Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3 Coursebook 9781108407199

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3 Coursebook
 9781108407199

Table of contents :
Cover
Title page
Copyright
Contents
Series introduction
How to use this book
Acknowledgements
1 Algebra
2 Logarithmic and exponential functions
3 Trigonometry
Cross-topic review exercise 1
4 Differentiation
5 Integration
6 Numerical solutions of equations
Cross-topic review exercise 2
7 Further algebra
8 Further calculus
Cross-topic review exercise 3
9 Vectors
10 Differential equations
11 Complex numbers
Cross-topic review exercise 4
Pure Mathematics 2 Practice exam-style paper
Pure Mathematics 3 Practice exam-style paper
Answers
Glossary
Index

Citation preview

Cambridge International AS & A Level Mathematics:

Pure Mathematics 2 & 3 Coursebook

Copyright Material - Review Only - Not for Redistribution

Copyright Material - Review Only - Not for Redistribution

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Cambridge International AS & A Level Mathematics: C op

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Sue Pemberton Julianne Hughes Series Editor: Julian Gilbey

Coursebook y op y op -R s es

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Pure Mathematics 2 & 3

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University Printing House, Cambridge CB2 8BS, United Kingdom

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One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia

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314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India

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79 Anson Road, #06–04/06, Singapore 079906

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Cambridge University Press is part of the University of Cambridge.

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© Cambridge University Press 2018

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www.cambridge.org Information on this title: www.cambridge.org/9781108407199

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.

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First published 2018

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It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence.

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20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Printed in the United Kingdom by Latimer Trend

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A catalogue record for this publication is available from the British Library

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ISBN 978-1-108-40719-9 Paperback

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® 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.

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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.

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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.

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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.

Copyright Material - Review Only - Not for Redistribution

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Contents

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Contents

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1 Algebra

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Acknowledgements

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How to use this book

1.1 The modulus function

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1.2 Graphs of y = f( x ) where f( x ) is linear 1.3 Solving modulus inequalities

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Series introduction

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1.4 Division of polynomials

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End-of-chapter review exercise 1

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2.2 Logarithms to base a

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2.1 Logarithms to base 10 2.3 The laws of logarithms

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2 Logarithmic and exponential functions

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2.4 Solving logarithmic equations

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2.5 Solving exponential equations

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3.3 Double angle formulae

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3.2 Compound angle formulae

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3.5 Expressing a sin θ + b cos θ in the form R sin(θ ± α ) or R cos(θ ± α )

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3.4 Further trigonometric identities

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End-of-chapter review exercise 3

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Cross-topic review exercise 1

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3.1 The cosecant, secant and cotangent ratios

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3 Trigonometry

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End-of-chapter review exercise 2

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2.8 Transforming a relationship to linear form

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2.7 Natural logarithms

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2.6 Solving exponential inequalities

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1.6 The remainder theorem

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1.5 The factor theorem

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4.3 Derivatives of exponential functions

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End-of-chapter review exercise 4

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4.7 Parametric differentiation

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6.1 Finding a starting point 6.2 Improving your solution

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6 Numerical solutions of equations

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End-of-chapter review exercise 5

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6.3 Using iterative processes to solve problems involving other areas of mathematics

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End-of-chapter review exercise 6

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7.3 Binomial expansion of (1 + x ) for values of n that are not positive integers

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7.4 Binomial expansion of ( a + x ) n for values of n that are not positive integers

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7.5 Partial fractions and binomial expansions

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End-of-chapter review exercise 7

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8.1 Derivative of tan −1 x 1 8.2 Integration of 2 x + a2 k f ′( x ) 8.3 Integration of f( x )

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8 Further calculus

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7.2 Partial fractions

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7.1 Improper algebraic fractions

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7 Further algebra

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Cross-topic review exercise 2 P3

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P2 5.5 The trapezium rule

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5.1 Integration of exponential functions 1 5.2 Integration of ax + b 5.3 Integration of sin( ax + b ), cos( ax + b ) and sec 2 ( ax + b ) 5.4 Further integration of trigonometric functions

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4.6 Implicit differentiation

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4.5 Derivatives of trigonometric functions

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4.4 Derivatives of natural logarithmic functions

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4.2 The quotient rule

5 Integration

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4.1 The product rule

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4 Differentiation

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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184 186 187 189

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Contents

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8.7 Further integration

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9.1 Displacement or translation vectors

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Cross-topic review exercise 3

9.2 Position vectors

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9.3 The scalar product

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End-of-chapter review exercise 9

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10.1 The technique of separating the variables

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10 Differential equations

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11 Complex numbers

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End-of-chapter review exercise 11

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Cross-topic review exercise 4

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11.5 Loci

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11.4 Solving equations

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Glossary

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Answers

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Pure Mathematics 3 Practice exam-style paper

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Pure Mathematics 2 Practice exam-style paper

Index

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11.3 The complex plane

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11.2 Complex numbers

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11.1 Imaginary numbers

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10.2 Forming a differential equation from a problem End-of-chapter review exercise 10

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9.5 Intersection of two lines

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9.4 The vector equation of a line

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End-of-chapter review exercise 8

9 Vectors

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8.6 Integration by parts

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8.5 The use of partial fractions in integration

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8.4 Integration by substitution

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308 310 353 355

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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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.

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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.

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Series introduction

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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.

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In addition to the mathematical tools that you will learn in this course, the problem-solving skills that you 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.

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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.

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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’.

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Series introduction

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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.

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To support you in your learning, these coursebooks have a variety of new features, for example:

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■ 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 problemsolving 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 used to being active in your mathematics, you will stand a better chance of being able to successfully handle such challenges.

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We wish you every success as you embark on this course.

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Julian Gilbey London, 2018

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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.

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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.

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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.

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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How to use this book Pr es s

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Throughout this book you will notice particular features that are designed to help your learning. This section provides a brief overview of these features.

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Where it comes from

IGCSE / O Level Mathematics

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x + 3 = 10 − x + 5

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x + 3 = x + 5 − 10

(2)

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Split the equation into two parts.

x + 3 = 10 − x + 5

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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 explaining each step in the working.

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f(3.1) . 0 and f(3.2) . 0. There is no change of sign so there is no root of tan x = 0 between 3.1 and 3.2.

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Mika states:

y = tan x and it meets the x-axis where x = π radians so Anil has made a calculation error.

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Solving trigonometric inequalities is not in the Cambridge syllabus. You should, however, have the skills necessary to tackle these more challenging questions.

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Explore boxes contain enrichment activities for extension work. These activities promote group-work and peerto-peer discussion, and are intended to deepen your understanding of a concept. (Answers to the Explore questions are provided in the Teacher’s Resource.)

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Discuss Anil and Mika’s statements.

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Subtract x + 5 from both sides.

Using equation (1):

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Anil is trying to find the roots of f( x u) = xtan x = 0 between 0 and 2π radians.

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x + 3 + x + 5 = 10

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He states:

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Answer

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EXPLORE 6.1

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Solve x + 3 + x + 5 = 10.

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WORKED EXAMPLE 1.3

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Key terms are important terms in the topic that you are learning. They are highlighted in orange bold. The glossary contains clear definitions of these key terms.

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2 Sketch the graph of y = 2 x − 5.

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C Key point boxes contain a l AS & A L v l M t m ti s: Pure M t summary of the most important methods, facts and formulae.

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c 4283 ÷ 32 Sketch straight-line graphs.

Prerequisite knowledge exercises identify prior learning that you need to have covered before starting the chapter. Try the questions to identify any areas that you need to review before continuing with the chapter.

e double angle formulae. W 2 ≡ − 2

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a 4998 ÷ 14 b 10 287 ÷ 27

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1 Using long division, calculate:

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KEY POINT 8.4

The formula for integration by parts: dv du u dx = uv − v dx dx dx



Check your skills

Perform long division on numbers and find the remainder where necessary.

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Learning objectives indicate the important concepts within each chapter and help you to v uv u navigate through the coursebook.

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What you should be able to do

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IGCSE® / O Level Mathematics

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■ ■ ■ ■

PREREQUISITE KNOWLEDGE

formulate a simple statement involving a rate of change as a differential equation find, by integration, a general form of solution for a first order differential equation in which the variables are separable use an initial condition to find a particular solution interpret the solution of a differential equation in the context of a problem being modelled by the equation.

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In this chapter you will learn how to:

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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.

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Tip boxes contain helpful guidance about calculating or checking your answers.

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These questions focus on proofs.

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These questions focus on modelling.

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DID YOU KNOW?

You should not use a calculator for these questions.

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The relationship between the Fibonacci numbers and the golden ratio produces the golden spiral.

These questions are taken from past examination papers.

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This can be seen in many natural settings, one of which is the spiralling pattern of the petals in a rose flower.

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Did you know? boxes contain interesting facts showing how Mathematics relates to the wider world.

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1 sin(ax + b ) + c a

END-OF-CHAPTER REVIEW EXERCISE 7

1 sin(ax + b ) dx = − cos(ax + b ) + c a

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Expand

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Expand (2 − x )(1 + 2 x ) 2 in ascending powers of x, up to and including the term in x 2, simplifying the coefficients.

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Expand

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Expand

31−

Cambridge International A Level Mathematics 9709 Paper 31 Q1 June 2011 −

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1 + 3x in ascending powers of x up to and including the term in x 2, simplifying the coefficients. [4] (1 + 2 x ) Cambridge International A Level Mathematics 9709 Paper 31 Q2 June 2013 1 in ascending powers of x, up to and including the term in x 3 , simplifying the coefficients. 1 + 3x x x

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This exercise is for Pure Mathematics 3 students only. 1

Relative to the origin O, the position vectors of the points A and B are given by  −5 OA =  0   3

Web link boxes contain links to useful resources on the internet.

  1  and OB =  7  .     2 



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Find a vector equation of the line AB.

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The line AB is perpendicular to the line L with vector equation:  m  4 r = 2 +µ 3     −3   9

 .  

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[3]

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Cross-topic review exercises appear after several chapters, and cover topics from across the preceding chapters.

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Cambridge International A Level Mathematics 9709 Paper 31 Q2 November 2016

CROSS-TOPIC REVIEW EXERCISE 4 2

Try the Equation or identity?(II) resource on the Underground Mathematics website

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(6x ) in ascending powers of x up to and including the term in x 3, simplifying the coefficients. [4]

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WEB LINK

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Expand (1 − 2 x )−4 in ascending powers of x, up to and including the term in x 3, simplifying the coefficients. [4]

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Extension material goes beyond the syllabus. It is highlighted by a red line to the left of the text.

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At the end of each chapter there is a Checklist of learning and understanding. The checklist contains a summary of the concepts that were covered in the chapter. You can use this to quickly check that you have covered the main topics.

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cos(ax + b ) dx =

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1 sin x dx = − cos x + c

ax + b

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1 dx = ln x + c x 1 1 cos x dx = sin x + c

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ax + b

dx = e x + c

The End-of-chapter review contains exam-style questions covering all topics in the chapter. You can use this to check your understanding of the topics you have covered. The number of marks gives an indication of how long you should be spending on the question. You should spend more time on questions with higher mark allocations; questions with only one or two marks should not need you to spend time doing complicated calculations or writing long explanations.

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∫ 1 ∫ e ∫ dx = a e + c 1 1 ∫ ax +∫ b dx = a ln ax + b

Integration formulae

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Checklist of learning and understanding ●

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From a tight centre, petals flow outward in a spiral, growing wider and larger as they do so. Nature has made the best use of an incredible structure. This perfect arrangement of petals is why the small, compact rose bud grows into such a beautiful flower.

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Throughout each chapter there are multiple exercises containing practice questions. The questions are coded:

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Rewind and Fast forward boxes direct you to related learning. Rewind boxes refer to earlier learning, in case you need to revise a topic. Fast forward boxes refer to topics that you will cover at a later stage, in case you would like to extend your study.

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You need to have studied the skills in Chapter 8 before attempting questions 13 and 14.

You might need to look back at Chapter 3 before attempting question 11.

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FAST FORWARD

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REWIND

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How Series to use introduction this book

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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Acknowledgements

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The authors and publishers acknowledge the following sources of copyright material and are grateful for the permissions granted. While every effort has been made, it has not always been possible to identify the sources of all the material used, or to trace all copyright holders. If any omissions are brought to our notice, we will be happy to include the appropriate acknowledgements on reprinting.

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Thanks to the following for permission to reproduce images: Cover image Mint Images – Frans Lanting/Getty Images

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Past examination questions throughout are reproduced by permission of Cambridge Assessment International Education.

