2 years.

Answer $ ................................................. [2]__________________________________________________________________________________________

12 Calculate 5.32 2.153.27 0.84

-

# .

Give your answer correct to 4 signifi cant fi gures.

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

5


13A

C B36°

8.7 cm

NOT TOSCALE

Use trigonometry to calculate AC.

Answer AC = .......................................... cm [2]__________________________________________________________________________________________

14

x°

286°

NOT TOSCALE

The diagram shows an isosceles triangle.

Find the value of x.

Answer x = ................................................ [2]__________________________________________________________________________________________

6

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15 (a) Calculate 19% of $461.

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

(b) A computer costs $485. The cost is reduced by 24% in a sale.

Calculate the cost of the computer in the sale.

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

16 Solve the simultaneous equations. 3x – y = 10 x + 2y = 1

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

y = ................................................ [3]__________________________________________________________________________________________

7


17C

B A

10 cm

7 cm

NOT TOSCALE

Calculate the length of BC.

Answer BC = .......................................... cm [3]__________________________________________________________________________________________

18 Work out 81

32+` j ÷ 4

5 , giving your answer as a fraction.

Do not use a calculator and show all the steps of your working.

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

8

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19 Ilde leaves her home at 05 20 and drives to her friend’s house. Her average speed is 96 km/h. She arrives at her friend’s house at 09 05.

Calculate the distance she drives.

Answer .......................................... km [3]__________________________________________________________________________________________

20 p = 42e o q =

41

-e o r =

05-

e o

Find

(a) 3p,

Answer(a) f p [1]

(b) 2r – q.

Answer(b) f p [2]

__________________________________________________________________________________________

21 A cylinder has radius 6 cm and height 15 cm.

(a) Calculate the volume of the cylinder.

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

(b) Luigi has a jug containing 2 litres of water. He fi lls the empty cylinder with water from the jug.

How much water is left in the jug? Give the units of your answer.

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

9


22 (a) Here are the fi rst four terms in a sequence.

4 7 10 13

(i) Write down the next term in the sequence.

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

(ii) Work out the eighth term of the sequence.

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

(b) The n th term of a different sequence is 5n + 4.

Find the fi rst three terms of this sequence.

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

(c) Here are the fi rst four terms of another sequence.

–5 –1 3 7

Find the n th term of this sequence.

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

10

0580/17/M/J/14© UCLES 2014

23

70

60

50

40

30

20

10

10 20 30 40 50 60 700

Historyscore

Mathematics score

14 students take tests in mathematics and history. Their scores are plotted on the scatter diagram.

(a) Another 4 students take both tests. Their scores are shown in the table.

Mathematics score 30 61 17 37

History score 25 5 53 18

Plot these scores on the scatter diagram. [2]

11


(b) (i) On the scatter diagram, draw a line of best fi t. [1]

(ii) A different student scores 40 on the history test.

Use your line of best fi t to estimate a mathematics score for this student.

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

(iii) What type of correlation is shown on the scatter diagram?

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

12

0580/17/M/J/14© UCLES 2014

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

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.

BLANK PAGE





MATHEMATICS 0580/27

Paper 2 (Extended) May/June 2014

1 hour 30 minutes






*5174884275*


2

0580/27/M/J/14© UCLES 2014

1 In a desert the noon temperature was 28 °C. At midnight the temperature was 33 °C lower than the noon temperature.

Find the temperature at midnight.

Answer ........................................... °C [1]__________________________________________________________________________________________

2 A L G E B R A

(a) A letter is chosen at random from the list.

Find the probability that the letter chosen is A.

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

(b) A letter is chosen at random from the list and then replaced. This is done 63 times.

Work out the number of times the letter A is expected to be chosen.

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

3 During a football match a player ran 7.8 km, correct to 1 decimal place.

Complete the statement about the distance, d km, the player ran during the football match.

Answer ....................... Y d < ....................... [2]__________________________________________________________________________________________

4 Sara invests $600 at a rate of 4% per year compound interest.

Calculate the total amount Sara has after 2 years.

Answer $ ................................................. [2]__________________________________________________________________________________________

3


5 (a) Calculate 0.58115 233

3+ 4

c m.

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

(b) Write your answer to part (a) in standard form.

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

6 (a)A B

On the Venn diagram, shade the region A ∪ B'. [1]

(b)P Q

R

Use set notation to describe the region shaded on the Venn diagram.

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

7 w varies directly as v . When v = 9, w = 24.

Find w in terms of v.

Answer w = ................................................ [2]__________________________________________________________________________________________

4

0580/27/M/J/14© UCLES 2014

8 Solve the simultaneous equations. 3x – y = 10 x + 2y = 1

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

y = ................................................ [3]__________________________________________________________________________________________

9C

B A

10 cm

7 cm

NOT TOSCALE

Calculate the length of BC.

Answer BC = .......................................... cm [3]__________________________________________________________________________________________

10 Work out 81

32+` j ÷ 4

5 , giving your answer as a fraction.

Do not use a calculator and show all the steps of your working.

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

5


11 x = p2 – q2

(a) Find the value of x when p = 7 and q = 9.

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

(b) Make q the subject of the formula.

Answer(b) q = ................................................ [2]__________________________________________________________________________________________

12 A cone has a volume of 21 cm3 and a height of 4 cm.

Calculate the radius of the cone. [The volume, V, of a cone with radius r and height h is V = 3

1 πr 2h.]

Answer .......................................... cm [3]__________________________________________________________________________________________

13 (a) Simplify (3p3)4.

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

(b) (p2)n = 1p6

Find the value of n.

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

6

0580/27/M/J/14© UCLES 2014

14

OBA 10 cm

NOT TOSCALE

The diagram shows a shape made with two semicircles. AO = OB = 10 cm.

Calculate the perimeter of the shape.

Answer .......................................... cm [3]__________________________________________________________________________________________

15 Solve the equation. 1

2 3xx

+- = 2

1

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

7


16

AB

NOT TOSCALE

12 cm x cm

A and B are two similar pentagons. The area of A is 126 cm2 and the area of B is 56 cm2.

Calculate the value of x.

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

17 The scale of a map is 1: 20 000. On the map the area of a lake is 60 cm2.

Calculate the actual area of the lake, giving your answer in square kilometres.

Answer ......................................... km2 [3]__________________________________________________________________________________________

8

0580/27/M/J/14© UCLES 2014

18

O

124°C

BA

NOT TOSCALE

In the diagram, O is the centre of the circle which passes through A, B and C. OC is parallel to AB. Angle AOB = 124°.

Find

(a) angle BOC,

Answer(a) Angle BOC = ................................................ [2]

(b) angle OBC,

Answer(b) Angle OBC = ................................................ [1]

(c) angle CAB.

Answer(c) Angle CAB = ................................................ [1]__________________________________________________________________________________________

9


19 (a) Factorise completely.

(i) a2 – b2

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

(ii) 2a + 2b + 3ay + 3by

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

(b) Simplify 2 2 3 3a b

a b ay by2 2

+ + +-

.

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

20 The probability that a train arrives at station A on time is 107 .

If it is on time the probability that it arrives at station B on time is 87 .

If it is not on time the probability that it arrives at station B on time is 32 .

(a) Complete the tree diagram.

On time

Not on time

On time

Station A Station B

Not on time

Not on time

On time........

........

........

78

23

710

[1]

(b) Calculate the probability that the train arrives at station B on time.

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

10

0580/27/M/J/14© UCLES 2014

21 M = 23

14

f p

(a) Find M2.

Answer(a) f p [2]

(b) Find M–1.

Answer(b) f p [2]

__________________________________________________________________________________________

22 f(x) = 4 – 3x g(x) = x2 + 5

(a) Find fg(2).

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

(b) Find gf(x). Give your answer in its simplest form.

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

(c) Find x when f –1(x) = 3

2 .

Answer(c) x = ................................................ [1]__________________________________________________________________________________________

11


23

0

110

74

0.5 4.5 5Time (minutes)

Speed(km/h)

NOT TOSCALE

The diagram shows the speed-time graph of a car which slows down to pass through road works. The car slows down from a speed of 110 km/h to a speed of 74 km/h in 0.5 minutes. It then travels at a speed of 74 km/h for 4 minutes. The car then accelerates for 0.5 minutes to return to its speed of 110 km/h.

(a) Calculate the acceleration of the car between 4.5 and 5 minutes. Give your answer in m/s2.

Answer(a) ........................................ m/s2 [2]

(b) Calculate the total distance travelled by the car during the journey shown in the diagram. Give your answer in kilometres.

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

12

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MATHEMATICS 0580/37


2 hours




This document consists of 16 printed pages.


*1223789761*


2

0580/37/M/J/14© UCLES 2014

1 (a) Mr and Mrs Da Silva fl y from Manchester to Orlando. The plane takes off at 11 10 and arrives in Orlando 8 hours 20 minutes later. The time in Orlando is 5 hours behind the time in Manchester.

Work out the local time in Orlando when the plane arrives.

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

(b) Mr and Mrs Da Silva stay in a hotel for 16 nights. The cost of their room is $115 per night.

Work out the total cost.

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

(c) At the end of their holiday Mr Da Silva changes $862 into pounds (£) at a rate of £1 = $1.5972 .

(i) Calculate how many pounds he receives. Give your answer correct to the nearest pound.

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

(ii) Mr Da Silva invests £430 of this money for 3 years at a rate of 4% per year simple interest.

Calculate the total amount of money he has at the end of 3 years.

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

3


(d) On holiday Mr and Mrs Da Silva went to the places listed in the table. The total time spent in these places was 216 hours.

Activity Time (hours) Angle in pie chart

Shops 54 90°

Theme Park 160°

Water Park 48

Beach

(i) Complete the table. [3]

(ii) Complete the pie chart. Label each of the sectors.

Shops

Theme Park

[2]

(iii) Write down the percentage of time they spent in shops.

Answer(d)(iii) ............................................ % [1]__________________________________________________________________________________________

4

0580/37/M/J/14© UCLES 2014

2 Ricardo owns a restaurant.

(a) Ricardo has a piece of lamb of mass 4.5 kg. He cooks it for 20 minutes per kilogram and then for a further 20 minutes.

For how long does he cook the piece of lamb?

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

(b) Ricardo serves three different types of potatoes. Their masses are in the ratio mashed : roasted : boiled = 3 : 7 : 2 .

Find the mass of roasted potatoes when the total mass is 4.8 kg.

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

(c) On Wednesday evening there are 72 guests in the restaurant.

83 of them order vegetarian lasagne.

Work out how many guests order vegetarian lasagne.

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

5


(d) Ricardo serves two types of pizza. One is rectangular and the other is circular.

Pizza A Pizza B

28 cm

30 cm24 cm

NOT TOSCALE

Complete the statement below.

The area of Pizza .......... is larger than the area of Pizza .......... by ...................... cm2. [5]__________________________________________________________________________________________

6

0580/37/M/J/14© UCLES 2014

3

A

B

YX

W

10

9

8

7

6

5

4

3

2

1

–1

–2

–3

–4

–5

–6

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

y

(a) (i) Refl ect triangle A in the line x = –1. [2]

(ii) Rotate triangle A through 180° about (0, 0). [2]

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

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

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

7


(b) The squares on the grid each have area 1 cm2.

Work out the area of triangle B.

Answer(b) ......................................... cm2 [2]

(c) (i) Write down the co-ordinates of point W.

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

(ii) Plot point Z (2, 4). [1]

(iii) WXYZ is a quadrilateral.

Write down the mathematical name of this quadrilateral.

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

4 (a) Find

(i) 163,

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

(ii) 2.25 ,

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

(iii) 20.

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

(b) Write down a prime number between 50 and 60.

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

8

0580/37/M/J/14© UCLES 2014

5 Patrice records the number of goals scored by his football team in each of 20 matches.

2 4 8 3 9 2 11 8 9 0

1 2 5 0 4 2 1 2 3 7

(a) (i) Find the median.

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

(ii) Write down the mode.

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

(iii) Find the range.

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

(iv) Calculate the mean.

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

9


(b) Two football teams play the same number of matches. The mean number of goals scored by XR United is 4.5 and the range is 2. The mean number of goals scored by Pool City is 4.5 and the range is 8.

(i) What does the information tell you about the number of goals scored by each team?

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

(ii) What does the difference in the ranges tell you?

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

(c) The attendance at a football match was 75 546.

Write 75 546 correct to

(i) the nearest ten,

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

(ii) two signifi cant fi gures.

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

(d) Mikhail buys 4 child tickets at $c each. He also spends $152 on other tickets. Juan buys 9 child tickets at $c each. He also spends $86 on other tickets. Mikhail and Juan both pay the same total amount of money for their tickets.

Write an equation and solve it to calculate the value of c.

Answer(d) c = ................................................ [3]__________________________________________________________________________________________

10

0580/37/M/J/14© UCLES 2014

6 Cartons of drink are sold in boxes of 12. Each box contains,

• 5 blackcurrant drinks • 3 orange drinks • 2 lemon drinks • 2 peach drinks.

(a) (i) Zahira buys a box and chooses a drink at random. Write down the drink she is most likely to choose.

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

(ii) Draw an arrow on the probability scale to show the probability that she chooses an orange drink.

0 0.5 1[1]

(b) A shop buys 500 boxes. Work out the number of peach drinks.

