Cambridge IGCSE and O Level Additional Mathematics Coursebook

Transcription

1 Cambridge IGCSE and O Level Additional Mathematics Coursebook Second edition

2 University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia , 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi , India 79 Anson Road, 06-04/06, Singapore Cambridge University Press is part of the University of Cambridge. It furthers the University s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. Information on this title: cambridge.org/ Cambridge University Press 2018 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. First published Printed in Malaysia by Vivar Printing A catalogue record for this publication is available from the British Library ISBN Paperback ISBN Cambridge Elevate Edition ISBN Paperback + Cambridge Elevate Edition 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. Information regarding prices, travel timetables, and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter. IGCSE is a registered trademark. Past exam paper questions throughout are reproduced by permission of Cambridge Assessment International Education. Cambridge Assessment International Education bears no responsibility for the example answers to questions taken from its past question papers which are contained in this publication. All exam-style questions and sample answers in this title were written by the authors. In examinations, the way marks are awarded may be different. notice to teachers in the uk It is illegal to reproduce any part of this work in material form (including photocopying and electronic storage) except under the following circumstances: (i) where you are abiding by a licence granted to your school or institution by the Copyright Licensing Agency; (ii) where no such licence exists, or where you wish to exceed the terms of a licence, and you have gained the written permission of Cambridge University Press; (iii) where you are allowed to reproduce without permission under the provisions of Chapter 3 of the Copyright, Designs and Patents Act 1988, which covers, for example, the reproduction of short passages within certain types of educational anthology and reproduction for the purposes of setting examination questions.

3 Contents Acknowledgements Introduction How to use this book 1 Functions Mappings Definition of a function Composite functions Modulus functions Graphs of y = f(x) where f(x) is linear Inverse functions The graph of a function and its inverse 15 Summary 18 Examination questions 19 2 Simultaneous equations and quadratics Simultaneous equations (one linear and one non-linear) Maximum and minimum values of a quadratic function Graphs of y = f(x) where f(x) is quadratic Quadratic inequalities Roots of quadratic equations Intersection of a line and a curve 42 Summary 44 Examination questions 46 3 Indices and surds Simplifying expressions involving indices Solving equations involving indices Surds Multiplication, division and simplification of surds Rationalising the denominator of a fraction Solving equations involving surds 63 Summary 67 Examination questions 67 4 Factors and polynomials Adding, subtracting and multiplying polynomials Division of polynomials The factor theorem Cubic expressions and equations The remainder theorem 82 Summary 86 Examination questions 87 5 Equations, inequalities and graphs Solving equations of the type ax b = cx d Solving modulus inequalities Sketching graphs of cubic polynomials and their moduli Solving cubic inequalities graphically Solving more complex quadratic equations 103 Summary 105 Examination questions Logarithmic and exponential functions Logarithms to base Logarithms to base a The laws of logarithms 118 vi vii viii iii

4 Cambridge IGCSE and O Level Additional Mathematics iv 6.4 Solving logarithmic equations Solving exponential equations Change of base of logarithms Natural logarithms Practical applications of exponential equations The graphs of simple logarithmic and exponential functions The graphs of y = k e nx + a and y = k ln (ax + b) where n, k, a and b are integers The inverse of logarithmic and exponential functions 133 Summary 134 Examination questions Straight-line graphs Problems involving length of a line and midpoint Parallel and perpendicular lines Equations of straight lines Areas of rectilinear figures Converting from a non-linear equation to linear form Converting from linear form to a non-linear equation Finding relationships from data 159 Summary 165 Examination questions Circular measure Circular measure Length of an arc Area of a sector 177 Summary 180 Examination questions Trigonometry Angles between 0 and The general definition of an angle Trigonometric ratios of general angles Graphs of trigonometric functions Graphs of y = f(x), where f(x) is a trigonometric function Trigonometric equations Trigonometric identities Further trigonometric equations Further trigonometric identities 218 Summary 220 Examination questions Permutations and combinations Factorial notation Arrangements Permutations Combinations 234 Summary 237 Examination questions Series Pascal s triangle The binomial theorem Arithmetic progressions Geometric progressions Infinite geometric series Further arithmetic and geometric series 267 Summary 270 Examination questions 271 iv

5 Contents 12 Differentiation The gradient function The chain rule The product rule The quotient rule Tangents and normals Small increments and approximations Rates of change Second derivatives Stationary points Practical maximum and minimum problems 305 Summary 310 Examination questions Vectors Further vector notation Position vectors Vector geometry Constant velocity problems 327 Summary 331 Examination questions Differentiation Derivatives of exponential functions Derivatives of logarithmic functions Derivatives of trigonometric functions Further applications of differentiation 350 Summary 356 Examination questions Integration Differentiation reversed Indefinite integrals Integration of functions of the form (ax + b)n Integration of exponential functions Integration of sine and cosine functions Integration of functions of the form 1 x and ax + b 15.7 Further indefinite integration Definite integration Further definite integration Area under a curve Area of regions bounded by a line and a curve 392 Summary 397 Examination questions Kinematics Applications of differentiation in kinematics Applications of integration in kinematics 412 Summary 418 Examination questions 419 Answers 422 Index 454 v

