Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3 Coursebook Sue Pemberton Julianne Hughes Series Editor: Julian Gilbey Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3 Coursebook 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 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 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. www.cambridge.org Information on this title: www.cambridge.org/9781108407199 © Cambridge University Press 2018 This publication is in copyright. 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Contents Contents Series introduction vi How to use this book viii Acknowledgements x 1 Algebra 1 1.1 The modulus function 2 1.2 Graphs of y = f(x) where f(x) is linear 7 1.3 Solving modulus inequalities 8 1.4 Division of polynomials 11 1.5 The factor theorem 14 1.6 The remainder theorem 18 End-of-chapter review exercise 1 22 2 Logarithmic and exponential functions 25 2.1 Logarithms to base 10 26 iii 2.2 Logarithms to base a 30 2.3 The laws of logarithms 33 2.4 Solving logarithmic equations 35 2.5 Solving exponential equations 38 2.6 Solving exponential inequalities 40 2.7 Natural logarithms 42 2.8 Transforming a relationship to linear form 44 End-of-chapter review exercise 2 50 3 Trigonometry 52 3.1 The cosecant, secant and cotangent ratios 53 3.2 Compound angle formulae 58 3.3 Double angle formulae 61 3.4 Further trigonometric identities 66 3.5 Expressing asinθ+ bcosθ in the form Rsin(θ±α) or Rcos(θ±α) 68 End-of-chapter review exercise 3 74 Cross-topic review exercise 1 76 Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3 4 Differentiation 80 4.1 The product rule 81 4.2 The quotient rule 84 4.3 Derivatives of exponential functions 87 4.4 Derivatives of natural logarithmic functions 91 4.5 Derivatives of trigonometric functions 95 4.6 Implicit differentiation 99 4.7 Parametric differentiation 103 End-of-chapter review exercise 4 108 5 Integration 111 5.1 Integration of exponential functions 112 1 5.2 Integration of 115 ax +b 5.3 Integration of sin(ax +b), cos(ax +b) and sec2(ax +b) 118 5.4 Further integration of trigonometric functions 121 P2 5.5 The trapezium rule 126 End-of-chapter review exercise 5 131 iv 6 Numerical solutions of equations 133 6.1 Finding a starting point 135 6.2 Improving your solution 140 6.3 Using iterative processes to solve problems involving other areas of mathematics 149 End-of-chapter review exercise 6 155 Cross-topic review exercise 2 158 7 Further algebra 165 P3 7.1 Improper algebraic fractions 166 7.2 Partial fractions 168 7.3 Binomial expansion of (1+ x)n for values of n that are not positive integers 174 7.4 Binomial expansion of (a + x)n for values of n that are not positive integers 177 7.5 Partial fractions and binomial expansions 179 End-of-chapter review exercise 7 182 P3 8 Further calculus 184 8.1 Derivative of tan−1 x 186 1 8.2 Integration of 187 x2 + a2 kf′(x) 8.3 Integration of 189 f(x) Contents 8.4 Integration by substitution 191 8.5 The use of partial fractions in integration 194 8.6 Integration by parts 197 8.7 Further integration 201 End-of-chapter review exercise 8 203 Cross-topic review exercise 3 205 P3 9 Vectors 209 P3 9.1 Displacement or translation vectors 211 9.2 Position vectors 220 9.3 The scalar product 225 9.4 The vector equation of a line 231 9.5 Intersection of two lines 236 End-of-chapter review exercise 9 240 10 Differential equations 243 P3 10.1 The technique of separating the variables 245 10.2 Forming a differential equation from a problem 252 v End-of-chapter review exercise 10 262 11 Complex numbers 265 P3 11.1 Imaginary numbers 267 11.2 Complex numbers 269 11.3 The complex plane 273 11.4 Solving equations 283 11.5 Loci 288 End-of-chapter review exercise 11 297 Cross-topic review exercise 4 300 P3 Pure Mathematics 2 Practice exam-style paper 307 Pure Mathematics 3 Practice exam-style paper 308 Answers 310 Glossary 353 Index 355 Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3 Series introduction Cambridge International AS & A Level Mathematics can be a life-changing course. On the one hand, it is a facilitating subject: there are many university courses that either require an A Level or equivalent qualification in mathematics or prefer applicants who have it. On the other hand, it will help you to learn to think more precisely and logically, while also encouraging creativity. Doing mathematics can be like doing art: just as an artist needs to master her tools (use of the paintbrush, for example) and understand theoretical ideas (perspective, colour wheels and so on), so does a mathematician (using tools such as algebra and calculus, which you will learn about in this course). But this is only the technical side: the joy in art comes through creativity, when the artist uses her tools to express ideas in novel