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

(c) (i) A lorry delivers a total of 10 250 boxes.

Work out the number of drinks delivered. Write your answer in standard form.

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

(ii) The lorry travels 375 km in 7.5 hours.

Calculate the average speed of the lorry.

Answer(c)(ii) ....................................... km/h [1]

(d) Each box costs $8.75 . Carmen buys 5 boxes and pays with a $50 note.

Calculate how much change she receives.

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

11


7

53°

x°

A

BC

F

E

D

O

NOT TOSCALE

41°

A, B, C, D, E and F are points on the circumference of a circle centre O. AE is a diameter of the circle and AE is parallel to BD.

(a) Write down the mathematical name of the line

(i) OB,

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

(ii) BD.

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

(b) Find angle FAE.

Answer(b) Angle FAE = ................................................ [2]

(c) Find the value of x. Give a reason for your answer.

Answer(c) x = ................. because .....................................................................................................

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

12

0580/37/M/J/14© UCLES 2014

8 The scale drawing shows a farmyard ABCF and a fi eld CDEF. The scale is 1 centimetre represents 10 metres.

A

F C

B

E D

Scale: 1 cm to 10 m

Pond

13


(a) Find the actual length of BD.

Answer(a) BD = ............................................ m [1]

(b) Calculate the actual area of the farmyard.

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

(c) Horses are kept in the fi eld CDEF. The horses graze in a region

• more than 50 m from EF and • more than 20 m from the edge of the pond.

Construct these two loci on the scale drawing. Shade the region where the horses can graze. [5]

(d) Tarik walks across the farmyard from B to the fence CF. His path is equidistant from AB and BC.

Using a straight edge and compasses only construct his path. Show all your construction lines clearly. [2]

(e) Tarik buys 3 cows costing $495 each. He later sells the cows for a total of $2836.35 .

Work out the percentage profi t.

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

(f) The diameter of the pond is 20 m.

Calculate the circumference of the pond.

Answer(f) ............................................ m [2]__________________________________________________________________________________________

14

0580/37/M/J/14© UCLES 2014

9 (a) (i) Complete the table for y = x2 – 2x – 4.

x –3 –2 –1 0 1 2 3

y 4 –4 –4 –1[3]

(ii) On the grid, draw the graph of y = x2 – 2x – 4 for –3 Y x Y 3.

y

x

12

11

10

9

8

7

6

5

4

3

2

1

–1

–2

–3

–4

–5

–6

0–1–2–3 321

[4]

15


(iii) Use your graph to solve the equation x2 – 2x – 4 = –2.

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

(b)

L

4

3

2

1

–1

–2

–3

–4

–4 –3 –2 –1 10 2 3 4x

y

Write down the equation of a line which is parallel to line L.

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

Question 10 is printed on the next page.

16

0580/37/M/J/14© UCLES 2014



10 (a) Solve.

(i) 3x = 10.5

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

(ii) 4x – 3 = 17

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

(b) Simplify. 4p – 5p + 3p

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

(c) Factorise. 5x + 15y

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

(d) Expand the brackets and simplify.4(x – 2) – 3(x – 7)

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

(e) Make a the subject of the formula.3(a + b) = a + 2.

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





MATHEMATICS 0580/47


2 hours 30 minutes






*3434783675*


2

0580/47/M/J/14© UCLES 2014

1 Ahmed and Ali went on a fi shing trip. They left home at 06 45 and took 1 hour 50 minutes to travel the 88 km to their destination.

(a) (i) Write down the time they arrived at their destination.

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

(ii) Calculate the average speed of their journey.

Answer(a)(ii) ....................................... km/h [2]

(b) Ahmed caught 12 fi sh. The numbers of fi sh caught by Ahmed and Ali are in the ratio Ahmed : Ali = 6 : 7.

Find the number of fi sh that Ali caught.

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

(c) (i) The total mass of the fi sh Ahmed and Ali caught is 35 kg. The fi sh that Ahmed caught have a mass of 12.4 kg.

Calculate the mass of the fi sh Ahmed caught as a percentage of 35 kg.

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

(ii) The 12.4 kg is 20% less than the mass of fi sh Ahmed caught on a previous trip.

Calculate the mass of fi sh Ahmed caught on the previous trip.

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

(d) Ahmed and Ali left for home at 15 40. The average speed for their 88 km journey was 55 km/h.

Work out the time at which Ahmed and Ali arrived home.

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

3


2

T

W

y

x

10

8

6

4

2

–2

–4

–6

0–2 2 4 6 8–4–6–8

(a) On the grid, draw the image of

(i) triangle T after a translation by 2

8-e o, [2]

(ii) triangle T after a refl ection in the line y = –x. [2]

(b) Describe fully the single transformation represented by the matrix 1

0 10-

f p.

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

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

(c) (i) Describe fully the single transformation that maps triangle T onto triangle W.

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

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

(ii) Find the 2 × 2 matrix which represents the transformation that maps triangle T onto triangle W.

Answer(c)(ii) f p [2]

__________________________________________________________________________________________

4

0580/47/M/J/14© UCLES 2014

3T

B

A

C

24.5 m

29.5 m

11.6 m 53°

NOT TOSCALE

The diagram shows a vertical fl agpole TC. A, B and C are on horizontal ground. AC = 11.6 m, BC = 24.5 m and AB = 29.5 m. The angle of elevation of T from A is 53°.

(a) Calculate the angle of elevation of T from B.

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

5


(b) Calculate angle ACB and show that it rounds to 104°, correct to the nearest degree.

Answer(b)

[4]

(c) Calculate the area of triangle ABC.

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

(d) D is on CB so that angle CDA = 60°.

Calculate the length of AD.

Answer(d) AD = ............................................ m [3] __________________________________________________________________________________________

6

0580/47/M/J/14© UCLES 2014

4

10 cm

NOT TOSCALE

5 cm

7 cm

A solid sphere of radius 5 cm is placed inside a cylinder of radius 7 cm. A liquid is poured into the cylinder to a depth of 10 cm, as shown in the diagram.

(a) Calculate the volume of liquid in the cylinder and show that it rounds to 1016 cm3, correct to the nearest cubic centimetre.

[The volume, V, of a sphere with radius r is V = 34 πr

3.]

Answer(a)

[3]

(b) The sphere is made of metal and 1 cm3 of the metal has a mass of 7.85 g. 1 cm3 of the liquid has a mass of 0.85 g. The mass of the cylinder is 1.14 kilograms.

Calculate the total mass of the cylinder, the sphere and the liquid. Give your answer in kilograms.

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

7


(c) The sphere is removed from the cylinder.

Calculate the new depth of the liquid in the cylinder.

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

(d) The sphere is melted down and all the metal is used to make a cuboid with a square base of side 6.5 cm.

(i) Calculate the height, h, of the cuboid.

Answer(d)(i) h = .......................................... cm [2]

(ii)

NOT TOSCALE

h

The cuboid is placed inside the cylinder. More liquid is poured into the cylinder until the liquid just reaches the top of the cuboid.

Calculate the volume of liquid that must be added to the liquid already in the cylinder.

Answer(d)(ii) ......................................... cm 3 [3]__________________________________________________________________________________________

8

0580/47/M/J/14© UCLES 2014

5 y = x1 – x2, x ≠ 0.

(a) Complete the table of values for y.

x –2 –1.5 –1 –0.5 –0.3 0.3 0.5 1 1.5 2

y –2.9 –2.3 –3.4 3.2 1.8 0 –1.6[3]

(b) On the grid, draw the graph of y = x1 – x2 for –2 Y x Y –0.3 and 0.3 Y x Y 2.

y

x

4

3

2

1

–1

–2

–3

–4

–5

0–1–2 21

[5]

9


(c) (i) On the grid, draw the line y = x – 1. [1]

(ii) Complete the statement.

The line y = x – 1 is a ............................ to the graph of y = x1 – x2 at the point (.......... , ..........).

[2]

(d) (i) Complete the table of values for y = 2x2.

x –1 –0.5 0 0.5 1

y 2 0.5 0[1]

(ii) On the grid, draw the graph of y = 2x2 for –1 Y x Y 1. [2]

(iii) Use your graphs to solve the equation x1 – x2 = 2x2.

Answer(d)(iii) x = ................................................ [1]

(iv) The equation x1 – x2 = 2x2 can be simplifi ed to kx3 – 1 = 0.

Find the value of k.

Answer(d)(iv) k = ................................................ [2]__________________________________________________________________________________________

10

0580/47/M/J/14© UCLES 2014

6 100 students estimate the length, l metres, of a sports fi eld. The cumulative frequency diagram shows the results.

100

90

80

70

60

50

40

30

20

10

10 20 30 40 50 60 70 80

Length (metres)

90 100 110 120 130 1501400 l

Cumulativefrequency

(a) Find

(i) the median,

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

(ii) the inter-quartile range,

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

(iii) the number of students who give estimates of more than 80 m.

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

11


(b) (i) Use the cumulative frequency diagram to complete the frequency table.

Length(l metres) 30 < l Y 60 60 < l Y 80 80 < l Y 90 90 < l Y 100 100 < l Y 150

Frequency 18 20 20

[2]

(ii) Calculate an estimate of the mean value of l.

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

(iii) Use the frequency table in part (b)(i) to complete the frequency density table.

Length(l metres) 30 < l Y 80 80 < l Y 100 100 < l Y 150

Frequency density

[3]__________________________________________________________________________________________

12

0580/47/M/J/14© UCLES 2014

7 (a) In the diagrams below, all the lengths are measured in centimetres.

AB

x + 1 x – 1

2x + 1x + 1

NOT TOSCALE

The area of rectangle B is 8 cm2 more than the area of square A.

(i) Show that x2 – 3x – 10 = 0.

Answer(a)(i)

[3]

(ii) Factorise x2 – 3x – 10.

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

(iii) Find the perimeter of square A.

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

13


(b) (i) Lia cycles 20 km at an average speed of x km/h.

Write down an expression, in terms of x, for the time it takes Lia to complete the journey.

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

(ii) Lia cycles another 20 km at an average speed of (x + 1) km/h. This journey takes 4

1 hour less than the journey in part (b)(i).

Show that x2 + x – 80 = 0.

Answer(b)(ii)

[3]

(iii) Solve the equation x2 + x – 80 = 0. Show your working and give your answers correct to 2 decimal places.

Answer(b)(iii) x = ..................... or x = ..................... [4]

(iv) Find the total time taken by Lia to complete both journeys. Give your answer in hours and minutes correct to the nearest minute.

Answer(b)(iv) ................. h ................. min [2]__________________________________________________________________________________________

14

0580/47/M/J/14© UCLES 2014

8 In a class of 24 students,

16 students play football (F ), 12 students play hockey (H ), 3 students do not play either football or hockey.

[In this question you may use the Venn diagram to help you.]

(a) Work out how many students play

(i) football or hockey,

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

(ii) football and hockey.

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

(b) Find n (F' ∩ H).

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

(c) Two students from the class of 24 are chosen at random.

Find the probability that they both play football. Give your answer as a fraction in its lowest terms.

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

(d) A student who plays hockey is chosen at random.

Find the probability that this student plays football.

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

(e) A student who plays football or hockey is chosen at random.

Find the probability that this student plays football.

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

F H

15


9 (a) p = 2

3e o q =

12

5e o

Find

(i) p – 2q,

Answer(a)(i) f p [2]

(ii) the value of k when kp + q = 16e o.

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

(b)C

AO

Ec

a

NOT TOSCALE

In triangle OAC, = a and = c. E lies on AC so that AE : EC = 2 : 1.

Find the following, in terms of a and c, in their simplest form.

(i)

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

(ii)

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

(iii)

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


16

0580/47/M/J/14© UCLES 2014



10 (a) (i) Complete the table for the 5th term and the n th term of each sequence.

Term 1 2 3 4 5 n

Sequence A 3 8 13 18

Sequence B 1 3 9 27[6]

(ii) Find which term in sequence A is equal to 633.

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

(iii) Find the 9th term in sequence B. Give your answer in standard form.

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

(b) The fi rst four terms of a sequence are –1, 4, 11, 20.

The n th term of this sequence is n2 + pn + q.

(i) Find the values of p and q.

Answer(b)(i) p = ................................................

q = ................................................ [4]

(ii) Find the value of the 100th term of this sequence.

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





MATHEMATICS 0580/11


1 hour





[Turn overIB14 06_0580_11/2RP© UCLES 2014

*1477753275*


2

0580/11/M/J/14© UCLES 2014

1 Work out. 10 – 3 × 2

Answer ................................................ [1]__________________________________________________________________________________________

2 Write down the prime numbers between 20 and 30.

Answer ................................................ [1]__________________________________________________________________________________________

3

NOT TOSCALE

163°59° x°

(a) Find the value of x.

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

(b) One of the angles is 163°.

What type of angle is this?

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

4 A city has a population of fi ve hundred and six thousand.

Write the size of the population

(a) in fi gures,

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

(b) in standard form.

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

3


5 p = 16.834.8 1.98276#

(a) In the spaces provided, write each number in this calculation correct to 1 signifi cant fi gure.

Answer(a)............ × ............

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

(b) Use your answer to part (a) to estimate the value of p.

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

6 Solve the equation. 8n

2- = 11

Answer n = ................................................ [2]__________________________________________________________________________________________

7 a = 3-

4e o b =

15

-e o

Work out a – 2b.

