Maths Paper 2 Examinations With simplified answers

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Somaliland National Examination and Certification Board

Certificate of Secondary Education Maths paper 2 examinations

REPUBLIC OF

SOMALILAND Preface

The administration of Somaliland national examination and certification board (SLNECB) authorized the compilation, production and publication of Somaliland national maths paper 1 examination books with simplified answers and solutions for secondary students. The production of these books came after a success through great and selfless effort, made by SLNECB and the teacher who compiled it. This was necessitated by the tremendous needs of the whole Somaliland secondary students in order that they get equal rights and experience in performing national examinations in particular and the international exanimations in general. I hope all Somaliland students and teachers will enjoy using these valuable books.

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Somaliland National Examination and Certification Board

Certificate of Secondary Education Maths paper 2 examinations

Republic of

Somaliland SLNECB

Maths Paper 1 Examinations With simplified answers

First edition 2014 All rights reserved. No part of this book can be reproduced, stored, recorded or transformed in any means, electronic, photocopying or otherwise without permission of the Somaliland National Examination and certification board.

Compiled by: Khadar X. Yusuf C/laahi (Khadar Al-jazeera)

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Somaliland National Examination and Certification Board

Certificate of Secondary Education Maths paper 2 examinations

Acknowledgement Somaliland National Examination and Certification Board (SLNECB) is very grateful to Khadar X. Yusuf C/laahi (Khadar Al-jazeera) for his selfless effort in the compilation of national maths paper 2 examinations and for answering all maths paper 2 examination questions in a clear and understandable manner. SLNECB would like to thank Muktar Awmuse Hassan (Cawl) for his crucial role in organizing and improving this book. SLNECB would also like to thank Khadar Abdi Ibrahim (bukhaari) for his overall typing, designing and contribution to this book. In general, SLNECB would also like to appreciate any person who provided direct or indirect contribution and effort of typing, suggestions, encouragement and improvement of this book.

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Certificate of Secondary Education Maths paper 2 examinations

Contents Chapter 1: Algebra ........................................................................................................... 5 1.1. Multiple choices: ................................................................................................... 5 1.2. Structured questions .......................................................................................... 19 Chapter 2: Functions ..................................................................................................... 30 2.1. Multiple choices .................................................................................................. 30 2.2. Structured questions .......................................................................................... 33 Chapter 3: Inequality...................................................................................................... 37 3.1. Multiple choices .................................................................................................. 37 3.2. Structured questions .......................................................................................... 40 Chapter 4: Plane geometry ........................................................................................... 42 4.1. Multiple choices .................................................................................................. 42 4.2. Structured questions .......................................................................................... 53 Chapter 5: Area .............................................................................................................. 60 5.1. Multiple choices .................................................................................................. 60 5.2. Structured questions .......................................................................................... 64 Chapter 6: Analytic geometry ....................................................................................... 70 6.1. Multiple choices .................................................................................................. 70 6.2. Structured questions .......................................................................................... 74 Chapter7: Solid geometry ............................................................................................. 80 7.1. Multiple choices .................................................................................................. 80 7.2. Structured questions .......................................................................................... 82 Chapter 8: Vectors and transformation ....................................................................... 88 8,1. Multiple choices .................................................................................................. 88 Chapter 9: Trigonometry ............................................................................................... 92 9.1. Multiple choices .................................................................................................. 92 9.2. Structured questions .......................................................................................... 97 Chapter 10: Calculus ................................................................................................... 110 10.1. Multiple choices .............................................................................................. 110 10.2. Structured questions ...................................................................................... 115 ANSWERS .................................................................................................................... 136 THE END ...................................................................................................................... 200

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Somaliland National Examination and Certification Board

Certificate of Secondary Education Maths paper 2 examinations

Chapter 1: Algebra 1.1. Multiple choices: 1. I start with x, then square it, multiply by 3 and finally subtract 4. The final result is A

(3x)2 - 4

B

( 3x – 4 )2

C

3x2 - 4

D

3( x – 4 )2

2. Given that

2 x ( x − 1) 11 = , − 15 3 5

then x is

A. 6 B. 10 C. 12 D. 16 4 5

1 3

3. The exact value of 4 × 8 ÷

1 2

is

A. 80 B. 64

8 15

C. 20 D. 16

2 15

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Somaliland National Examination and Certification Board

4. Given that h=

Certificate of Secondary Education Maths paper 2 examinations

k (1 − p ) , then making p the subject, 2 p +1

k (1 − p ) − h 2h k −h B. p = 2h + k k C. p = 2h + k k+h D. p = 2p − k

A. p =

5. ( 3y – 2)(2y+3) is the same as A. 6y2 + 5y – 6 B. 5y2 + 5y + 1 C. 6y2 – 13y - 6 D. 5y2 –13y + 1 6.

1 3

2

+ = 5

2

A.

