Page 1


Cambridge O Level

Mathematics Vol. 2

Audrey Simpson

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CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Tokyo, Mexico City Cambridge University Press 4381/4 Ansari Road, Daryaganj, New Delhi 110002, India www.cambridge.org Information on this title: www.cambridge.org/9780521186056 Š Cambridge University Press 2012 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. &IRST0UBLISHED Printed in India at A catalogue for this publication is available from the British Library

ISBN 978-0-521-18605-6 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Every effort has been made to trace the owners of copyright material included in this book. The publishers would be grateful for any omissions brought to their notice for acknowledgement in future editions of the book.

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Contents

Introduction Acknowledgements Note 1

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vii viii ix

Real Numbers Essential Skills More about Sets Defining a Set Notation and Symbols used with Sets More Notation and Symbols Venn Diagrams More about Square Roots and Irrational Numbers Fractions, Decimals and Percentages Mixed Exercise Examination Questions

1 1 2 3 4 5 7 12 13 14 14

2

Algebra I Essential Skills Expanding Products of Algebraic Factors Factorising Quadratic Expressions Factorising a Difference of Squares Factorising by Pairing Factorising Systematically More about Indices Algebraic Fractions Mixed Exercise Examination Questions

22 22 23 25 29 29 30 31 34 41 42

3

Working with Numbers Essential Skills Upper and Lower Bounds in Calculations Reverse Percentages Mixed Exercise Examination Questions

47 47 49 50 52 53

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iv

Contents

4

Algebra II Essential Skills More Equations Quadratic Equations More Sequences More Simultaneous Equations Inequalities Variation Rearranging Formulae Mixed Exercise Examination Questions

5

Geometry and Shape Essential Skills Symmetry in Three Dimensional Shapes Further Circle Facts Investigations Finding the Centre of a Circle Irregular Polygons Mixed Exercise Examination Questions

102 102 104 107 110 112 114 115 117

6

Algebra and Graphs Essential Skills Distance/Time Graphs Speed/Time Graphs More Graphs of Curves Straight Line Segments Mixed Exercise Examination Questions

127 127 130 132 138 147 151 153

7

Length, Area and Volume Essential Skills Arc Lengths in Circles Sector Areas in Circles More Volumes and Surface Areas of Solids Curved Surface Areas Volumes Volume and Surface Area of a Frustum Similar Shapes Similar Triangles Congruent Triangles

169 169 170 171 171 172 173 174 176 180 186

61 61 62 63 73 76 80 83 86 89 90

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Contents v Mixed Exercise Examination Questions

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189 191

8 Further Algebra Beginning Matrices Operations on Matrices Multiplication of Matrices The Zero Matrix The Identity Matrix and Inverse Matrices Functions Inverse Functions Graphs of Inequalities Mixed Exercise Examination Questions

204 204 205 207 212 213 219 224 226 235 236

9

246 246 247 252 258 263 265 268 270

Trigonometry Essential Skills Trigonometry with Angles between 0째 and 180째 The Sine Rule The Cosine Rule Area of a Triangle 3-D Trigonometry Mixed Exercise Examination Questions

10 Transformations, Vectors and Matrices Essential Skills More about Vectors More about Transformations Matrices and Transformations Mixed Exercise Examination Questions

286 286 289 301 312 324 326

11 Statistics Essential Skills More Histograms The Mean from a Grouped Frequency Table Cumulative Frequency Quartiles, Inter-Quartile Range and Percentiles Mixed Exercise Examination Questions