6 Cambridge IGCSE and O Level Additional Mathematics Acknowledgements Past examination paper questions throughout are reproduced by permission of Cambridge Assessment International Education. Thanks to the following for permission to reproduce images: Cover artwork: Shestakovych/Shutterstock Chapter 1 Fan jianhua/shutterstock; Chapter 2 zhu difeng/shutterstock; Chapter 3 LAGUNA DESIGN/Getty Images; Chapter 4 Michael Dechev/Shutterstock; Fig. 4.1 Steve Bower/Shutterstock; Fig. 4.2 Laboko/Shutterstock; Fig. 4.3 irin-k/shutterstock; Chapter 5 zentilia/shutterstock; Chapter 6 Peshkova/Shutterstock; Chapter 7 ittipon/shutterstock; Chapter 8 Zhu Qiu/EyeEm/Getty Images; Chapter 9 paul downing/getty Images; Fig. 9.1 aarrows/shutterstock; Chapter 10 Gino Santa Maria/ Shutterstock; Fig. 10.1snake3d/Shutterstock; Fig Keith Publicover/Shutterstock; Fig Aleksandr Kurganov/Shutterstock; Fig Africa Studio/Shutterstock; Chapter 11 elfinadesign/ Shutterstock; Chapter 12 AlenKadr/Shutterstock; Chapter 13 muratart/shutterstock; Chapter 14 Neamov/Shutterstock; Chapter 15 Ahuli Labutin/Shutterstock; Chapter 16 AlexLMX/Getty vi

7 Introduction This highly illustrated coursebook covers the Cambridge IGCSE and O Level Additional Mathematics syllabuses (0606 and 4037). The course is aimed at students who are currently studying or have previously studied Cambridge IGCSE Mathematics (0580) or Cambridge O Level Mathematics (4024). Where the content in one chapter includes topics that should have already been covered in previous studies, a recap section has been provided so that students can build on their prior knowledge. Class discussion sections have been included to provide students with the opportunity to discuss and learn new mathematical concepts with their classmates, with their class teacher acting as the facilitator. The aim of these class discussion sections is to improve the student s reasoning and oral communication skills. Challenge questions have been included at the end of most exercises to challenge and stretch highability students. Towards the end of each chapter, there is a summary of the key concepts to help students consolidate what they have just learnt. This is followed by a Past paper questions section, which contains real questions taken from past examination papers. A Practice Book is also available in the Cambridge IGCSE Additional Mathematics series, which offers students further targeted practice. This book closely follows the chapters and topics of the coursebook offering additional exercises to help students to consolidate concepts learnt and to assess their learning after each chapter. A Teacher s Resource, to offer support and advice, is also available. vii

8 Cambridge IGCSE and O Level Additional Mathematics How to use this book Chapter each chapter begins with a set of learning objectives to explain what you will learn in this chapter. viii Recap check that you are familiar with the introductory skills required for the chapter. Class Discussion additional activities to be done in the classroom for enrichment.

9 How to use this book Worked Example detailed step-by-step approaches to help students solve problems. Note quick suggestions to remind you about key facts and highlight important points. Challenge challenge yourself with tougher questions that stretch your skills. ix Summary at the end of each chapter to review what you have learnt.

10 Cambridge IGCSE and O Level Additional Mathematics Examination questions exam-style questions for you to test your knowledge and understanding at the end of each chapter. Examination questions Worked example 3 2 The function f is such that f( x) = 4x 8 x + ax + b, where a and b are constants. It is given that 2x 1 is a factor of f(x) and that when f(x) is divided by x + 2 the remainder is 20. Find the remainder when f(x) is divided by x 1. [6] Answer 3 2 f( x) = 4x 8x + ax + b 1 If 2x 1 is a factor, then f = a + b = a + b = a + b = a + 2b = (1) Cambridge IGCSE Additional Mathematics 0606 Paper 11 Q2 Nov 2011 x Remainder = 20 when divided by x + 2, means that f( 2) = ( 2) 8( 2) + a( 2) + b = a + b = 20 2a + b = (2) From (1) a = 3 2b. Substituting in (2), gives: 23 ( 2b) + b = b + b = 84 5b = 90 b = 18 So a = 33, b = Remainder when f( x) = 4x 8x 33x + 18 is divided by (x 1) is f(1). 3 2 Remainder = 41 () 81 () 33() = = 19 Activity for you to apply your theoretical knowledge to a practical task. y = cos x + 1 is a translation of y = cos x by the vector 0 1 y = cos x + 2 is a translation of y = cos x by the vector y = cos x 3 is a translation of y = cos x by the vector 3 and y = tan x + 1 is a translation of y = tan x by the vector y = tan x 2 is a translation of y = tan x by the vector 2