Answer f p [2]

__________________________________________________________________________________________

8 The width, w cm, of a carpet is 455 cm, correct to the nearest centimetre.

Complete the statement about the value of w.

Answer ............................ Ğ w < ............................ [2]__________________________________________________________________________________________

4

0580/11/M/J/14© UCLES 2014

9 y = 2x

x2+2

2

Find the value of y when x = 6. Give your answer as a mixed number in its simplest form.

Answer y = ................................................ [2]__________________________________________________________________________________________

10 Use your calculator to work out 43 + 2–1.

Give your answer correct to 2 decimal places.

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

11 The diagram shows a cuboid.

8 cm

15 cmh

NOT TOSCALE

The volume of this cuboid is 720 cm3. The width is 8 cm and the length is 15 cm.

Calculate h, the height of the cuboid.

Answer h = .......................................... cm [2]__________________________________________________________________________________________

5


12 The scatter diagram shows the rainfall and the average temperature in a city for the month of June, over a period of 10 years.

30

25

20

15

10

5

0 5 10 15

Rainfall (cm)

Temperature (°C)

20 25 30

(a) What type of correlation does this scatter diagram show?

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

(b) Describe the relationship between the rainfall and the average temperature.

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

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

6

0580/11/M/J/14© UCLES 2014

13 The graph can be used to convert between miles and kilometres.

80

70

60

50

40

30

20

10

0 10 20 30

Miles

Kilometres

40 50

A train travels 24 miles in 20 minutes.

Find its average speed in kilometres per hour.

Answer ....................................... km/h [2]__________________________________________________________________________________________

7


14

127°

a°b°

A

D

E

BC

NOT TOSCALE

The diagram shows an isosceles triangle ABC. DCB is a straight line and is parallel to AE. Angle DCA = 127°.

Find the value of

(a) a,

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

(b) b.

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

15 Carlo changed 800 euros (€) into dollars for his holiday when the exchange rate was €1 = $1.50 . His holiday was then cancelled. He changed all his dollars back into euros and he received €750.

Find the new exchange rate.

Answer €1 = $ ................................................. [3]__________________________________________________________________________________________

8

0580/11/M/J/14© UCLES 2014

16 (a) Simplify the expressions.

(i) p 3 × p

7

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

(ii) t 5 ÷ t

8

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

(b) (h3)k = h12

Find the value of k.

Answer(b) k = ................................................ [1]__________________________________________________________________________________________

17

OP R

Q

17 cm9 cm

NOT TOSCALE

The diagram shows a circle, centre O. P, Q and R are points on the circumference. PQ = 17 cm and QR = 9 cm.

(a) Explain why angle PQR is 90°.

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

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

(b) Calculate the length PR.

Answer(b) PR = .......................................... cm [2]__________________________________________________________________________________________

9


18 In this question, do not use your calculator and show all the steps in your working.

(a) Show that 3 51 – 2 8

5 = 4023 .

Answer(a)

[2]

(b) Work out 87 ÷ 40

23 .

Give your answer as a mixed number in its simplest form.

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

19 The table shows the average monthly temperature (°C) for Fairbanks, Alaska.

Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Temperature (°C) –23.4 –19.8 –11.7 –0.8 9.2 15.4 16.9 13.8 7.5 –5.8 –21.4 –21.8

(a) Find

(i) the difference between the highest and the lowest temperatures,

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

(ii) the median.

Answer(a)(ii) ........................................... °C [2]

(b) A month is chosen at random from the table.

Find the probability that its average temperature is below zero.

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

10

0580/11/M/J/14© UCLES 2014

20 A bus company in Dubai has the following operating times.

Day Starting time

Finishing time

Saturday 06 00 24 00

Sunday 06 00 24 00

Monday 06 00 24 00

Tuesday 06 00 24 00

Wednesday 06 00 24 00

Thursday 06 00 24 00

Friday 13 00 24 00

(a) Calculate the total number of hours that the bus company operates in one week.

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

(b) Write the starting time on Friday in the 12-hour clock.

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

11


21

The diagram shows a circle inside a square. The circumference of the circle touches all four sides of the square.

(a) Calculate the area of the circle when the side of the square is 15 cm.

Answer(a) ......................................... cm2 [2]

(b) Draw all the lines of symmetry on the diagram. [2]__________________________________________________________________________________________


12

0580/11/M/J/14© UCLES 2014



22

B

C

A27 m

34 m

NorthNOT TOSCALE

In the diagram, B is 27 metres due east of A. C is 34 metres from A and due south of B.

(a) Using trigonometry, calculate angle ACB.

Answer(a) Angle ACB = ................................................ [2]

(b) Find the bearing of C from A.

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





MATHEMATICS 0580/12


1 hour





*5359060919*



2

0580/12/M/J/14© UCLES 2014

1 Simplify the expression. p + p + p + p

Answer ................................................ [1]__________________________________________________________________________________________

2 Calculate 216

1.3

3

.

Answer ................................................ [1]__________________________________________________________________________________________

3 Write down in fi gures

(a) three hundred and forty thousand,

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

(b) the number that is one less than one million.

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

4 Write the following numbers in order, starting with the smallest.

115 0.2 45.4% 20

9

Answer ...................... < ...................... < ...................... < ...................... [2]__________________________________________________________________________________________

3


5 (a) The temperature on Monday was –6°C. On Tuesday the temperature was 3 degrees lower.

Write down the temperature on Tuesday.

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

(b) The temperature on Saturday was –2°C. The temperature on Sunday was 8°C.

Write down the difference in these two temperatures.

Answer(b) ........................................... °C [1]__________________________________________________________________________________________

6 (a) Write 569 000 correct to 2 signifi cant fi gures.

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

(b) Write 569 000 in standard form.

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

7 Find three numbers which have a mode of 4 and a mean of 6.

Answer ...................... , ...................... , ...................... [2]__________________________________________________________________________________________

4

0580/12/M/J/14© UCLES 2014

8

lP

NOT TOSCALE

y

x0

The equation of the line l in the diagram is y = 5 – x .

(a) The line cuts the y-axis at P.

Write down the co-ordinates of P.

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

(b) Write down the gradient of the line l.

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

9 Solve the simultaneous equations. 2x – y = 7 3x + y = 3

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

y = ................................................ [2]__________________________________________________________________________________________

5


10C

B A

8 cm

28°

NOT TOSCALE

Calculate the length of AB.

Answer AB = .......................................... cm [2]__________________________________________________________________________________________

11 The height of Mount Everest is 8800 m, correct to the nearest hundred metres.

Complete the statement about the height, h metres, of Mount Everest.

Answer ......................... Ğ h < ......................... [2]__________________________________________________________________________________________

12 Colin is travelling from Sydney, Australia, to Auckland, New Zealand.

(a) Colin’s bus leaves for Sydney airport at 12 38. The bus arrives at the airport at 13 24.

How many minutes does the bus journey take?

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

(b) Colin’s fl ight from Sydney to Auckland leaves at 14 45 local time and takes 3 hours 20 minutes. The time in Auckland is 2 hours ahead of the time in Sydney.

What is the local time in Auckland when his fl ight arrives?

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

6

0580/12/M/J/14© UCLES 2014

13 (a) The scale drawing shows the positions of two villages, A and B. The scale is 1 centimetre represents 200 metres.

North

North

B

A Scale: 1 cm to 200 m

(i) Measure the bearing of B from A.

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

(ii) Work out the actual distance from A to B.

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

(b) The post box in Village A has a volume of 84 000 cm3. The post box in Village B has a volume of 0.1 m3.

Which post box has the greater volume? Show how you decide.

Answer(b) Post box in Village ............... [1]__________________________________________________________________________________________

7


14 V = 31 Ah

(a) Find V when A = 15 and h = 7 .

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

(b) Make h the subject of the formula.

Answer(b) h = ................................................ [2]__________________________________________________________________________________________

15 At the beginning of July, Kim had a mass of 63 kg. At the end of July, his mass was 61 kg.

Calculate the percentage loss in Kim’s mass.

Answer ............................................ % [3] __________________________________________________________________________________________

16 Without using your calculator, work out 65 – 2 2

1 11#` j.

Write down all the steps of your working.

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

8

0580/12/M/J/14© UCLES 2014

17 A plane is travelling at 180 metres per second.

How many minutes will it take the plane to travel 800 km? Give your answer correct to the nearest minute.

Answer ......................................... min [4]__________________________________________________________________________________________

18 (a) The probability that FC Victoria wins the cup is 0.18 .

Work out the probability that they do not win the cup.

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

(b) After training, the shirts are washed. There are 5 red, 3 blue and 6 green shirts. One shirt is taken from the washing machine at random.

Find the probability that it is

(i) red,

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

(ii) blue or green,

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

(iii) white.

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

9


19 similar acute line perpendicular radius

refl ex obtuse parallel congruent isosceles

Choose the correct word from this box to complete each of these statements.

(a)

Angle A is ..................................... [1]

(b)

Angle B is ..................................... [1]

(c)

These lines are ..................................... [1]

(d)

These lines are ..................................... [1]__________________________________________________________________________________________

A

B

10

0580/12/M/J/14© UCLES 2014

20

6.7 cm

NOT TOSCALE

Each edge of this cube is 6.7 cm long.

Work out

(a) the volume,

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

(b) the surface area.

Answer(b) ......................................... cm2 [2]__________________________________________________________________________________________

11


21

O63°

A

B

C

NOT TOSCALE

The diagram shows a circle, centre O with diameter AB = 15 cm. AC is a tangent to the circle at A and angle AOC = 63°.

(a) Calculate the area of the circle.

Answer(a) ......................................... cm2 [2]

(b) (i) Work out the size of angle ACO.

Answer(b)(i) Angle ACO = ................................................ [2]

(ii) Give one geometrical reason for your answer to part (b)(i).

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

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

12

0580/12/M/J/14© UCLES 2014



BLANK PAGE





MATHEMATICS 0580/13


1 hour




[Turn overIB14 06_0580_13/RP© UCLES 2014

*7662998175*



2

0580/13/M/J/14© UCLES 2014

1–3°C 8°C –19°C 42°C –7°C

Write down the lowest temperature from this list.

Answer ........................................... °C [1]__________________________________________________________________________________________

2 Change 6450 cm into metres.

Answer ............................................ m [1]__________________________________________________________________________________________

3

52°

x°

NOT TOSCALE

In the diagram, a straight line intersects two parallel lines.

Find the value of x.

Answer x = ................................................ [1]__________________________________________________________________________________________

4 Calculate.

0.256.2 34.8

-

-

Answer ................................................ [1]__________________________________________________________________________________________

5 Write down the value of 70.

Answer ................................................ [1]__________________________________________________________________________________________

3


6 Write 45 000 in standard form.

Answer ................................................ [1]__________________________________________________________________________________________

7 Four faces of a cube are drawn on the grid.

Complete the net of this cube.

[1]__________________________________________________________________________________________

8 Write down all the prime numbers that are greater than 30 and less than 40.

Answer ................................................ [1]__________________________________________________________________________________________

9 a =

43-

e o b = 26e o

Write each of the following as a single vector.

(a) 2a

Answer(a) f p [1]

(b) a – b

Answer(b) f p [1]

__________________________________________________________________________________________

4

0580/13/M/J/14© UCLES 2014

10 (a)1 4 8 12 27 40

Write down the number from this list which is both a cube number and has a factor of 4.

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

(b) 1258 is a multiple of 34.

Write down a different multiple of 34 between 1200 and 1300.

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

11–3 –5 1 0 3

Three different numbers from the list are added together to give the smallest possible total.

Complete the sum below.

................. + ................. + ................. = .................[2]

__________________________________________________________________________________________

12 The area of a square is 36 cm2.

Calculate the perimeter of this square.

Answer .......................................... cm [2]__________________________________________________________________________________________

13 The mean of fi ve numbers is 6. Four of the numbers are 3, 4, 5, and 10.

Work out the number that is missing from the list.

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

5


14 Find the value of 3a – 5b when a = –4 and b = 2 .

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

15 Celine buys a bag of 24 tulip bulbs. There are 8 red bulbs and 5 white bulbs. All of the other bulbs are yellow.

Celine chooses a bulb at random from the bag.

(a) Write down the probability that the bulb is red or white.

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

(b) Write down the probability that the bulb is yellow.

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

16 Find the fraction that is half-way between 21 and 3

2 .

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

6

0580/13/M/J/14© UCLES 2014

17 Using a straight edge and compasses only, construct the perpendicular bisector of AB. All construction arcs must be clearly shown.

A

B

[2]__________________________________________________________________________________________

18 Michelle sells ice cream. The table shows how many of the different fl avours she sells in one hour.

Flavour Vanilla Strawberry Chocolate Mango

Number sold 6 8 9 7

Michelle wants to show this information in a pie chart.

Calculate the sector angle for mango.

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

7


19 Chris changes $1350 into euros (€) when €1 = $1.313 .

Calculate how much he receives.

Answer € ................................................. [2]__________________________________________________________________________________________

20

A

y

x

7

6

5

4

3

2

1

–1

–2

–3

0–1 1 2 3 4 5–2–3–4–5–6–7

Draw the image of triangle A after a translation by the vector 43

-e o. [2]

__________________________________________________________________________________________

8

0580/13/M/J/14© UCLES 2014

21 Each exterior angle of a regular polygon is 30°.

Work out the number of sides the polygon has.