345 345 346 354 356 358 366 368

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vi

Contents

12 Probability Essential Skills Tree Diagrams Venn Diagrams Using Set Notation Mixed Exercise Examination Questions

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382 382 382 387 387 391 393

Revision and Examination Hints

405

Answer key

408

Index

483

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Introduction

This book continues the work started in O Level Vol. 1, preparing students for the O-Level Mathematics examinations from University of Cambridge International Examinations. It is intended to form the second year of a two-year course and develops the ideas introduced in O Level Vol. 1. As explained in more detail in the introduction to Volume 1, students will find that the structure of the book allows them to work through at their own pace, checking their progress as they go. Each chapter starts with a short Essential Skills exercise to remind students of the work they did in the corresponding chapter in O Level Vol. 1, and ends with examination questions taken from past O-Level papers, and, where appropriate, from past Extended Level IGCSE papers. A final section alerts the reader to common mistakes made by students in the examinations and lists particular items that need to be committed to memory. Note to students L The text in each chapter in O Level Vol. 1 and 2 introduces you to essential mathematical tools. L The exercises help you gain facility in using these tools. L Finally, at the end of each chapter you will find a set of questions from past examination papers. In some of these examination questions you will need to apply the skills you have acquired to solve problems in a less familiar context. It is therefore essential that you work through all the examination questions and satisfy yourself that you understand the questions and can obtain the correct answers. L If you work through Volume 1 and Volume 2 in this way you will have covered all the work you need to be successful in your final examination.

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Acknowledgements

I would like to thank Professor Gordon Kirby for his invaluable advice and encouragement. I am also grateful for his efforts to check my work patiently for errors, both mathematical and stylistic. I am also indebted to my sister, Pat Victor, for the times she sorted out frustrating problems with both my computer and the software needed for the production of the manuscript. Audrey Simpson

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Note

A note about the currencies used in this book: 1 1 1 1

dollar ($) is worth 100 cents. rupee (`) is worth 100 paise (singular paisa). pound (£) is worth 100 pence. euro (€) is worth 100 cents.

Unless greater accuracy is required, answers to money questions which are not whole numbers, should be given to 2 decimal places. For example, $12.7 would be given as $12.70, which is read ‘twelve dollars and seventy cents’.

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Chapter

1

Real Numbers The set of Real Numbers includes all the numbers we need in everyday life. Your course for Cambridge O Level Mathematics is based entirely on the set of real numbers. You may wonder what other numbers there could possibly be. If you study Mathematics further you will eventually meet the set of Imaginary Numbers, which are the square roots of negative numbers. As you know, it is not possible to multiply a number by itself to make negative number, so the square root of a negative number does not exist. However, Mathematicians need to work with the square roots of negative numbers, so they are called imaginary numbers. Imaginary numbers form the basis of a very interesting branch of Mathematics and you might like to ďŹ nd out more about them. This Chapter follows on from Chapters 1 and 2 in Cambridge O Level Mathematics Vol. 1. Before starting the new work you should work through the Essential Skills exercise and check your answers. You should be able to complete the exercise with conďŹ dence. If you are unsure of anything you will need to go back to Chapters 1 and 2 in O Level Vol. 1.

Essential Skills

NO CALCULATOR IN THIS EXERCISE

1. 100 P 22.5 1 9 18 5 24 6 0 25 2 2 49 2 7 From the list above select (a) the natural numbers (b) the integers (c) the rational numbers (d) the irrational numbers (e) the prime numbers (f ) the factors of 45 (g) the multiples of 6 (h) the cube root of 8 ( i) the square of 7 ( j) the square root of 81 2. (a) List the set of factors of 144. (b) List the set of prime factors of 144. (c) Write 144 as a product of its prime factors. 3. Find the Highest Common Factor of 198 and 110. 4. Find the Lowest Common Multiple of 15 and 18. 5. Arrange the following in order of size, starting with the smallest. 0.513 0.5006 0.52 0.5111

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2

6. Insert a correct symbol between each of the following pairs of numbers. 7. 8. 9. 10. 11. 12. 13.

14.

15. 16. 17.

64 , 23 64 , 32 19, 18 Write the following in standard form (a) 60137.1 (b) 0.005401 Use the correct order of working to calculate the following (a) 20 4 s 5 (b) 15 w (5 2) (c) (6 2) w 2 3 List the set of prime numbers between 20 and 40. Write 37 as a mixed number. 3 3 Write 5 as an improper fraction. 5 Fill in the spaces in this set of equivalent fractions 1   4 75 3 Without using a calculator, calculate the following 2 4 2 4 2 4 (a) 2 (b) 2 s (c) 2 w 7 7 7 7 7 7 (a) Change 0.71 to a percentage. (b) Change 15% to a fraction. (c) Change 7 to a decimal. 50 Find 16% of 54. Find 28 as a percentage of 63. 4 5 Find a fraction that lies between and . 7 7

More about Sets We have already studied some sets of numbers. These were the sets of natural numbers, integers, rational numbers, real numbers and prime numbers. A set is a collection of objects, ideas or numbers that can be clearly defined. Clearly defined means that we can tell with certainty whether or not something belongs to the set. For example, (a) The set of colours of the rainbow is {red, orange, yellow, green, blue, indigo and violet}. Brown is a colour, but it does not belong to this set. (b) The set of domestic pets may be quite large, but it would not include a brontosaurus! However a brontosaurus would be included in the set of dinosaurs. (c) The set of factors of 10 would include 1, 2, 5 and 10, but not 4. In each of these cases, the sets are well defined because it is possible to tell whether or not something belongs to the set.

