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A Prelims i-xiv.qxd 15/6/04 10:19 am Page i Advancing Maths for AQA CORE MATHS 1 Sam Boardman, Tony Clough and David Evans ● 1 ● 2 ● 3 ● 4 ● 5 ● 6 ● 7 ● 8 ● 9 ● 10 ● 11 ● 12 Series editors Sam Boardman Roger Williamson Ted Graham David Pearson Core Maths 1 1 Advancing from GCSE maths: algebra review 2 Surds 3 Coordinate geometry of straight lines 4 Quadratics and their graphs 5 Polynomials 6 Factors, remainders and cubic graphs 7 Simultaneous equations and quadratic inequalities 8 Coordinate geometry of circles 9 Introduction to differentiation: gradient of curves 10 Applications of differentiation: tangents, normals and 1 13 28 52 70 80 99 114 137 153 rates of change 11 Maximum and minimum points and optimisation problems 166 12 Integration 185 Exam style practice paper 204 Answers 388 Index 437 A Prelims i-xiv.qxd 15/6/04 10:19 am Page ii A Prelims i-xiv.qxd 15/6/04 10:19 am Page iii Advancing Maths for AQA CORE MATHS 2 Sam Boardman,Tony Clough, David Evans and Maureen Nield ● 1 ● 2 ● 3 ● 4 ● 5 ● 6 ● 7 ● 8 ● 9 ● 10 ● 11 ● 12 Series editors Sam Boardman Roger Williamson Ted Graham David Pearson Core Maths 2 1 Indices 2 Further differentiation 3 Further integration and the trapezium rule 4 Basic trigonometry 5 Simple transformation of graphs 6 Solving trigonometrical equations 7 Factorials and binomial expansions 8 Sequences and series 9 Radian measure 10 Further trigonometry with radians 11 Exponentials and logarithms 12 Geometric series 207 221 239 253 279 288 303 317 337 349 359 372 Exam style practice paper 385 Answers 413 Index 437 A Prelims i-xiv.qxd 15/6/04 10:19 am Page iv Heinemann Educational Publishers Halley Court, Jordan Hill, Oxford OX2 8EJ Part of Harcourt Education Heinemann is the registered trademark of Harcourt Education Limited © Sam Boardman, Tony Clough, David Evans and Maureen Nield 2000, 2004 Complete work © Harcourt Education Limited 2004 First published 2004 08 07 06 05 04 10 9 8 7 6 5 4 3 2 1 British Library Cataloguing in Publication Data is available from the British Library on request. ISBN 0 435 51330 3 Copyright notice All rights reserved. No part of this publication may be reproduced in any form or by any means (including photocopying or storing it in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright owner, except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1T 4LP. Applications for the copyright owner’s written permission should be addressed to the publisher. Edited by Alex Sharpe, Standard Eight Limited Typeset and illustrated by Tech-Set Limited, Gateshead, Tyne & Wear. Original illustrations © Harcourt Education Limited, 2004 Cover design by Miller, Craig and Cocking Ltd Printed in Great Britain by The Bath Press, Bath Acknowledgements The publishers’ and authors’ thanks are due to the AQA for permission to reproduce questions from past examination papers. The answers have been provided by the authors and are not the responsibility of the examining board. Every effort has been made to contact copyright holders of material reproduced in this book. Any omissions will be rectified in subsequent printings if notice is given to the publishers. A Prelims i-xiv.qxd 15/6/04 10:19 am Page v About this book This book is one in a series of textbooks designed to provide you with exceptional preparation for AQA’s 2004 Mathematics Specification. The series authors are all senior members of the examining team and have prepared the textbooks specifically to support you in studying this course. Finding your way around The following are there to help you find your way around when you are studying and revising: ● edge marks (shown on the front page) – these help you to get to the right chapter quickly; ● contents list – this identifies the individual sections dealing with key syllabus concepts so that you can go straight to the areas that you are looking for; ● index – a number in bold type indicates where to find the main entry for that topic. Key points Key points are not only summarised at the end of each chapter but are also boxed and highlighted within the text like this: An equation of the form ax b 0, where a and b are constants, is said to be a linear equation with variable x. Exercises and exam questions Worked examples and carefully graded questions familiarise you with the syllabus and bring you up to exam standard. Each book contains: ● Worked examples and Worked exam questions to show you how to tackle typical questions; Examiner’s tips will also provide guidance; ● Graded exercises, gradually increasing in difficulty up to exam-level questions, which are marked by an [A]; ● Test-yourself sections for each chapter so that you can check your understanding of the key aspects of that chapter and identify any sections that you should review; ● Answers to the