8

3

B. C. D. 𝑚𝑚

8

3

15

11 15

7. �� � = 𝑎𝑎, 𝑥𝑥 = 𝑥𝑥

A. a2m

B. a -m C. D.

𝒎𝒎

𝒂𝒂𝟐𝟐

𝒂𝒂𝟐𝟐 𝒎𝒎

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Somaliland National Examination and Certification Board

Certificate of Secondary Education Maths paper 2 examinations

8. 𝑡𝑡he equation 𝑎𝑎𝑥𝑥 2 + 𝑥𝑥 − 6 = 0 ℎ𝑎𝑎𝑎𝑎𝑎𝑎ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

𝑥𝑥 = −2 𝑤𝑤ℎ𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎?

A. 2 B. −2 C. √2 D. 1

9. How many prime numbers are there between 30 and 40? A. 0 B. 1 C. 2 D. 3 10.

Work out 24÷ 8 + 4 × 5 − 2 A. 8

B. 21 C. 4 D. 48 11.

The solutions of 2x+3y=7 and 3x+2y=8 are A. 2, 1 B. 4, 2 C. 6,3 D. 1,3

12.

The solutions for the equation x 2 - 2x = 15 are A. (-5, 3) B. (-3, 5) C. -15 D. (-5, 0) 7

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13.

Simplify: A. B. C.

14.

D.

2 4

Certificate of Secondary Education Maths paper 2 examinations 1 2

𝑥𝑥+

4𝑥𝑥+2

2𝑥𝑥+1

4𝑥𝑥+2 1 4 1 2

The factors of 𝑓𝑓(𝑥𝑥) = 5𝑥𝑥 2 − 20 are A. 5(𝑥𝑥 + 2)(𝑥𝑥 − 2) B. 5(𝑥𝑥 2 − 20)

C. 5(𝑥𝑥 2 − 10) D. 5(𝑥𝑥 − 2)2 .

15. The sum of 4 consecutive numbers is 38. What are the numbers. A. 11,10,9,8

B. 9,7,12,10 C. 14,10,8,6

D. 7,9,10,12. 16. is:

The coefficient of "X" in the expansion of (𝑥𝑥 + 3)(𝑥𝑥 − 1) A. 2 B. 4 C. -1 D. 3.

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17.

Certificate of Secondary Education Maths paper 2 examinations

If 𝑚𝑚 = √2 and 𝑛𝑛 = √8 , what is the value of (𝑚𝑚 + 𝑛𝑛)2 ? A. 54 B. 36 C. 18 D. 9.

18. Solving 28 − 4𝑛𝑛 = 56, hence, what is the value of n ? A. 7

B. −7 C. 14

19.

D. −14.

The value of x in the equation 2x + 4= 6 is A.

1

B. 2 C. 5 D. 10

20.

One of the factors of 25x2 + 5x – 2 is: A.

5x - 2

B. 2x – 1 C. 5x+1 D. 21.

5x – 1

Simplifying 3 + 7( 2x - 5) is A. 14x – 31 B. -14x + 32 C. 14x -32 D. 14x + 32 9

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22.

Certificate of Secondary Education Maths paper 2 examinations

making x the subject of the equation 2√𝑥𝑥 = a , x will be

equal to:

A. B.

𝑎𝑎2 2

𝑎𝑎 2 4

C. √𝑎𝑎

D. 4a2

23.

The value of 2x2 - 3y when x = 3 and y = 2 is: A. 14 B. -14 C. -12 D. 12

24.

The solution of 23x + 6 = 10: A. 6

B. 10 C. 18 D. 24

25.

If x is an integer, for what set of values is 4 – 2 x > x – 5 A. x > 3 B. – x < - 3 C. x < 3 D. x > - 3

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26.

Certificate of Secondary Education Maths paper 2 examinations

What value of x makes the equation below true? X +6=8 9

A. 2 B. 18 C. 66 D. 126

27.

The solution set of x2 –5x –24 = 0 is A. {8, –3} B. {5, –3} C. {–3, 8, } D. {4, 6}

28.

If a = 3 and b = - 2, what is the value of a2 + 3ab – b2 A. 5 B. -5 C. 13 D. -13

29.

Factor the expression: x2+ x - 12. A. (x – 4 ) (x + 4 ) B. (x – 2 ) (x + 6 ) C. ( x + 4) ( x – 3) D. ( x - 4) ( x + 3 ) 11

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30.

Certificate of Secondary Education Maths paper 2 examinations

51.92𝑥𝑥202

The value of

4.93

is closest to

A. 20. B. 200. C. 2,000. D. 20,000. 31.

If - 3xy = 45, which of the following statements is true? A. Only x is negative B. Only y is negative C. Both x and y are negative D. Either x or y is negative.

32.

Compute: 10 - (3)2 + (-3) X 2 A. -5 B. 4 C. -2 D. 2

33. If the expression x2– kx – 12 is equal to zero when x= 4, what is the value of k ? A. 7 B. 1 C. -1 D. -3 34.

If x + 2y = 1, and 2x + y = 5, then x + y= ? A. 2 B. 3 C. 4 D. 5 12

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35.