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Defining a Set There are various ways of defining a set. We can list the members of the set in curly brackets. The set of factors of 10  {1, 2, 5, 10} L If there is no end to the set, we use dots to represent ‘and so on’. The set of natural numbers  {1, 2, 3, 4, 5, 6, 7…} L If there is an end to the set, but it is too long to list all the numbers, we can still use dots, but include one or more of the last numbers in the set. The set of even numbers between 1 and 99  {2, 4, 6, 8, 10, … 96, 98} L Sets can also be defined by a description. For example, {planets in our solar system} or {capital cities of the world}. L Sets are often labelled by using capital letters. The letters N, Z, Q and R are reserved for the sets we have already met, namely natural numbers, integers, rational numbers and real numbers. L If we use a capital letter to represent another set we must define it clearly. For example, P  {the prime numbers between 1 and 20} or B  {boys in your class} L

Example 1 (a) Write down a description of each of the following sets (there could be more than one suitable description). (i) {1, 3, 5, 7} (ii) {5, 10, 15, 20, 25 …} (iii) {1, 2, 3, 4, 6, 12} (b) List the following sets (i) The set of square numbers less than 30. (ii) The set of vowels in the English alphabet. (iii) The set of natural numbers less than 80. (iv) The set of cube numbers. Answer 1 (a) (i) The set of odd numbers between 0 and 8, or simply {odd numbers between 0 and 8} (ii) The set of multiples of 5 (iii) The set of factors of 12 (b) (i) {1, 4, 9, 16, 25} (ii) {a, e, i, o, u} (iii) {1, 2, 3, 4, 5, 6, 7 … 76, 77, 78, 79} (iv) {1, 8, 27, 64, 125 …}

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Cambridge O Level Mathematics Vol. 2

Notation and Symbols used with Sets The members of the set are also called elements of the set. If we define a set C  {primary colours}, then the elements of the set C are red, blue and green. The number of elements or members in the set is referred to as n(C ). In this case, n(C )  3. Of course n( Z ) (the number of elements in the set of integers) is undefined, or infinitely large, because the integers go on forever. The Greek letter Œ is used to mean ‘is a member of’ or ‘is an element of’. So Monday Œ {days of the week}, should be read as ‘Monday is an element of the set of days of the week’. If this letter has a cross through it, i.e. œ, it reads ‘is not an element of’. Hence, April Ž {days of the week}. We also need an ‘empty’ set. An empty set has no members. For example, {even prime numbers greater than 2} is an empty set because 2 is the only even prime number. The symbol used for the empty set is ∆. In some textbooks, you may see the empty set written as {}, which is an alternative for …. Notice that we cannot use {0} as the empty set, because it has a member, which is zero. So far the symbols we have met are: Number of elements in set A n(A) ‘is an element of’ Œ ‘is not an element of’ œ The empty set ∆

Exercise 1.1 1. List the elements of these sets (a) {square numbers between 10 and 40} (b) {months of the year beginning with M} (c) {natural numbers , 10} 2. Describe these sets (a) {1, 3, 5, 7, 9} (b) {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday} (c) {a, b, c, d, e … x, y, z} 3. Which of the following are empty sets? (a) {people over 5 metres tall} (b) {birds that swim} (c) {0} (d) {integers between 1.1 and 1.9} 4. Insert the correct symbol in the following statements. (a) 1 … {prime numbers}

(b) 1000 … {even numbers}

5. A  {letters used in the word ‘mathematics’}. (a) List set A

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(b) Find n(A)

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Real Numbers

5

More Notation and Symbols You will find that when we talk about sets we need to limit the possibilities in some way. For example, if you were to make up a school team to play football, it would be no good including students from another school. The team would be chosen from the set of all the students in your school. This is called the universal set. The symbol used to denote the universal set is - So we could have: -  {students in your school} F  {students in your school’s football team} M  {students in your maths class} Both F and M belong to the universal set -. Another situation we need to be able to describe is that some students may be in your maths class and in the school football team. This is called an intersection of sets, and has the symbol †. NOTE: This symbol, ‡ , could be likened to a bridge between the two sets.

