MACMILLAN MASTER SERIES Work Out Mathematics GCSE MACMILLAN WORKOUT SERIES G.D. Buckwell M MACMILLAN The titles in this. series For examinations at 16+ Accounting Human Biology Biology Mathematics Chemistry Numeracy Computer Studies Physics Economics Sociology English Spanish French Statistics German For examinations at A' level Applied Mathematics Physics Biology Pure Mathematics Chemistry Statistics English Literature For examinations at college level Dynamics Operational Research Elements of Banking Engineering Thermodynamics Mathematics for Economists © G. D. Buckwell 1986, 1987 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright Act 1956 (as amended). Any person who does any unauthorised act in relation to this publication may be liable to criminal prosecution and civil claims for damages. First published 1986 Reprinted 1986 Reprinted (with corrections) 1986 This edition 1987 Published by MACMILLAN EDUCATION LTD Houndmills, Basingstoke, Hampshire RG2l 2XS and London Companies and representatives throughout the world Typeset by TecSet Ltd, Wallington, Surrey Printed and bound in Great Britain by Scotprint Ltd., Musselburgh, Scotland British Library Cataloguing in Publication Data Buckwell, G. D. (Macmillan work out series) 1. Mathematics-Examinations, questions, etc. I. Title 510'.76 QA43 ISBN 978-0-333-44013-1 ISBN 978-1-349-09462-2 (eBook) DOI 10.1007/978-1-349-09462-2 Acknowledgements ix Organizations Responsible for GCSE Examinations ix Introduction 1 1 Set Theory 3 1.1 Definition 3 1.2 Equality 3 1.3 Standard Sets 3 1.4 Alternative Notation 3 1.5 Universal Set, Complement 4 1.6 Intersection, Null Set 4 1.7 Union 4 1.8 Subsets 5 1.9 Venn Diagrams 5 1.10 Number of Elements 5 1.11 Sets of Points 6 1.12 Logic 6 1.13 De Morgan's Laws, Distributive Laws 7 2 The Number System 17 2.1 The Number System 17 2.2 Decimal Places and Significant Figures 18 2.3 Number Sequences and Patterns 20 2.4 Number Bases 21 2.5 Manipulation in Different Bases 22 2.6 Fractions 23 2. 7 Fractions to Decimals 24 2. 8 Fraction as a Ratio 24 2.9 Rate and Proportion 25 2.10 Percentages 25 2.11 Using Percentages, Profit and Loss 25 2.12 Exchange Rates 26 2.13 Map Scales 27 2.14 Interest 27 2.15 Large and Small Numbers, Standard Form 29 2.16 Manipulation in Standard Form 30 2.17 Approximations 31 2.18 Surds 32 2.19 Rates 32 2.20 Wages and Salaries 33 v 3 Algebra 40 3.1 Directed Numbers 40 3.2 Commutative, Associative, Distributive, Brackets 41 3.3 Substitution 42 3.4 Indices 43 3.5 Like and Unlike Terms 44 3.6 Removal of Brackets 44 3.7 Factors 45 3.8 LCM and HCF 47 3.9 LCM and HCF in Algebra 47 3.10 Algebraic Fractions 48 3.11 Forming Expressions 49 3.12 Linear Equations 50 3.13 Quadratic Equations 51 3.14 Simultaneous Equations (Linear) 52 3.15 Forming Equations 53 3.16 Changing the Subject of Formulae 55 3.1 7 Remainder Theorem 57 3.18 Variation and Proportion 59 3.19 Simultaneous Equations, One Linear, One Non-linear 60 3.20 Flow Charts and Iterative Methods 61 4 The Metric System 74 4.1 Units 74 4.2 Changing Units in Area and Volume 74 4.3 Conversion of Units Involving Rates 75 4.4 Area 75 4.5 Volume 77 4.6 Density 78 4.7 Surface Area 78 4.8 Approximating Areas, Trapezium Rule 83 4.9 Sectors and Segments 84 4.10 Speed, Velocity and Acceleration 86 4.11 Displacement-Time (or Distance-Time) Graphs 86 4.12 Velocity (Speed)-Time Graphs 86 5 Functions and Graphs 97 5.1 Definition 97 5.2 Domain, Codomain, Range, Image 97 5.3 Composition of Functions 98 5.4 Inverse 98 5.5 Graphical Representation 99 5.6 The Equation of a Straight Line 101 5.7 Simple Coordinate Geometry (Distance, Mid-point) 102 5.8 Graphical Solution of Equations 102 5.9 Curve Sketching 106 5.10 Variation and Proportion 107 6 Inequalities 114 6.1 Properties of< and ~ 114 6.2 Inequations 114 6.3 Errors 115 6.4 Regions 116 6.5 Linear Programming 117 6.6 Locus 121 vi 7 Vectors 124 7.1 Definition 124 7.2 Multiplication by a Scalar 124 7.3 Addition and Subtraction 125 7.4 Length 125 7.5 Equations 126 7.6 Geometrical Proofs 126 8 Transformation Geometry 136 8.1 Translation 136 8.2 Reflection 136 8.3 Rotation 137 8.4 Glide Reflection 138 8.5 Isometries 138 8.6 Dilatation 138 8.7 Shear 139 8.8 Stretch 140 