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Inside (in order of appearance) sakkmesterke/Getty Images, Don Johnston/Getty Images, Photo by marianna armata/Getty Images, DEA PICTURE LIBRARY/Getty Images, malcolm park/Getty Images, agsandrew/ Shutterstock, Stephan Snyder/Shutterstock, alengo/Getty Images, Moritz Haisch/EyeEm/Getty Images, Rosemary Calvert/Getty Images, bluecrayola/Shutterstock, PinkCat/Shutterstock, Maurice Rivera/EyeEm/Getty Images, PASIEKA/Getty Images, Alfred Pasieka/Science Photo Library/Getty Images

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Chapter 1 Algebra

1

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understand the meaning of x , sketch the graph of y = ax + b and use relations such as a = b ⇔ a 2 = b2 and x − a , b   ⇔   a − b , x , a + b in the course of solving equations and inequalities divide a polynomial, of degree not exceeding 4, by a linear or quadratic polynomial, and identify the quotient and remainder (which may be zero) use the factor theorem and the remainder theorem.

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■ ■ ■

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In this chapter you will learn how to:

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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PREREQUISITE KNOWLEDGE

What you should be able to do

IGCSE® / O Level Mathematics

Perform long division on numbers and find the remainder where necessary.

1 Using long division, calculate: a 4998 ÷ 14

Pr es s

-C y

b 10 287 ÷ 27 c 4283 ÷ 32

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Sketch straight-line graphs.

2 Sketch the graph of y = 2 x − 5.

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IGCSE / O Level Mathematics

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Why do we need to study algebra?

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−b ± b2 − 4ac , for solving quadratic 2a equations. Versions of this formula were known and used by the Babylonians nearly 4000 years ago.

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Check your skills

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Where it comes from

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You have previously learnt the general formula, x =

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It took until the 16th century for mathematicians to discover formulae for solving cubic equations, ax 3 + bx 2 + cx + d = 0 , and quartic equations, ax 4 + bx 3 + cx 2 + dx + e = 0. (These formulae are very complicated and there is insufficient space to include them here but you might wish to research them on the internet.)

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In this chapter you will develop skills for factorising and solving cubic and quartic equations. These types of equations have various applications in the real world. For example, cubic equations are used in thermodynamics and fluid mechanics to model the pressure/volume/temperature behaviour of gases and fluids.

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Mathematicians then spent many years trying to discover a general formula for solving quintic equations, ax5 + bx 4 + cx 3 + dx 2 + ex + f = 0. Eventually, in 1824, a mathematician managed to prove that no general formula for this exists.

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You will also learn about a new type of function, called the modulus function.

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1.1 The modulus function

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The modulus of a number is the magnitude of the number without a sign attached.

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The modulus of 3 is written 3 .

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3 = 3 and −3 = 3.

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The modulus of a number is also called the absolute value.

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 x if x ù 0 x =  − x if x , 0

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The modulus of x, written as x , is defined as:

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It is important to note that the modulus of any number (positive or negative) is always a positive number.

Copyright Material - Review Only - Not for Redistribution

WEB LINK Explore the Polynomials and rational functions station on the Underground Mathematics website.

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EXPLORE 1.1

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Chapter 1: Algebra

a−b = a − b

3

ab = a × b

Pr es s

5

a

6

a

7

a+b ø a + b

8

a−b ø a−c + c−b

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a = a ÷ b , if b ≠ 0 b = a2

n

= a n , where n is a positive integer

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4

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2

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a+b = a + b

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1

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You are given these eight statements:

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You must decide whether a statement is

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If you think that a statement is either always true or never true, you must give a clear explanation to justify your answer. 3

If you think that a statement is sometimes true you must give an example of when it is true and an example of when it is not true.

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Never true

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Sometimes true

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Always true

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This property is used to solve equations that involve modulus functions.

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The statement x = k , where k ù 0 , means that x = k or x = − k .

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and ax + b = − k

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ax + b = k

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If you are solving equations of the form ax + b = k , you can solve the equations using

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and ax + b = −( cx + d )

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  ax + b = cx + d

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If you are solving harder equations of the form ax + b = cx + d , you can solve the equations using

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When solving these more complicated equations you must always check that your answers satisfy the original equation.

Copyright Material - Review Only - Not for Redistribution

FAST FORWARD You will see why it is necessary to check your answers when you learn how to sketch graphs of modulus functions in Section 1.2.

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Solve:

2x − 1 = 3

Pr es s

Answer

2x − 1 = 3

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2 x − 1 = 3 or 2 x − 1 = −3 2 x = −2 2x = 4 x = −1  x = 2

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a

x − 4 = 2x + 1

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WORKED EXAMPLE 1.1

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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CHECK: 2  ×  2 − 1 = 3 ✓ and 2  ×  ( −1) − 1 = 3  ✓

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x − 4 = 2x + 1

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The solution is: x = −1 or x = 2

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   x − 4 = 2 x + 1 or x − 4 = −(2 x + 1) x = −5 3x = 3 x =1 x =1

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CHECK: −5 − 4 = 2  ×  ( −5) + 1  ⨯ and 1 − 4 = 2  ×  1 + 1  ✓

Pr ity

= a 2 you can write a 2 − b2 as a

2

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rs ve

2

2

− b .

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Using a

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EXPLORE 1.2

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The solution is: x = 1

C

4

)( a

+ b

C w

− b

)

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(a

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a 2 − b2 =

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Using the difference of two squares you can now write:

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Using the previous statement, explain how these three important results can be obtained:

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(The symbol ⇔ means ‘is equivalent to’.)

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a . b   ⇔   a 2 . b2



a , b   ⇔   a 2 , b2 , if b ≠ 0 .

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Pr



w

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we can use the rule:

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a = b ⇔a =b

2

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To solve equations of the form cx + d = ex + f

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KEY POINT 1.1

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Worked example 1.2 shows you how to solve equations of the form cx + d = ex + f .

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a = b ⇔   a 2 = b2

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Copyright Material - Review Only - Not for Redistribution

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Chapter 1: Algebra

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WORKED EXAMPLE 1.2

Answer

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Solve the equation 3x + 4 = x + 5 .

Pr es s

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Method 1

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1 9   or   x = − 2 4

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Solution is: x = Method 2

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    (3x + 4) = ( x + 5) 2

2

 9x + 24x + 16 = x 2 + 10 x + 25

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WORKED EXAMPLE 1.3

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Solve x + 3 + x + 5 = 10. Answer

s es (2)

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−7 + x + x + 5 = 0

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Using equation (1):

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x + 3 = x + 5 − 10

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(1)

x + 3 = 10 − x + 5

   x + 5 = 7 − x x + 5 = 7 − x or x + 5 = −(7 − x )  2 x = 2 or 0 = −12 x =1

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Split this equation into two parts.

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0 = −12 is false.

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Subtract x + 5 from both sides. Split the equation into two parts.

Pr

x + 3 = 10 − x + 5

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x + 3 + x + 5 = 10

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(2 x − 1)(4x + 9) = 0 9 1 x= or x = − 2 4

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Factorise.

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y

8x 2 + 14x − 9 = 0

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Use a = b ⇔ a 2 = b2.

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       3x + 4 = x + 5

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1 1 9 9 +4 = +5  ✓, 3×−  +4 = − +5  ✓   2 2 4 4

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CHECK: 3 ×

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3x + 4 = x + 5 or 3x + 4 = −( x + 5) 2x = 1 4x = −9 or 9 1 x=  x = − 2 4

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3x + 4 = x + 5

Copyright Material - Review Only - Not for Redistribution

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Using equation (2):

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

  x + 5 = x + 13

Split this equation into two parts.

Pr es s

2 x = −18

or

  x = −9

y

e

x + 2 2x − =2 3 5

b

3x − 1 =1 x+5

3x − 5 = x + 1

e

-R

f

x2 − 2 = 7

d

x2 − 3 = 2x + 1

b

5 − x2 = 3 − x

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2 x 2 − 5x = 4 − x

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x + 2y = 8

2

+ 9 x − 1 − 2 = 0.

7 a Solve the equation x 2 − 5 x + 6 = 0 .

3x + y = 0 y = 2x2 − 5

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op y

x2 − 7x + 6 = 6 − x

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6 Solve the equation 5 x − 1

f

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x+2 + y=6

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8 Solve the equation 2 x + 1 + 2 x − 1 = 3.

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9 Solve the equation 3x − 2 y − 11 + 2 31 − 8x + 5 y = 0 .

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Name the equation of the line of symmetry of the curve.

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b Use graphing software to draw the graph of y = x 2 − 5 x + 6 .

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1 x−3 2

x2 + 2x = x + 2

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5 Solve the simultaneous equations.

3 2x − 1 =

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a

y

x −5 = 2 x +1

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3x + 5 = 1 + 2 x

8 − 1 − 2x = x

c   2x − 5 = 1 − x

C

d

a

f

x+2 =5 x−3

3 − 2 x = 3x

b

4 Solve.

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x+ x+4 =8

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2x + 1 = x

C

a

3x − 2 =4 5

c  2−

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3 Solve.

6

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f   2 x + 7 = 3x

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x +2 =3 3 2 Solve. 2x + 1 =5 a x−2 d

d

y 1 − 2x = 5

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4x − 3 = 7

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1 Solve.

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EXERCISE 1A

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The solution is x = 1 or x = −9.

0 = 8 is false.

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0=8

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x + 5 = x + 13 or x + 5 = −( x + 13)

Copyright Material - Review Only - Not for Redistribution

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Chapter 1: Algebra

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1.2 Graphs of y = f(x) where f(x) is linear

y = |x|

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Consider drawing the graph of y = x .

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First draw the line y = x .

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Then reflect, in the x-axis, the part of the line that is below the x-axis.

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1 x − 1 , showing the points where the graph meets the axes. 2 1 x − 1 in an alternative form. Use your graph to express 2 Sketch the graph of y =

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WORKED EXAMPLE 1.4

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Answer

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1 First sketch the graph of y = x − 1. 2 1 The line has gradient and a y-intercept of −1. 2 You then reflect in the x-axis the part of the line that is below the x-axis. 1 x − 1 can be written as: The graph shows that 2  1 x − 1 if x ù 2  2 1 x −1 =  2  −  1 x − 1  if x , 2    2

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y=

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2

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7

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1 –1 O –1

1

2

3

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2

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1

1

2

3

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–1 O –1

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br am

y = |x – 4|

4

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y = 2x + 1

C

op y

y 6

This root can be found graphically by finding the x-coordinates of the points of intersection of the graphs of y = x − 4 and y = 2 x + 1 as shown.

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y=3

3

–2

Also in Worked example 1.1, we found that there was only one root, x = 1, to the equation x − 4 = 2 x + 1.

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y = |2x – 1|

4

These can also be found graphically by finding the x-coordinates of the points of intersection of the graphs of y = 2 x − 1 and y = 3 as shown.

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x

−1

In Worked example 1.1, we found that there were two roots, x = −1 and x = 2, to the equation 2 x − 1 = 3.

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1 x –1 2

1

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EXERCISE 1B

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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1 Sketch the graphs of each of the following functions showing the coordinates of the points where the graph meets the axes. Express each function in an alternative form. 1 a y= x+2 b y = 3−x c y = 5− x 2 0 5

1

2 3

3

4

5

6

b Draw the graph of y = x − 3 + 2 for 0 ø x ø 6 . Describe the transformation that maps the graph of y = x onto the graph of y =   x −  3 +  2.

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2 a Complete the table of values for y = x − 3 + 2 .

b

y = x−5 −2

d

y = 2x − 3

e

y = 1− x + 2

c

y=2− x

f

y =5−2 x

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y = x +1 + 2

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3 Describe fully the transformation (or combination of transformations) that maps the graph of y = x onto each of these functions.

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4 Sketch the graphs of each of the functions in question 3. For each graph, state the coordinates of the vertex.

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5 f( x ) = 5 − 2 x + 3 for 2 ø x ø 8

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Find the range of function f .

6 a Sketch the graph of y = 2 x − 2 + 1 for −2 , x , 6, showing the coordinates of the vertex and the y-intercept.

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Use your graph to solve the equation 2 x − 2 + 1 = x + 2 .

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b On the same diagram, sketch the graph of y = x + 2.

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Use your graph to solve the equation x − 2 = 1 − 2 x .

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b On the same diagram, sketch the graph of y = 1 − 2 x .

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7 a Sketch the graph of y = x − 2 for −3 , x , 6 , showing the coordinates of the vertex and the y-intercept.

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8 a Sketch the graph of y = x + 1 + x − 1 .