ways. Mathematics is very similar: the tools are needed, but the deep joy in the subject comes through solving problems. You might wonder what a mathematical ‘problem’ is. This is a very good question, and many people have offered different answers. You might like to write down your own thoughts on this question, and reflect on how they change as you progress through this course. One possible idea is that a mathematical problem is a mathematical question that you do not immediately know how to answer. (If you do know how to answer it immediately, then we might call it an ‘exercise’ instead.) Such a problem will take time to answer: you may have to try different approaches, using different tools or ideas, on your own or with others, until you finally discover a way into it. This may take minutes, hours, days or weeks to achieve, and your sense of achievement may well grow with the effort it has taken. In addition to the mathematical tools that you will learn in this course, the problem-solving skills that you vi will develop will also help you throughout life, whatever you end up doing. It is very common to be faced with problems, be it in science, engineering, mathematics, accountancy, law or beyond, and having the confidence to systematically work your way through them will be very useful. This series of Cambridge International AS & A Level Mathematics coursebooks, written for the Cambridge Assessment International Education syllabus for examination from 2020, will support you both to learn the mathematics required for these examinations and to develop your mathematical problem-solving skills. The new examinations may well include more unfamiliar questions than in the past, and having these skills will allow you to approach such questions with curiosity and confidence. In addition to problem solving, there are two other key concepts that Cambridge Assessment International Education have introduced in this syllabus: namely communication and mathematical modelling. These appear in various forms throughout the coursebooks. Communication in speech, writing and drawing lies at the heart of what it is to be human, and this is no less true in mathematics. While there is a temptation to think of mathematics as only existing in a dry, written form in textbooks, nothing could be further from the truth: mathematical communication comes in many forms, and discussing mathematical ideas with colleagues is a major part of every mathematician’s working life. As you study this course, you will work on many problems. Exploring them or struggling with them together with a classmate will help you both to develop your understanding and thinking, as well as improving your (mathematical) communication skills. And being able to convince someone that your reasoning is correct, initially verbally and then in writing, forms the heart of the mathematical skill of ‘proof’. Series introduction Mathematical modelling is where mathematics meets the ‘real world’. There are many situations where people need to make predictions or to understand what is happening in the world, and mathematics frequently provides tools to assist with this. Mathematicians will look at the real world situation and attempt to capture the key aspects of it in the form of equations, thereby building a model of reality. They will use this model to make predictions, and where possible test these against reality. If necessary, they will then attempt to improve the model in order to make better predictions. Examples include weather prediction and climate change modelling, forensic science (to understand what happened at an accident or crime scene), modelling population change in the human, animal and plant kingdoms, modelling aircraft and ship behaviour, modelling financial markets and many others. In this course, we will be developing tools which are vital for modelling many of these situations. To support you in your learning, these coursebooks have a variety of new features, for example: ■ Explore activities: These activities are designed to offer problems for classroom use. They require thought and deliberation: some introduce a new idea, others will extend your thinking, while others can support consolidation. The activities are often best approached by working in small groups and then sharing your ideas with each other and the class, as they are not generally routine in nature. This is one of the ways in which you can develop problem- solving skills and confidence in handling unfamiliar questions. ■ Questions labelled as P , M or PS: These are questions with a particular emphasis on ‘Proof’, ‘Modelling’ or ‘Problem solving’. They are designed to support you in preparing for the new style of examination. They may or may not be harder than other questions in the exercise. ■ The language of