8.9 Similarity and Congruence 140 8.10 Geometrical Problems 140 8.11 Area and Volume of Similar Figures 145 9 Matrices 153 9.1 Array Storage 153 9.2 Sum Difference, Multiplication by a Scalar 153 9.3 Multiplication 154 9.4 Algebraic Manipulation 1 154 9.5 Identity 155 9.6 Inverse Matrices 155 9.7 Algebraic Manipulation 2 156 9.8 Simultaneous Equations 156 9.9 Networks and Route Matrices 158 9.1 0 Incidence Matrices 159 9.11 Euler's Theorem 160 9.12 Traversability 160 9.13 Transformations 161 9.14 Combining Transformations 164 9.15 Inverse and Identity Transformations 164 10 Straight Lines and Circles 172 10.1 Parallel Lines 172 10.2 The Triangle 172 10.3 Polygons 173 10.4 Regular Polygons 173 10.5 Symmetry 174 10.6 Geometric Properties of Simple Plane Figures 175 10.7 Tesselations 178 10.8 Regular Tessellations 178 10.9 Combined or Modified Polygons 178 10.10 Semi-regular Tessellations 179 10.11 The Circle (Definitions) 179 10.12 The Properties of the Circle 179 11 Probability and Statistics 188 11.1 Sample Space 188 11.2 Definition of Probability 188 11.3 Complement, Certain and Impossible Events 189 vii 11.4 Set Notation 189 11.5 Combined Events, Tree Diagrams 191 11.6 Statistical Diagrams 194 11.7 Averages 196 11.8 Frequency Tables 196 11.9 Cumulative Frequency Curves (Ogives) 198 11.10 Information from the Cumulative Frequency Curve 198 11.11 Assumed Mean 199 11.12 Histograms 200 12 Trigonometry 200 12.1 Trigonometrical Ratios (Sine, Cosine, Tangent) 209 12.2 Pythagoras' Theorem 211 12.3 Relationship Between sin x, cos x and tan x 213 12.4 Solution of Right-angled Triangles 214 12.5 Special Triangles 215 12.6 Angles of Elevation and Depression 216 12.7 Bearings 216 12.8 Reduction to Right-angled Triangles 218 12.9 Perpendicular Height of a Triangle 220 12.10 The Sine Rule 220 12.11 Sine Rule (Ambiguous Case) 221 12.12 The Cosine Rule 222 12.13 Three-dimensional Problems 224 12.14 Latitude and Longitude 226 13 Drawing and Construction 23 7 13.1 Constructions Using Ruler and Compass 237 13.2 Plans and Elevations 244 13.3 Nets 246 13.4 Locus 247 14 Aural Examinations 253 Mental Arithmetic Test 1 254 Mental Arithmetic Test 2 255 Mental Arithmetic Test 3 255 Mental Arithmetic Test 4 256 15 Coursework 260 Answers 267 Index 279 viii Acknowledgements The author and publishers wish to thank the following who have kindly given permission for the use of copyright material: The Associated Examining Board, the East Anglian Regional Examinations Board, the Joint Matriculation Board, the Northern Ireland Schools Examination Council, the Oxford and Cambridge Schools Examination Board, the Scottish Examination Board, the Southern Universities' Joint Board, the University of Cambridge Local Examinations Syndicate, the University of London School Examinations Board, the University of Oxford Delegacy of Local Examinations and the Welsh Joint Education Committee for questions from past examination papers. Every effort has been made to trace all the copyright holders but if any have been inadvertently overlooked the publishers will be pleased to make the necessary arrangement at the first oppor­ tunity. The University of London Entrance and Schools Examinations Council accepts no responsi­ bility whatsoever for the accuracy or method in the answers given in this book to actual questions set by the London Board. Acknowledgement is made to the Southern Universities' Joint Board for School Examinations for permission to use questions taken from their past papers but the Board is in no way respon­ sible for answers that may be provided and they are solely the responsibility of the authors. The Associated Examining Board, the University of Oxford Delegacy of Local Examinations, the University of Cambridge Local Examinations Syndicate, the Northern Ireland Schools Examination Council and the Scottish Examination Board wish to point out that worked examples included in the text are entirely the responsibility of the author and have neither been provided nor approved by the Board. Organizations Responsible for GCSE Examinations In the United Kingdom, examinations are administered by the following organi­ zations. 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