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b Use your graph to solve the equation x + 1 + x − 1 = 4.

Pr

1.3 Solving modulus inequalities

a ù b ⇔ a ø − b or a ù b

and

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a ø b ⇔ −b ø a ø b

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The following examples illustrate the different methods that can be used when solving modulus inequalities.

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Two useful properties that can be used when solving modulus inequalities are:

Copyright Material - Review Only - Not for Redistribution

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Chapter 1: Algebra

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WORKED EXAMPLE 1.5

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Pr es s

Method 1: Using algebra −3 , 2 x − 5 , 3  2 , 2 x , 8  1 , x , 4

ni

Method 2: Using a graph

y

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2x − 5 , 3

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The graphs of y = 2 x − 5 and y = 3 intersect at the points A and B.

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y=3

B

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1 1

2

3

4

5

x

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O –1

9

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To solve the inequality 2 x − 5 , 3 you must find where the graph of the function y = 2 x − 5 is below the graph of y = 3.

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op y

Solve the inequality 2 x − 1 ù 3 − x .

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Method 1: Using algebra

U

(2 x − 1)2 ù (3 − x )2

Use a ù b ⇔ a 2 ù b2.

y

  2x − 1 ù 3 − x

op

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Factorise.

id g

3x 2 + 2 x − 8 ù 0

C

4x 2 − 4x + 1 ù 9 − 6x + x 2

-R s

-C

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(3x − 4)( x + 2) ù 0

es

C

Answer

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A

Pr

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op C ev

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2x − 5 = 3 2x = 8 x=4

WORKED EXAMPLE 1.6

ev

4

At B, the line y = 2 x − 5 intersects the line y = 3.

Hence, 1 , x , 4.

R

y = |2x – 5|

2

ev

id

br

am =3 =3 =2 =1

-C

−(2 x − 5)    −2 x + 5 2x    x

y 5

3

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 2 x − 5 if x ù 2 1 2  2x − 5 =   −(2 x − 5) if x , 2 1 2  At A, the line y = −(2 x − 5) intersects the line y = 3.

C op

Answer

-R

Solve 2 x − 5 , 3.

Copyright Material - Review Only - Not for Redistribution

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y = 3x2 + 2x – 8

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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4 Critical values are −2 and . 3 4 Hence, x ø −2 or x ù . 3

-C

-R

+

−2



Pr es s

Method 2: Using a graph

+ 4 3

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A

Pr

y

rs

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–1 O

–2

2

C

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br am

s es Pr

5

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4

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3 2

2

3

4

5

6

7

8

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1

U

–1 O

y

y = 2|x – 4|

1

9 x

br

w

ev

Write down the solution to the inequality x − 1 . 2 x − 4 .

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The graphs of y = x − 1 and y = 2 x − 4 are shown on the grid.

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s

b Solve the inequality 2 x − 1 . 4 − x − 1 .

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2 a On the same axes, sketch the graphs of y = 2 x − 1 and y = 4 − x − 1 .

-C

C

op y

6

w

B

1 –3

y = |x – 1|

-C

y 8 7

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2

U

4 Hence, x ø − 2 or x ù . 3

R

3

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op C w

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of y = 3 − x .

1

ev

4

the graph of the function y = 2 x − 1 is above the graph

EXERCISE 1C

R

y = |3 – x| 5

4  x = 3 To solve the inequality 2 x − 1 ù 3 − x you must find where

10

y = |2x – 1|

8

6

s

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2 x − 1 = −( x − 3) 3x = 4

y

7

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br

2x − 1 = x − 3 x = −2 At B, the line y = 2 x − 1 intersects the line y = −( x − 3).

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if x ù 3 x − 3 3−x = x−3 =   −( x − 3) if x , 3 At A, the line y = −(2 x − 1) intersects the line y = −( x − 3).

C

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C

ve rs ity

  2 x − 1 if x ù 1  2 2x − 1 =    1  −(2 x − 1) if x , 2 

op

op

y

The graphs of y = 2 x − 1 and y = 3 − x intersect at the points A and B.

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3

4

5

6

7

8

x

ve rs ity

C

U

ni

op

y

Chapter 1: Algebra

w

b

2−x ,4

ev ie

2x + 1 ø 3

am br id

a

ge

3 Solve. (You may use either an algebraic method or a graphical method.) c

3x − 2 ù 7

c

x − 7 − 2x ø 4

a

2x − 5 ø x − 2

b

-R

4 Solve. (You may use either an algebraic method or a graphical method.) 3 + x . 4 − 2x

x + 4 , 2x

d

x −1 .2 x − 4

op

y

a

b

2x − 5 . 3 − x

c

3x − 2 ø 1 − 3x

e

3 3 − x ù 2x − 1

f

3x − 5 , 2 2 − x

y

1.4 Division of polynomials

ve rs ity

6 Solve 2 x + 1 + 2 x − 1 . 5.

C w ev ie

Pr es s

-C

5 Solve. (You may use either an algebraic method or a graphical method.)

U

R

ni

C op

A polynomial is an expression of the form

ie

ev

br

-R

Pr

y



s



es



am



x is a variable n is a non-negative integer the coefficients an , an −1, an − 2 , … , a2 , a1, a0 are constants an is called the leading coefficient and an ≠ 0 a0 is called the constant term.

-C



id

where:

w

ge

an x n + an −1x n −1 + an − 2 x n − 2 +  … + a2 x 2 + a1x1 + a0

op

The highest power of x in the polynomial is called the degree of the polynomial.

11

rs

In this section you will learn how to use long division to divide a polynomial by another polynomial. Alternative methods are possible, this is shown in Worked example 1.10.

y

ge

C

U

DID YOU KNOW?

R

op

ni

ev

ve

ie

w

C

ity

For example, 5x 3 − 6x 2 + 2 x − 6 is a polynomial of degree 3.

w y op

x2 x − 2 x + 2 x 2 − 11x + 6

Divide the first term of the polynomial by x: x3 ÷ x = x2 .

e

C

U

3

x3 − 2x2

Pr

ni ve rs

Answer Step 1:

ity

Divide x 3 + 2 x 2 − 11x + 6 by x − 2 .

w

Multiply ( x − 2) by x 2 : x 2 ( x − 2) = x 3 − 2 x 2 .

ie

id g

2

4x − 11x

ev

Subtract: ( x 3 + 2 x 2 ) − ( x 3 − 2 x 2 ) = 4x 2 .

Bring down the −11x from the next column.

es

s

-R

br am -C

R

ev

ie

w

C

op y

WORKED EXAMPLE 1.7

)

ie

es

s

-C

-R

am

br

ev

id

Polynomial functions can be used to predict the mass of an animal from its length.

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ve rs ity

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

ev ie

w

Step 2: Repeat the process x 2 + 4x x − 2 x + 2 x 2 − 11x + 6

-R

Divide 4x 2 by x: 4x 2 ÷ x = 4x.

Multiply ( x − 2) by 4x: 4x( x − 2) = 4x 2 − 8x .

x3 − 2x2

-C

Pr es s

4x 2 − 8x

C

x 2 + 4x − 3 x − 2 x + 2 x 2 − 11x + 6

ni

x3 − 2x2

w

Multiply ( x − 2) by −3: −3( x − 2) = −3x + 6.

id

4x 2 − 8x

Divide −3x by x: −3x ÷ x = −3.

ge

4x 2 − 11x

ie

R

3

U

ev ie

w

Step 3: Repeat the process

)

Subtract: ( −3x + 6) − ( −3x + 6) = 0.

-R

am

br

ev

− 3x + 6 − 3x + 6

op

Pr

y

es

s

-C

0

Hence, ( x 3 + 2 x 2 − 11x + 6) ÷ ( x − 2) = x 2 + 4x − 3. 12

Bring down the 6 from the next column.

ve rs ity

− 3x + 6

Subtract: (4x 2 − 11x ) − (4x 2 − 8x ) = −3x .

y

op

y

4x 2 − 11x

C op

)

3

y

quotient

op

divisor

ni

ev

There was no remainder in this calculation so we say that x − 2 is a factor of x 3 + 2 x 2 − 11x + 6.

-R

s

-C

Answer

am

Find the remainder when 2 x 3 − x + 52 is divided by x + 3.

ev

ie

w

ge

br

id

WORKED EXAMPLE 1.8

C

U

R

dividend

ve

rs

( x 3 + 2 x 2 − 11x + 6) ÷ ( x − 2) = ( x 2 + 4x − 3)

ie

w

C

ity

Each part in the calculation in Worked example 1.7 has a special name:

Pr

U

y Subtract: (2 x 3 + 0 x 2 ) − (2 x 3 + 6x 2 ) = −6x 2 . Bring down the − x from the next column.

-R s es

-C

am

br

ev

ie

id g

e

op

− 6x 2 − x

Multiply ( x + 3) by 2 x 2 : 2 x 2 ( x + 3) = 2 x 3 + 6x 2 .

C

2 x 3 + 6x 2

Divide the first term of the polynomial by x: 2 x 3 ÷ x = 2 x 2 .

w

ity

)

ni ve rs

2x2 x + 3 2 x 3 + 0 x 2 − x + 52

R

ev

ie

w

C

op y

es

There is no x 2 term in 2 x 3 − x + 52 so we write it as 2 x 3 + 0 x 2 − x + 52 . Step 1:

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ve rs ity

w

ge

C

U

ni

op

y

Chapter 1: Algebra

am br id

ev ie

Step 2: Repeat the process

2 x 2 − 6x x + 3 2 x + 0 x 2 −    x + 52

)

3

-R

Divide −6x 2 by x: −6x 2 ÷ x = −6x . Multiply ( x + 3) by −6x : −6x( x + 3) = −6x 2 − 18x .

x

Subtract: ( −6x 2 − x ) − ( −6x 2 − 18x ) = 17 x . Bring down the 52 from the next column.

17 x + 52 Step 3: Repeat the process 2 x 2 −  6x + 17 x + 3 2 x + 0 x 2 − x + 52

ni

−  6x 2 −

U

R

2 x 3 + 6x 2

ge

x

Divide 17x by x: 17 x ÷ x = 17.

w

ev ie

)

3

y

ve rs ity

w

C

op

y

−  6x 2 − 18x

C op

−  6x 2 −

Pr es s

-C

2 x 3 + 6x 2

id

ie

−  6x 2 − 18x

Multiply ( x + 3) by 17: 17( x + 3) = 17 x + 51.

-R

Subtract: (17 x + 52) − (17 x + 51) = 1 .

1

-C

am

br

ev

17 x + 52 17 x + 51

Pr

y

es

s

The remainder is 1.

ity

remainder

y

quotient

op

divisor

ve

U

KEY POINT 1.2

C

ie ev

rs

(2 x 3 − x + 52) = ( x + 3)  ×  (2 x 2 − 6x + 17) + 1 dividend

R

13

ni

w

C

op

The calculation in Worked example 1.8 can be written as:

w

ge

The division algorithm for polynomials is:

-R

am

br

ev

id

ie

dividend = divisor × quotient + remainder

s

-C

EXERCISE 1D

es

c

(3x 3 − 7 x 2 + 6x − 2) ÷ ( x − 1)

e

(5x 3 − 13x 2 + 10 x − 8) ÷ (2 − x )

Pr

( x 3 + 5x 2 + 5x − 2) ÷ ( x + 2)

ni ve rs

ity

a

b

( x 3 − 8x 2 + 22 x − 21) ÷ ( x − 3)

d

(2 x 3 − 3x 2 + 8x + 5) ÷ (2 x + 1)

f

(6x 4 + 13x − 19) ÷ (1 − x )

op

y

2 Find the quotient and remainder for each of the following. b

(6x 3 + 7 x 2 − 6) ÷ ( x − 2)

c

(8x 3 + x 2 − 2 x + 1) ÷ (2 x − 1)

d

(2 x 3 − 9x 2 − 9) ÷ (3 − x )

e

( x 3 + 5x 2 − 3x − 1) ÷ ( x 2 + 2 x + 1)

f

(5x 4 − 2 x 2 − 13x + 8) ÷ ( x 2 + 1)

w

ie

e

id g

C

( x 3 + 2 x 2 + 5x − 4) ÷ ( x + 1)

U

a

b

x 3 + y3 = ( x + y )( x 2 − xy + y2 )

es

s

x 3 − y3 = ( x − y )( x 2 + xy + y2 )

am

a

-R

br

ev

3 Use algebraic division to show these two important factorisations.

-C

R

ev

ie

w

C

op y

1 Simplify each of the following.

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DID YOU KNOW? The number 1729 is the smallest number that can be expressed as the sum of two cubes in two different ways.