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

22

46°

74° 60°46°

x°

9.65 cm9.65 cm

8.69 cm

7.22 cmy cm

NOT TOSCALE

These two triangles are congruent. Write down the value of

(a) x,

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

(b) y.

Answer(b) y = ................................................ [1]__________________________________________________________________________________________

9


23 Without using a calculator, work out 1 41 – 9

7 .

Write down all the steps in your working.

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

24 Solve the simultaneous equations. 2x + 3y = 29 5x + y = 27

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

y = ................................................ [3]__________________________________________________________________________________________

10

0580/13/M/J/14© UCLES 2014

25

10 00 10 04 10 08 10 12 10 16

Time

Distance(km)

10 20 10 24 10 28 10 32

4

3

2

1

0

Town

Home

William Toby

Toby and William cycled into town. Their journeys are shown on the travel graph.

(a) For how many minutes did Toby stop on his journey into town?

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

(b) Explain what happened at 10 20.

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

(c) Work out how long William took to cycle into town.

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

(d) Calculate William’s speed in km/h.

Answer(d) ....................................... km/h [2]__________________________________________________________________________________________

11


26 (a) Factorise completely. 15a3 – 5ab

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

(b) Simplify. 3x2y3 × x4y

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

(c) Multiply out the brackets and simplify. 3(x – 2) – 4(2x – 3)

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

(d) Solve the equation. 8x + 9 = 3(x + 8)

Answer(d) x = ................................................ [3]__________________________________________________________________________________________

12

0580/13/M/J/14© UCLES 2014



BLANK PAGE





MATHEMATICS 0580/21


1 hour 30 minutes






*1585864014*


2

0580/21/M/J/14© UCLES 2014

1 Use your calculator to work out 43 + 2–1.

Give your answer correct to 2 decimal places.

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

2 y = 2x

x2+2

2

Find the value of y when x = 6. Give your answer as a mixed number in its simplest form.

Answer y = ................................................ [2]__________________________________________________________________________________________

3 Solve the equation. 8n

2- = 11

Answer n = ................................................ [2]__________________________________________________________________________________________

3


4 p = 16.834.8 1.98276#

(a) In the spaces provided, write each number in this calculation correct to 1 signifi cant fi gure.

Answer(a)............ × ............

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

(b) Use your answer to part (a) to estimate the value of p.

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

5 Write the following in order of size, smallest fi rst.

0.52 0.5 0.53 0.53

Answer .................. < .................. < .................. < .................. [2]__________________________________________________________________________________________

6 Carlo changed 800 euros (€) into dollars for his holiday when the exchange rate was €1 = $1.50 . His holiday was then cancelled. He changed all his dollars back into euros and he received €750.

Find the new exchange rate.

Answer €1 = $ ................................................. [3]__________________________________________________________________________________________

4

0580/21/M/J/14© UCLES 2014

7 Make x the subject of the formula. y = (x – 4)2 + 6

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

8 Write as a single fraction in its simplest form.

x2 – 1x +

2

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

5


9 A bus company in Dubai has the following operating times.

Day Starting time

Finishing time

Saturday 06 00 24 00

Sunday 06 00 24 00

Monday 06 00 24 00

Tuesday 06 00 24 00

Wednesday 06 00 24 00

Thursday 06 00 24 00

Friday 13 00 24 00

(a) Calculate the total number of hours that the bus company operates in one week.

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

(b) Write the starting time on Friday in the 12-hour clock.

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

6

0580/21/M/J/14© UCLES 2014

10 Factorise completely.

(a) ax + ay + bx + by

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

(b) 3(x – 1)2 + (x – 1)

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

11 A triangle has sides of length 2 cm, 8 cm and 9 cm.

Calculate the value of the largest angle in this triangle.

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

7


12 p = 4 × 105 q = 5 × 104

Find, giving your answer in standard form,

(a) pq,

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

(b) pq .

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

13

O

C

D

A

B58°

23°

NOT TOSCALE

A, B, C and D lie on a circle centre O. Angle ABC = 58° and angle CAD = 23°.

Calculate

(a) angle OCA,

Answer(a) Angle OCA = ................................................ [2]

(b) angle DCA.

Answer(b) Angle DCA = ................................................ [2]__________________________________________________________________________________________

8

0580/21/M/J/14© UCLES 2014

14

A(5, 10)

B(13, –2)

NOT TOSCALE

A(5, 10) and B(13, –2) are two points on the line AB. The perpendicular bisector of the line AB has gradient 3

2 .

Find the equation of the perpendicular bisector of AB.

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

9


15 Solve the inequality for positive integer values of x.

521 x+ > x + 1

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

16 (a) (224)1

2 = p4

Find the value of p.

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

(b) Simplify 2 2q q41

41

+

#q q .

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

10

0580/21/M/J/14© UCLES 2014

17

150°

ThailandHongKong

Malaysia

Singapore

NOT TOSCALE

A travel brochure has 72 holidays in four different countries. The pie chart shows this information.

(a) There are 24 holidays in Thailand.

Show that the sector angle for Thailand is 120°.

Answer(a)

[2]

(b) The sector angle for Malaysia is 150°. The sector angle for Singapore is twice the sector angle for Hong Kong.

Calculate the number of holidays in Hong Kong.

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

11


18

4 cm

10 cm

NOT TOSCALE

A solid cone has base radius 4 cm and height 10 cm. A mathematically similar cone is removed from the top as shown in the diagram. The volume of the cone that is removed is 8

1 of the volume of the original cone.

(a) Explain why the cone that is removed has radius 2 cm and height 5 cm.

Answer(a)

[2]

(b) Calculate the volume of the remaining solid.

[The volume, V, of a cone with radius r and height h is V = 31 πr

2h.]

Answer(b) ......................................... cm3 [4]__________________________________________________________________________________________


12

0580/21/M/J/14© UCLES 2014



19E

A

C

B

D

8 cm

8 cm30°

60°

NOT TOSCALE

The diagram shows a rectangle ABCE. D lies on EC. DAB is a sector of a circle radius 8 cm and sector angle 30°.

Calculate the area of the shaded region.

Answer ......................................... cm2 [7]





MATHEMATICS 0580/22


1 hour 30 minutes





*9522292004*



2

0580/22/M/J/14© UCLES 2014

1 Calculate 216

1.3

3

.

Answer ................................................ [1]__________________________________________________________________________________________

2 (a) Write 569 000 correct to 2 signifi cant fi gures.

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

(b) Write 569 000 in standard form.

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

3 Solve the simultaneous equations. 2x – y = 7 3x + y = 3

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

y = ................................................ [2]__________________________________________________________________________________________

3


4C

B A

8 cm

28°

NOT TOSCALE

Calculate the length of AB.

Answer AB = .......................................... cm [2]__________________________________________________________________________________________

5

lP

NOT TOSCALE

y

x0

The equation of the line l in the diagram is y = 5 – x .

(a) The line cuts the y-axis at P.

Write down the co-ordinates of P.

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

(b) Write down the gradient of the line l.

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

4

0580/22/M/J/14© UCLES 2014

6 The mass of 1 cm3 of copper is 8.5 grams, correct to 1 decimal place.

Complete the statement about the total mass, T grams, of 12 cm3 of copper.

Answer .............................. Y T < .............................. [2]__________________________________________________________________________________________

7 Write the following in order, smallest fi rst.

0.1 20143 2

21 % 0.2

Answer .................. < .................. < .................. < .................. [2]__________________________________________________________________________________________

8 Without using your calculator, work out 65 – 2 2

1 11#` j.

Write down all the steps of your working.

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

5


9 At the beginning of July, Kim had a mass of 63 kg. At the end of July, his mass was 61 kg.

Calculate the percentage loss in Kim’s mass.

Answer ............................................ % [3] __________________________________________________________________________________________

10 V = 31 Ah

(a) Find V when A = 15 and h = 7 .

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

(b) Make h the subject of the formula.

Answer(b) h = ................................................ [2]__________________________________________________________________________________________

6

0580/22/M/J/14© UCLES 2014

11 Anita buys a computer for $391 in a sale. The sale price is 15% less than the original price.

Calculate the original price of the computer.

Answer $ ................................................ [3]__________________________________________________________________________________________

12 Solve the equation.

23

11

x x+ + = 0

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

7


13 w varies inversely as the square root of x. When x = 4, w = 4.

Find w when x = 25.

Answer w = ................................................ [3]__________________________________________________________________________________________

14R

O P

Q

Mr

p

NOT TOSCALE

OPQR is a trapezium with RQ parallel to OP and RQ = 2OP. O is the origin, = p and = r. M is the midpoint of PQ.

Find, in terms of p and r, in its simplest form

(a) ,

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

(b) , the position vector of M.

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

8

0580/22/M/J/14© UCLES 2014

15 M = 43

25

e o

Find

(a) M2,

Answer(a) [2]

(b) the determinant of M.

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

16 Factorise completely.

(a) 4p2q – 6pq2

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

(b) u + 4t + ux + 4tx

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

9


17 (a) Simplify (3125t 125)

15 .

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

(b) Find the value of p when 3p = 91 .

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

(c) Find the value of w when x72 ÷ xw = x8.

Answer(c) w = ................................................ [1]__________________________________________________________________________________________

18

NOT TOSCALE

The two containers are mathematically similar in shape. The larger container has a volume of 3456 cm3 and a surface area of 1024 cm2. The smaller container has a volume of 1458 cm3.

Calculate the surface area of the smaller container.

Answer ......................................... cm2 [4]__________________________________________________________________________________________

10

0580/22/M/J/14© UCLES 2014

19 Simplify.

2

3 216 7

xx x

++ -

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

20 32 25 18 11 4

These are the fi rst 5 terms of a sequence.

Find

(a) the 6th term,

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

(b) the n th term,

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

(c) which term is equal to –332.

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

11


21P

D C

BA

M

6 cm

4 cm

4 cm

NOT TOSCALE

The diagram shows a pyramid on a square base ABCD with diagonals, AC and BD, of length 8 cm. AC and BD meet at M and the vertex, P, of the pyramid is vertically above M. The sloping edges of the pyramid are of length 6 cm.

Calculate

(a) the perpendicular height, PM, of the pyramid,

Answer(a) PM = .......................................... cm [3]

(b) the angle between a sloping edge and the base of the pyramid.

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


12

0580/22/M/J/14© UCLES 2014



22

P Q

k

m n

i

j

f

g

h

(a) Use the information in the Venn diagram to complete the following.

(i) P ∩ Q = {........................................................} [1]

(ii) P' ∪ Q = {........................................................} [1]

(iii) n(P ∪ Q)' = .................................................... [1]

(b) A letter is chosen at random from the set Q.

Find the probability that it is also in the set P.

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

(c) On the Venn diagram shade the region P' ∩ Q. [1]

(d) Use a set notation symbol to complete the statement.

{f, g, h} ........ P[1]





MATHEMATICS 0580/23


1 hour 30 minutes





*3753884750*



2

0580/23/M/J/14© UCLES 2014

1 In March 2011, the average temperature in Kiev was 3°C. In March 2012, the average temperature in Kiev was 19°C lower than in March 2011.

Write down the average temperature in Kiev in March 2012.

Answer ........................................... °C [1]__________________________________________________________________________________________

2 Michelle sells ice cream. The table shows how many of the different fl avours she sells in one hour.

Flavour Vanilla Strawberry Chocolate Mango

Number sold 6 8 9 7

Michelle wants to show this information in a pie chart.

Calculate the sector angle for mango.

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

3 Chris changes $1350 into euros (€) when €1 = $1.313 .

Calculate how much he receives.

Answer € ................................................. [2]__________________________________________________________________________________________

4 Factorise completely. 15a3 – 5ab

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

3


5 (a) Use your calculator to fi nd the value of 7.5–0.4 ÷ 57 . Write down your full calculator display.

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

(b) Write your answer to part (a) in standard form.

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

6 Simplify. 3x2y3 × x4y

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

7

46°

74° 60°46°

x°

9.65 cm9.65 cm

8.69 cm

7.22 cmy cm

NOT TOSCALE

These two triangles are congruent. Write down the value of

(a) x,

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

(b) y.

Answer(b) y = ................................................ [1]__________________________________________________________________________________________

4

0580/23/M/J/14© UCLES 2014

8 Hans draws a plan of a fi eld using a scale of 1 centimetre to represent 15 metres. The actual area of the fi eld is 10 800 m2.

Calculate the area of the fi eld on the plan.

Answer ......................................... cm2 [2]__________________________________________________________________________________________

9 Solve the inequality. 5t + 23 < 17 – 2t

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

10 Without using a calculator, work out 1 41 – 9

7 .

Write down all the steps in your working.

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

5


11 y varies as the cube root of (x + 3). When x = 5, y = 1.

Find the value of y when x = 340.

Answer y = ................................................ [3]__________________________________________________________________________________________

12 (a) Factorise 3x2 + 2x – 8.

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

(b) Solve the equation 3x2 + 2x – 8 = 0.

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

13 Find the equation of the line passing through the points with co-ordinates (5, 9) and (–3, 13).

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

6

0580/23/M/J/14© UCLES 2014

14

66°

77°37°

P

RQ 12.5 cm

NOT TOSCALE

Calculate PR.