We write {students both in the school football team and in your maths class}  F † M. If there is no one in your maths class who is also in the football team, then F † M  …. In this case, the intersection of F and M has no members, so it is the empty set. It is perhaps easier to see this using sets with numbers. We will use the following sets: A  {1, 2, 3, 4, 5, 6, 7, 8, 9} B  {10, 11, 12, 13, 14, 15} C  {2, 4, 6, 8, 10, 12} D  {1, 3, 5, 7} From these sets it will be seen that: A † C  {2, 4, 6, 8} because 2, 4, 6 and 8 are in both A and C and A†B… What about A † D? A † D  {1, 3, 5, 7}  D The whole of set D is in set A. We say that D is a proper subset of A, and we use the symbol ‹, so D ‹ A. You will usually hear a proper subset being referred to as just a subset. Strictly speaking a subset is different from a proper subset, which can be shown using the following example. V  {vowels in the English language) E  {letters of the English language} L  {a, e, i, o, u} F  {a, b, c} V is a proper subset of E because there are elements in E that are not in V. However, V could be thought of either as equal to L, or as a subset of L, so V is not a proper subset of L.

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Cambridge O Level Mathematics Vol. 2

Therefore, V is a proper subset of E. V is a subset of L ( and V is equal to L).

V‹E VŒL

NOTE: Think of  as ‘is a subset or equal to’ in the same way as , means ‘less than or equal to’. Also think of ‘Œ is a proper subset’, and ‘ is strictly less than’.

The same symbol with a line through it Š means ‘is not a proper subset of’. V is not a proper subset of L . VŠL We can also say: F‹E FŠL and FzL Suppose we wanted to use the whole of A and B. This would be called the union of A and B, and we would use the symbol ‡. NOTE: You could think of this as a U, standing for ‘Union’.

A ‡ B  {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} A ‡ C  {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12} Notice that the elements that appear in both sets, are not listed twice. We can now add to our list of symbols. Universal set Intersection † Proper subset ‹ Subset Œ Not a subset Š Union ‡ Example 2 F  {10, 20, 30, 40}

G  {11, 13, 17, 19}

H  {10, 11, 12, 13} J  {11, 13} Use these sets to answer the questions. (a) List (i) G † H (ii) F ‡ H (b) J is a subset of two of the sets. Which are those two sets? (c) Suggest a suitable universal set for F, G, H and J. (d) What can you say about F † J? Answer 2 (a) (i) G † H  {11, 13} (notice that this is also equal to J ) (ii) F ‡ H  {10, 11, 12, 13, 20, 30, 40} (b) J ‹ G and J ‹ H

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Real Numbers

7

(c) A universal set could be N (the set of natural numbers), or we could restrict it more, say  -  {10, 11, 12, 13, 14, 15 … 38, 39, 40} or -  {natural numbers between 10 and 40 inclusive} (d) F † J  … There is one more symbol left to add to the list. Look at these sets: -  {1, 2, 3, 4, 5} A  {1, 4} B  {2, 4, 5} The new symbol is the letter for the set followed by a dash, for example, Aa. This means everything in the universal set but not in set A. It is called the complement of A. So Aa  {2, 3, 5} and Ba {1, 3}. Since A † B  {4} then (A † B)a  {1, 2, 3, 5} In the same way, can you show that (A ‡ B)a  {3}?

Exercise 1.2 -  {3, 4, 5, 6, 7, 8, 9}

A  {4, 5, 6} B  {odd numbers between 2 and 8} C  {square numbers , 10} Using the above sets, 1. List (a) B (b) A † B (c) A ‡ B (e) the complement of B (f ) Aa † B 2. List the intersection of B and C. 3. List the union of A and C. 4. Write down n(C). 5. Find n(A ‡ C). 6. Find n(B a).