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

w

b Hence, factorise 2 x 3 + 7 x 2 − 43x + 42 completely.

ev ie

ge

4 a Use algebraic division to show that x − 2 is a factor of 2 x 3 + 7 x 2 − 43x + 42 .

-R

5 a Use algebraic division to show that 2 x + 1 is a factor of 12 x 3 + 16x 2 − 3x − 4.

-C

b Hence, solve the equation 12 x 3 + 16x 2 − 3x − 4 = 0 .

Pr es s

6 a Use algebraic division to show that x + 1 is a factor of x 3 − x 2 + 2 x + 4 .

ve rs ity

7 The general formula for solving the quadratic equation ax 2 + bx + c = 0 is x =

w

C

op

y

b Hence, show that there is only one real root for the equation x 3 − x 2 + 2 x + 4 = 0 . −b ± b2 − 4ac . 2a

y

C op

ge

U

R

ni

ev ie

There is a general formula for solving a cubic equation. Find out more about this formula and use it to solve 2 x 3 − 5x 2 − 28x + 15 = 0 .

w

1.5 The factor theorem

br

ev

id

ie

In Worked example 1.7 you found that x 3 + 2 x 2 − 11x + 6 is exactly divisible by x − 2 .

-R

am

( x 3 + 2 x 2 − 11x + 6) ÷ ( x − 2) = x 2 + 4x − 3

-C

This can also be written as:

es

s

x 3 + 2 x 2 − 11x + 6 = ( x − 2)( x 2 + 4x − 3)

op

Pr

y

If a polynomial P(x ) divides exactly by a linear factor x − c to give the polynomial Q(x ), then 14

C

ity

P(x ) = ( x − c ) Q(x )

op

ni U

R

ev

KEY POINT 1.3

y

ve

ie

w

rs

Substituting x = c into this formula gives P(c ) = 0 . This is known as the factor theorem.

ie

w

ge

br

ev

id

For example, when x = 2,

C

If for a polynomial P(x ) , P(c ) = 0 then x − c is a factor of P(x ) .

-R

am

x 3 + 2 x 2 − 11x + 6 = (2)3 + 2(2)2 − 11(2) + 6 = 8 + 8 − 22 + 6 = 0

-C

Therefore ( x − 2) is a factor of x 3 + 2 x 2 − 11x + 6.

s es Pr

op y

The factor theorem can be extended to:

ity

op

w

ie ev es

s

-R

br

am

C

U

e

id g

Hence 3x − 2 is a factor of 3x 3 − 2 x 2 − 3x + 2.

-C

ev

R

2 , 3 3 2 2 2 2 8 8 3x 3 − 2 x 2 − 3x + 2 = 3   − 2   − 3   + 2 = − − 2 + 2 = 0  3  3  3 9 9

For example, when x =

y

ni ve rs

b If for a polynomial P(x ) , P   = 0 then ax − b is a factor of P(x ) .  a

ie

w

C

KEY POINT 1.4

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ve rs ity ge

C

U

ni

op

y

Chapter 1: Algebra

am br id

ev ie

w

WORKED EXAMPLE 1.9

-C

Let f( x ) = x 3 + 2 x 2 − 16x + 3

If f(3) = 0 , then ( x − 3) is a factor.

C op

y

ve rs ity

ni

ev ie

w

C

op

y

f(3) = (3)3 + 2(3)2 − 16(3) + 3 = 27 + 18 − 48 + 3 =0 Hence x − 3 is a factor of x 3 + 2 x 2 − 16x + 3 .

Pr es s

Answer

-R

Use the factor theorem to show that x − 3 is a factor of x 3 + 2 x 2 − 16x + 3 .

w

ge

U

R

EXPLORE 1.3

br

ev

id

ie

B 2 x 3 + 7 x 2 + x − 10

A x 3 + x 2 − 3x + 1

E 6x 4 + x 3 − 8x 2 − x + 2

F 3x 3 + x 2 − 10 x − 8

-R

am

D 6x 3 + x 2 − 17 x + 10

C 3x 3 + 7 x 2 − 4

C

Pr

x + 2 as a factor

2

y w

ge

C

U

WORKED EXAMPLE 1.10

R

op

ni

ev

ve

ie

rs

3 2 x − 1 as a factor.

w

15

ity

x − 1 as a factor

y

1

op

es

s

-C

Discuss with your classmates which of the expressions have:

id

ie

a Given that 3x 2 + 5x − 2 is a factor of 3x 3 + 17 x 2 + ax + b , find the value of a and the value of b.

-R

-C

Answer

am

br

ev

b Hence, factorise the polynomial completely.

s es

Pr

Factorising 3x 2 + 5x − 2 gives (3x − 1)( x + 2).

ni ve rs

ity

Hence, 3x − 1 and x + 2 are also factors of f( x ). 1 Using the factor theorem, f   = 0 and f( −2) = 0.  3 2

y

3

es

w ie

s

-R

br am

(1)

ev

id g

e

C

U

op

1 1 1 1 f   = 0 gives 3   + 17   + a   + b = 0  3  3  3  3 1 17 a + + +b = 0 9 9 3 a + 3b = −6

-C

R

ev

ie

w

C

op y

a Let f( x ) = 3x 3 + 17 x 2 + ax + b .

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ve rs ity

y

ev ie

ve rs ity

op C w

Substituting in (2) gives b = −8.

Hence, a = 18 and b = −8.

(2)

Pr es s

-C

(1) + 3 × (2) gives 7 a = 126 a = 18

-R

am br id

f( −2) = 0 gives 3( −2)3 + 17( −2)2 + a ( −2) + b = 0 −24 + 68 − 2 a + b = 0 2 a − b = 44 

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

y

ev ie

b Method 1: Using long division

ni

U

R

)

C op

x+4 3x 2 + 5x − 2 3x 3 + 17 x 2 + 18x − 8

w

ge

3x 3 +   5x 2 − 2 x

ev

12 x 2 + 20 x − 8

-C

-R

0

am

br

id

ie

12 x 2 + 20 x − 8

es

s

Hence, 3x 3 + 17 x 2 + 18x − 8 = (3x 2 + 5x − 2)( x + 4)

Pr

Method 2: Equating coefficients

If 3x 2 + 5x − 2 is a factor, 3x 3 + 17 x 2 + 18x − 8 can be written as

ve

ie

w

rs

C

16

ity

op

y

= (3x − 1)( x + 2)( x + 4)

y op C

U

R

ni

ev

3x 3 + 17 x 2 + 18x − 8 = (3x 2 + 5x − 2)( cx + d )

coefficient of x 3 is 3, so c = 1 since 3 × 1 = 3

es

s

-C

WORKED EXAMPLE 1.11

C

ity

Pr

op y

Solve x 3 +  4x 2 − 19x + 14 = 0.

y

ni ve rs

Let f( x ) = x 3 + 4x 2 − 19x + 14.

op

The positive and negative factors of 14 are ±1, ± 2, ±7 and ±14.

-R s es

am

br

ev

ie

id g

The other factors of f( x ) can be found by any of the following methods.

w

e

C

U

f(1) = (1)3 + 4(1)2 − 19(1) + 14 = 0, so x − 1 is a factor of f( x ).

-C

ev

ie

w

Answer

R

ie

-R

am

br

Hence, 3x 3 + 17 x 2 + 18x − 8 = (3x − 1)( x + 2)( x + 4)

ev

id

w

ge

constant term is −8, so d = 4 since −2 × 4 = −8

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ve rs ity

w

ge

C

U

ni

op

y

Chapter 1: Algebra

am br id

ev ie

Method 1: Trial and error

f(2) = (2)3 + 4(2)2 − 19(2) + 14 = 0, so x − 2 is a factor of f( x )

-R

f( −7) = ( −7)3 + 4( −7)2 − 19( −7) + 14 = 0, so x + 7 is a factor of f( x ).

Pr es s

-C

Hence, f( x ) = ( x − 1)( x − 2)( x + 7).

)

5x 2 − 19x

ni U

− 14x + 14 − 14x + 14

w ie

0

br

id

  

ge

R

5x 2 −   5x

C op

ev ie

w

x 3 −    x 2

y

ve rs ity

C

x 2 + 5x − 14 x − 1 x 3 +  4x 2 − 19x + 14

ev

op

y

Method 2: Long division

-R

am

Hence, f( x ) = ( x − 1)( x 2 + 5x − 14)

s

-C

         = ( x − 1)( x − 2)( x + 7)

es

Method 3: Equating coefficients

op

Pr

y

Since x − 1 is a factor, x 3 + 4x 2 − 19x + 14 can be written as: 17

y

ev

ve

ie

w

rs

C

ity

x 3 + 4x 2 − 19x + 14 = ( x − 1)( ax 2 + bx + c )

constant term is 14, so c = −14 since −1 ×  −14 = 14

C

U

R

ni

op

coefficient of x 3 is 1, so a = 1 since 1  ×  1 = 1

w

ge

x 3 + 4x 2 − 19x + 14 = ( x − 1)( x 2 + bx − 14)

br

ie ev

id

Equating the coefficients of x 2 gives 4 = −1 + b, so b = 5 .

-R

am

Hence, f( x ) = ( x − 1)( x 2 + 5x − 14)

-C

= ( x − 1)( x − 2)( x + 7)

Pr

op y

es

s

The solutions of the equation x 3 + 4x 2 − 19x + 14 = 0 are: x = 1,  x = 2 and x = −7.

C

ity

EXERCISE 1E

ie

w

ni ve rs

1 Use the factor theorem to show that x + 1 is a factor of x 4 − 3x 3 − 4x 2 + 5x + 5.

y

op

U

3 Given that x + 4 is a factor of x 3 + ax 2 − 29x + 12 , find the value of a.

PS

4 Given that x − 3 is a factor of x 3 + ax 2 + bx − 30, express a in terms of b.

PS

5 Given that 2 x 2 + x − 1 is a factor of 2 x 3 − x 2 + ax + b , find the value of a and the value of b.

-R s es

am

br

ev

ie

id g

w

e

C

PS

-C

R

ev

2 Use the factor theorem to show that 2 x − 5 is a factor of 2 x 3 − 7 x 2 + 9x − 10.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

6 Given that x − 3 and 2 x + 1 are factors of 2 x 3 + px 2 + (2q − 1)x + q:

am br id

a Find the value of p and the value of q.

7 It is given that x + a is a factor of   x 3 + 4x 2 + 7 ax + 4a . Find the possible values of a.

PS

8 a It is given that x + 1 is a common factor of x 3 + px + q and x 3 + (1 − p )x 2 + 19x − 2q.

Pr es s

Find the value of p and the value of q.

b When p and q have these values, factorise x 3 + px + q and x 3 + (1 − p )x 2 + 19x − 2q completely.

ve rs ity

w

C

op

y

-C

PS

y

9 Given that x − 1 and x + 2 are factors of x 4 − x 3 + px 2 − 11x + q :

PS

ni

C op

a Find the value of p and the value of q.

b When p and q have these values, factorise x 4 − x 3 + px 2 − 11x + q completely.

ie

id

10 Solve.

w

ge

U

R

x 3 − 5x 2 − 4x + 20 = 0

c

2 x 3 − 5x 2 − 13x + 30 = 0

e

x 4 + 2 x 3 − 7 x 2 − 8x + 12 = 0

br

-R

am

3x 3 + 17 x 2 + 18x − 8 = 0

f

2 x 4 − 11x 3 + 12 x 2 + x − 4 = 0

s

-C

es

Pr

y op

rs

ity

y

Consider f( x ) = 2 x 3 − 5x 2 − 2 x + 12.

ve

1.6 The remainder theorem

U

ni

ie ev

R

d

12 Find the set of values for k for which the equation 3x 4 + 4x 3 − 12 x 2 + k = 0 has four real roots.

w

C

PS

-R

am

es Pr

−   x2 + 2x

ni ve rs

ity

− 4x + 12 − 4x +   8

C

op y

−   x2 − 2x

w

4

op

y

The remainder is 4. This is the same value as f(2).