Answer PR = .......................................... cm [3]__________________________________________________________________________________________

15 A rectangle has length 127.3 cm and width 86.5 cm, both correct to 1 decimal place.

Calculate the upper bound and the lower bound for the perimeter of the rectangle.

Answer Upper bound = .......................................... cm

Lower bound = .......................................... cm [3]__________________________________________________________________________________________

7


16H G

C

BA

E F

D3 cm

4 cm

12 cm

NOT TOSCALE

ABCDEFGH is a cuboid. AB = 4 cm, BC = 3 cm and AG = 12 cm.

Calculate the angle that AG makes with the base ABCD.

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

8

0580/23/M/J/14© UCLES 2014

17 = {x : 1 Y x Y 10, where x is an integer}

A = {square numbers}

B = {1, 2, 3, 4, 5, 6}

(a) Write all the elements of in their correct place in the Venn diagram.

A B

[2]

(b) List the elements of (A ∪ B)'.

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

(c) Find n(A ∩ B' ).

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

9


18 A =

54

23

e o

(a) Calculate A2.

Answer(a) [2]

(b) Calculate A–1, the inverse of A.

Answer(b) [2]__________________________________________________________________________________________

19 Robbie pays $10.80 when he buys 3 notebooks and 4 pencils. Paniz pays $14.50 when she buys 5 notebooks and 2 pencils.

Write down simultaneous equations and use them to fi nd the cost of a notebook and the cost of a pencil.

Answer Cost of a notebook = $ .................................................

Cost of a pencil = $ ................................................. [5]__________________________________________________________________________________________

10

0580/23/M/J/14© UCLES 2014

20 Jenna draws a cumulative frequency diagram to show information about the scores of 500 people in a quiz.

500

400

300

200

100

100

20 30

Score

Cumulativefrequency

40 50 60

Use the diagram to fi nd

(a) the median score,

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

(b) the inter-quartile range,

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

(c) the 40th percentile,

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

(d) the number of people who scored 30 or less but more than 20.

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

11


21

NOT TOSCALE

The diagram shows two concentric circles and three radii. The diagram has rotational symmetry of order 3.

A club uses the diagram for its badge with some sections shaded. The radius of the large circle is 6 cm and the radius of the small circle is 4 cm.

NOT TOSCALE

Calculate the total perimeter of the shaded area.

Answer .......................................... cm [5]__________________________________________________________________________________________


12

0580/23/M/J/14© UCLES 2014



22

A

B

y

x

7

6

5

4

3

2

1

–1

–2

–3

–4

–5

–6

0–1 1 2 3 4 5 6 7–2–3–4–5–6–7

(a) Draw the image of triangle A after a translation by the vector 43

-e o. [2]

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

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

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

(c) Draw the image of triangle A after the transformation represented by the matrix 2 0-

10e o. [3]





MATHEMATICS 0580/31


2 hours






*0224327052*


2

0580/31/M/J/14© UCLES 2014

1 (a) The angles in a triangle are in the ratio 3 : 4 : 8 .

(i) Show that the smallest angle of the triangle is 36°.

Answer(a)(i)

[2]

(ii) Work out the other two angles of the triangle.

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

(b) Another triangle ABC has angle BAC = 35° and angle ABC = 65°.

(i) Using a protractor and straight edge complete an accurate drawing of the triangle ABC. The side AB has been drawn for you.

A B[2]

(ii) Measure the length, in centimetres, of the shortest side of your triangle.

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

(c) A different triangle has base 7.0 cm and height 5.6 cm. Calculate the area of this triangle, giving the units of your answer.

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

3


2 (a) From the integers 50 to 100, fi nd

(i) a multiple of 43,

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

(ii) a factor of 165,

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

(iii) an odd number that is also a square number,

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

(iv) a number which is a square number and also a cube number.

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

(b) (i) Find the square root of 5929.

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

(ii) Find the lowest common multiple of 24 and 30.

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

(c) Elena goes on a journey to the North Pole. She leaves home at 7 am on 15 July and arrives at the North Pole at 10 pm on 27 July.

How long, in days and hours, did her journey take?

Answer(c) ....................... days ....................... hours [2]__________________________________________________________________________________________

4

0580/31/M/J/14© UCLES 2014

3

S

P

T

y

x–2 20 4 6 81 3 5 7–4–6–8 –1–3–5–7

10

9

8

7

6

5

4

3

2

1

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10

The diagram shows two shapes, S and T, on a 1 cm2 grid. P is the point (–2, 0).

5


(a) (i) Write down the mathematical name of shape S.

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

(ii) How many lines of symmetry does shape S have?

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

(b) Describe the single transformation that maps shape S onto shape T.

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

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

(c) On the grid,

(i) draw the refl ection of shape S in the y-axis, [2]

(ii) draw the rotation of shape S about (0, 0) through 90° anti-clockwise. [2]

(d) On the grid, draw the enlargement of shape S with scale factor 2 and centre P (–2, 0). Label the image E. [2]

(e) (i) Work out the area of shape S.

Answer(e)(i) ......................................... cm2 [2]

(ii) How many shapes, identical to shape S, will fi ll shape E completely?

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

(iii) Work out the area of shape E.

Answer(e)(iii) ......................................... cm2 [1]__________________________________________________________________________________________

6

0580/31/M/J/14© UCLES 2014

4 Denzil grows tomatoes. He selects a random sample of 25 tomatoes. The mass of each tomato, to the nearest 5 grams, is shown below.

55 65 50 75 6580 70 70 55 6070 60 65 50 7565 70 75 80 7055 65 70 80 55

(a) (i) Complete the frequency table. You may use the tally column to help you.

Mass(grams) Tally Frequency

50

55

60

65

70

75

80

[2]

(ii) Write down the mode.

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


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

(iv) Show that the mean mass is 66 g.

Answer(a)(iv)

[2]

7


(b) Denzil picks 800 tomatoes. 4% of the 800 tomatoes are damaged.

How many of these tomatoes are not damaged?

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

(c) Denzil sells 750 of his tomatoes.

(i) The mean mass of a tomato is 66 g.

Calculate the mass of the 750 tomatoes in kilograms.

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

(ii) Denzil sells his tomatoes at $1.40 per kilogram.

Calculate the total amount he receives from selling all the 750 tomatoes.

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

(iii) The cost of growing these tomatoes was $33.

Calculate his percentage profi t.

Answer(c)(iii) ............................................ % [3]__________________________________________________________________________________________

8

0580/31/M/J/14© UCLES 2014

5 Use a ruler and compasses only in parts (a), (c) and (d) of this question. Show all your construction arcs.

A

B

C

D

E

P

100 m

100 m

120 m

150 m

Scale: 1 cm to 20 m

Maria owns a farm. The scale drawing shows part of the boundary of the farm. The scale is 1 centimetre represents 20 metres.

9


(a) The point F is such that AF = 140 m and EF = 160 m. Angle BAF and angle DEF are both obtuse angles.

Complete the scale drawing of the farm boundary ABCDEF. [2]

(b) Write down the name of the polygon ABCDEF.

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

(c) (i) Construct the perpendicular bisector of the side CD. [2]

(ii) Construct the bisector of angle ABC. [2]

(iii) All the farm buildings are within a region that is

● nearer to C than to D and ● nearer to BC than to BA.

Shade the region containing the farm buildings. [1]

(d) A fence post, P, is shown on the boundary DE.

(i) Construct the locus of points that are 50 m from P and also inside the farm boundary. [2]

(ii) A region for keeping pigs is within 50 m of P and inside the farm boundary.

Calculate the actual area for keeping pigs.

Answer(d)(ii) ........................................... m2 [2]__________________________________________________________________________________________

10

0580/31/M/J/14© UCLES 2014

6 (a) (i) Complete the table of values for y = x8 , x ≠ 0 .

x –8 –4 –2 –1 1 2 4 8

y –2 2

[3]

(ii) On the grid, draw the graph of y = x8 for –8 Ğ x Ğ –1 and 1 Ğ x Ğ 8 .

y

x

8

6

4

2

–2

–4

–6

–8

0–2–4–6–8 6 842

[4]

11


(iii) Write down the order of rotational symmetry of your graph.

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

(b) (i) Complete this table of values for y = 1.5x + 3 .

x –6 –4 –2 0 2

y –6 3

[2]

(ii) On the grid, draw the graph of y = 1.5x + 3 . [1]

(c) Use your graphs to solve the equation x8 = 1.5x + 3 .

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

(d) Write down the gradient of the graph of y = 1.5x + 3 .

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

12

0580/31/M/J/14© UCLES 2014

7 120 people are asked how they travel to work. The pie chart shows the results.

Bus

Car

Cycle

Walk

(a) (i) Show that 45 people travel by car.

Answer(a)(i)

[2]

(ii) A person is chosen at random from the 120 people.

Find the probability that this person travels to work by bus or by car.

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

13


(b) One year later, the same 120 people were again asked how they travel to work.

Here is the information.

Number of people

Walk x

Cycle 31

Bus 17 more than the number of people who walk

Car 2 times the number of people who walk

(i) Use this information to complete the following equation, in terms of x.

............................................................................................. = 120 [3]

(ii) Solve the equation to fi nd the number of people who walk to work.

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

14

0580/31/M/J/14© UCLES 2014

8 (a) Write down an expression for the total mass of c cricket balls, each weighing 160 grams, and f footballs, each weighing 400 grams.

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

(b) Expand and simplify. 3(2x – 5y) – 4(x – 2y)

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

(c) Factorise completely. 5x2y – 20x

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

(d) Solve the simultaneous equations. 3x + 4y = 7 4x – 3y = 26

Answer(d) x = ................................................

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

15


9 (a) For these sequences, write down the next two terms and the rule for fi nding the next term.

(i) 84, 75, 66, 57, . . .

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

(ii) 2, 6, 18, 54, . . .

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

(b) For the sequence in part (a)(i),

(i) write down an expression, in terms of n, for the n th term,

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

(ii) fi nd the 21st term.

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

16

0580/31/M/J/14© UCLES 2014



BLANK PAGE





MATHEMATICS 0580/32


2 hours




[Turn overIB14 06_0580_32/RP© UCLES 2014

*4942783219*



2

0580/32/M/J/14© UCLES 2014

1 (a) Here is a list of numbers.

2 4 5 8 9 12

Write down all the numbers from this list which are

(i) odd,

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

(ii) square,

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

(iii) cube,

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

(iv) prime.

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

(b) Write one of these symbols >, < or = to make each statement true.

π .................... 722

2^ h2 .................... 2

1 11+ .................... 2

(–1)2 .................... –1[2]

(c) Put one pair of brackets in each statement to make it true.

(i) 16 + 8 ÷ 4 – 2 = 4 [1]

(ii) 16 + 8 ÷ 4 – 2 = 20 [1]

3


(d) (i) Write 84 as a product of its prime factors.

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

(ii) Find the highest common factor of 84 and 24.

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

(iii) Find the lowest common multiple of 84 and 24.

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

(e) Here are the fi rst four terms of a sequence.

3 7 11 15

(i) Write down the next term in this sequence.

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

(ii) Explain how you found your answer.

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

(iii) Write down an expression for the n th term of this sequence.

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

(iv) Explain why 125 is not in this sequence.

Answer(e)(iv) ..............................................................................................................................

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

4

0580/32/M/J/14© UCLES 2014

2

A

D C

B

180 cm

120 cm

240 cm

NOT TOSCALE

The diagram shows the cross section ABCD of a shed. AD = 180 cm, DC = 120 cm and BC = 240 cm.

(a) (i) Write down the mathematical name of the cross section ABCD.

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

(ii) Calculate the area of the cross section ABCD. Give the units of your answer.

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

(iii) The shed is a prism of length 2.5 metres.

Calculate the volume of the shed. Give your answer in cubic metres.

Answer(a)(iii) ........................................... m3 [2]

5


(iv) Calculate the length AB.

Answer(a)(iv) AB = .......................................... cm [3]

(b) Here is a scale drawing of a garden, GHIJ. The scale is 1 centimetre represents 5 metres.

I

H

G J

Scale: 1 cm to 5 m

The shed is placed in the garden so that it is

● nearer to GJ than to IJ and ● within 20 m of H.

Using a ruler and compasses only, construct and shade the region where the shed can be placed. Show all your construction arcs. [5]__________________________________________________________________________________________

6

0580/32/M/J/14© UCLES 2014

3 (a) Draw the line of symmetry on the shape below.

[1]

(b) Write down the order of rotational symmetry of the shape below.

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

(c) (i)

x°157°

72° NOT TOSCALE

Work out the value of x.

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

(ii)

y°

49°

54°

NOT TOSCALE

Work out the value of y.

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

7


(d)A

B C

O34°

NOT TOSCALE

AC is a diameter of the circle, centre O.

Calculate angle ACB.

Answer(d) Angle ACB = ................................................ [2]

(e) The diagram below shows parts of shape P and shape Q. Shape P is a regular hexagon and shape Q is another regular polygon. The two shapes have one side in common.

100°

100°

QP

NOT TOSCALE

Find the number of sides in shape Q. Show each step of your working.