(d) Aa (g) (A ‡ B)a

Venn Diagrams A Venn diagram is a very useful method of visualising the relationship between sets. A rectangular box is drawn to show the universal set, and within this, other shapes, usually ovals or circles, represent sets within the universal set. If there is an intersection between sets, it will be shown by an overlap of the ovals or circles. The following example should help you to understand the concept of a Venn diagram.

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8

Cambridge O Level Mathematics Vol. 2 Example 3 -  {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A  {even numbers} B  {1, 2, 3, 4, 5} C  {6, 7, 8} (a) Draw a Venn diagram to represent the above sets. Show the elements in each set. (b) List the following sets. (i) A † B (ii) A ‡ B (iii) A ‡ C (iv) (A ‡ C )a (v) A ‡ B ‡ C (vi) (A ‡ B ‡ C )a (vii) B † C Answer 3 (a) 

   

  

(b)

(i) (iii) (v) (vii)

 



NOTE: The symbol (-) for the Universal Set may be placed either inside or outside the rectangle. Either is correct. You may find different symbols used for the Universal Set in other books, but it should always be clear what is intended.

  

A † B  {2, 4} (ii) A ‡ B  {1, 2, 3, 4, 5, 6, 8, 10} A ‡ C  {2, 4, 6, 7, 8, 10} (iv) (A ‡ C)a  {1, 3, 5, 9} A ‡ B ‡ C  {1, 2, 3, 4, 5, 6, 7, 8, 10} (vi) (A ‡ B ‡ C)a  {9} B†C…

NOTE: You may find that shading or hatching can help to make the distinction between union and intersection between sets in the Venn diagram more clear.

If you look at the three Venn diagrams given in the diagrams below, you will see that in the first, one set has been hatched in a different direction from the other. In the second Venn diagram, the two sets show an intersection, where the hatching appears in both directions (cross hatching). In the third Venn diagram, the union is shown by all the shaded areas, in either direction, taken together. 

X



Y

X∩Y∅



P

Q

P ∩ Q (shaded) P

Q

P ∪ Q (shaded)

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9

Example 4 (a) Draw three identical Venn diagrams to illustrate the following sets.  -  {students in your school} C  {students in your class} B  {students who come to school by bus} (b) (i) In the first diagram shade and clearly label B † C. (ii) In the next diagram shade and clearly label B ‡ C. (iii) In the last diagram shade and clearly label C † Ba (c) How would you describe B † C and C † B in words? Answer 4 (a) and (b) 

C



B

(i) B C

C



B

(ii) B C

C

B

(iii) C B¢

(c) B † C  {students in the class who come to school by bus} C † Ba  {students in the class who do not come to school by bus} You should be able to see what a powerful tool Venn diagrams can be in the study of sets. We will use them again later in the course. Study the next four diagrams carefully to see how hatching and shading can clarify the required areas of Venn diagrams. 



A

C

D

B

C D

BA



C

D

CD



C

D

CD

NOTE: The hatching used above is to help you understand which area of the Venn diagram is required. In your examination you should only shade the area specified in the question.

Venn diagrams can also show the number of elements in each set as the next example shows.

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10

Example 5 







You are given the following information. n(-)  30, n(A)  11, n(B)  12, n(C)  10, n(A †B †C)  3, n(A †B)  5, n(A †C )  4, n(B †C )  6 (a) Copy and complete the Venn diagram. (b) Find (i) n(A ‡B ‡C)a (ii) n(A ‡B) Answer 5 (a) Using the information given, it is possible to fill in the Venn diagram in steps. STEP 1: n(A †B †C)  3, so 3 goes right in the centre of the diagram. STEP 2: n(A †B)  5, so take away the 3 that is already there, which leaves 2 to go in the rest of the intersection. n(A † C)  4 and n(B †C )  6 means we can fill in the other two intersections in the same way. (See the diagram below for steps 1 and 2) 





  

 

STEP 3: n(A)  11 so take away the 1, 2 and 3 which are already in A, leaving 5 to be written in the remaining part of A. Use n(B) 12 and n(C )  10 in the same way. STEP 4: Find n(A ‡B ‡C ) by adding all the numbers in your diagram. n(A ‡B ‡C )  5 1 3 2 4 3 3  21 It was given that n(-)  30, so n(A ‡B ‡C )a  30 21  9 Write in the 9 to complete the Venn diagram. (See the diagram given on the next page for step 4.)

