Topical Revision Notes Mathematics O Level ( PDFDrive )

SHINGLEE PUBLISHERS PTE LTD 120 Hillview Avenue #05-06/07 Kewalram Hillview Singapore 669594 Tel: 6760 1388 Fax: 6762 5684 e-mail: info@shinglee.com.sg http://www.shinglee.com.sg All rights reserved. No part of this publication may be reproduced in any form or stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission in writing of the Publishers. First Published 2016 ISBN 978 981 288 015 4 Printed in Singapore

PREFACE O Level Mathematics Topical Revision Notes has been written in accordance with the latest syllabus issued by the Ministry of Education, Singapore. This book is divided into 21 units, each covering a topic as laid out in the syllabus. Important concepts and formulae are highlighted in each unit, with relevant worked examples to help students learn how to apply theoretical knowledge to examination questions. To make this book suitable for N(A) Level students, sections not applicable for the N(A) Level examination are indicated with a bar ( ). We believe this book will be of great help to teachers teaching the subject and students preparing for their O Level and N(A) Level Mathematics examinations. Preface iii

CONTENTS Unit 1.1 Numbers and the Four Operations 1 Unit 1.2 Ratio, Rate and Proportion 15 Unit 1.3 Percentage 21 Unit 1.4 Speed 24 Unit 1.5 Algebraic Representation and Formulae 28 Unit 1.6 Algebraic Manipulation 33 Unit 1.7 Functions and Graphs 41 Unit 1.8 Solutions of Equations and Inequalities 51 Unit 1.9 Applications of Mathematics in Practical Situations 64 Unit 1.10 Set Language and Notation 77 Unit 1.11 Matrices 85 Unit 2.1 Angles, Triangles and Polygons 91 Unit 2.2 Congruence and Similarity 103 Unit 2.3 Properties of Circles 113 Unit 2.4 Pythagoras’ Theorem and Trigonometry 119 Unit 2.5 Mensuration 129 Unit 2.6 Coordinate Geometry 138 Unit 2.7 Vectors in Two Dimensions 143 Unit 3.1 Data Handling 151 Unit 3.2 Data Analysis 156 Unit 3.3 Probability 167 Mathematical Formulae 176 iv Contents

1Numbers and the Four Operations Numbers 1. The set of natural numbers,  = {1, 2, 3, …} 2. The set of whole numbers, W = {0, 1, 2, 3, …} 3. The set of integers, Z = {…, –2, –1, 0, 1, 2, …} 4. The set of positive integers, Z+ = {1, 2, 3, …} 5. The set of negative integers, Z– = {–1, –2, –3, …} 6. The set of rational numbers, Q = { a b , a, b  Z, b ≠ 0} 7. An irrational number is a number which cannot be expressed in the form a b , where a, b are integers and b ≠ 0. 8. The set of real numbers R is the set of rational and irrational numbers. 9. Real Numbers Rational Numbers Irrational Numbers e.g. π 2, Integers Fractions e.g. 22 7 Zero Negative {..., –3, –2, –1} Positive {1, 2, 3, ...} Natural Numbers Whole Numbers UNIT 1.1 Numbers and the Four Operations

2 Numbers and the Four OperationsUNIT 1.1 Example 1 The temperature at the bottom of a mountain was 22 °C and the temperature at the top was –7 °C. Find (a) the difference between the two temperatures, (b) the average of the two temperatures. Solution (a) Difference between the temperatures = 22 – (–7) = 22 + 7 = 29 °C (b) Average of the temperatures = 22 + (−7) 2 = 22 − 7 2 = 15 2 = 7.5 °C ……………………………………………………………………………… Prime Factorisation 10. A prime number is a number that can only be divided exactly by 1 and itself. However, 1 is not considered as a prime number. e.g. 2, 3, 5, 7, 11, 13, … 11. Prime factorisation is the process of expressing a composite number as a product of its prime factors.

3Numbers and the Four OperationsUNIT 1.1 Example 2 Express 30 as a product of its prime factors. Solution Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 Of these, 2, 3, 5 are prime factors. 2 30 3 15 5 5 1 ∴ 30 = 2 × 3 × 5 Example 3 Express 220 as a product of its prime factors. Solution 220 2 110 25 5 51 1  220 = 22 × 5 × 11 ……………………………………………………………………………… Factors and Multiples 12. The highest common factor (HCF) of two or more numbers is the largest factor that is common to all the numbers.

4 Numbers and the Four OperationsUNIT 1.1 Example 4 Find the highest common factor of 18 and 30. Solution 18 = 2 × 32 30 = 2 × 3 × 5 HCF = 2 × 3 = 6 Example 5 Find the highest common factor of 80, 120 and 280. Solution Method 1 2 80, 120, 280 2 40, 60, 140 2 20, 30, 70 5 10, 15, 35 2, 3, 7 HCF = 2 × 2 × 2 × 5 (Since the three numbers cannot be divided further by a common prime factor, we stop here) = 40 Method 2 Express 80, 120 and 280 as products of their prime factors 80 = 23 × 5 120 = 23 × 5 280 = 23 × 5 × 7 HCF = 23 × 5 = 40