the explanatory sections makes much more use of the words ‘we’, ‘us’ and ‘our’ than in previous coursebooks. This language invites and encourages you to be an active participant rather than an observer, simply following instructions (‘you do this, then you do that’). It is also the way that professional mathematicians usually write about mathematics. The new examinations may well present you with unfamiliar questions, and if you are vii used to being active in your mathematics, you will stand a better chance of being able to successfully handle such challenges. At various points in the books, there are also web links to relevant Underground Mathematics resources, which can be found on the free undergroundmathematics.org website. Underground Mathematics has the aim of producing engaging, rich materials for all students of Cambridge International AS & A Level Mathematics and similar qualifications. These high-quality resources have the potential to simultaneously develop your mathematical thinking skills and your fluency in techniques, so we do encourage you to make good use of them. We wish you every success as you embark on this course. Julian Gilbey London, 2018 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. The questions, example answers, marks awarded and/or comments that appear in this book were written by the author(s). In examination, the way marks would be awarded to answers like these may be different. Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3 How to use this book Throughout this book you will notice particular features that are designed to help your learning. This section provides a brief overview of these features. This book covers both Pure Mathematics 2 and Pure Mathematics 3. One topic (5.5 The trapezium rule) is only covered in Pure Mathematics 2 and this section is marked with the icon P2 . Chapters 7–11 are only covered in Pure Mathematics 3 and these are marked with the icon P3 . The icons appear in the Contents list and in the relevant sections of the book. In this chapter you will learn how to: PREREQUISITE KNOWLEDGE ■ ■ formulate a simple statement involving a rate of change as a differential equation fi nd, by integration, a general form of solution for a fi rst order differential equation in which the Where it comes from What you should be able to do Check your skills ■ variables are separable IGCSE® / O Level Mathematics Perform long division on numbers 1 Using long division, calculate: ■ u instee rapnr eint itthiael scoolnudtiiotino no ft oa fid nifdfe are pnatiratilc euqluara tsioolnu tiino tnh e context of a problem being modelled by the annecde fissnadr yt.he remainder where ba 140992887÷÷1427 equation . c 4283÷32 IGCSE / O Level Mathematics Sketch straight-line graphs. 2 Sketch the graph of y=2x−5. Learning objectives indicate the important concepts within each chapter and help you to v nauvvigaute through the coursebook. Prerequisite knowledge exercises identify prior learning that you need to have covered before starting the chapter. KEY POINT 8.4 Try the questions to identify any areas that you need to viii The∫ fourmdvuldax fo=r iunvte−gr∫avtiodnu bdyx parts: review before continuing with the chapter. dx dx Key point boxes contain a m ri ge I t r ti l AS & A L v l M t m ti s: Pure M t m ti s 2 & 3 WORKED EXAMPLE 1.3 summary of the most important Solve x+3+x+5 =10. methods, facts and formulae. Answer x+3+x+5 =10 Subtract x+5 from both sides. e double angle formulae. W x+3 =10−x+5 Split the equation into two parts. ≡ 2 − 2 x+3=10−x+5 (1) x+33= x++55 −−10 (2) Key terms are important terms in the topic that you Using equation (1): are learning. They are highlighted in orange bold. The x glossary contains clear definitions of these key terms. Worked examples provide step-by-step approaches to answering questions. The left side shows a fully worked solution, while the right side contains a commentary EXPLORE 6.1 u explaining each step in the working. Anil is trying to find the roots of f(xu)=xtanx =0 between 0 and 2π radians. He states: v f(3.1).0 and f(3.2).0. There is no change of sign so there is no root of tanx =0 between 3.1 and 3.2. TIP Mika states: y= tanx and it meets the x-axis where x=π radians so Anil has made a Solving trigonometric inequalities is not in the calculation error. Cambridge syllabus. You should, however, have the Discuss Anil and Mika’s statements. skills necessary to tackle these more challenging questions. Explore boxes contain enrichment activities for extension work. These activities promote group-work and peer- to-peer discussion, and are intended to deepen your Tip boxes contain helpful understanding of a concept. (Answers to the Explore guidance about calculating questions are provided in the Teacher’s Resource.) or checking your answers. Solv ° ° s r