Certificate of Secondary Education Maths paper 2 examinations

The solution to the following pair of simultaneous

equations is: 9 x − 4 y = −14

4x − 9 y = 1 A. (−2, −1) B. (2, 1) C. (1, 2) D. (−1, −2 )

36.

The graph which best represents y = x2 – 2x – 1 is:

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37.

Certificate of Secondary Education Maths paper 2 examinations

Solution of the equation

A.

3x = 2 is 4

3 8

B. 8 C. 2

2 3

D. 3 38.

The solution to the following pair of simultaneous

equations

is:

3 x − y = 18 4 x + y = 10

A. (5, −3) B. (−3, 5) C. (4, −6) D. ( 4, 6)

39.

When factorized 36j – 48 becomes:

A. 12(3j – 4) B. 12(24j – 36) C. 9(4j – 7) D. 8(4j – 6)

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Certificate of Secondary Education Maths paper 2 examinations

2x x + When simplified 5 2 becomes:

40.

41.

A.

3x 7

B.

3x 10

C.

9x 7

D.

9x 10

What is the slope of the line 5y = –3x + 10 ? A. -3 B. − C.

3 5

5 3

D. 2 42.

The value of 23.1+ 0.231+6.32 is A. 2.9651 B. 96.531 C. 7.782 D. 29.651

43.

Each band member had 12 tickets to sell to the spring

concert. •

Jama sold 2/3 of his tickets.

•

Kule sold 1/2 of hers.

•

Mohamed sold 3/4 of his.

•

Suad sold 7/12 of hers. 15

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Certificate of Secondary Education Maths paper 2 examinations

Who sold the most tickets? A. Jama B. Kule C. Mohamed D. Suad 44.

Solve for x,

x−9 = 5 5

A. x = 45 B. x = 25 C. x = 9 D. x = 34 45. Solving 2(3a – 7) + 5(1 – 2a) = 2(a + 6) + 3(1 – 3a) yields: A. 4 B. 6 C. 7 D. 8 46. Solving 28 – 4n = 56 yields: A. −7 B. 7 C. −4 D. 4 47. Solve for x:

3 ( x + 2) = x − 4 5

A. 8 B. 15 C. 13 D. 23 16 ©SLNECB

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Certificate of Secondary Education Maths paper 2 examinations

48. The solutions to the equation x 2 − 6 x + 5 = 0 are: A. x = 1 and x = 5 B. x = 1 and x = −5 C. x = −1 and x = 5 D. x = −1 and x = −5 49.

If x − 4 = 7 , what is the value of X? A. 11 B. 45 C. 18 D. 53

50. What is the x-intercept of the line defined by -2x+3y=12? A. 6 B. 4 C. -4 D. -6 51.

If e = 3

3 1 and f = 4 then 5e – 3f equals: 3 4

A. 2

7 12

B. 3

1 12

C. 2

5 12

D. 2

1 12

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Certificate of Secondary Education Maths paper 2 examinations

52.

53. If 3ax+b=c, then x equals A. c − b + 3a B.

c −b 3a

C. c + b − 3a D.

b−c 3a

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Certificate of Secondary Education Maths paper 2 examinations

1.2. Structured questions 1. Simplify : 3 – 9 + 14 with the help of this number line.

…………………………………………………………………………… …………………………………………………………………………… …………………………………………………………………………… …………………………………………………………………………… …………………………………………………………………………… …………………………………………………………………………… 2. Factorize: 3 x2 + 10 x + 3 ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……..………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… …………………………………………………………………………

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Certificate of Secondary Education Maths paper 2 examinations

3. Solve : 4x - 1

- 2x - 3 3

= 1 2

…………………………………………………………………………… …………………………………………………………………………… …………………………………………………………………………… ………………………………………………………………………..…… …………………………………………………………………………… …………………………………………………………………………… …………………………………………………………………………… ………………………………………………………………….. 4. I think of a number and multiply it by 5. Then I add 15 and the result is 100. Let the number be x and so find the number I thought of ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……..………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………….

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Certificate of Secondary Education Maths paper 2 examinations

5. a) A man is now five times as old as his son. In five years time he will be three times as old as his son. Find their present ages. ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ………………..…………………………………………………… ……………………………………………………………………… ……………………………………………………………………… b) Solve the quadratic 5x2 + x – 4 = 0 ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ………………..…………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… …………………………………..………………………………… ……………………………………………………………………… ………………………………………………………………………

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Certificate of Secondary Education Maths paper 2 examinations

6. Solve the simultaneous equations

3x + 2y = 13

5x – 3y = 9 ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……..………………………………………………………………… …………………………………………………………………………

7. Solve for x in the equation:

6x− 4 3

-

2x− 1 2

=

6 − 5x 6

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… …………………………………………………………………………

8. Aysha received X marks in test, Zahra received 4 more marks than aysha. a) i)

Write down zahra’s marks in terms of X i.

