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This document consists of 23 printed pages and 1 blank page.

DC (KN/SG) 115632/3© UCLES 2016� [Turn over

Cambridge International ExaminationsCambridge Ordinary Level

*9653730277*

MATHEMATICS (SYLLABUS D) 4024/21Paper 2 May/June 2016

2 hours 30 minutes

Candidates answer on the Question Paper.

Additional Materials: Geometrical instruments Electronic calculator

READ THESE INSTRUCTIONS FIRST

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

Section AAnswer all questions.

Section BAnswer any four questions.

If working is needed for any question it must be shown in the space below that question.Omission of essential working will result in loss of marks.You are expected to use an electronic calculator to evaluate explicit numerical expressions.If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place.For π, use either your calculator value or 3.142, unless the question requires the answer in terms of π.

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 100.

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Section A�[52�marks]

Answer�all�questions�in�this�section.

1 A�shopkeeper�buys�some�plates�from�a�manufacturer�for�$12�each.

(a) The�manufacturer�makes�a�profit�of�60%.

Calculate�the�cost�of�manufacturing�each�plate.

Answer $��........................................ [2]

(b) The�shopkeeper�sells�each�plate�for�$17.40.

Calculate�the�percentage�profit�made�by�the�shopkeeper.

Answer �.......................................%�[2]

(c) In�a�sale,�each�plate�is�reduced�from�$17.40�to�$11.31.

Calculate�the�percentage�discount�given.

Answer �.......................................%�[2]

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(d) The�shopkeeper�buys�100�plates�at�$12�each. He�sells�60�plates�at�$17.40�each�and�x�plates�at�$11.31�each. The�shopkeeper�makes�a�profit�of�at�least�10%.

� � Find�the�least�possible�value�of�x.

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

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2 (a) Solve�the�equation���� pp7

15

--= .

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

(b) Simplify����a bab93 2

6 21J

LKK

N

POO .

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

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(c) Simplify���� qq q3 3

2 3

--

.

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

(d) (i) Factorise���� t t4 35 92 + - .

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

(ii) Hence�solve�the�equation���� t t4 35 9 02 + - = .

Answer �........................................... [1]

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3 The�table�below�is�for����� y x x 32= + - .

x –3 –2 –1 0 1 2

y 3 –1 –3 –3 –1 3

(a) Using�a�scale�of�2�cm�to�1�unit�on�the�x-axis�for���� x3 2G G- � and�a�scale�of�1�cm�to�1�unit�on�the�y-axis�for���� y4 4G G- ,� plot�the�points�from�the�table�and�join�them�with�a�smooth�curve.

y

x

� [2]

(b) (i) Use�your�graph�to�estimate�the�solutions�of�the�equation���x x 3 02 + - = .

Answer x�=�................�or��................ [1]

(ii) Use�your�graph�to�estimate�the�solutions�of�the�equation���x x 5 02 + - = .

Answer x�=�................�or��................ [2]

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(c) By�drawing�a�tangent,�estimate�the�gradient�of�the�curve�at�� ,1 1–^ h.

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

(d) The�equation���x x 1 02 - - = ��can�be�solved�by�drawing�a�straight�line�on�the�graph�of� y x x 32= + - .

(i) Find�the�equation�of�this�straight�line.

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

(ii) Draw�this�straight�line�and�hence�solve���x x 1 02 - - = .

Answer x�=�................�or��................ [2]

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4A

B

N M

CL

ANB,�BLC�and�CMA�are�straight�lines.�NM�is�parallel�to�BC�andLN�is�parallel�to�CA.

(a) Prove�that�triangle�ANM�is�similar�to�triangle�NBL. Give�a�reason�for�each�statement�you�make.

�............................................................................................................................................................

�............................................................................................................................................................

�............................................................................................................................................................

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

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(b) AN�:�NB�=�2�:�3

(i) Find���NM�:�BC.

Answer �................... �:�....................�[2]

(ii) Find������area�ANM�:�area�NBL.