C

U

f( x ) = 2 x 3 − 5x 2 − 2 x + 12 can be written as:

ev

br

In general:

ie

id g

w

e

f( x ) = ( x − 2)(2 x 2 − x − 4) + 4

-R s

P( x ) = ( x − c )Q( x ) + R

es

am

If a polynomial P(x ) is divided by x − c to give the polynomial Q( x ) and a remainder R, then:

-C

ie ev

s

-C

2 x 3 − 4x 2    

w

ev

id br

2 x 2 − x  −  4 x − 2 2 x 3 −  5x 2 − 2 x + 12

)

ie

ge

C

Substituting x = 2 in the polynomial gives f(2) = 2(2)3 − 5(2)2 − 2(2) + 12 = 4 . When 2 x 3 − 5x 2 − 2 x + 12 is divided by x − 2 , there is a remainder:

R

x 3 + 5x 2 − 17 x − 21 = 0

11 You are given that the equation x 3 + ax 2 + bx + c = 0 has three real roots and that these roots are consecutive terms in an arithmetic progression. Show that 2 a 3 + 27c = 9ab.

P 18

b

ev

a

op

ev ie

-R

b Explain why x + 2 is also a factor of the expression.

ev ie

w

ge

PS

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ve rs ity

C

U

ni

op

y

Chapter 1: Algebra

w

ge

Substituting x = c into this formula gives P( c ) = R .

am br id

ev ie

This leads to the remainder theorem.

-R

KEY POINT 1.5

Pr es s

-C

The remainder theorem:

op

y

If a polynomial P( x ) is divided by x − c, the remainder is P( c ).

ev ie

KEY POINT 1.6

y

w

C

ve rs ity

The remainder theorem can be extended to:

id

ie

w

ge

U

R

ni

C op

b If a polynomial P( x ) is divided by  ax − b , the remainder is P   .  a

am

br

ev

WORKED EXAMPLE 1.12

s

-C

-R

Use the factor theorem to find the remainder when 4x 3 − 2 x 2 + 5x − 8 is divided by x + 1.

y

es

Answer

= 4( −1)3 − 2( −1)2 + 5( −1) − 8 = −4 − 2 − 5 − 8 = −19

ity

Remainder = f( −1)

19

y op w

ge

C

U

R

ni

ev

ve

ie

w

rs

C

op

Pr

Let f( x ) = 4x 3 − 2 x 2 + 5x − 8

br

ev

id

ie

WORKED EXAMPLE 1.13

es

op y

When f( x ) is divided by 3x + 2 , the remainder is −2.

Pr ity

f( x ) = 6x 3 + ax 2 + bx − 4

ni ve rs

Answer

op w

(1)

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e

a+b=1

C

U

6(1)3 + a (1)2 + b(1) − 4 = 3

y

When f( x ) is divided by x − 1  the remainder is 3, hence f(1) = 3 .

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 2 When f( x ) is divided by 3x + 2 the remainder is −2, hence f  −  = −2.  3

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C

Find the value of a and the value of b.

R

-R

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When f( x ) is divided by x − 1, the remainder is 3.

s

am

f( x ) = 6x 3 + ax 2 + bx − 4

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2

am br id

2 2 2 6  −  + a  −  + b  −  − 4 = −2  3  3  3 2 a − 3b = 17

-R

(2)

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3

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Pr es s

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2 × (1) − (2) gives 5b = −15 b = −3

op

y

Substituting b = −3 in equation (1) gives a = 4.

y C op

EXERCISE 1F

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ev ie

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C

ve rs ity

a = 4 and b = −3 .

U

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1 Find the remainder when:

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a 6x 3 + 3x 2 − 5x + 2 is divided by x − 1

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br

ev

x 4 + 2 x 3 − 5x 2 − 2 x + 8 is divided by x + 1

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c

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b x 3 + x 2 − 11x + 12 is divided by x + 4

-R

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d 4x 3 − x 2 − 18x + 1 is divided by 2 x − 1

es

s

2 a When x 3 − 3x 2 + ax − 7 is divided by x + 2 , the remainder is −37. Find the value of a.

Pr

3

f( x ) = x 3 + ax 2 + bx − 5

f( x ) has a factor of x − 1 and leaves a remainder of −6 when divided by x + 1.

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20

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b When 9x 3 + bx − 5 is divided by 3x + 2 , the remainder is −13. Find the value of b.

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Find the value of a and the value of b.

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4 The polynomial 3x 3 + ax 2 + bx + 8 , where a and b are constants, is denoted by f( x ) . It is given that x + 2 is a factor of f( x ), and that when f( x ) is divided by x − 1 the remainder is 15. Find the value of a and the value of b.

-R

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5 The function f( x ) = ax 3 + 7 x 2 + bx − 8, where a and b are constants, is such that 2 x + 1 is a factor. The remainder when f( x ) is divided by x + 1 is 7.

s

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a Find the value of a and the value of b.

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op y

b Factorise f( x ) completely.

Pr

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ni ve rs

a Find the value of p.

op

y

b Hence find the remainder when the expression is divided by x − 2 .

e

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U

7 The polynomial x 3 + ax 2 + bx + 2, where a and b are constants, is denoted by f( x ) . It is given that x − 2  is a factor of f( x ), and that when f( x ) is divided by x + 1 the remainder is 21.

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a Find the value of a and the value of b.

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b Solve the equation f( x ) = 0, giving the roots in exact form.

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C

6 The polynomial x 3 + 6x 2 + px − 3 leaves a remainder of R when divided by x + 1 and a remainder of −10R when divided by x − 3.

Copyright Material - Review Only - Not for Redistribution

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Chapter 1: Algebra

a Find the value of k.

Find the remainder when P( x ) is divided by ( x − 1).

op

y

9

Pr es s

P( x ) = 2 x + 4x 2 + 6x 3 + … + 100 x50

-C

PS

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b Find the remainder when f( x ) is divided by x + 2 .

ev ie

am br id

w

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8 The polynomial 2 x 3 + ax 2 + bx + c is denoted by f( x ). The roots of f( x ) = 0 are −1, 2 and k. When f( x ) is divided by x − 1, the remainder is 6.

C

ve rs ity

10 P( x ) = 3( x + 1)( x + 2)( x + 3) + a ( x + 1)( x + 2) + b( x + 1) + c

PS

y C op

br

-R

am

The modulus function

The modulus function x is defined by:

-C



 x if x ù 0

s

x =

C

= D( x ) ×

Q( x )

R( x )

+

Pr

P( x )

21

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op

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es

 − x if x , 0 The division algorithm for polynomials ●

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ev

id

Checklist of learning and understanding

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It is given that when P( x ) is divided by each of x + 1, x + 2 and x + 3 the remainders are −2, 2 and 10, respectively. Find the value of a, the value of b and the value of c.

op

ni



If for a polynomial P( x ) , P( c ) = 0 then x − c is a factor of P( x ).



If for a polynomial P( x ), P

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ev

id

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The remainder theorem

w

ge

 b  = 0 then ax − b is a factor of P( x ).  a

C

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The factor theorem

y

ve

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rs

dividend  =  divisor  ×  quotient  + remainder

If a polynomial P( x ) is divided by x − c, the remainder is P( c ).  b . ● If a polynomial P( x ) is divided by  ax − b, the remainder is P  a

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op y

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s

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am



Copyright Material - Review Only - Not for Redistribution

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Solve the inequality 5x − 3 ù 7.

3

Solve the inequality 2 x − 3 , 2 − x .

y

Solve the equation x 2 − 14 = 11.

Given that ( x − 4) is a factor of p( x ) , find the value of a.

[2]

b

When a has this value, factorise p( x ) completely.

[3]

ve rs ity

a

ie

-R

ev

id

br

Find the quotient and remainder when p( x ) is divided by ( x 2 − 2 x − 1).

b

Use the factor theorem to show that ( x − 3) is a factor of p( x ).

s

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a

[5]

[4]

es

Pr

The polynomial 4x 4 + 4x 3 − 7 x 2 − 4x + 8 is denoted by p( x ) . Find the quotient and remainder when p( x ) is divided by ( x 2 − 1).

[3]

b

Hence solve the equation 4x 4 + 4x 3 − 7 x 2 − 4x + 3 = 0.

[3]

rs

ity

a

y

ve

ie

Find the value of the constant k for which f( x ) = ( x 2 + kx + 2)( x 2 − kx − 1) .

b

Hence solve the equation f( x ) = 0. Give your answers in exact form.

ni

op

a

[3]

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11 The polynomial 2 x 4 + 3x 3 − 12 x 2 − 7 x + a is denoted by p( x ) .

[3]

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ev

[1]

[2]

10 The polynomial x 4 − 48x 2 − 21x − 2 is denoted by f( x ).

R

[4]

w

( ) = 0.

y op C

y

Write down the roots of f x

The polynomial x 3 − 5x 2 + 7 x − 3 is denoted by p( x ).

8

w

C op

ni

b

U

Show that ( x − 5) is a factor of f( x ) and hence factorise f( x ) completely.

ge

R

a

The polynomial   x 3 − 5x 2 + ax + b is denoted by f( x ). It is given that ( x + 2) is a factor of f( x ) and that when f( x ) is divided by ( x − 1) the remainder is −6. Find the value of a and the value of b.

7

9

[4]

The polynomial 6x 3 − 23x 2 − 38x + 15 is denoted by f( x ).

6

22

[3]

The polynomial ax 3 − 13x 2 − 41x − 2 a , where a is a constant, is denoted by p( x ).

am

ev ie

w

C

op

5

[3]

Pr es s

2

[3]

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am br id

Solve the equation 2 x − 3 = 5x + 1 .

-C

1

4

ev ie

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END-OF-CHAPTER REVIEW EXERCISE 1

Given that (2 x − 1) is a factor of p( x ), find the value of a.

[2]

b

When a has this value, verify that ( x + 3) is also a factor of p( x ) and hence factorise p( x ) completely.

[4]

-C

-R

am

br

ev

a

3

2

es

Given that ( x − 2) is a factor of p( x ) , find the value of a.

[2]

b

When a has this value, solve the equation p( x ) = 0 .

[4]

3

2

ity

Pr

a

op y

ni ve rs

13 The polynomial 2 x + 5x − 7 x + 11 is denoted by f( x ). a

Find the remainder when f( x ) is divided by ( x − 2).

b

Find the quotient and remainder when f( x ) is divided by ( x 2 − 4x + 2) .

op

y

[2]

2

U

3

e

Given that ( x − 3) and ( x + 1) are factors of p( x ) , find the value of a and the value of b.

b

When a and b take these values, find the other linear factor of p( x ) .

es

s

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a

am

[4]

C

14 The polynomial ax + bx − x + 12 is denoted by p( x ) .

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R

s

12 The polynomial 3x + ax − 36x + 20 is denoted by p( x ) .

Copyright Material - Review Only - Not for Redistribution

[4] [2]

ve rs ity am br id

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Chapter 1: Algebra

Find the value of a and hence verify that (2 x + 1) is a factor of P( x ) .

When a has this value, solve the equation P( x ) = 0.

y

b

Pr es s

-C

a

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15 The polynomial 6x 3 + x 2 + ax − 10 , where a is a constant, is denoted by P( x ) . It is given that when P( x ) is divided by ( x + 2) the remainder is −12. [3] [4]

a

Given that ( x + 2) and ( x − 3) are factors of p( x ) , find the value of a and the value of b.

[4]

b

When a and b take these values, factorise p( x ) completely.

[3]

y

ve rs ity

ev ie

w

C

op

16 The polynomial 2 x 3 + ax 2 + bx + 6 is denoted by p( x ) .

ni

C op

17 The polynomials P( x ) and Q( x ) are defined as:

ge

U

R

P( x ) = x 3 + ax 2 + b and Q( x ) = x 3 + bx 2 + a.

id

ie

w

It is given that ( x − 2) is a factor of P( x ) and that when Q( x ) is divided by ( x + 1) the remainder is −15. a

Find the value of a and the value of b.

b

When a and b take these values, find the least possible value of P( x ) − Q(x ) as x varies.

ev

br

-R

am

[5] [2]

s

Find the quotient when p( x ) is divided by ( x − 1), and show that the remainder is 2.

[4]

b

Hence show that the polynomial 5x 3 − 13x 2 + 17 x − 9 has exactly one real root.

[3]

ity

Pr

es

a

Given that ( x + 2) is a factor of f( x ), find the value of k.

b

When k has this value, solve the equation f( x ) = 0.

c

Write down the roots of f( x 2 ) = 0 .

[2]

ni

op

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ve

a

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rs

19 The polynomial 4x 3 + kx 2 − 65x + 18 is denoted by f( x ).

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C

op

y

-C

18 The polynomial 5x 3 − 13x 2 + 17 x − 7 is denoted by p( x ) .

[4] [1]

br

-R

[5]

When a and b have these values, find the quotient and remainder when f( x ) is divided by x 2 − x + 2.