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Real Numbers 



5 1

11

 2 3

4 3

3 

9

(b) (i) n(A ‡B ‡C)a  9 (ii) n(A ‡B)  5 1 2 3 4 3  18 The answers to these and many other similar questions can now be easily obtained from the diagram.

Exercise 1.3 1. Draw Venn diagrams to illustrate the following. (a) -  N, P  {prime numbers less than 10}, E  {2, 4, 6} (Mark and label each element in its correct place). (b) -  Z, S  {square numbers}, E  {even numbers} (The individual elements cannot be labelled because there are an infinite number of elements). 2. Copy these Venn diagrams and in each case, shade the required areas.     





 ∪



 





 ∩



 





( ∩)

3. Draw Venn diagrams to illustrate each of the following. (a) A † B  … (b) A ‹ B 4. Draw a Venn diagram to illustrate the following. -  {natural numbers from 1 to 10} P  {1, 2, 3, 4, 5, 6, 7, 8}

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R  {2, 4, 6, 8} S  {1, 2, 3, 4} Mark and label the correct position for each element in the diagram. 5. (a) Draw a Venn diagram to show two sets, A and B, where B ‹A. (b) Given that n(-)  48, n(A)  25 and n(B)  10, find (i) n(A † Ba) (ii) n(Aa) (c) Complete the Venn diagram showing the numbers in each area. Check that all your numbers add up to 48. 6. (a) Draw a Venn diagram showing two sets, P and S, with an intersection. (b) Given that n(-)  20, n(P)  7, n(S) 16 and n(P ‡ S)a  0, find n(P †S). (c) Complete the diagram. NOTE: Since n(P ˆ S)a = 0, all 20 of the elements must lie in P and S, but adding 7 and 16 makes too many, so how many must be in the intersection?

More about Square Roots and Irrational Numbers In this section of this chapter we will see how we can find the square root of any perfect square, or express irrational square roots in their simplest form, without using a calculator. There are two possible methods. (a) In the first, we express the number, as a product of its prime factors (by using a factor tree if necessary). For example, 900  2 s 2 s 3 s 3 s 5 s 5 900  2 s 2 s 3 s 3 s 5 s 5

so This is the same as

2s 2s 3s 3s 5s 5

We know

2 s 2  4  2,

3 s 3  3 and 5 s 5  5

This means

900  2 s 2 s 3 s 3 s 5 s 5  2 s 3 s 5  30 900  30.

so

This is also the best method for finding cube roots. 216  3 2 s 2 s 2 s 3 s 3 s 3  3 2 s 2 s 2 s 3 3 s 3 s 3  2 s 3  6 (b) In the second method, we look for factors of the number that are perfect squares themselves, such as 4, 9, 16, 25 and so on. 3

So 900  4 s 9 s 25  2 s 3 s 5  30 Either method will also work for numbers whose square roots are irrational. For example, by the first method, 450  2 s 3 s 3 s 5 s 5  2 s 3 s 3 s 5 s 5  2 s 3 s 5  15 s 2 and 15 s

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2 is usually written 15 2 .

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13

By the second method, 450  9 s 25 s 2  3 s 5 s 2  15 2 . This is as far as you can go easily without using a calculator. Square roots that are irrational are also called surds. The number 15 2 is a surd because 2 is irrational. Example 6 (a) Find 225 without using a calculator. (b) Simplify 432 by writing it as a surd, in terms of 2 or 3 or 5. Answer 6 (a)

225  3 s 3 s 5 s 5  9 s 25  3 s 5  15

(b)

432  2 s 2 s 2 s 2 s 3 s 3 s 3  16 s 9 s 3 4 s 3 s 3  12 3

Exercise 1.4

NO CALCULATOR IN THIS EXERCISE

1. Without using a calculator (and showing all your working) find the following. (a) 784 (b) 1600 (c) 625 2. Express the following square roots as surds in terms of 2 3 or 5 (a) 180 (b) 98 (c) 192 3. In each case, show that the following square roots are either rational or irrational. (a) 50 (b) 144 (c) 45

Fractions, Decimals and Percentages The basic work on fractions, decimals and percentages was covered in Cambridge O Level Mathematics Vol. 1. The questions you will be asked at this level may be just a little more difficult. Work through the next exercise and check your answers before attempting the final exercise.