Answer �................... �:�....................�[1] (iii) Find�����area�ANM�:�area�NMCL.

Answer �................... �:�....................�[2]

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5 (a)

A

BC

31

115

AB�is�vertical�and�CB�is�horizontal. AB�=�31�m�and�CB�=�115�m.

� � Calculate�the�angle�of�depression�of�C�from�A.

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

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(b)

3541100

L

K

J

J�and�K�are�two�positions�at�sea. The�base�of�a�lighthouse�is�at�L. J�is�due�East�of�L�and�K�is�due�South�of�L. KL�=�354�m�and�KJ�=�1100�m.

(i) Calculate�LJKt .

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

(ii) Hence�find�the�bearing�of�K�from�J.

Answer �........................................... [1]

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6 �41

13A =-J

LKK

N

POO���������

27

05B =-

J

LKK

N

POO

(a) Evaluate�����2A�–�B.

Answer �J

L

KKK

N

P

OOO�������������[2]

(b) Find��A2.

Answer �J

L

KKK

N

P

OOO�������������[2]

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(c) Find��B–1.

Answer �J

L

KKK

N

P

OOO�������������[2]

(d) A�+�Z�=�A

Find�Z.

Answer �J

L

KKK

N

P

OOO�������������[1]

(e) M�+�2I�=�B,�where�I�is�the�2 2# �identity�matrix.

Find�M.

Answer �J

L

KKK

N

P

OOO�������������[2]

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Section B�[48�marks]

Answer�four�questions�in�this�section.

Each�question�in�this�section�carries�12�marks.

7 (a) AC�is�a�diameter�of�the�circle,�centre�O,�radius�5�cm. ACBt �=�64°.

� � Calculate�the�length�of�the�minor�arcBC.

Answer �..................................... cm�[4]

(b)

15.5

16.5rim

A�baking�tray�is�an�open�cylinder�of�radius�15.5�cm�with�a�rim. The�outer�edge�of�the�rim�is�a�circle�of�radius�16.5�cm.

A

B

CO

64°

5

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(i) Calculate�the�area�of�the�top�surface�of�the�rim.

Answer �....................................cm2�[2]

(ii) 44�identical�circular�holes�are�cut�out�of�the�bottom�of�the�baking�tray. The�area�of�the�bottom�that�remains�is�650�cm2.

Calculate�the�radius�of�each�circular�hole.

Answer �..................................... cm�[3]

(iii)

15.5 cm

d mm

To�make�a�pizza,�the�baking�tray�is�completely�filled�with�dough�to�a�depth�of�d�mm.� The�open�cylinder�holds�500�cm3�of�dough.

Calculate�the�depth�of�the�dough,�d�mm,�giving�your�answer�correct�to�the�nearest��millimetre.

Answer �....................................mm�[3]

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8 (a) �� p qq8 5

=-

(i) Find�p�when�q�=�2.6�.

Answer �........................................... [1]

(ii) Express�q�in�terms�of�p.

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

(b) x – 2

x + 3H

h

x x

Trapezium A Trapezium B

The�lengths�of�the�parallel�sides�of�trapezium�A�are�x�cm�and� x 2-^ h�cm. The�lengths�of�the�parallel�sides�of�trapezium�B�are�x�cm�and� x 3+^ h�cm. The�height�of�trapezium�A�is�H�cm�and�the�height�of�trapezium�B�is�h�cm. The�area�of�each�trapezium�is�15�cm2.

(i) Show�that��H x 115

=-

��and��h x2 330

=+

.

� [2]

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(ii) Find�an�expression�in�terms�of�x�for�the�difference�in�height,�H�–�h,�between�trapezium�A�and�� trapezium�B,�and�show�that�it�simplifies�to��

x x1752 3- +^ ^h h

�.

� [3]

(iii) The�difference�in�height�is�1.5�cm.

(a) Show�that��� x x2 53 02 + - = .

� [2]

(b) Find�x,�giving�your�answer�correct�to�2�decimal�places.