……………………………………………………… ……………………………………………………… ……………………………………………………… ………………………………………………………

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ii.

Certificate of Secondary Education Maths paper 2 examinations

When aysha subtracts 7 from her marks and squares the result, her answer is 1 more than Zahra’s mark.write down an equation in X, show that it simplifies to X2 -15X+44 =0 ……………………………………………………… ……………………………………………………… ……………………………………………………… ……………………………………………………… ……………………………………………………… ………………………..……………………………… ………………………………………………………

b) Solve the equation X2 -15X+44 =0 ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ………………..…………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………..

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Certificate of Secondary Education Maths paper 2 examinations

9. Solve each of the following x a) 3(x – 1 ) = 18

b) x : 12 = 6 : 8

c) 2x = 16

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……..………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……………..………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……………………..………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……………………………..………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… …………………………………………………………………………

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10.

Certificate of Secondary Education Maths paper 2 examinations

2x ̶ 5y = 24

Solve algebraically

7x + 8y = 24

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……..………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……………………………………………………………………….. ………………………………………………………………………… ………………………………………………………………………… 11.

Change the subject of the formula P = 2(L + B ) to L

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……………………………………………………………………….. 12.

4 3 2 Evaluate  3 − 2  × 1  5

4

3

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……………………………………………………………………… 25 ©SLNECB

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13.

4 Evaluate   3

Certificate of Secondary Education Maths paper 2 examinations

−2

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… …………………………………………………………………………. 14.

The graph below models the cost of manufacturing

calculators.

Derive an equation that shows the relationship between the number of calculators, n and the total cost, C? ……………………………………………………………………… ……………………………………………………………………… 26 ©SLNECB

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Certificate of Secondary Education Maths paper 2 examinations

……………………………………………………………………… ……………………………………………………………………… …………………………………………………………….. 15.

p+q S = A formula was given as p . Rearrange the formula

to make p the subject ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……………………………………………………………………… 16.

Solve for x in each of the following

2(x - 1) = 6 ……………………………………………………………………………… ……………………………………………………………………………… ……………………………………………………………………………… ……………………………………………………………………………… ………………………………………………………………………………

17.

Solve the equations i)

3y+7=28 ………………………………………………………… ………………………………………………………… ………………………………………………………… ………………………………………………………… ………………………………………………………… 27

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ii)

Certificate of Secondary Education Maths paper 2 examinations

2(3p+2)=19 ………………………………………………………… ………………………………………………………… …………………………………………………………

18.

Solve the quadratic 6 x + 13 x − 5 = 0 2

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……..………………………………………………………………… ………………………………………………………………………… 19.

Make h the subject of the formula D =

3h 2

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……………………………………………………………………….. 20.

Solve the simultaneous equations

4x + y = 5 2 x − 3 y = 13

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……..………………………………………………………………… ………………………………………………………………………… 28 ©SLNECB

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21.

Certificate of Secondary Education Maths paper 2 examinations

Factorize the expression x + x − 12 2

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……..………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… 22.

Solve for x:

2x 1 7x − 2 + = ? 5 3 15

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……..………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… 23.

Simplify

25 x − 125 x 2 − 25

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… 29 ©SLNECB

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Certificate of Secondary Education Maths paper 2 examinations

Chapter 2: Functions 2.1. Multiple choices 1. If f(x) = 3x + 1 and g(x) = 1 / 3x then fg (½) equals A. 2½ B. 1/6 C. 3½ D. 3 2. If f(x) =

x +1 , then f 3

-1

( 3 ) equals

A

6

B

7

C

8

D

9

3. f ( x) = x2 – x + c

and f(6)= 18, then f(- 4) is

A. 8 B. – 2 C. - 6 D. - 32. 4. If f(x) = x 2 − 3, then f(3) − f(−1) is A. 8

B. 10 C. 5 D. 9 30 ©SLNECB

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Certificate of Secondary Education Maths paper 2 examinations

5. If f(x) = 3x 2 − 2x find out f(−2) A. 16 B. 12 C. -8 D. -16. 6. If f(x) =

1−x x

, find f −1 (0) ?

A.

1 2

B. 1 C. 0

D. −1.

7. f(x) = 3x + 5 and g(x) = 2x – 1 so

f(g(2)) is

A. 11 B. 3 C. 21 D. 14 8. Given that f ( x) = 3 x 2 − 4 and g ( x) = 2 x − 6 ,what is g ( f ( 2) ? A. -2 B. 6 C. 8 D. 10 31 ©SLNECB

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Certificate of Secondary Education Maths paper 2 examinations

9. If f(x)=3x+2, then f

−1

(3) is

A. 11 B.

1 2

C.

1 3

D. 1

10.