[3]

s

-C

b

Find the value of a and the value of b.

am

a

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id

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w

20 The polynomial 2 x 3 − 5x 2 + ax + b , where a and b are constants, is denoted by f( x ). It is given that when f( x ) is divided by ( x + 2) the remainder is 8 and that when f( x ) is divided by ( x − 1) the remainder is 50.

es

Pr

a

Find the value of a and the value of b.

b

When a and b have these values, solve the equation f( x ) = 0.

ity

[5]

ii

Use the factor theorem to show that ( x + 1) is a factor of f( x ).

C

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id g

[4] [2]

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q4 June 2010

es

s

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br am

op

Find the quotient and remainder when f( x ) is divided by x 2 + x − 1.

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i

y

ni ve rs

[4]

22 The polynomial x 3 + 3x 2 + 4x + 2  is denoted by f( x ).

-C

R

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C

op y

21 The polynomial 2 x 3 − 9x 2 + ax + b , where a and b are constants, is denoted by f( x ). It is given that ( x + 2) is a factor of f( x ), and that when f( x ) is divided by ( x + 1) the remainder is 30.

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23

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

-R

23 The polynomial 4x 3 + ax 2 + 9x + 9, where a is a constant, is denoted by p( x ). It is given that when p( x ) is divided by (2 x − 1) the remainder is 10. Find the value of a and hence verify that ( x − 3) is a factor of p( x ) .

ii

When a has this value, solve the equation p( x ) = 0 .

y op

ve rs ity

i

Find the values of a and b.

ii

When a and b have these values, factorise p( x ) completely.

C op

y

[5]

ni

C

24 The polynomial ax 3 − 5x 2 + bx + 9, where a and b are constants, is denoted by p( x ) . It is given that (2 x + 3) is a factor of p( x ) , and that when p( x ) is divided by ( x + 1) the remainder is 8.

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[4]

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q5 November 2011

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[3]

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q4 June 2013

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[3]

Pr es s

-C

i

Copyright Material - Review Only - Not for Redistribution

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ve rs ity ni

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Pr es s

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ni

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y

ve rs ity

op C w ev ie

Chapter 2 Logarithmic and exponential functions y

op

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Pr

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25

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understand the relationship between logarithms and indices, and use the laws of logarithms (excluding change of base) understand the definition and properties of e x and ln x, including their relationship as inverse functions and their graphs use logarithms to solve equations and inequalities in which the unknown appears in indices use logarithms to transform a given relationship to linear form, and hence determine unknown constants by considering the gradient and/or intercept.

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ni ve rs

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Pr

op y

■ ■ ■ ■

ie

In this chapter you will learn how to:

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

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PREREQUISITE KNOWLEDGE

What you should be able to do

IGCSE / O Level Mathematics

Use and interpret positive, negative, fractional and zero indices.

1 Evaluate. a 5−2

Pr es s

-C y

2

b 83 c 70

ve rs ity

op C

2 Solve 32 x = 2 . 3 Simplify.

y

Use the rules of indices.

a 3x −4 ×

ie es

s

-C

y

FAST FORWARD

Pr

ity

op

rs

C

C

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ev

-R

s es Pr

op y

Consider the exponential function f( x ) = 10 x .

w

ni ve rs

C

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To solve the equation 10 x = 6, we could make an estimate by noting that 10 0 = 1 and 101 = 10 , hence 0 , x , 1. We could then try values of x between 0 and 1 to find a more accurate value for x.

y

y = 10x

C w ie 1 x

ev

0.78

es

s

0.5

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e id g br am

2 O

op

y 10 8 6 4

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ev

The graph of y = 10 x could also be used to give a more accurate value for x when 10 x = 6.

–0.5

You will encounter logarithmic functions again when studying calculus in Chapters 4, 5 and 8.

op

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2.1 Logarithms to base 10

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3

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 2 x5  c   3 

At IGCSE / O Level we learnt about exponential growth and decay and its applications to numerous real-life problems such as investments and population sizes. Logarithmic functions also have real-life applications such as the amount of energy released during an earthquake (the Richter scale). One of the most fascinating occurrences of logarithms in nature is that of the logarithmic spiral that can be observed in a nautilus shell. We also see logarithmic spiral shapes in spiral galaxies, and in plants such as sunflowers.

ie

2 21 x 3

2 21 x ÷ 2 x −2 5

At IGCSE / O Level we learnt how to solve some simple exponential equations. For example, 32 x = 2 can be solved by first writing 32 as a power of 2. In this chapter we will learn how to solve more difficult exponential equations such as a x = b where there is no obvious way of writing the numbers a and b in index form with the same base number. In order to be able to do this we will first learn about a new type of function called the logarithmic function. A logarithmic function is the inverse of an exponential function.

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IGCSE / O Level Mathematics

What is a logarithm? 26

Check your skills

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Where it comes from

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WEB LINK Explore the Exponentials and logarithms station on the Underground Mathematics website.

ve rs ity am br id

TIP

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If 10 x = 6 then x = log10 6.

w

There is, however, a function that gives the value of x directly:

ev ie

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From the graph, x ≈ 0.78 .

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Chapter 2: Logarithmic and exponential functions

log10 6 can also be written as log 6 or lg 6.

Pr es s

-C

log10 6 is read as ‘log 6 to base 10’.

y

log is short for logarithm.

op

On a calculator, for logs to the base 10, we use the log or lg key.

x = log10 6

then

ev ie

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C

ve rs ity

So if 10 x = 6

U

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The rule for base 10 is:

ge

KEY POINT 2.1

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y

x = 0.778 to 3 significant figures.

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If y = 10 x then x = log10 y

y = 10x

-R

This rule can be described in words as:

y

y=x

s

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log10 y is the power that 10 must be raised to in order to obtain y.

op

x

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O

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This is read as: y = 10 x is equivalent to saying that x = log10 y. y

y w

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EXPLORE 2.1

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C

We say that y = 10 ⇔ x = log10 y.

es

y = 10 x and y = log10 x   are inverse functions.

1

Pr

y

For example, log10 100 = 2 since 100 = 10 2.

x

= x for x . 0

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br

10

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10 log

log10 10 x = x for x ∈ ℝ

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1 Discuss with your classmates why each of these two statements is true.

2 a Discuss with your classmates why each of these two statements is true. log10 10 = 1

es

s

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log10 1 = 0

Pr

0

log10 100

log10 1000

1

…..

…..

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-C

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log10 10

y

…..

log10 1

op

…..

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log10 0.1

ni ve rs

log10 0.01

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b Discuss what the missing numbers should be in this table of values, giving reasons for your answers.

Copyright Material - Review Only - Not for Redistribution

y = log10 x 1

x

27

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WORKED EXAMPLE 2.1

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

-R

a Convert 10 x = 58 to logarithmic form.

Pr es s

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b Solve 10 x = 58 giving your answer correct to 3 significant figures.

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10 x = 58

base = 10 , index = x

Step 2: Start to write in log form

x = log10 ?

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TIP

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Step 1: Identify the base and index

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Answer a Method 1

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x = log10 58

Step 3: Complete the log form

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Method 2

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10 x = 58 ⇔ x = log10 58

10 x = 58

s

es

Use log10 10 x = x.

  x = log10 58

ity

10 x = 58

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WORKED EXAMPLE 2.2

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x = log10 58 x ≈ 1.76

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C

Take logs to base 10 of both sides.

Pr

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log10 10 x = log10 58 28

In log form the index is always on its own and the base goes at the base of the logarithm.

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a Convert log10 x = 3.5 to exponential form.

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am

b Solve log10 x = 3.5 giving your answer correct to 3 significant figures.

es

s

-C

Answer a Method 1

Pr

TIP

y

103.5 =  ?

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-C

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base = 10, index = 3.5

op

Step 2: Start to write in exponential form

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Step 1: Identify the base and index

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C

op y

log10 x = 3.5

Copyright Material - Review Only - Not for Redistribution

In log form the index is always on its own.

ve rs ity

x = 103.5

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ev ie

Step 3: Complete the exponential form log10 x = 3.5 ⇔ x = 103.5

-R

Method 2

= 103.5

Use 10 log

= x.

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x = 103.5 x ≈ 3160

y

log10 x = 3.5

ev ie

x

ve rs ity

op b

10

C op

y

x

x = 103.5

w

C

10

Write each side as an exponent of 10.

Pr es s

-C

log10 x = 3.5

10 log

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Chapter 2: Logarithmic and exponential functions

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log10 10 000

b log10 0.0001

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Find the value of:

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WORKED EXAMPLE 2.3

es Pr

Write 10 000 as a power of 10, 10 000 = 10 4.

ity

29

c

log10 1000 10 = log10 103.5 = 3.5

Write 0.0001 as a power of 10, 0.0001 = 10 −4 .

rs

log10 0.0001 = log10 10 −4 = −4

ve

y

Write 1000 10 as a power of 10. 1000 10 = 103 × 10 0.5 = 103.5.

br

EXERCISE 2A

b

-R

102 = 100

10 x = 200

c

10 x = 0.05

es

s

-C

a

am

1 Convert from exponential form to logarithmic form.

ev

id

ie

w

ge

C

U

R

ni

op

ie ev

log10 10 000 = log10 10 4 =4

b

w

C

op

y

a

s

Answer

c log10 1000 10.

b

Pr

10 x = 52

10 x = 250

c

10 x = 0.48

c

log10 x = −0.6

ity

a

b

log10 x = 1.2

y

log10 10 000 = 4

op

a

ni ve rs

3 Convert from logarithmic form to exponential form.

log10 x = 2.76

c

log10 x = −1.4

ie ev es

s

-R

br am

C

b

w

e

log10 x = 1.88

id g

a

U

4 Solve each of these equations, giving your answers correct to 3 significant figures.

-C

R

ev

ie

w

C

op y

2 Solve each of these equations, giving your answers correct to 3 significant figures.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

)

e

(

log10

100 3 10

)

(

log10 10 10

ev ie

(

d log10

3 10

c

w

b log10 0.0001

)

 100  log10  1000 

f

-R

log10 100

am br id

a

ge

5 Without using a calculator, find the value of:

y

7 Given that p and q are positive and that 4(log10 p )2 + 2(log10 q )2 = 9, find the greatest possible value of p.

2.2 Logarithms to base a

ni

The same principles can be applied to define logarithms in other bases.

ie

w

ge br

ev

id

KEY POINT 2.2

-R s

a log a x = x

y

rs

y op

ni

w

ge

C

U

WORKED EXAMPLE 2.4

ev

id

-R

am

s

-C

es

base = 5, index = 3 3 = log5 ?

Take logs to base 5 of both sides.

ie

Use log5 5x = x .

-R s es

am

br

 3 = log5 125

id g

log5 53 = log5 125

e

U

53 = 125

3 = log5 125

y

Method 2

In log form the index is always on its own and the base goes at the base of the logarithm.

op

5 = 125 ⇔ 3 = log5 125

-C

w ie ev

3

ni ve rs

C

ity

Step 2: Start to write in log form Step 3: Complete the log form

TIP

Pr

op y

Step 1: Identify the base and index

ev

53 = 125

br

Answer Method 1

ie

Convert 53 = 125 to logarithmic form.

C

R

ve

We say that y = a x ⇔ x = log a y.

ev

ie

w



Pr ity

a . 0 and a ≠ 1 x.0

C



op

The conditions for log a x to be defined are:

w

-C

log a a x = x

log a 1 = 0

es

log a a = 1

am

If y = a x then x = log a y

R

C op

y

In the last section you learnt about logarithms to the base of 10.

U

R

ev ie

w

C

ve rs ity

op

PS

30

Try the See the power resource on the Underground Mathematics website.

Pr es s

-C

6 Given that the function f is defined as f : x ֏ 10 x − 3 for x ∈ ℝ, find an expression for f −1( x ) .

WEB LINK

Copyright Material - Review Only - Not for Redistribution

ve rs ity ge

C

U

ni

op

y

Chapter 2: Logarithmic and exponential functions

am br id

ev ie

w

WORKED EXAMPLE 2.5

Pr es s

-C

Answer

-R

Convert log3 x = 2.5 to exponential form.

Method 1

32.5 =   ?

Step 3: Complete the exponential form

32.5 =   x

=x

-R

am -C

y

es

s

WORKED EXAMPLE 2.6

ity

a log2 16

y

ve

op

C

log3

ni

b

Write 16 as a power of 2, 16 = 2 4 .

U

log2 16 = log2 2 4  = 4

w

ge

a

1 = log3 3−2 9 = −2

ie

Write

1 1 as a power of 3, = 3−2. 9 9

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

am

br

ev

id

ie

Answer

ev

31

1 b log3 9

rs

w

C

op

Pr

Find the value of:

R

w

x

ie

3

ev

 x = 3

2.5

Use 3log

id

= 32.5

br

x

In log form the index is always on its own.