Answer(e) ................................................ [5]__________________________________________________________________________________________

8

0580/32/M/J/14© UCLES 2014

4 Paolo’s football team played 46 games. The pictogram shows some information about the number of goals scored by Paolo’s football team. They did not score any goals in fi ve games.

Numberof goals Number of games

0

1

2

3

4

5

6

Key: = .................. games

(a) (i) Complete the key. [1]

(ii) Paolo’s team scored 2 goals in each of nine games.

Complete the pictogram. [1]

(b) (i) Write down the modal number of goals.

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

(ii) Find the median number of goals.

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


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

(iv) One of the 46 games is chosen at random.

Work out the probability that Paolo’s team scored at least 4 goals.

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

9


(c) The table shows the total goals scored and the total points gained by 10 teams.

Team A B C D E F G H I J

Goals 31 40 46 50 43 92 60 84 68 87

Points 36 35 52 56 72 78 59 70 61 75

(i) Complete the scatter diagram. The fi rst six points have been plotted for you. [2]

80

70

60

50

40

3030 40 50 60 70

Goals

80 90 100

Points

(ii) Draw the line of best fi t. [1]

(iii) What type of correlation is shown?

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

(iv) Use your line of best fi t to estimate the total points gained by a team scoring 75 goals.

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

(v) Which team only scores a few goals but gains a lot of points?

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

10

0580/32/M/J/14© UCLES 2014

5 (a) Jasmine works for 38 hours each week and she earns $12.15 each hour.

(i) Calculate her earnings in one week.

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

(ii) Jasmine pays 14% of her earnings in tax.

Calculate how much money she has left after tax is paid.

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

(iii) She pays 31 of the money she has left after tax in rent.

Calculate how much rent she pays in one year (52 weeks).

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

(iv) In one week she spends $140 on food and electricity in the ratio

food : electricity = 3 : 2 .

Calculate how much she spends on food.

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

(b) Jasmine buys a watch for 10 000 Japanese Yen (¥). The exchange rate is $1 = ¥ 80.4 .

Calculate the cost of this watch in dollars, giving your answer correct to the nearest dollar.

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

11


6 (a) Complete the table of values for y = x2 + 2x – 3 .

x –4 –3 –2 –1 0 1 2 3 4

y 0 –3 –4 –3 0 5 21[2]

(b) On the grid, draw the graph of y = x2 + 2x – 3 for –4 Ğ x Ğ 4 .

y

x

25

20

15

10

5

–5

0 1 2 3 4–1–2–3–4

[4]

(c) On the grid, draw the line y = 10 . [1]

(d) Use your graphs to solve the equation x2 + 2x – 3 = 10 for –4 Y x Y 4 .

Answer(d) x = ................................................ [1]__________________________________________________________________________________________

12

0580/32/M/J/14© UCLES 2014

7 (a)

5p + 3r7p – 6r

p + 2r

NOT TOSCALE

Write an expression for the perimeter of this triangle. Give your answer in its simplest form.

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

(b) Another triangle has a perimeter 12w – 2z .

Calculate this perimeter when w = 16 and z = –3.

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

(c) Solve.

(i) 5a = 32

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

(ii) 5b + 23 = 8

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

(iii) 5c + 7 = 2(c – 10)

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

13


(d) (i) Multiply out the brackets. 8(2x + 3)

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

(ii) Factorise completely. 6x2 – 12x

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

(e) Write each expression in its simplest form.

(i) 3q4 × 5q2

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

(ii) t 8 ÷ t

2

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

14

0580/32/M/J/14© UCLES 2014

8 (a) Work out.

(i) 5 3-

2e o


(ii) 5

4

-e o +

1

3

-e o

Answer(a)(ii) f p [1]

(b) A translation moves the point (6, 3) to the point (2, 8).

Work out the vector which represents this translation.

Answer(b) f p [1]

15


(c) A point P is translated by the vector 3

4-e o to the point (7, –2).

Find the co-ordinates of P. You may use the grid below to help you.

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

__________________________________________________________________________________________


16

0580/32/M/J/14© UCLES 2014



9

A

10

9

8

7

6

5

4

3

2

1

–1

–2

–3

–4

–5

–6–7 –5 –4 –3 –2 –1 10 2 3 4 5 6 7x

y

B

(a) On the grid, draw the image of triangle A after the following transformations.

(i) Refl ection in the x-axis. [1]

(ii) Rotation about (0, 0) through 180°. [2]

(iii) Translation by the vector 5-

3e o. [2]

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

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

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





MATHEMATICS 0580/33


2 hours





*9994985227*



2

0580/33/M/J/14© UCLES 2014

1 (a)

D

A

B

C

y

x

7

6

5

4

3

2

1

–1

–2

–3

–4

0–1 1 2 3 4 5 6–2–3–4

Four shapes, A, B, C and D, are shown on the grid.

Describe fully the single transformation that maps shape A onto

(i) shape B,

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

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

(ii) shape C,

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

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

(iii) shape D.

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

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

3


(b) (i)

Shade in one more square so that this shape has rotational symmetry of order 2. [1]

(ii)

Refl ect this shape in the line of symmetry shown. [2]__________________________________________________________________________________________

4

0580/33/M/J/14© UCLES 2014

2 A group of students take part in their school’s sports day.

(a) (i) The length, l m, that Anna throws the javelin is 23.6 metres correct to the nearest 10 centimetres.

Complete the statement about l.

Answer(a)(i) .......................... Y l < .......................... [2]

(ii) Billy throws the hammer a distance of 8 metres on his fi rst throw. His second throw is 15% further.

Calculate the distance of his second throw.

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

(iii) Carl runs 100 metres at a speed of 8 m/s.

Calculate the time it takes him to run 100 m.

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

(iv) Change Carl’s speed of 8 m/s into km/h.

Answer(a)(iv) ....................................... km/h [2]

(b) Ten students take part in both the long jump and 100 m hurdles competitions. The results are shown in the table below.

Student A B C D E F G H I J

Distance in long jump (metres) 3.25 3.60 3.75 3.90 4.10 4.20 4.30 4.40 4.65 4.70

Time for 100 m hurdles (seconds) 17.3 17.4 16.7 16.1 16.5 15.8 15.3 14.8 15.5 15.0

5


(i) Complete the scatter diagram. The fi rst six points have been plotted for you.

18.0

17.0

16.0

15.0

14.03.5 4.0 4.5 5.0 5.53.0

Distance in long jump (metres)

Time for100 m hurdles(seconds)

[2]

(ii) What type of correlation does this scatter diagram show?

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

(iii) Describe the relationship between the distance in the long jump and the time for the 100 m hurdles.

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

(iv) On the grid, draw the line of best fi t. [1]

(v) Another student jumps 3.50 m in the long jump.

Use your line of best fi t to estimate the time for this student in the 100 m hurdles.

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

(vi) A different student jumps 5.20 m in the long jump.

Explain why you should not use your scatter diagram to estimate their time in the 100 m hurdles.

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

6

0580/33/M/J/14© UCLES 2014

3 The Wong family spend the day at the zoo.

(a) The Wong family has 2 adults and 3 children aged 2, 5 and 11 years old.

Admission

Adults $8.50Children 11-16 years $6.00Children 3-10 years $4.50Children under 3 years FREE

Mr Wong pays for his family to go into the zoo using a $50 note.

Work out the change he receives.

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

(b) The dolphin show fi nishes at 11 05. It lasts for 1 hour and 20 minutes.

Write down the time the dolphin show starts.

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

(c) Torty the tortoise was born on 27 December 1898.

Work out how many years old she was on 3 January 2003.

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

(d) Last year, the ratio snakes : lizards = 3 : 5 . There were 45 lizards.

(i) Work out how many snakes there were last year.

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

(ii) This year, there are 3 more snakes and the same number of lizards.

Write down the new ratio snakes : lizards. Give your answer in its simplest form.

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

(e) Mr Wong hires a vehicle to drive around the zoo. The cost is $25 for the fi rst hour and $7.50 for every extra half hour. He pays $85 altogether.

For how long does he hire the vehicle?

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

7


(f) Mrs Wong wants to buy some food for the giraffes.

Small Bag

225g

60 cents

Medium Bag

250g

70 cents

Large Bag

325g

90 cents

Work out which bag is the best value for money. Show how you decide.

Answer(f) ................................................ [3]

(g) The diagram shows a map of the zoo. The scale is 1 centimetre represents 50 metres.

North

Entrance

Flamingos

North

Exit

Scale: 1 cm to 50 m

(i) Measure the bearing of the fl amingos from the entrance.

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

(ii) Xanthe looks after all the animals within 200 m of the exit.

Draw accurately the locus of points inside the zoo which are 200 m from the exit. [2]

(iii) A shop, S, is on a bearing of 212° from the entrance and a bearing of 293° from the exit.

Mark the point S on the map. [3]__________________________________________________________________________________________

8

0580/33/M/J/14© UCLES 2014

4 The ages of 15 children who go to a swimming club are shown below.

10 11 10 12 12 13 11 12 12 12 12 10 11 11 11

(a) Complete the frequency table. You may use the tally column to help you.

Age Tally Frequency

10

11

12

13[2]

(b) For the ages of the 15 children, fi nd

(i) the range,

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

(ii) the mode,

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

(iii) the median,

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

(iv) the mean.

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

(c) One child is chosen at random from the group.

Write down the probability that the child’s age is

(i) 10,

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

(ii) more than 13.

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

9


5 (a) (i) Write down the name of a solid which is not a prism.

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

(ii) A prism has a cross-sectional area, A, and height, h.

Write down an expression, in terms of A and h, for the volume of the prism.

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

(b) The volume, V, of a cylinder with radius r and height h is V = πr2h .

(i) Calculate the volume of a cylinder with radius 3 cm and height 12 cm.

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

(ii) Ravi puts 150 identical marbles in the cylinder. He fi lls the cylinder to the top with 160 cm3 of water.

Find the volume of one marble. Give your answer correct to 2 signifi cant fi gures.

Answer(b)(ii) ......................................... cm3 [4]

(iii) Make r the subject of the formula V = πr2h .

Answer(b)(iii) r = ................................................ [2]__________________________________________________________________________________________

10

0580/33/M/J/14© UCLES 2014

6y

x

6

5

4

3

2

1

–1

–2

–3

–4

–5

–6

0–1 1 2 3 4 5 6–2–3–4–5–6

(a) On the grid, draw the graphs of

(i) y = 5, [1]

(ii) x = –3. [1]

(b) (i) Write down the co-ordinates of the point of intersection of y = 5 and x = –3.

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

(ii) Write down the equation of a line parallel to y = 5.

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

11


(c) (i) Complete the table of values for the function y = x2 – 3x .

x –2 –1 0 1 2 3 4 5

y 4 0 0 4[2]

(ii) On the grid, draw the graph of y = x2 – 3x for –2 Y x Y 5 .

y

x

11

10

9

8

7

6

5

4

3

2

1

–1

–2

–3

–4

0–1 1 2 3 4 5 6–2–3

[4]

(iii) Write down the co-ordinates of the lowest point of the graph.

Answer(c)(iii) (...................... , ......................) [1]__________________________________________________________________________________________

12

0580/33/M/J/14© UCLES 2014

7 Today it is Simon’s birthday.

(a) Simon is x years old. Katy is twice as old as Simon. Bob is 8 years younger than Simon.

(i) Write expressions, in terms of x, for the ages of Katy and Bob.

Answer(a)(i) Katy ................................................

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

(ii) The sum of their three ages is 40 years.

Write an equation in terms of x.

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

(iii) Solve your equation for x.

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

(b) Simon’s birthday cake weighs 600 grams.

He eats 81 of the cake.

Katy eats 25% of the cake. Bob eats 0.3 of the cake.

Find the weight of the cake that is left.

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

13


(c) Aunty Millie gives Simon $150 for his birthday. He invests the money in a bank at a rate of 6% per year compound interest.

Calculate the total amount Simon will have after 3 years.

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

(d) One of Simon’s presents is a bag of sweets. He decides to eat the sweets in a sequence. On day 1 he eats 1 sweet, on day 2 he eats 5 sweets, on day 3 he eats 9 sweets and so on.

(i) Describe in words the rule for continuing the sequence 1, 5, 9, 13, 17 ..... .

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

(ii) Write down an expression for the number of sweets he eats on day n.

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

14

0580/33/M/J/14© UCLES 2014

8 (a)

h

10 cm

NOT TOSCALE

The triangle has an area of 30 cm2 and a base of 10 cm.

Calculate the perpendicular height h of the triangle.

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

(b)NOT TOSCALE

D C

A B

8 cm

14 cm

7 cm

AB is parallel to CD. AB is 14 cm and CD is 8 cm. The perpendicular distance between AB and CD is 7 cm.

(i) Write down the mathematical name for the quadrilateral ABCD.

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

(ii) Calculate the area of ABCD.

Answer(b)(ii) ......................................... cm2 [2]

15


(c) An isosceles triangle has an angle of 40°. Tikka draws the triangle with angles 40°, 70° and 70°. Kanwarpreet draws a different correct triangle.

What angles did Kanwarpreet use?

Answer(c) 40°, .............. , .............. [2]__________________________________________________________________________________________


16

0580/33/M/J/14© UCLES 2014



9

A

C

B

O

NOT TOSCALE

The diagram shows a circle with diameter AB and centre O. C is a point on the circumference of the circle.