Answer x�=�...................................... [2]

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9 (a)D

B

F

A

C

E15°

52

ABCD�represents�the�rectangular�sloping�surface�of�a�triangular�prism. ABEF�is�a�horizontal�rectangle.�CE�and�DF�are�vertical. CBEt �=�15°,�DC�=�5�m�and�AD�=�2�m.�

(i) Calculate�AC.

Answer �....................................... m�[2]

(ii) Calculate�CE.

Answer �....................................... m�[2]

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(iii) Calculate�FAEt .

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

(b) (i)

9

10

6

θ °

A�triangle�has�sides�of�10�cm,�9�cm�and�6�cm,�and�an�angle�of�θ °,�as�shown�in�the�diagram.

Calculate�θ.

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

(ii) The�triangle�KGH�has�sides�of�a�cm,�b�cm�and�c�cm� as�shown�in�the�diagram.

It�is�given�that�KGHt �is�an�obtuse�angle.

Complete�the�statement�below�using�one�of�the�symbols���1 2G H= .

�� � c2�……� a b2 2+^ h� [1]

G

HK

a

c

b

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10 100�electric�light�bulbs�of�Brand�A�were�tested�to�find�how�long�each�bulb�lasted. The�results�are�summarised�in�the�table�below.

Time�(t�hours) t�G�50 50�1 t�G�100 100�1�t�G�150 150�1�t�G�200 200�1�t�G�250 250�1�t�G�300 300�1�t�G�350

Number�of�bulbs 2 2 10 40 30 14 2

(a) Complete�the�cumulative�frequency�table.

Time�(t�hours) t�G�50 t�G�100 t�G�150 t�G�200 t�G�250 t�G�300 t�G�350

Cumulative�frequency 2 4 100

� [1]

(b) On�the�grid,�draw�a�smooth�cumulative�frequency�curve�to�represent�this�information. Label�this�curve�Brand�A.

100

80

60

40

20

00 50 100 150 200 250 300 350

Cumulativefrequency

Time (t hours)

� [2]

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(c) (i) Use�your�graph�to�estimate�the�median.

Answer �................................. hours�[1]

(ii) Use�your�graph�to�estimate�the�interquartile�range.

Answer �................................. hours�[2]

(d) 100�Brand�B�bulbs�gave�the�following�results. 4�bulbs�lasted�50�hours�or�less. The�longest�time�any�bulb�lasted�was�300�hours. The�median�is�250�hours. The�upper�quartile�is�275�hours. The�interquartile�range�is�75�hours.

On�the�grid,�draw�and�label�the�cumulative�frequency�curve�for�the�Brand�B�bulbs.� [4]

(e) Using�your�graph,�estimate�the�number�of�Brand�A�bulbs�that�lasted�275�hours�or�less.

Answer �........................................... [1]

(f) Complete�the�statement�below.

� � Brand�............�had�............�more�bulbs�that�lasted�longer�than�275�hours�than�Brand�.............�.� [1]

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11 (a) Triangle�ABC�has�vertices�A(2,�2),�B(3,�5)�and�C(4,�1). Triangle�AB Cl l l�has�vertices�Al(–4,�4),�Bl(–3,�7)�and�C l(–2,�3).

� � Write�down�the�column�vector�of�the�translation�that�maps�triangle�ABC�onto�triangle�AB Cl l l�.

Answer���������

J

L

KKKK

N

P

OOOO���������[1]

(b) PQRS�is�a�parallelogram.

The�position�vector�of�P�relative�to�O�is�given�by�OP �=�42-J

LKKN

POO.

The�position�vector�of�Q�relative�to�O�is�given�by�OQ �=�46

J

LKKN

POO.

(i) Express�PQ �as�a�column�vector.

Answer���������

J

L

KKKK

N

P

OOOO���������[2]

(ii) Find�RS .

Answer���������

J

L

KKKK

N

P

OOOO���������[1]

(iii) Find� RS .

Answer �...................................units�[2]

Q

RP

S

O

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(c)

10

5

00 5 10 15

D

y

x

The�diagram�shows�triangle�D.