Exercise 1.5

NO CALCULATOR IN THIS EXERCISE

1. Write 2.23 as (a) a mixed number, (b) an improper fraction. 2. Arrange in order of size, starting with the smallest 1 21 0.21% 2 s 10 3 250 500 3. (a) Calculate 65 as a percentage of 50. (b) Calculate 38.2 as a percentage of 40. (c) Calculate 6 as a percentage of 1200.

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4. Arrange the following in order of size, starting with the smallest. 1 9 9 19 77 , , , and 2 16 20 40 160 5. Find a fraction halfway between 17 9 4 5 and and (a) (b) 20 10 5 6

Exercise 1.6 Mixed Exercise 1. -  {a, b, c, d, e, f, g, h} L  {a, c, e} M  {b, c, d, e, f} (a) Draw a Venn Diagram showing these sets. Shade L in one direction and M in another. (b) List (i) L † M (ii) L ‡ M (iii) La (iv) Ma (v) (L † M)a (c) Draw another Venn diagram, Shade L in one direction and Ma in another. (d) List (i) L † Ma (ii) L ‡ Ma 2. List the integers from 2 to 2 inclusive. (Inclusive means including 2 and 2.) 3. Describe the following (a) {1, 8, 27, 64} (b) {1, 4, 9, 16, 25} 4. Without using a calculator simplify as far as possible. (a) 4096 (b) 2450 5. Which of the following are irrational numbers? (a) 500 (b) 121 6. You are given the following information. -  {integers from 10 and 10 inclusive} F  {2, 4, 6} G  { 2, 1, 0, 1, 2} H  {1, 2, 3, 4, 5, 6, 7, 8} By first drawing a Venn diagram, find (a) n(F †G) (b) n(F ‡G) (c) n(F †G †H ) (d) n(F ‡G ‡H ) (e) n(F ‡G ‡H )a

Examination Questions 7. From the list of numbers 22 , P, 14, 16 , 27.4, 65 write down 13 7 (a) one integer, (b) one irrational number. (0580/02 Oct/Nov 2004 q 3)

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8. -  {40, 41, 42, 43, 44, 45, 46, 47, 48, 49} A  {prime numbers} B  {odd numbers} (a) Place the ten numbers in the correct places on the Venn diagram. 

 

(b) State the value of n(B † Aa) (0580/02 Oct/Nov 2004 q 11) Ž 1 Žº 9. -  2 , 1, 2 , 3.5, 30 , 36  ŽŽ­ 2 ŽŽŸ X  {integers} Y  {irrational numbers} List the members of (a) X, (b) Y. (0580/02 Oct/Nov 2003 q 5) 10. (a) Express in set notation, as simply as possible, the subset shaded in the Venn diagram. 





(b) It is given that n(-)  20 and P and S are sets such that n(P)  7 and n(S)  16. Find the smallest possible value of n(P † S). (4024/01 Oct/Nov 2004 q 18 a and d) 11. In a group of language students, 24 studied Spanish, 23 studied French and 15 studied German. 12 studied Spanish and French, 10 studied German and French, 6 studied Spanish and German and 4 studied all three languages. By drawing a Venn diagram or otherwise, calculate the number of students who studied (a) both Spanish and French, but not German, (b) only one language. (4024/01 May/June 2003 q 18 a)

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12. Rearrange the quantities in order with the smallest first. 1 3 , , 0.00126 8 2500 (0580/02 Oct/Nov 2003 q 4) 13. Lin scored 18 marks in a test and Jon scored 12 marks. Calculate Lin’s mark as a percentage of Jon’s mark. (0580/02 May/June 2008 q 3) 14. 0.0008

8 s 10 5

0.8%

1 125000

Write the above numbers in order, smallest first. (0580/02 Oct/Nov 2006 q 6) 15. (a) Shade the region A † B.  



(b) Shade the region (A ‡ B)a.  



(c) Shade the complement of set B.  



(0580/92 Oct/Nov 2006 q 11) 16. Write down the next prime number after 89. (0580/02 May/June 2006 q 2)

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17. n(A)  18, n(B)  11 and n(A‡ B)a  0. (a) Label the Venn diagram to show the set A and B where n(A ‡ B)  18. Write down the number of elements in each region. 