questions are included at the end of the book. A Prelims i-xiv.qxd 15/6/04 10:19 am Page vi Contents C1 1 Advancing from GCSE maths: algebra review Learning objectives 1.1 Advancing from GCSE 1.2 Solving linear equations An historical problem 1.3 Solving simultaneous linear equations 1.4 Linear inequalities 1.5 Function notation 1.6 Use of calculators Key point summary Test yourself 1 1 2 4 5 7 9 10 11 12 2 Surds Learning objectives 2.1 Special sets of numbers 2.2 Surds Order of a surd 2.3 Simplest form of surds 2.4 Manipulating square roots 2.5 Use in geometry 2.6 Like and unlike surds 2.7 Adding and subtracting surds 2.8 Multiplying surds 2.9 Rationalising the denominator 2.10 Equations and inequalities involving surds Key point summary Test yourself 13 13 14 14 15 16 17 17 18 19 21 23 26 26 3 Coordinate geometry of straight lines Learning objectives 3.1 Cartesian coordinates 3.2 The distance between two points 3.3 The coordinates of the mid-point of a line segment joining two known points 3.4 The gradient of a straight line joining two known points 3.5 The gradients of perpendicular lines 3.6 The y mx c form of the equation of a straight line 28 28 30 33 35 37 38 A Prelims i-xiv.qxd 15/6/04 10:19 am Page vii Contents 3.7 The y y1 m(x x1) form of the equation of a straight line 3.8 The equation of a straight line passing through two given points 3.9 The coordinates of the point of intersection of two lines Key point summary Test yourself 40 42 43 50 51 4 Quadratic functions and their graphs Learning objectives 4.1 Parabolas and quadratic functions 4.2 Factorising quadratics and sketching graphs 4.3 Completing the square 4.4 Translations parallel to the x-axis 4.5 General translations of quadratic graphs 4.6 General quadratic equation formula 4.7 The discriminant Key point summary Test yourself 52 52 53 55 61 62 63 64 68 69 5 Polynomials Learning objectives 5.1 Introduction and definitions 5.2 Adding and subtracting polynomials Notation 5.3 Multiplying polynomials 5.4 Finding coefficients Key point summary Test yourself 70 70 71 72 74 75 78 79 6 Factors, remainders and cubic graphs Learning objectives 6.1 Factorisation 6.2 The factor theorem 6.3 Further factorisation 6.4 Graphs of cubic functions 6.5 Dividing a polynomial by a linear expression 6.6 The remainder theorem Key point summary Test yourself 80 80 80 84 88 91 94 97 98 7 Simultaneous equations and quadratic inequalities Learning objectives 7.1 Simultaneous equations 7.2 The intersection of a line and a curve 99 99 101 vii A Prelims i-xiv.qxd viii 15/6/04 10:19 am Page viii Contents 7.3 The condition for a line to be a tangent to the graph of a quadratic function 7.4 Quadratic inequalities 7.5 Quadratics with irrational roots 7.6 Completing the square 7.7 The discriminant revisited Key point summary Test yourself 103 105 107 108 109 112 113 8 Coordinate geometry of circles Learning objectives 8.1 The Cartesian equation of a circle 8.2 Sketching and applying translations on circles 8.3 Finding the equation of a circle using circle properties 8.4 Conditions for a line to meet a circle 8.5 The length of the tangents from a point to a circle 8.6 The equation of the tangent and the equation of the normal at a point on a circle Key point summary Test yourself 114 114 118 121 125 128 129 134 136 9 Introduction to differentiation: gradient of curves Learning objectives 9.1 Introduction 9.2 Gradients of chords 9.3 The tangent to a curve at a point 9.4 The gradient of a curve as the limit of the gradient of the chord 9.5 The derivative or derived function 9.6 Differentiation notation 9.7 General rules for differentiation 9.8 Finding gradients at specific points on a curve 9.9 Finding points on a curve with a given gradient 9.10 Simplifying expressions before differentiating Key point summary Test yourself 137 137 138 139 141 143 144 145 147 149 150 151 152 10 Applications of differentiation: tangents, normals and rates of change Learning objectives 10.1 The equation of a tangent to a curve 10.2 The equation of a normal to a curve 153 153 155 A Prelims i-xiv.qxd 15/6/04 10:19 am Page ix Contents 10.3 Rates of change 10.4 Increasing and decreasing functions Key point summary Test yourself 157 161 164 165 11 Maximum and minimum points and optimisation problems Learning objectives 11.1 The least value of a quadratic function 11.2 Maximum and minimum points Maximum points Minimum points 11.3 Second derivatives 11.4 The second derivative test 11.5 Stationary points of inflection 11.6 Optimisation Key point summary Test yourself 166 166 168 168 168 171 171 176 177 182 184 12 Integration Learning objectives 12.1 Finding a function from its derivative 12.2 Use of an arbitrary constant 12.3 Finding the equation of a curve 12.4 Simplifying expressions before integrating 12.5 Indefinite integrals 12.6 Integration