What is the inverse of the function y = 3x − 2 ? A. y = −3x + 2 B. y =

x−2 3

C. y =

x+2 3

D. 3 y = 2 x

32 ©SLNECB

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Certificate of Secondary Education Maths paper 2 examinations

2.2. Structured questions Show that f {g (x )} ≡ g { f (x )}, when f(x) =

1.

and g (x) =

2x + 1 3

3x − 1 2

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……..………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… 2. f( x ) = 3-2X

and

X+1

g(x) =

a) Find the value of f (-3/4)

4

……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… …………………………………………………………………… b) Find the value of g(-1) ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ………………………………………………………………………

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Certificate of Secondary Education Maths paper 2 examinations

c) Find the inverse function f −1 (x).

……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………

d) find the composite function fg(x) ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… …………………………………………………………………….. e) find the composite function gf(x) ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… …………………………………………………………………….. 3. Given f(x) = 4X2+3 , find the inverse function f-1 (x)

………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………..……………………… ………………………………………………………………………… ………………………………………………………………………… 34

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Certificate of Secondary Education Maths paper 2 examinations

4. Given f(x) = 3x 2 + 5x and g(x) = 2x − 5.calculate a. f(g(x))

………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………

b. f(g(2))

………………………………………………………………… ………………………………………………………………… ………………………………………………………………… …………………………………………………………………

c. g(f(x))

………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………………………………….

d. g�f(−1)�

………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… …………………………………………………………………

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5. Given that f(x) = x2 + 3 a. Evaluate f(-4) ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… …………………………………………………………… b. Find x when f(x) = 54 ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… …………………………………………………………………

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Chapter 3: Inequality 3.1. Multiple choices 1. Solve the inequality =4𝑥𝑥 − 4 < 20. A. 𝑥𝑥 < 4

B. 𝑥𝑥 > 4

C. 𝑥𝑥 < 6 D. > 6

2. Muna tried to solve the inequality 13 -3p−15 is: A. q>−1 B. q 1 , 2x – 3y ≤ 5 and x < 6. ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… …………………………………………………………………………

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3. Solve the inequality

Certificate of Secondary Education Maths paper 2 examinations

2x 3

+2 ≤

5x 2

- 9

………………………………………………………………………… ………………………………………………………………………… …………………………………………………………………………

4. Solve the inequality 4x -2 < -2x+10? ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……..………………………………………………………………… ………………………………………………………………………… …………………………………………………………………………

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Chapter 4: Plane geometry 4.1. Multiple choices 1. A trapezium of area 105cm2 has a parallel sided of length 5cm and 9cm. how far apart are the parallel sides A. 15𝑐𝑐𝑐𝑐

B. 12𝑐𝑐𝑐𝑐

C. 21

D. 10𝑐𝑐𝑐𝑐

2. AD and CE are parallel, AC = CB . Find X.

A. 40 0 B. 50 0 C. 70 0 D. 110 0 3. A point P lies inside a square EFGH of side 4 cm. P is

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• within 3.5 cm of E, • nearer to HG than to EF, • nearer to E than to G. Then P lies in region A. 1 B. 2 C. 3 D. 4 4. The sum of degrees of interior angles of a regular octagon is: A. 360° B. 540° C. 720° D. 1080°

5.

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6.

This wooden frame is cut from a 2 meters length of wood 2 cm by 1 cm. The minimum length of wood which must be used is A. 28 cm B. 36 cm C. 44 cm D. 48 cm 7. The rectangle shown below has a width of 2.5 m and a perimeter of 13 m. 2.5 m

What is the area of the rectangle? A. 4 m2 B. 8 m2 C. 10 m2 D. 10.5 m2

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8. What values of aandb make quadrilateral MNOP a parallelogram?

A.a= 1, b= 5 B. a=5, b= 1 C. a =

11 34 , b= 7 7

D. a =

34 11 , b= 7 7

9. The value of the pronumeral in the pair of similar triangles in the figure below is:

A. 36 B. 32 C. 35 D. 37

1 3

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10. What is the measure of angle A in the given figure?

A. 60o B. 120o C. 50o D. 30o

11. In the figure shown on the right, lines l and m are parallel, and triangle ABC is isosceles. What is the measure of angle ABC?

A. 40° B. 50° C. 60° D. 70° 46 ©SLNECB

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12. If AD = BD = BC and the measure of angle ADB is twice the measure of angle BDC, what is the measure of angle ABC?

A. 70° B. 80° C. 90° D. 100°

13. What is the length x of the altitude of the triangle shown below?

A. 20 cm B. 25 cm C. 30 cm D. 40 cm

14. In the trapezium, the size of the angle xis

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A. 45 B. 30 C. 20 D. 15 15. What is the mathematical name given to the line AB?

A. radius B. chord C. diameter D. arc 16. From the diagram calculates the angle e.

A.

250

B.

350

C.

600

D.

300 48

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17.

Certificate of Secondary Education Maths paper 2 examinations

Calculate the size of the angle marked c and give a

reason for your answer

A. 95° as vertically opposite angles add up to 180°. B. 85° as the angles are corresponding angles. C. 95° as the angles are corresponding angles. D. 85° as alternate angles add up to 180°. 18. Layla made the game board shown below. Small marbles start at the top and always roll down the marked paths until they land in one of the circles at the bottom.