Write each side as an exponent of 3.

ge

log3 x = 2.5 3

C op

ni U

Method 2 3log

TIP

y

Step 2: Start to write in exponential form

log3 x = 2.5 ⇔ 32.5 = x

R

base = 3, index = 2.5

ve rs ity

ev ie

w

C

op

y

log3 x = 2.5 Step 1: Identify the base and index

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

w

ge

am br id

WORKED EXAMPLE 2.7

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Pr es s

( )

2 3

U

ni

EXERCISE 2B

R

x as a power of x. x

y

ev ie

w

C

=−

3

Write

ve rs ity

op

y

 3x −2 = log x x 3 log x    x 

C op

-C

Answer

-R

 3x . Simplify log x   x 

w h

x5y

log8 1 = 0

d

log16 4 =

g

log a 1 = 0

h

1 2 log x 5 = y

c

log 4 x = 0

d

log x 49 = 2

f

xy = 6

g

-R

am

c

3−5 =

1 1024 =7

2 −10 =

2 4 = 16

ie

8x = 15

1 243 ab = c

d

b

ev

e

id

52 = 25

br

a

ge

1 Convert from exponential form to logarithmic form.

f

log2 y = 4

log2 x = 3

b

log3 x = 2

ity rs

y

log2 (3x − 1) = 5

c

ge

b

op

ni

log3 ( x + 5) = 2

U

R

ev

4 Solve. a

c

s

1 3

log8 2 =

3 Solve. a

ie

log3 81 = 4

es

e

b

ve

w

C

32

log2 8 = 3

log y (7 − 2 x ) = 0

C

op

y

a

Pr

-C

2 Convert from logarithmic form to exponential form.

am

w (

log5 5 5

)

-C

f

C

 3 log3   3 

f

log x

(

x

es 5

)

(

c

log x x 2 x

g

 x  log x  3   x

Pr

2

ity

e

1 log x  3  x 

b log x 3 x

ni ve rs

log x x 3

op y

a

w ie

g

d log2 0.125  37  h log7   7 

2

)

d log x

1 x4

 x2  h log x   x 

3

op

C

U

b

log 4 23− 5 x = x 2

ev

log2 (log3 x ) = −1

-R s es

am

br

a

ie

id g

w

e

8 Solve.

-C

PS

y

7 Given that the function f is defined as f : x ֏ 1 + log2 ( x − 3) for x ∈ ℝ, x > 3, find an expression for f −1( x ).

ev

R

log 4 2

s

6 Simplify.

ie

1 log 4    16 

c

ev

e

b log6 36

-R

log3 27

br

a

id

5 Find the value of:

Copyright Material - Review Only - Not for Redistribution

ve rs ity log2 8

log3 20

log3 4

log2 2

log 4 3

log2 3

w

log3 9

ev ie

9

ge

PS

C

U

ni

op

y

Chapter 2: Logarithmic and exponential functions

am br id

Without using a calculator, arrange these logarithms in ascending order.

-R

Add some more logarithms of your own in the correct position in your ordered list.

Try the Logarithm lattice resource on the Underground Mathematics website.

Pr es s

-C

EXPLORE 2.2

WEB LINK

log10 12 + log10 4 = log10 (12 + 4) = log10 16

Prya says that:

ve rs ity

w

C

op

y

1 Prya and Hamish are asked what they think log10 12 + log10 4 simplifies to. Hamish says that: log10 12 + log10 4 = log10 (12  ×  4) = log10 48

y

C op

U

R

ni

ev ie

Use a calculator to see who is correct and check that their rule works for other pairs of numbers.

w

ge

2 Can you predict what log10 12 − log10 4 simplifies to?

-R

am

br

ev

id

ie

Use a calculator to check that you have predicted correctly and check for other pairs of numbers.

es

s

-C

2.3 The laws of logarithms

ity

KEY POINT 2.3

ve

Division law

Power law

log a (xy ) = log a x + log a y

 x log a   = log a x − log a y  y

log a ( x ) m = m  log a x

log a x   +  log a y

a

)

)

C w ie

a

a



 

x   −  log a y

C

ity

= log a ( a log

(

= log a ( a log = log a ( a

a

x m

)

m log a x

)

)

= m log a x

)

y

ni ve rs

= log a x − log a y

w

-R s es

am

br

ev

ie

id g

w

e

C

U

op

1 Using the power law, log a   = log a x −1  x = − log a x

-C

ie ev

ev

a = log a  log a

log a x  

= log a x + log a y

R

-R

y

s

×   a log

log a ( x ) m

 x log a    y

op y

= log a ( a

a

Power law

es

x   

-C

= log a ( a log

Division law

Pr

am

Multiplication law log a ( xy )

op

ni U

br

id

Proofs:

y

Multiplication law

ge

R

ev

ie

w

rs

C

op

Pr

y

If x and y are both positive and a . 0 and a ≠ 1, then the following laws of logarithms can be used.

Copyright Material - Review Only - Not for Redistribution

33

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

ev ie

w

ge

Another useful rule to remember: KEY POINT 2.4

Pr es s

-C

-R

1 log a   = − log a x  x

op

y

WORKED EXAMPLE 2.8

b

log3 8 – log3 4

ni br

w

2 log5 2 +  log5 3 = log5 22 + log5 3 = log5 (4 × 3)

-R

= log5 12

s

-C

op

Pr

y

es

WORKED EXAMPLE 2.9

ity

log 4  p5 − log 4 q 2

y op

= 5 log 4  p − 2 log 4 q = 5x − 2 y

id

1 5 log 4 p + log 4 q 2 3 1 5 = x+ y 2 3

ev

= log 4 43 − log 4 p = 3−x

es Pr ity

c

3 log5 2 − log5 4

e 1 + 2 log2 3

f

2 − log 4 2

c

log2 60 − log2 5

op C

ie ev es

s

-R

br am

w

b log6 20 + log6 5

id g

e

U

2 Write as a single logarithm, then simplify your answer. log2 40 − log2 5

y

b log6 20 − log6 4

ni ve rs

log2 7 + log2 11

d 2 log3 8 − 5 log3 2

a

 64  log 4   p 

s

-C 1 Write as a single logarithm.

-C

ev

ie

w

C

op y

EXERCISE 2C

a

c

-R

am

br

=

w

log 4  p + 5 log 4 3 q

ie

b

ge

log 4  p5 − log 4 q 2

C

U

R

ni

Answer

ev

 64  c log 4  .  p 

b log 4  p + 5 log 4 3 q

rs

ie

w

a

ve

C

Given that log 4 p = x and log 4 q = y, express in terms of x and/or y

a

R

c

ie

8 = log3    4 = log 3 2

am

= log 2 15

34

c 2 log5 2 +  log5 3

y

log3 8 − log3 4

id

= log2 (3 × 5)

U

log2 3 + log2 5

a

ge

R

Answer

b

C op

a log2 3 + log2 5

ev

ev ie

w

C

ve rs ity

Use the laws of logarithms to simplify these expressions.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

w

1 1 log3 8 + log3 18 − 1 2 2

b

log5 8 . log5 0.25

-C

5 Simplify. log2 27   a log2 3

op

y

b

log3 128   log3 16

Pr es s

-R

4 Express 8 and 0.25 as powers of 2 and hence simplify

ev ie

1 log5 36 − log5 12 2

am br id

a 3 log5 2 +

ge

3 Simplify.

C

U

ni

op

y

Chapter 2: Logarithmic and exponential functions

log3 25   log3 0.04

c

d

log5 1000 log5 0.01 

ve rs ity

b log5 (25x )

x

x x log5   125 

c

U

R

a

ni

8 Given that y = log5 x, find in terms ofy:

y

7 Given that log3 ( z − 1) − log3 z = 1 + 3 log3 y, express z in terms of y.

C op

ev ie

w

C

6 Given that log8 y = log8 ( x − 2) − 2 log8 x, express y in terms of x.

d log x 125

ie

id

ev

br

-R

(

d log 4 p2 − log 4 4 q

pq

-C

c

 2 b log 4   q

log 4 (64 p )

am

a

w

ge

9 Given that x = log 4 p and y = log 4 q, express in terms of x and/or y:

)

s

11

c

 y  d log a  3  .  x

)

y w

C

U ge

ie es

Pr

ni ve rs

For example, x + y = − log2 5.

ity

op y C

U

op

y

Find the value of x, the value of y and the value of z.

e

C

2.4 Solving logarithmic equations

br

ev

ie

id g

w

We have already learnt how to solve simple logarithmic equations. In this section we will learn how to solve more complicated equations.

-R

s

-C

am

Since log a x only exists for x . 0 and for a . 0 and a ≠ 1, it is essential when solving equations involving logs that all roots are checked in the original equation.

es

w

WEB LINK

The number in the rectangle on the side of the triangle is the sum of the numbers at the adjacent vertices.

ev

ie

z

s

-C

-R

am

br

ev

id

log3 2 − log2 5

2 log2 5 − 3 log3 2

y

35

op

ni

ev

R

−log2 5

R

(

log a x 2 y

ve

ie

w

x

es

 x b log a  2   y 

Pr

 x log a    y

ity

PS

a

rs

C

op

y

10 Given that log a x = 7 and log a y = 4 , find the value of:

Copyright Material - Review Only - Not for Redistribution

Try the Summing to one and the Factorial fragments resources on the Underground Mathematics website.

ve rs ity

Solve:

2 log8 ( x + 2) = log8 (2 x + 19)

Pr es s

Answer

2 log8 ( x + 2) = log8 (2 x + 19) log8 ( x + 2)2 = log8 (2 x + 19)

Expand brackets.

y

x 2 + 4x + 4 = 2 x + 19

ni U

R

Use equality of logarithms.

C op

ev ie

w

( x + 2)2 = 2 x + 19 x 2 + 2 x − 15 = 0

Use the power law.

ve rs ity

C

op

y

a

4 log x 2 − log x 4 = 2

-R

b

-C

a

ev ie

w

ge

am br id

WORKED EXAMPLE 2.10

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ge

( x − 3)( x + 5) = 0

id

ie

w

x = 3 or x = −5

-R

am

log8 (2 x + 19) = log8 25 is defined.

ev

br

Check when x = 3:  2 log8 ( x + 2) = 2 log8 5 = log8 25 is defined

es

s

-C

So x = 3 is a solution, since both sides of the equation are defined and equivalent in value.

Pr

So x = −5 is not a solution of the original equation.

ve

2

=2

y

C

U

log x 2

Use the division law.

ge

R

log x 2 4 − log x 22 = 2 4−2

Use the power law.

op

4 log x 2 − log x 4 = 2

w

ev

b

ni

ie

w

Hence, the solution is x = 3

ity

C

36

rs

op

y

Check when x = −5: 2 log8 ( x + 2) = 2 log8 ( −3) is not defined.

id

ie

log x 2 = 2 2

Convert to exponential form.

-R

x =4 x = ±2

s

-C

am

br

ev

log x 4 = 2

y

ni ve rs

ity

Pr

Check when x = 2: 4 log2 2 − log2 4 = 4 − 2 log2 2 = 4−2 =2 So x = 2 satisfies the original equation.