as an area 12.7 Definite integrals 12.8 Regions bounded by lines and curves 12.9 Regions below the x-axis Key point summary Test yourself 185 185 186 187 188 189 191 192 194 195 202 203 Exam style practice paper 204 Answers 388 Index 437 ix A Prelims i-xiv.qxd 15/6/04 10:19 am Page x Contents C2 1 Indices Learning objectives 1.1 Introduction 1.2 Index notation 1.3 The laws of indices 1.4 Negative indices 1.5 The zero index 1.6 Fractional indices 1.7 Solving equations with indices Key point summary Test yourself 207 207 208 208 211 212 214 217 220 220 2 Further differentiation Learning objectives 2.1 Review of differentiation techniques 2.2 Negative and fractional powers 2.3 Simplifying expressions before differentiating 2.4 Finding gradients of curves and equations of tangents and normals 2.5 Stationary points 2.6 Optimisation Key point summary Test yourself 221 221 221 225 227 229 233 237 237 3 Further integration and the trapezium rule Learning objectives 3.1 Basic rule for the integration of xn 3.2 Simplifying expressions before integrating 3.3 Indefinite integrals 3.4 Definite integrals 3.5 Interpretation of an area 3.6 Estimation of area using the trapezium rule Key point summary Test yourself 239 239 241 243 244 245 247 251 252 4 Basic trigonometry Learning objectives 4.1 Introduction 4.2 The sine function – an informal approach 253 253 253 A Prelims i-xiv.qxd 15/6/04 10:19 am Page xi Contents 4.3 4.4 The cosine function – an informal approach Definitions of the sine and cosine functions Sine Cosine 4.5 The graph of the sine function Key features of the graph of y sin 4.6 The graph of the cosine function Key features of the graph of y cos 4.7 Finding angles with the same sine or cosine 4.8 The tangent function 4.9 The graph of the tangent function Key features of the graph of y tan 4.10 Solving trigonometrical equations 4.11 Sine rule Ambiguous case 4.12 Cosine rule 4.13 The area of a triangle Key point summary Test yourself 255 256 256 256 257 257 257 258 259 261 262 262 263 267 270 271 272 275 278 5 Simple transformations of graphs Learning objectives 5.1 Translations parallel to the y-axis 5.2 Translations parallel to the x-axis 5.3 General translations of graphs 5.4 Reflections 5.5 Stretches in the y-direction 5.6 Stretches in the x-direction 5.7 Scale factors Key point summary Test yourself 279 279 280 281 282 283 284 284 286 287 6 Solving trigonometrical equations Learning objectives 6.1 Solving equations of the form sin bx k, cos bx k and tan bx c 6.2 Solving equations of the form sin(x b) k, cos(x b) k and tan(x b) c 6.3 Solving quadratic equations in sin x, cos x and tan x 6.4 The relationship cos2 sin2 1 6.5 The relationship between sin , cos and tan Key point summary Test yourself 288 288 291 292 295 295 300 301 xi A Prelims i-xiv.qxd xii 15/6/04 10:19 am Page xii Contents 7 Factorials and binomial expansions Learning objectives 7.1 Factorial notation 7.2 Inductive definition of factorial and 0! n 7.3 The notation r 7.4 Pascal’s triangle 7.5 Binomial coefficients Key point summary Test yourself 303 303 304 305 306 310 315 316 8 Sequences and series Learning objectives 8.1 Sequences 8.2 Suffix notation 8.3 Inductive definition 8.4 Limit of a sequence 8.5 Arithmetic sequences 8.6 Arithmetic series 8.7 Sum of the first n natural numbers 8.8 Sum of the first n terms of an arithmetic series Alternative formula for Sn 8.9 Sigma notation 8.10 Use of the sigma notation for arithmetic series 8.11 Review of arithmetic series Key point summary Test yourself 317 317 318 318 319 322 323 324 326 327 329 329 332 334 335 9 Radian measure Learning objectives 9.1 Radians as a unit of measure of angles 9.2 Changing between degrees and radians 9.3 Arc length of a circle 9.4 Area of a sector of a circle Key point summary Test yourself 337 337 337 340 342 348 348 10 Further trigonometry with radians Learning objectives 10.1 Introduction 10.2 Solving simple trigonometrical equations in terms of radians 10.3 Using trigonometrical identities to solve equations where answers are given in radians Key point summary Test yourself 349 349 350 354 358 358 A Prelims i-xiv.qxd 15/6/04 10:19 am Page xiii Contents 11 Exponentials and logarithms Learning objectives 11.1 Exponential functions 11.2 Introduction to logarithms and the notation used 11.3 Laws of logarithms 11.4 Solving equations of the form ax b Key point summary Test yourself 359 359 361 363 365 370 370 12 Geometric series Learning objectives 12.1 Would you like to be a millionaire? 12.2 Geometric series 12.3 Sum of the first n terms of a geometric series 12.4 Sum to infinity of a geometric series 12.5 Use of logarithms to find number of terms Key point summary Test yourself 372 372 373 375 376 379 384 384 Exam style practice paper 385 Answers 413 Index 417 xiii A Prelims i-xiv.qxd 15/6/04 10:19 am Page xiv