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How many different paths can a marble take to land in Circle 2? A. 2 B. 3 C. 4 D. 6 19. What is the value of x in the given triangle? A. 20 B. 10 C. 30 D. 6.7 20. The diagram below shows the construction of the perpendicular bisector of AB .

Which statement is NOT true? A. AC=CB B. AC=2AB 50 ©SLNECB

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1 2

C. CB = AB D. AC+CB=AB

21. In the diagram below, JL = KM .

Based on the angle measures in the diagram, what is the value of y?

A. 75 B. 90 C. 95 D. 120 22. In the diagram below, EH intersects FJ at point G. Based on the angle measures in the diagram, what is the value of x ? A.

34

B.

38

C.

43

D.

56

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23. What is the measure of an interior angle of a regular octagon A. 450 B. 600 C. 1200 D. 1350 24. In the diagram below, circle O has a radius of 5, and CE = 2. Diameter AC is

perpendicular to chord BD at E.

What is the length of BD ? A. 12 B. 10 C. 8 D. 4

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4.2. Structured questions 1. Find the marked angles “ x and y ”

……………………………………………………………………………… ……………………………………………………………………………… ……………………………………………………………………………… ………………………………………………………………………….......... ............................................................................................................... 2. In the diagram, OS = 4s, OT = 4t and P is the point on ST such that

1 TP = TS . 3

T P

4t

O

S 4s

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Express in terms of s and t, a) ST ………………………………………………………… ………………………………………………………… ………………………………………………………… b) SP ………………………………………………………… ………………………………………………………… ………………………………………………………… c) OP . ………………………………………………………… ………………………………………………………… ………………………………………………………… 3. The interior angle of regular polygon is 480 more than the interior angle of a regular hexagon? a) Find the size of the interior angle of the polygon? ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… b) Find the number of sides? ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… 54 ©SLNECB

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4. A mason on the top o roof at A measures sides AB=10m and AC=12m. And angle A as 600.

a) Calculate BC …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… b) If angle A is decreased to 450, what is the decrease in BC? …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………… …………………………………………………………………….

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5. The length of a classroom is 3m longer than its breadth. If its area is 270 m2, find the dimensions of the room. …………………………………………………………………………… …………………………………………………………………………… …………………………………………………………………………… …………………………………………………………………………… 6. The line l is a line of symmetry for figure MNOPQ. What is the measure of angle

NOP?

……………………………………………………………………………… ……………………………………………………………………………… ……………………………………………………………………………… ……………………………………………………………….. 7. Write down the special names of the following shapes

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8. A, B, C and D are four points on the circumference of a circle centre O. the tangent FG meets the circle at C. Angle BCF 720 and angle ABC = 500.

Find each of the following angles a. Angle AOC ……………………………………………………………………………… ……………………………………………………………………………… b. Angle ADC ……………………………………………………………………………… ……………………………………………………………………………… ……… ……………………………………………………………………… c.

Angle BAC

……………………………………………………………………………… ……………………………………………………………………………… ……………………………………………………………………………… 57 ©SLNECB

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d. Angle OAB ……………………………………………………………………………… ……………………………………………………………………………… …………………………………………………………………………… 9. What is the perimeter of the figure shown below?

……………………………………………………………………………… ……………………………………………………………………………… ………………………………………………………………………………

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10.

In the diagram E, F and G are points on the circle, centre O. Angle 0

EFG is 54 . MN is a tangent to the circle at point G. i) Calculate the size of angle FGE ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ii) Calculate the size of angle EGN

………………………………………………………………… ………………………………………………………………… ………………………………………………………………… …………………………………………………………………

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Chapter 5: Area 5.1. Multiple choices 1. The shaded area, in cm2,is

A

16 - 2 π

B

16 - 4 π

C

4 π

D

64 - 8 π

8 cm

8 cm

2. In the figure all angles are right angles. 1

2 4 2 1 1

1

The area, in square units, is A. 7 B. 8 C. 9 D. 1 60 ©SLNECB

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3. The area, in square units, of the quadrilateral with vertices (0,0), (6,0), (6,6), (2,5) is A. 20 B. 27 C. 29 D. 36 4. The figure shows two concentric circles, of radii 10 cm and 20 cm, together with parts of a pair of perpendicular diameters.

4

3 5

1

2

The area of the region marked 3 is A. 5 π B. 75 π C. 100 π D.

1 (400 π –100) 4

5. Given the area of the rectangle is 350cm2, what is the shaded area?

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A. 56cm2 B. 126cm2 C. 147cm2 D. 210 cm2 6. The rectangle shown below has a width of 2.5 m and a perimeter of 13 m. 2.5 m

What is the area of the rectangle? A. 4 m2 B. 8 m2 C. 10 m2 D. 10.5 m2 7. What is the area, in square units, of the trapezoid shown below?

A. 37.5 B. 42.5 62 ©SLNECB

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C. 50 D. 100 8.