(a) Explain how you know that angle ACB is 90° without having to measure it.

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

(b) AB = 13 cm and AC = 5 cm.

Calculate the length BC.

Answer(b) BC = .......................................... cm [3]

(c) Calculate angle ABC.

Answer(c) Angle ABC = ................................................ [2]





MATHEMATICS 0580/41


2 hours 30 minutes






*3415255704*


2

0580/41/M/J/14© UCLES 2014

1 A = 1

21-

3f p B = (–2 5) C =

25

-e o D =

20

02

f p

(a) Work out, when possible, each of the following. If it is not possible, write ‘not possible’ in the answer space.

(i) 2A

Answer(a)(i) [1]

(ii) B + C

Answer(a)(ii) [1]

(iii) AD

Answer(a)(iii) [2]

(iv) A–1, the inverse of A.

Answer(a)(iv) [2]

(b) Explain why it is not possible to work out CD.

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

(c) Describe fully the single transformation represented by the matrix D.

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

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

3


2 Ali leaves home at 10 00 to cycle to his grandmother’s house. He arrives at 13 00. The distance-time graph represents his journey.

10 00 11 00 12 00 13 00 14 00 15 00 16 00 17 00

40

30

20

10

0

Time

Distance fromhome (km)

(a) Calculate Ali’s speed between 10 00 and 11 30. Give your answer in kilometres per hour.

Answer(a) ...................................... km/h [2]

(b) Show that Ali’s average speed for the whole journey to his grandmother’s house is 12 km/h.

Answer(b)

[2]

(c) Change 12 kilometres per hour into metres per minute.

Answer(c) ..................................... m/min [2]

(d) Ali stays for 45 minutes at his grandmother’s house and then returns home. He arrives home at 16 42.

Complete the distance-time graph. [2]__________________________________________________________________________________________

4

0580/41/M/J/14© UCLES 2014

3 (a) The running costs for a papermill are $75 246. This amount is divided in the ratio labour costs : materials = 5 : 1.

Calculate the labour costs.

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

(b) In 2012 the company made a profi t of $135 890. In 2013 the profi t was $150 675.

Calculate the percentage increase in the profi t from 2012 to 2013.

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

(c) The profi t of $135 890 in 2012 was an increase of 7% on the profi t in 2011.

Calculate the profi t in 2011.

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

(d)2 cm

NOT TOSCALE

21 cm

30 cm

Paper is sold in cylindrical rolls. There is a wooden cylinder of radius 2 cm and height 21 cm in the centre of each roll. The outer radius of a roll of paper is 30 cm.

(i) Calculate the volume of paper in a roll.

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

5


(ii) The paper is cut into sheets which measure 21 cm by 29.7 cm. The thickness of each sheet is 0.125 mm.

(a) Change 0.125 millimetres into centimetres.

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

(b) Work out how many whole sheets of paper can be cut from a roll.

Answer(d)(ii)(b) ................................................ [4]__________________________________________________________________________________________

6

0580/41/M/J/14© UCLES 2014

4

BT

P

x

4

911

6 – x

In the Venn diagram, = {children in a nursery}

B = {children who received a book for their birthday} T = {children who received a toy for their birthday} P = {children who received a puzzle for their birthday}

x children received a book and a toy and a puzzle. 6 children received a toy and a puzzle.

(a) 4 children received a book and a toy. 5 children received a book and a puzzle. 7 children received a puzzle but not a book and not a toy.

Complete the Venn diagram above. [3]

(b) There are 40 children in the nursery.

Using the Venn diagram, write down and solve an equation in x.

Answer(b)

[3]

7


(c) Work out

(i) the probability that a child, chosen at random, received a book but not a toy and not a puzzle,

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

(ii) the number of children who received a book and a puzzle but not a toy,

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

(iii) n(B),

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

(iv) n(B ∪ P),

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

(v) n(B ∪ T ∪ P)'.

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

(d)

BT

P

Shade the region B ∩ (T ∪ P)'. [1]__________________________________________________________________________________________

8

0580/41/M/J/14© UCLES 2014

5

North

Scale: 2 cm to 3 km

P

S

L

In the scale drawing, P is a port, L is a lighthouse and S is a ship. The scale is 2 centimetres represents 3 kilometres.

(a) Measure the bearing of S from P.

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

(b) Find the actual distance of S from L.

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

(c) The bearing of L from S is 160°.

Calculate the bearing of S from L.

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

9


(d) Work out the scale of the map in the form 1 : n.

Answer(d) 1 : ................................................ [2]

(e) A boat B is

● equidistant from S and L and ● equidistant from the lines PS and SL.

On the diagram, using a straight edge and compasses only, construct the position of B. [5]

(f) The lighthouse stands on an island of area 1.5 cm2 on the scale drawing.

Work out the actual area of the island.

Answer(f) ......................................... km2 [2]__________________________________________________________________________________________

10

0580/41/M/J/14© UCLES 2014

6 (a) A square spinner is biased. The probabilities of obtaining the scores 1, 2, 3 and 4 when it is spun are given in the table.

Score 1 2 3 4

Probability 0.1 0.2 0.4 0.3

(i) Work out the probability that on one spin the score is 2 or 3.

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

(ii) In 5000 spins, how many times would you expect to score 4 with this spinner?

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

(iii) Work out the probability of scoring 1 on the fi rst spin and 4 on the second spin.

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

(b) In a bag there are 7 red discs and 5 blue discs. From the bag a disc is chosen at random and not replaced. A second disc is then chosen at random.

Work out the probability that at least one of the discs is red. Give your answer as a fraction.

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

11


7

A

y

x

4

3

2

1

–1

–2

–3

–4

–5

0–1 1 2 3 4 5 6–2–3–4–5–6

(a) On the grid,

(i) draw the image of shape A after a translation by the vector 4

5

-

-e o, [2]

(ii) draw the image of shape A after a rotation through 90° clockwise about the origin. [2]

(b) (i) On the grid, draw the image of shape A after the transformation represented by the matrix 20

01

f p.

[3]

(ii) Describe fully the single transformation represented by the matrix 20

01

f p.

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

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

12

0580/41/M/J/14© UCLES 2014

8 (a) Complete the table of values for y = x3 – 3x + 1 .

x –2.5 –2 –1.5 –1 –0.5 0 0.5 1 1.5 2 2.5

y –7.125 –1 3 1 –0.375 –1 –0.125 3 9.125

[2]

(b) Draw the graph of y = x3 – 3x + 1 for –2.5 Ğ x Ğ 2.5 .

y

x

10

9

8

7

6

5

4

3

2

1

–1

–2

–3

–4

–5

–6

–7

–8

0–1–2–3 321

[4]

13


(c) By drawing a suitable tangent, estimate the gradient of the curve at the point where x = 2.

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

(d) Use your graph to solve the equation x3 – 3x + 1 = 1 .

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

(e) Use your graph to complete the inequality in k for which the equation

x3 – 3x + 1 = k has three different solutions.

Answer(e) ........................ < k < ........................ [2]__________________________________________________________________________________________

14

0580/41/M/J/14© UCLES 2014

9

80

70

60

50

40

30

20

10

10 20 30

Time (minutes)

40 500

Cumulativefrequency

t

The times (t minutes) taken by 80 people to complete a charity swim were recorded. The results are shown in the cumulative frequency diagram above.

(a) Find

(i) the median,

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

(ii) the inter-quartile range,

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

15


(iii) the 70th percentile.

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

(b) The times taken by the 80 people are shown in this grouped frequency table.

Time (t minutes) 0 < t Ğ 20 20 < t Ğ 30 30 < t Ğ 45 45 < t Ğ 50

Frequency 12 21 33 14

(i) Calculate an estimate of the mean time.

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

(ii) Draw a histogram to represent the grouped frequency table.

4

3

2

1

10 20 30

Time (minutes)

40 500

Frequencydensity

t

[4]__________________________________________________________________________________________

16

0580/41/M/J/14© UCLES 2014

10 (a) f(x) = 2x – 3 g(x) = 11

x + + 2 h(x) = 3x

(i) Work out f(4).

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

(ii) Work out fh(–1).

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

(iii) Find f –1(x), the inverse of f(x).

Answer(a)(iii) f –1(x) = ................................................ [2]

(iv) Find ff(x) in its simplest form.

Answer(a)(iv) ff(x) = ................................................ [2]

17


(v) Show that the equation f(x) = g(x) simplifi es to 2x2 – 3x – 6 = 0 .

Answer(a)(v)

[3]

(vi) Solve the equation 2x2 – 3x – 6 = 0 .

Give your answers correct to 2 decimal places. Show all your working.

Answer(a)(vi) x = ..................... or x = ..................... [4]

(b) Simplify 2

2

3 103 2

x xx x

++-

- .

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

18

0580/41/M/J/14© UCLES 2014

11 (a) = 3

4

-e o

(i) P is the point (–2, 3).

Work out the co-ordinates of Q.

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

(ii) Work out , the magnitude of .

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

19


(b)

A

N

O

Y

C

Ba

b

NOT TOSCALE

OACB is a parallelogram. = a and = b. AN : NB = 2 : 3 and AY = 5

2 AC.

(i) Write each of the following in terms of a and/or b. Give your answers in their simplest form.

(a)

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

(b)

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

(ii) Write down two conclusions you can make about the line segments NY and BC.

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

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

20

0580/41/M/J/14© UCLES 2014



BLANK PAGE





MATHEMATICS 0580/42


2 hours 30 minutes





*0048847567*



2

0580/42/M/J/14© UCLES 2014

1 Jane and Kate share $240 in the ratio 5 : 7 .

(a) Show that Kate receives $140.

Answer(a)

[2]

(b) Jane and Kate each spend $20.

Find the new ratio Jane’s remaining money : Kate’s remaining money. Give your answer in its simplest form.

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

(c) Kate invests $120 for 5 years at 4% per year simple interest.

Calculate the total amount Kate has after 5 years.

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

(d) Jane invests $80 for 3 years at 4% per year compound interest.

Calculate the total amount Jane has after 3 years. Give your answer correct to the nearest cent.

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

(e) An investment of $200 for 2 years at 4% per year compound interest is the same as an investment of $200 for 2 years at r % per year simple interest.

Find the value of r.

Answer(e) r = ................................................ [3]__________________________________________________________________________________________

3


2 f(x) = 21x – 2x , x ≠ 0

(a) Complete the table of values for f(x).

x –3 –2.5 –2 –1.5 –1 –0.5 0.4 0.5 1 1.5 2

f(x) 6.1 5.2 4.3 3.4 5 5.5 –2.6 –3.8

[3]

(b) On the grid, draw the graph of y = f(x) for –3 Y x Y – 0.5 and 0.4 Y x Y 2 .

y

x

7

6

5

4

3

2

1

–1

–2

–3

–4

0–1–2–3 21

[5]

(c) Solve the equation f(x) = 2 .

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

(d) Solve the equation f(x) = 2x + 3 .

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

(e) (i) Draw the tangent to the graph of y = f(x) at the point where x = –1.5 . [1]

(ii) Use the tangent to estimate the gradient of the graph of y = f(x) where x = –1.5 .

Answer(e)(ii) ................................................ [2]__________________________________________________________________________________________

4

0580/42/M/J/14© UCLES 2014

3

80 m

90 m

95 m

49°55°

D

A

B

C

NOT TOSCALE

The diagram shows a quadrilateral ABCD. Angle BAD = 49° and angle ABD = 55°. BD = 80 m, BC = 95 m and CD = 90 m.

(a) Use the sine rule to calculate the length of AD.

Answer(a) AD = ............................................ m [3]

(b) Use the cosine rule to calculate angle BCD.

Answer(b) Angle BCD = ................................................ [4]

5


(c) Calculate the area of the quadrilateral ABCD.

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

(d) The quadrilateral represents a fi eld. Corn seeds are sown across the whole fi eld at a cost of $3250 per hectare.

Calculate the cost of the corn seeds used. 1 hectare = 10 000 m2

Answer(d) $ ................................................ [3]__________________________________________________________________________________________

6

0580/42/M/J/14© UCLES 2014

4

Q

y

x

8

7

6

5

4

3

2

1

–1

–2

–3

–4

–5

–6

–7

–8

0–1 1 2 3 4 5 6 7 8–2–3–4–5–6–7–8

(a) Draw the refl ection of shape Q in the line x = –1 . [2]

(b) (i) Draw the enlargement of shape Q, centre (0, 0), scale factor –2 . [2]

(ii) Find the 2 × 2 matrix that represents an enlargement, centre (0, 0), scale factor –2 .

Answer(b)(ii) f p [2]

7


(c) (i) Draw the stretch of shape Q, factor 2, x-axis invariant. [2]

(ii) Find the 2 × 2 matrix that represents a stretch, factor 2, x-axis invariant.

Answer(c)(ii) f p [2]

(iii) Find the inverse of the matrix in part (c)(ii).

Answer(c)(iii) f p [2]

(iv) Describe fully the single transformation represented by the matrix in part (c)(iii).

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

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

8

0580/42/M/J/14© UCLES 2014

5

12 cm

10 cm

4 cm

8 cm

NOT TOSCALE

The diagram shows a cylinder with radius 8 cm and height 12 cm which is full of water. A pipe connects the cylinder to a cone. The cone has radius 4 cm and height 10 cm.