(b) Draw another Venn diagram to show the sets A and B where n(A ‡#  29. Write down the number of elements in each region. 

(0580/02 May/June 2006 q 17) 18. On the Venn diagrams shade the regions (a) Aa† Ca













(b) (A ‡ C ) † B. 



(0580/02 May/June 2007 q 8)

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19. n(-)  21, n(A ‡ B)  19, n(A † Ba)  8 and n(A)  12. Complete the Venn diagram to show this information. 





(0580/02 May/June 2005 q 11) 20. In a survey, 100 students were asked if they liked basketball (B), football (F ) and swimming (S). The Venn diagram shows the results. 



 



  









42 students liked swimming. 40 students liked exactly one sport. (a) Find the value of p, q and r. (b) How many students like (i) all three sports, (ii) basketball and swimming but not football? (c) Find (i) n(B) (ii) n((B ‡ F) † Sa ) (0580/04 Oct/Nov 2008 q 9 a, b and c) 21. Write the following in order of size, smallest first. 399 401

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689 701

598 601

(0580/02 May/June 2008 q 6)

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22. A and B are sets. Write the following sets in their simplest form. (a) A † Aa. (b) A ‡ Aa. (c) (A † B) ‡ (A † Ba . (0580/02 Oct/Nov 2007 q 12) 2

23. Work out the value of 1

3

4 5 6 (0580/21 Oct/Nov 2008 q 3)

24. In a group of 100 students, 80 study Spanish and 35 study French. x students study Spanish and French. y students study neither Spanish nor French. The Venn diagram illustrates this information. 



 







(a) Expressed in set notation, the value of x is n(S † F). Express the value of y in set notation. (b) Find, in its simplest form, an expression for y in terms of x. (c) Find (i) the least possible value of x, (ii) the greatest possible value of y. (4024/02 May/June 2004 q 5) 25. The results of a survey of 31 students are shown in the Venn diagram. - {students questioned in the survey} M  {students who study Mathematics} P  {students who study Physics} S  {students who study Spanish} 

 

 



 



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Cambridge O Level Mathematics Vol. 2 (i) Write down the value of (a) x, (b) n(M † P), (c) n(M ‡ S), (d) n(Pa). (ii) Write down a description, in words, of the set that has 16 members. (4024/02 May/June 2006 q 6 a)

26. (a) On a copy of the Venn diagram, shade the set A ‡ (B † C ). 







(b) Express in set notation the subset shaded in the Venn diagram. 





(c) In a class of 36 students, 25 study History, 20 study Geography and 4 study neither History nor Geography. Find how many students study both History and Geography. (4024/01 Oct/Nov 2006 q 18) 27. (a) The sets A and B are shown on the Venn Diagram. The element y is such that y  A and y Ž B. On a copy of the diagram, write y in the correct region. 





(b) -{x : x is an integer and 1 ,Œx ,Œ8}. P  {x : x Œ5}. Q  {x : x ,Œ3}. (i) Find the value of n(P ‡ Q). (ii) List the elements of P¢ † Q¢. (4024/01 May/June 2007 q 9)

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28. (a) Express, in set notation, as simply as possible, the subset shaded in the Venn diagram.  





(b) It is given that n(-)  40, n(P)  18, n(Q)  20 and n(P † Q) 7. Find (i) n(P ‡ Q), (ii) n(P¢ † Q¢ ). (4024/01 Oct/Nov 2007 q 9) 29. (a) On a copy of the Venn diagram, shade the set P ‡ Q¢ .  



(b) There are 27 children in a class. Of these children, 19 own a bicycle, 15 own a scooter and 3 own neither a bicycle nor a scooter. Using a Venn diagram, or otherwise, find the number of children who own a bicycle but not a scooter. (4024/01 Oct/Nov 2008 q 9) 30. (a) - = {1, 2, 3, 4, 5}, A  {1, 2, 3}, B  {5}, C  {3, 4}. List the elements of (i) A ‡ C, (ii) B¢ † C¢. (b) A group of 60 children attend an after school club. Of these, 35 children play football and 29 play hockey. 3 children do not play either football or hockey. By drawing a Venn diagram, or otherwise, find the number of children who play only hockey. (4024/01 May/June 2009 q 18)

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