Ali cut four congruent triangles off the corners of a rectangle to make an octagon shown below.

B

What is the area of the shaded octagon? A.

128 cm2

B.

136 cm2

C.

140 cm2

D.

152 cm2

9. In the figure below, the perimeter of the equilateral triangle is 24 cm. What is the area of the square? A. 32 sq. cm. B. 476 sq. cm. C. 12 sq. cm. D. 64 sq. cm.

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10. What is the area of the UNSHADED part of the rectangle below?

A.

19 000 m2

B. 45 0000 m2 C. 28 000 m2 D. 26 000 m2

5.2. Structured questions 1. Calculate the shaded area. ( Take π =

20 cm

22

7

)

14 cm

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14 cm ……………………………………………………………………………… ……………………………………………………………………………… ……………………………………………………………………………… ………………………………………………………………..……………… ……………………………………………………………………………… ……………………………………………………………………………… …………………………………………………………………………… ……………………………………………………………………………..

2. The cross – section of an oil drum is shown. The chord AB subtends an angle. of 120 0 at the center O. The radius of the circle is 30 cm. Calculate:

i) The shaded area of the cross – section. ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………. 65 ©SLNECB

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ii) The volume of oil ( in cm3 ) in the drum, if the drum is 2m long. ( Take π = 22 7 and Sin 1200 = Sin 600 = 0.8660 ) ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………..…………………………… …………………………………………………………………

3. A square is inscribed in a circle of radius 14 cm. Find: a) the area of the square ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… …………………………………………………………………

b) The area shaded. ( Take π =

22 ) 7

………………………………………………………………… ………………………………………………………………… ………………………………………………………………… …………………………………………………………………

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4.

Students at SOS school decide to have T-shirts made with a blue "V" inside a gold rectangle as shown in the diagram above. The costs are as follows: •

plain T-shirt $8.50

•

blue colouring $0.02 per square cm

•

gold colouring $0.04 per square cm

a) What is the area of the blue "V" in the diagram above? Show your work. ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… …………………………………………………………………….. b) What is the area coloured gold. 67 ©SLNECB

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……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………….. ………………………………………………….. c) What will be the total cost for each T-shirt shown above? Show your work. ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… …………………………………………………………………….

5. A square ABCD is inscribed in a circle of radius 6 cm. Find the shaded area. Take π = 22 7

………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… …………………………………………………………………

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………………………………………………………………… …………………………………………………………………

6. Find the area of the shaded region in the figure below, given that AD=15cm, BE =3cm, AB=8cm and angle DAB= angle EBC=900. D E 15cm 3cm A

C 8 cm

B

………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………..…………………………… ………………………………………………………………… ………………………………………………………………… …………………………………………………………………

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Chapter 6: Analytic geometry 6.1. Multiple choices 1. The line y = 2x – 1 cuts the x –axis at P. The coordinates of P are A

( 0, -1 )

C

(-2 ,0)

1

B

(

D

( - 1, 0 )

1 2

,0)

2. The equation of the line joining ( 12, 32) and ( -8, 2) is A. 15x – 2y –114 =0 B. 2y = 3x + 28 C. 2y + 3x + 20 = 0 D. 3x + 2y = 100 3. The gradient of the line 2x+y = 3 is A. 3 B. – 2 C. – D.

1

1 2

2

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4. The line y = 4x – 8 meets the x – axis at the point P. What are the coordinates of P? A. ( 0,8) B. (2,2) C. (2,0) D. (0, 2) 5. Given the points A(3,1,2) and B(1,3,3), find |AB|, the modulus of AB  − 2   A.  2   1   

 2    B.  − 2   −1  

C. 5 D. 2 6. What is the slope (gradient) of this line?

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7. The table of values for the rule y = 6x – 11 is

A x

–5

–4

–3

–2

–1

0

y

–41 –35 –29 –23 –19 –11

x

–3

y

–29 –23 –17 –11

c

x

–1

0

y

–17 –11

1

2

3

4

–6

0

7

13

0

1

2

3

–6

–1

7

B –2

–1

0

1

2

–5

1

D.

x

–2

–1

y

–23 –17 –11

8. Which of the following points does not lie on the line y = 3x – 5? A. (3, 4) B. (0, -5) C. (-1, -8) D. (2, 0) 9. Which one of the following represents the equation of the straight line shown?

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A. y = x – 2 B. y = 2x – 2 C. y = x – 1 D. y = 2x – 1 10.