(a) (i) Calculate the volume of water in the cylinder. Show that it rounds to 2410 cm3 correct to 3 signifi cant fi gures.

Answer(a)(i)

[2]

(ii) Change 2410 cm3 into litres.

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

9


(b) Water fl ows from the cylinder along the pipe into the cone at a rate of 2 cm3 per second.

Calculate the time taken to fi ll the empty cone. Give your answer in minutes and seconds correct to the nearest second.

[The volume, V, of a cone with radius r and height h is V = 31 πr

2h.]

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

(c) Find the number of empty cones which can be fi lled completely from the full cylinder.

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

10

0580/42/M/J/14© UCLES 2014

6

21°

117°

y°

x°

S

P

Q

R

T

NOT TOSCALE

(a) The chords PR and SQ of the circle intersect at T. Angle RST = 21° and angle STR = 117°.

(i) Find the values of x and y.

Answer(a)(i) x = ................................................

y = ................................................ [2]

(ii) SR = 8.23 cm, RT = 3.31 cm and PQ = 9.43 cm.

Calculate the length of TQ.

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

11


(b) EFGH is a cyclic quadrilateral. EF is a diameter of the circle. KE is the tangent to the circle at E. GH is parallel to FE and angle KEG = 115°.

Calculate angle GEH.

Answer(b) Angle GEH = ................................................ [4]

(c) A, B, C and D are points on the circle centre O. Angle AOB = 140° and angle OAC = 14°. AD = DC.

Calculate angle ACD.

Answer(c) Angle ACD = ................................................ [5]__________________________________________________________________________________________

115°

G

F

H

E

K

NOT TOSCALE

140°14°O

BA

D

C

NOT TOSCALE

12

0580/42/M/J/14© UCLES 2014

7 (a)

1.0

0.8

0.6

0.4

0.2

020 40 60

Mass (grams)80 10010 30 50 70 90

Frequencydensity

m

The histogram shows some information about the masses (m grams) of 39 apples.

(i) Show that there are 12 apples in the interval 70 < m Y 100 .

Answer(a)(i)

[1]

(ii) Calculate an estimate of the mean mass of the 39 apples.

Answer(a)(ii) ............................................. g [5]

(b) The mean mass of 20 oranges is 70 g. One orange is eaten. The mean mass of the remaining oranges is 70.5 g.

Find the mass of the orange that was eaten.

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

13


8 The distance a train travels on a journey is 600 km.

(a) Write down an expression, in terms of x, for the average speed of the train when

(i) the journey takes x hours,

Answer(a)(i) ....................................... km/h [1]

(ii) the journey takes (x + 1) hours.

Answer(a)(ii) ....................................... km/h [1]

(b) The difference between the average speeds in part(a)(i) and part(a)(ii) is 20 km/h.

(i) Show that x 2 + x – 30 = 0 .

Answer(b)(i)

[3]

(ii) Find the average speed of the train for the journey in part(a)(ii). Show all your working.

Answer(b)(ii) ....................................... km/h [4]__________________________________________________________________________________________

14

0580/42/M/J/14© UCLES 2014

9 If the weather is fi ne the probability that Carlos is late arriving at school is 101 .

If the weather is not fi ne the probability that he is late arriving at school is 31 .

The probability that the weather is fi ne on any day is 43 .

(a) Complete the tree diagram to show this information.

Fine

Not fine

Late

Weather Arriving at school

Not late

Not late

Late........

........

........

........

34

110

[3]

(b) In a school term of 60 days, fi nd the number of days the weather is expected to be fi ne.

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

(c) Find the probability that the weather is fi ne and Carlos is late arriving at school.

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

(d) Find the probability that Carlos is not late arriving at school.

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

(e) Find the probability that the weather is not fi ne on at least one day in a school week of 5 days.

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

15


10 f(x) = x1 , x ≠ 0 g(x) = 1 – x h(x) = x

2 + 1

(a) Find fg 21

` j.

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

(b) Find g–1(x), the inverse of g(x).

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

(c) Find hg(x), giving your answer in its simplest form.

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

(d) Find the value of x when g(x) = 7 .

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

(e) Solve the equation h(x) = 3x. Show your working and give your answers correct to 2 decimal places.

Answer(e) x = ......................... or x = ......................... [4]

(f) A function k(x) is its own inverse when k –1(x) = k(x).

For which of the functions f(x) , g(x) and h(x) is this true?

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


16

0580/42/M/J/14© UCLES 2014



11 The total area of each of the following shapes is X. The area of the shaded part of each shape is kX.

For each shape, fi nd the value of k and write your answer below each diagram.

A B C D

NOT TOSCALE

NOT TOSCALE

72°O

J

K

NOT TOSCALE

F

E

G

I

H

AB = BC = CD

k = .....................................

Angle JOK = 72°

k = .....................................

EF = FG and EI = IH

k = .....................................

NOT TOSCALE

NOT TOSCALE

A

O B

The shape is a regular hexagon.

k = .....................................

The diagram shows a sector of a circle centre O.Angle AOB = 90°

k = .....................................[10]





MATHEMATICS 0580/43


2 hours 30 minutes





*9468931136*



2

0580/43/M/J/14© UCLES 2014

1 In July, a supermarket sold 45 981 bottles of fruit juice.

(a) The cost of a bottle of fruit juice was $1.35 .

Calculate the amount received from the sale of the 45 981 bottles. Give your answer correct to the nearest hundred dollars.

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

(b) The number of bottles sold in July was 17% more than the number sold in January.

Calculate the number of bottles sold in January.

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

(c) There were 3 different fl avours of fruit juice. The number of bottles sold in each fl avour was in the ratio apple : orange : cherry = 3 : 4 : 2. The total number of bottles sold was 45 981.

Calculate the number of bottles of orange juice sold.

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

(d) One bottle contains 1.5 litres of fruit juice.

Calculate the number of 330 ml glasses that can be fi lled completely from one bottle.

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

(e) 95 of the 45 981 bottles are recycled.

Calculate the number of bottles that are recycled.

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

3


24

3

2

1

10 20 30

Amount ($x)

40 50 600

Frequencydensity

A survey asked 90 people how much money they gave to charity in one month. The histogram shows the results of the survey.

(a) Complete the frequency table for the six columns in the histogram.

Amount ($x) 0 < x Y 10

Frequency 4

[5]

(b) Use your frequency table to calculate an estimate of the mean amount these 90 people gave to charity.

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

4

0580/43/M/J/14© UCLES 2014

3 (a)P

X

Q

R

17 cm

12 cm NOT TOSCALE

The diagram shows triangle PQR with PQ = 12 cm and PR = 17 cm. The area of triangle PQR is 97 cm2 and angle QPR is acute.

(i) Calculate angle QPR.

Answer(a)(i) Angle QPR = ................................................ [3]

(ii) The midpoint of PQ is X.

Use the cosine rule to calculate the length of XR.

Answer(a)(ii) XR = .......................................... cm [4]

5


(b)

37°

42° a cm9.4 cmNOT TOSCALE

Calculate the value of a.

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

(c) sin x = cos 40°, 0° Y x Y 180°

Find the two values of x.

Answer(c) x = .................. or x = .................. [2]__________________________________________________________________________________________

6

0580/43/M/J/14© UCLES 2014

4 The table shows some values for the function y = 21x + x , x ≠ 0.

x –3 –2 –1 –0.5 0.5 1 2 3 4

y –2.89 –1.75 3.5 2 2.25 4.06

(a) Complete the table of values. [3]

(b) On the grid, draw the graph of y = 21x + x for –3 Y x Y – 0.5 and 0.5 Y x Y 4.

y

x

5

4

3

2

1

–1

–2

–3

0–1–2–3 3 421

[5]

7


(c) Use your graph to solve the equation 21x + x – 3 = 0 .

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

(d) Use your graph to solve the equation 21x + x = 1 – x.

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

(e) By drawing a suitable tangent, fi nd an estimate of the gradient of the curve at the point where x = 2.

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

(f) Using algebra, show that you can use the graph at y = 0 to fi nd 13 - .

Answer(f)

[3]__________________________________________________________________________________________

8

0580/43/M/J/14© UCLES 2014

5 (a)

5

4

3

2

1

01 2 3 4 5 6 7 8

x

y

A

B

(i) Write down the position vector of A.


(ii) Find ì ì , the magnitude of .

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

(b)

p

q

O

Q

S

P R

NOT TOSCALE

O is the origin, = p and = q. OP is extended to R so that OP = PR. OQ is extended to S so that OQ = QS.

(i) Write down in terms of p and q.

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

(ii) PS and RQ intersect at M and RM = 2MQ.

Use vectors to fi nd the ratio PM : PS, showing all your working.

Answer(b)(ii) PM : PS = ....................... : ....................... [4]__________________________________________________________________________________________

9


6 In this question, give all your answers as fractions.

N A T I O N

The letters of the word NATION are printed on 6 cards.

(a) A card is chosen at random.

Write down the probability that

(i) it has the letter T printed on it,

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

(ii) it does not have the letter N printed on it,

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

(iii) the letter printed on it has no lines of symmetry.

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

(b) Lara chooses a card at random, replaces it, then chooses a card again.

Calculate the probability that only one of the cards she chooses has the letter N printed on it.

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

(c) Jacob chooses a card at random and does not replace it. He continues until he chooses a card with the letter N printed on it.

Find the probability that this happens when he chooses the 4th card.

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

10

0580/43/M/J/14© UCLES 2014

7 (a)

Y XA B

F

E D

C

x°

x°

32°

t °

p°q°

NOT TOSCALE

ABCDEF is a hexagon. AB is parallel to ED and BC is parallel to FE. YFE and YABX are straight lines. Angle CBX = 32° and angle EFA = 90°.

Calculate the value of

(i) p,

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

(ii) q,

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

(iii) t,

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

(iv) x.

Answer(a)(iv) x = ................................................ [3]

11


(b)

63°

y°

x°

RQ

S

T UP

NOT TOSCALE

P, Q, R and S are points on a circle and PS = SQ. PR is a diameter and TPU is the tangent to the circle at P. Angle SPT = 63°.

Find the value of

(i) x,

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

(ii) y.

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

12

0580/43/M/J/14© UCLES 2014

8 (a) (i) Show that the equation 47

x + + 22 3x - = 1 can be simplifi ed to 2x2 + 3x – 6 = 0 .

Answer(a)(i)

[3]

(ii) Solve the equation 2x2 + 3x – 6 = 0 .

Show all your working and give your answers correct to 2 decimal places.

Answer(a)(ii) x = ........................... or x = ........................... [4]

(b) The total surface area of a cone with radius x and slant height 3x is equal to the area of a circle with radius r.

Show that r = 2x. [The curved surface area, A, of a cone with radius r and slant height l is A = πrl.]

Answer(b)

[4]__________________________________________________________________________________________

13


9 f(x) = 4 – 3x g(x) = 3–x

(a) Find f(2x) in terms of x.

Answer(a) f(2x) = ................................................ [1]

(b) Find ff(x) in its simplest form.

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

(c) Work out gg(–1). Give your answer as a fraction.

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

(d) Find f –1(x), the inverse of f(x).

Answer(d) f –1(x) = ................................................ [2]

(e) Solve the equation gf(x) = 1.

Answer(e) x = ................................................ [3]__________________________________________________________________________________________

14

0580/43/M/J/14© UCLES 2014

10 (a)

8 cm

r cm

NOT TOSCALE

The three sides of an equilateral triangle are tangents to a circle of radius r cm. The sides of the triangle are 8 cm long.

Calculate the value of r. Show that it rounds to 2.3, correct to 1 decimal place.

Answer(a)

[3]

(b)

8 cm

12 cm

NOT TOSCALE

The diagram shows a box in the shape of a triangular prism of height 12 cm. The cross section is an equilateral triangle of side 8 cm.

Calculate the volume of the box.

Answer(b) ......................................... cm3 [4]

15


(c) The box contains biscuits. Each biscuit is a cylinder of radius 2.3 centimetres and height 4 millimetres.

Calculate

(i) the largest number of biscuits that can be placed in the box,

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

(ii) the volume of one biscuit in cubic centimetres,

Answer(c)(ii) ......................................... cm3 [2]

(iii) the percentage of the volume of the box not fi lled with biscuits.

Answer(c)(iii) ............................................ % [3]__________________________________________________________________________________________


16

0580/43/M/J/14© UCLES 2014



11

Diagram 1 Diagram 2 Diagram 3

The fi rst three diagrams in a sequence are shown above. Diagram 1 shows an equilateral triangle with sides of length 1 unit.

In Diagram 2, there are 4 triangles with sides of length 21 unit.

In Diagram 3, there are 16 triangles with sides of length 41 unit.

(a) Complete this table for Diagrams 4, 5, 6 and n.

Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 Diagram 6 Diagram n

Length of side 1 21

41

Length of side as a power of 2 20 2–1 2–2

[6]

(b) (i) Complete this table for the number of the smallest triangles in Diagrams 4, 5 and 6.

Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 Diagram 6

Number of smallesttriangles 1 4 16

Number of smallesttriangles as a power of 2 20 22 24

[2]

(ii) Find the number of the smallest triangles in Diagram n, giving your answer as a power of 2.

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

(c) Calculate the number of the smallest triangles in the diagram where the smallest triangles have sides of

length 1281 unit.

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

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