(i) An�enlargement�with�centre�(5,�4),�scale�factor�2,�maps�triangle�D�onto�triangle�E.

Draw�and�label�triangle�E.� [2]

(ii) An�enlargement�with�centre�(5,�4),�scale�factor�0.5,�maps�triangle�D�onto�triangle�F.

Draw�and�label�triangle�F.� [1]

(iii) Triangle�G�has�vertices�(5,�4),�(4,�3)�and�(3,�5). Triangle�F�can�be�mapped�onto�triangle�G�using�a�single�enlargement. Triangle�F�can�also�be�mapped�onto�triangle�G�using�a�different single�transformation�T.

Describe�fully�the�single�transformation�T.

Answer��.......................................................................................................................................

�....................................................................................................................................................

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

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24

© UCLES 2016 4024/21/M/J/16

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

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

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

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® IGCSE is the registered trademark of Cambridge International Examinations.

This document consists of 6 printed pages.

© UCLES 2016 [Turn over

Cambridge International Examinations Cambridge Ordinary Level

MATHEMATICS (SYLLABUS D) 4024/21

Paper 2 May/June 2016

MARK SCHEME

Maximum Mark: 100

Published

This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 2016 series for most Cambridge IGCSE

®,

Cambridge International A and AS Level components and some Cambridge O Level components.

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Page 2 Mark Scheme Syllabus Paper

Cambridge O Level – May/June 2016 4024 21

© Cambridge International Examinations 2016

Question Answers Mark Part Marks

1 (a)

7.5(0)

2 M1 for x + 60

100

x

= 12 soi or

B1 for ÷ by 160

(b)

45

2 M1 for 17.40 12

10012

−

×

(c)

35

2 M1 for 17.4 11.31

10017.4

−

×

(d) 25 3 M1 for 60 × 17.4 + x × 11.31(⩾ 1320) or B1 276

A1 for 24.4(03...)

2 (a) 6 2 M1 for p − 1 = 5(7 − p) soi

(b) 2

3b

a

2 M1 for 4

2

9b

a oe

1

32

3

2

3a b

a b

oe

or B1 for 3b² as numerator or k

a

(c) 2

3

q

2

B1 for 2 (1 )q q− or 3(1 )q−

(d) (i) (4 1)( 9)t t− + 2 B1 for ( )( )at c bt d+ + with ab = 4 or cd = 9−

(ii) 1

94−

or ft

1ft

3 (a) Correct graph 2 B1 for correct scales and 4 points or wrong scales and all points.

(b) (i) 2.3 0.5− ± 1.3 ± 0.5 1

(ii) 2.8 0.5− ± 1.8 ± 0.5 2 M1 for 23 2x x+ − = soi

(c) 2.4 to 3.6 2 M1 for tangent at x =1

(d) (i) 2 2y x= − 2 B1 for 2x or 2−

(ii) 0.6− 1.6 2 Dependent on line drawn

B1 for their line having FT gradient or FT intercept

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Page 3 Mark Scheme Syllabus Paper

Cambridge O Level – May/June 2016 4024 21

© Cambridge International Examinations 2016

Question Answers Mark Part Marks

4 (a) Complete proof 3 B2 for 2 pairs of equal angles 1 pair with reason.

B1 for 1 pair of equal angles.

(b) (i) 2 : 5 2 B1 for NM : BL = 2 : 3 oe or NM = LC

(ii) 4 : 9 1

(iii) 1 : 3 2 B1 for such as ANM

ABC

∆

∆=

4

25or

9

25

NBL

ABC

∆=

∆

5 (a)

15.1 or 15.08(…..

3 M1 for 31

tan115

θ = or115

tan31

θ =

A1for 15.1θ = or 74.9θ =

(b) (i)

18.8 or 18.77……

2 M1 for sin354

1100θ =

(ii) 251 or 251.2(……. 1ft 270 − their LJK�

final ans.