The coordinate of the mid-points of the line (-4,6) and (1,-

6) are

A. (1.5, 0) B. (-1.5, 0) C. (-5, 1) D. (2, 6) 11.

Rectangle MNPQ has diagonals that intersect at point X. Which of the following represents point X? 7 A. ( , − 2) 2 13 B. ( , 4) 2 13 C. (− , − 4) 2 7 D. (− , 2) 2

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6.2. Structured questions 1. .(a) A toad is tied by a rope 10.5 m long to a fixed wire 15m long. A ring at the end of the rope can slide along the wire.

i) Sketch the locus of the toad’s head if the toad moves so as to keep as far as possible from the wire, within the given frame. ii) Use the value 3 1 7 for π (pie) to calculate approximate length of the locus. ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………..…………………………… …………………………………………………………………

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2. Write the equation of the line passing through ( 4,4 ) and parallel to the line 2x – y + 1 = 0. ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… …………………………………….…………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… 3. Find the equation o the straight line which passes tough the point (1,3)And I perpendicular to the line 2y-3x = 7. ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………..…………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… …………………………………………………………………

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4. Find the equation of the tangent of the curve 𝑦𝑦 = 𝑥𝑥 3 − 3𝑥𝑥 2 + 2𝑥𝑥 − 1 at the point (3,5)

………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………..…………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… …………………………………………………………… 𝑥𝑥 2

5. The equation is

16

−

𝑦𝑦 2 9

=1

a) Find the coordinates of the vertices of the hyperbola ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………..…………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… 76 ©SLNECB

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b) Find the coordinates of its foci? ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………..…………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………………………………… 6. The equation of the parabola is (x+2)2 =4 (y+1) Find a) The coordinates of the vertex ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………..…………………………… ………………………………………………………………… ………………………………………………………………… b) The equation of the directrix ……………………………………………………………………… ……………………………………………………………………… ……………………………………………………………………… ………………………………………………………………… 77 ©SLNECB

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7.

A straight line has been drawn on the grid. a) Find the equation of the straight line. ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……..………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… …………………………………………………………………

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b) Find the gradient of the line x+3y=6. ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… c) Find the equation of the line which is parallel to the line with the equation 4x+2y=8 and passes through the point with coordinates (0,-1). ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ……..………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… ………………………………………………………………………… …………………………………………………………………………

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Chapter7: Solid geometry 7.1. Multiple choices 1. Two similar cylinders have heights of 3 cm and 6 cm respectively. If the volume of the smaller cylinder is 30 cm2, find the volume of the larger cylinder.

H = 6 cm Vol= ? h =3cm vol= 30cm3

A

120 cm3

B

240 cm3

C

60 cm3

D

180 cm3

2.

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3. The surface area of a closed cube of side x cm

is:

A. 4x B. 4x4 C. X3 D. 6x2 4. The sector shown is assembled to form a cone. The base circumference of the cone is:

A. 47 B. 153

2700

7 cm

C. 37 D. 33

5. The cone shown below has a radius of 8 feet and a height of 12 feet. What is the volume of the cone?

A. 32π cubic feet B. 256π cubic feet C. 374π cubic feet D. 768π cubic feet

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7.2. Structured questions 1. Find the area scale factor and the volume scale factor of the above two similar tins

…………………………………………………………………………… …………………………………………………………………………… …………………………………………………………………………… ……………………………………………………………………… 2. A packet of soap powder is a cuboid of breadth of 6 cm, length 16cm and height 25 cm. Box Packet 25 cm 50 cm 6 cm 16 cm 30 cm 64 cm 82 ©SLNECB

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a) Calculate the volume of the packet

………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………..…………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… b) Calculate the number of packets which can be placed in rectangular box of breadth 30 cm, length 64 cm and height 50cm. ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………..…………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… …………………………………………………………………

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3. The figure shows a triangular prism. F 5 cm A D 3 cm 7cm B

4 cm

C

Find:

a)

The surface area. ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… ………………………………………………..…………… ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… ……………………………………………………………….

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b)

Certificate of Secondary Education Maths paper 2 examinations

volume ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… ………………………………………………..…………… ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… …………………………………………………………….

4. The figure shows a solid cone.

h

L

If the base radius ( r) is 5 cm and the vertical height ( h ) is 12 cm, find a) the slant height ( L ) ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… 85 ©SLNECB

Somaliland National Examination and Certification Board

Certificate of Secondary Education Maths paper 2 examinations

b) the total surface area of the cone ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………..…………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… c) the volume of the cone ( Take

π=

22 7

)

………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… ……………………………………..…………………………… ………………………………………………………………… ………………………………………………………………… ………………………………………………………………… …………………………………………………………………

5. Calculate the surface areas of the following solids a. A closed cylinder of 𝑟𝑟 = 14𝑐𝑐𝑐𝑐and ℎ𝑒𝑒𝑒𝑒𝑒𝑒ℎ𝑡𝑡 30𝑐𝑐𝑐𝑐

b. A frustum of a cone with radio , 𝑟𝑟1 = 7𝑐𝑐𝑚𝑚, 𝑟𝑟2 = 21𝑐𝑐𝑐𝑐 and total height is 28𝑐𝑐𝑐𝑐

©SLNECB

86

Somaliland National Examination and Certification Board

Certificate of Secondary Education Maths paper 2 examinations

6. The diagram shows a cone with base radius CD = 3 cm length of BC = 5cm and the Vertical