6 (a) 6 2

5 11

− −

2

B1 for at least 2 elements correct in a 2 x 2 matrix

(b) 15 7

7 8

−

2

B1 for at least 2 elements correct or

M1 for

4 1

1 3

−

4 1

1 3

−

soi

(c) 5 01

7 210

− −

− oe

isw

2

B1 for det B = 10− soi or 5 0

7 2

− −

(d) 0 0

0 0

1

(e) 0 0

7 7

−

2 B1 for 1 0

0 1

soi

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Page 4 Mark Scheme Syllabus Paper

Cambridge O Level – May/June 2016 4024 21

© Cambridge International Examinations 2016

Question Answers Mark Part Marks

7 (a) 4.53 to 4.54 4 B2 for BOC = 52 or after B0

B1 for ˆABC = 90 or triangle OBC isosceles or BAC�

= 26

M1 for

522 5

360π× ft

(b) (i) 101 or 32π or 100 to 100.6 2 M1 for π(16.52 ) or 215.5

(ii) 0.87 to 0.871 3 B1 for π15.52 or 44πr2 and

M1 for

2

2 15.5 650

44r

π

π

−

=

(iii) 7 3 M1 for π15.52d = 500

A1 for 0.66 to 0.663

8 (a) (i) 1.92− (3…… 1

(ii) 8

5p +

2 M1 for 8

5pq= + or 8 5pq q= − or

85p

q= −

(b) (i) H and h correctly derived 2 M1 for correct substitution in the formula for the area of a trapezium.

(ii) 75

( 1)(2 3)x x− +

correctly

derived

3 M1 for 15(2 3) 30( 1)

( 1)(2 3)

x x

x x

+ − −

− +

soi

B1 for 30 45 30 30x x+ − + soi

(iii) (a)

Equation correctly derived.

2 B1 for 75

1.5( 1)(2 3)x x

=

− +

(b)

4.90

2 B1 for 21 4 2 ( 53)− × × − soi or

B1 for

1 425

2 2

their− ±

×

soi

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Page 5 Mark Scheme Syllabus Paper

Cambridge O Level – May/June 2016 4024 21

© Cambridge International Examinations 2016

Question Answers Mark Part Marks

9 (a) (i) 5.38 to 5.39 or √29 2 M1 for 2 2 2( ) 2 5AC = +

(ii)

0.517 to 0.518

2 M1 for sin152

CE=

oe

(iii)

68.8 to 68.9

4 M1 for cos152

AF=

oe or BC²=BE²+ (their CE)² or

any complete alternative method

A1 for 1.932 and

M1 for tan ˆFAE =

5

2cos15 oe or

5

( )their AF

(b) (i) 80.9(4…. Or 81 3 B1 for 102 = 62 + 92 − 2×6×9×cosθ or

B2 for cosθ =

2 2 29 6 10

2 9 6

+ −

× ×

(ii) > 1

10 (a) (2) (4) 14 54 84 98 (100) 1

(b) Correct curve 2 P1 for at least 5 correct plots

(c) (i) 195 ft 190 ⩽ and < 200 1

(ii) 50 –75 2 B1 for one quartile correct in ranges 225 to 235 or 160 to 175

(d) Correct curve 4 P3 for at least 4 correct plots or

B1 + B1 for any two correct points soi.

(e) 92 ft 1

(f) B 15 ft A 1ft Their 90 – 75

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Page 6 Mark Scheme Syllabus Paper

Cambridge O Level – May/June 2016 4024 21

© Cambridge International Examinations 2016

Question Answers Mark Part Marks

11 (a) 6

2

−

1

(b) (i) 8

4

2 B1 for 8

k

or 4

k

(ii) 8

4

− −

ft

1

(iii)

8.94 or 8.94 to 8.95 or √80 oe

2 M1 for 2 2( 8) ( 4)− + − oe ft

(c) (i) Triangle vertices (5,4), (13,0), (9,8)

2 B1 for 2 correct

(ii) Triangle F (5,4), (7,3), (6,5) 1

(iii) Rotation

180 Centre (5,4)

3 B2 for Rotation with either centre or angle. B1 for Rotation.