-R s es

-C

am

br

ev

ie

id g

w

e

C

U

op

Hence, the solution is x = 2

R

ev

ie

w

C

op y

es

Since logarithms only exist for positive bases, x = −2 is not a solution.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

1 Solve.

log2 (2 − x ) + log2 9 = 2 log2 8

Pr es s

-C

c

log5 x + log5 3 = log5 30

a

log10 (2 x + 9) − log10 5 = 1

c

log2 ( 5 − 2 x ) = 3 + log2 ( x + 1 )

ve rs ity

log10 ( x − 4) = 2 log10 5 + log10 2

b

log3 2 x − log3 ( x − 1) = 2

d

log5 (2 x − 3) + 2 log5 2 = 1 + log5 (3 − 2 x )

2 log3 x − log3 ( x − 2) = 2

d

3 + 2 log2 x = log2 (3 − 10 x )

U

log x 40 − log x 5 = 1

c

log x 25 − 2 log x 3 = 2

am

br

a

b

log x 36 + log x 4 = 2

d

2 log x 32 = 3 + 2 log x 4

b

(log10 x )2 − log10 ( x 2 ) = 3

d

4(log2 x )2 − 2 log2 ( x 2 ) = 3

b

4x = 2 y

ev

id

ie

w

C op

c

ni

log5 ( x − 2) + log5 ( x − 5) = log5 4x

(log2 x )2 − 8 log2 x + 15 = 0

c

(log5 x )2 + log5 ( x 3 ) = 10

ity

Pr

a

es

s

-C

y op

y

ni

d

log 4( x − y ) = 2 log 4 x

log10 x = 2 log10 y log10 (2 x − 17 y ) = 2

ge

c

3 log10 y = log10 x + log10 2

U

R

ev

log x y = 3

op

xy = 81

C

ie

a

rs

w

6 Solve the simultaneous equations.

ve

C

d

b

5 Solve.

id

ie

w

log 4( x + y + 9) = 0

br

ev

 x4  7 Given that log10 ( x 2 y ) = 4 and log10  3  = 18 , find the value of log10 x and of log10 y.  y 

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

am

PS

log3 4x − log3 2 = log3 7

log2 x + log2 ( x − 1) = log2 20

4 Solve.

PS

b

a

ge

ev ie

3 Solve.

-R

w

C

op

y

2 Solve.

R

-R

a

y

am br id

EXERCISE 2D

w

ge

C

U

ni

op

y

Chapter 2: Logarithmic and exponential functions

Copyright Material - Review Only - Not for Redistribution

37

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

w

ge

2.5 Solving exponential equations

-R

am br id

ev ie

At IGCSE / O Level you learnt how to solve exponential equations whose terms could be converted to the same base. In this section you will learn how to solve exponential equations whose terms cannot be converted to the same base.

Pr es s

-C

WORKED EXAMPLE 2.11

5 2 x +1 = 7

y

Use the power rule. Expand the brackets.

w

ge

(2 x + 1) log 5 = log 7

Take logs to base 10 of both sides.

U

R

log 52 x +1 = log 7

32 x = 4x + 5

ni

52 x +1 = 7

a

b

C op

Answer

ev ie

w

C

a

ve rs ity

op

y

Solve, giving your answers correct to 3 significant figures.

2 x log 5 + log 5 = log 7

id

ie

Rearrange to find x.

op

-R s

Take logs to base 10 of both sides.

es

32 x = 4x + 5 log 32 x = log 4x + 5

Use the power rule.

ity

2 x log 3 = ( x + 5) log 4

Expand the brackets.

w

rs

C

y

Divide both sides by (2 log 3 − log 4).

op

ev

x(2 log 3 − log 4) = 5 log 4

Collect x terms on one side.

ve

ie

2 x log 3 = x log 4 + 5 log 4

ni

38

x = 0.105 to 3 significant figures.

Pr

y

b

-C

am

br

ev

log 7 − log 5 2 log 5

x=

w

ge

C

U

R

5 log 4 x= 2 log 3 − log 4

am

br

ev

id

ie

x = 8.55 to 3 significant figures.

-C

-R

WORKED EXAMPLE 2.12

op y

es

s

Solve the equation 2(32x ) + 7(3x ) = 15, giving your answers correct to 3 significant figures.

Pr

Answer

ni ve rs

C

y ev

ie

Take logs of both sides.

es

s

log1.5 = 0.369 … log 3

am

x=

-R

br

x log 3 = log1.5

id g

3x = 1.5

w

e

C

U

y = 1.5  or y = −5

Factorise.

op

(2 y − 3)( y + 5) = 0

-C

w ie ev

R

2 y2 + 7 y − 15 = 0

When y = 1.5 

Use the substitution y = 3x.

ity

2(32 x ) + 7(3x ) − 15 = 0

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

am br id

When y = −5

w

ge

C

U

ni

op

y

Chapter 2: Logarithmic and exponential functions

y

Pr es s

-C

Hence, the solution is x = 0.369 to 3 significant figures.

ve rs ity

op

EXERCISE 2E

1 Solve, giving your answers correct to 3 significant figures. b

2 x = 35

c

32x = 8

d

2 x +1 = 25

e

32 x − 5 = 20

f

3x = 2 x +1

g

5x + 3 = 7 4 − 3x

h

41− 3x = 3x − 2

i

2 x = 2(5x )

j

4x = 7(3x )

k

2 x +1 = 3(5x )

l

5(2 x − 3 ) = 4(3x + 4 )

C op

U

y

5x = 18

ni

a

ge

R

ev ie

w

C

There are no solutions to this equation since 3x is always positive.

-R

3x = −5

ie

w

2 a Show that 2 x +1 + 6(2 x −1 ) = 12 can be written as 2(2 x ) + 3(2 x ) = 12.

-R

am

br

ev

id

b Hence solve the equation 2 x +1 + 6(2 x −1 ) = 12, giving your answer correct to 3 significant figures. 3 Solve, giving your answers correct to 3 significant figures. b

3x +1 = 3x −1 + 32

d

5 x + 53 = 5 x + 2

e

4 x −1 = 4 x − 4 3

es

c

2 x +1 + 5(2 x −1 ) = 2 4

f

3x +1 − 2(3x − 2 ) =  5

Pr

op

y

s

2 x + 2 + 2 x = 22

-C a

39

ity

d

42x + 27 = 12(4x )

U

22x + 5 = 6  ×  2 x

y

32x = 6  ×  3x + 7

b

op

c

ve

52x − 5(5x ) + 6 = 0

ni

ev

R

a

rs

5 Solve, giving your answers correct to 3 significant figures.

ie

w

C

4 Use the substitution y = 2 x to solve the equation 22x + 32 = 12(2 x ).

id

w

7 Solve, giving your answers correct to 3 significant figures.

ie

ge

C

6 Use the substitution u = 2 x to solve the equation 22 x − 5(2 x +1 ) + 24 = 0.

b

32 x − 3x +1 = 10

c

2(5x +1 ) − 52 x = 16

d

42 x +1 = 17(4x ) − 15

-R

am

ev

22 x − 2 x +1 − 35 = 0

br

a

es

4x + 2 x − 12 = 0

b

d

2(3x ) − 4 = 8

y

d

3x + 4 = 3x − 9

e

5(2 x ) + 3 = 5(2 x ) − 10

f

2 x + 2 + 1 = 2 x + 12

32 x = 6 3 x

h

ev

-R 4

x

( ) + 14

=5 2

x

s

( ) + 16

es

g

ie

3 2x − 5 = 2x

am

c

br

id g

b

w

4x − 8 = 2

a

e

U

11 Solve, giving your answers correct to 3 significant figures.

C

10 Solve the equation x 2.5 = 20 x1.25, giving your answers in exact form.

op

ni ve rs

ity

Pr

4x + 15 =  4(2 x +1 )

3(16x ) − 10(4x ) + 3 = 0

2(9x ) =  3x +1 + 27 x correct to 3 significant figures. 9 For each of the following equations find the value of y b 7 x = (2.7) y c 42 x = 35 y a 3x = 7 y c

-C

R

ev

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op y

a

s

-C

8 Solve, giving your answers correct to 3 significant figures.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

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op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

-C

2.6 Solving exponential inequalities

w

Try the To log or not to log? resource on the Underground Mathematics website.

Pr es s

13 Solve the equation 8(8x −1 − 1) = 7(4x − 2 x +1 ).

PS

ev ie

b x

WEB LINK

-R

x

am br id

a 6

ge

12 Given that 2 4 x +1   ×  32 − x = 8x   ×  35 − 2 x , find the value of:

PS

w

C

ve rs ity

op

y

To solve inequalities such as 2 x . 3 we take the logarithm of both sides of the inequality. We can take logarithms to any base but care must be taken with the inequality symbol when doing so.

C op

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ev ie

R



y

If the base of the logarithm is greater than 1 then the inequality symbol remains the same. If the base of the logarithm is between 0 and 1 then the inequality symbol must be reversed. This is because y = log a x is an increasing function if a . 1 and it is a decreasing function when 0 , a , 1. ●

-R

am

br

ev

id

ie

Hence, when taking logarithms to base 10 of both sides of an inequality the inequality symbol is unchanged.

es

s

-C

It is also important to remember that if we divide both sides of an inequality by a negative number then the inequality symbol must be reversed.

op

Pr

y

WORKED EXAMPLE 2.13 40

rs

op

ni

Take logs to base 10 of both sides, inequality symbol is unchanged.

w

ge id

ie

Use the power rule for logs.

ev

Divide both sides by  log10 0.6.

br

log 0.6x , log 0.7

C

U

0.6x , 0.7

R

y

ve

Answer

ev

ie

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C

ity

Solve the inequality 0.6x , 0.7, giving your answer in terms of base 10 logarithms.

-R

log 0.7 log 0.6

y op -R s es

-C

am

br

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ni ve rs

C

ity

Pr

op y

x . 0.698 (correct to 3 significant figures)

log 0.6 is negative so the inequality symbol is reversed.

es

s

-C

x.

am

x log 0.6 , log 0.7

Copyright Material - Review Only - Not for Redistribution

ve rs ity ge

C

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op

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Chapter 2: Logarithmic and exponential functions

am br id

ev ie

w

WORKED EXAMPLE 2.14

Pr es s

Divide both sides by 4. Take logs to base 10 of both sides and use the power rule.

ve rs ity

5 4 5 ( 2 log 3 ) x − log 3 . log 4 5 log   +  log 3 4 x. 2 log 3

Expand brackets. Rearrange.

C op

(2 x − 1) log 3 . log

U

R

w

ge

Simplify.

ie

15   4 x. log 9

s

-C

-R

am

br

ev

id

log

y

es

EXERCISE 2F

Pr

1 Solve the inequalities, giving your answer in terms of base 10 logarithms. x 2 a 2x , 5 b 5x ù 7 c   .3 d 0.8x , 0.3  3

w

rs

ity

op C

y

4  ×  32 x −1 . 5 5 32 x −1 . 4

ni

ev ie

w

C

op

y

-C

Answer

-R

Solve the inequality 4  ×  32 x −1 . 5, giving your answer in terms of base 10 logarithms.

y

op

U

R

ni

ev

ve

ie

2 Solve the inequalities, giving your answer in terms of base 10 logarithms. 3− x 5 c  2  ×  52 x +1 ø 3 d  7  ×     .4 a 85 − x , 10 b 32 x + 5 . 20  6 2

w

ge

ev

id

ie

1− 3x

-R

am

PS

15 log    2    ×  2 ù 5 is x ù  . 4 Prove that the solution to the inequality 3 9 log    8 5 In this question you are not allowed to use a calculator. 2 x −1

br

P

C

3 Solve 5x . 2 x , giving your answer in terms of logarithms.

s

-C

You are given that log10 4 = 0.60206 correct to 5 decimal places and that 10 0.206 , 2 .

es

Pr

ity

b Find the first digit in the number 4100.

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

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ev

ie

w

ni ve rs

C

op y

a Find the number of digits in the number 4100.

Copyright Material - Review Only - Not for Redistribution

41

ve rs ity

ev ie

There is another type of logarithm to a special base called e.

am br id

FAST FORWARD

w

ge

2.7 Natural logarithms

C

U

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op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

The number e is a very important number in Mathematics as it has very special properties. You will learn about these special properties in Chapters 4 and 5.

-R

The number e is an irrational number and e ≈ 2.718.

-C

The function y = e x is called the natural exponential function.

Pr es s

Logarithms to the base of e are called natural logarithms.

op

y

ln x is used to represent log e x.

y w ie

x

s

-C

-R

am

1

O

br

y = ln x

ev

y=x 1

C op

U

y = ex

ge

R

y

ni

If y = e x then x = ln y  

id

ev ie

w

C

ve rs ity

KEY POINT 2.5

Pr

y

es

The curve y = ln x is the reflection of y = e x in the line y = x.

C

ity

op

y = ln x and   y = e x are inverse functions.

42

y op

ni

C

U

DID YOU KNOW?

R

ve

We say that y = e x ⇔ x = log e y.

ev

ie

w

rs

All the rules of logarithms that we have learnt so far also apply for natural logarithms.

ni ve rs

DID YOU KNOW?

-R s es

am

br

ev

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id g

w

e

C

U

op

y

The number e is also known as Euler’s number. It is named after Leonhard Euler (1707 −1783), who devised the following formula for calculating the value of e: 1 1 1 1 1 e = 1+ + + + + +  … 1 1× 2 1× 2 × 3 1× 2 × 3 × 4 1× 2 × 3 × 4 × 5

-C

R

ev

ie

w

C

ity

Pr

op y

es

s

-C

-R

am

br

ev

id

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w

ge

The Scottish mathematician John Napier (1550 −1617) is credited with the discovery of logarithms. His original studies involved logarithms to 1 the base of  . e

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

am br id

EXERCISE 2G

w

ge

C

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op

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Chapter 2: Logarithmic and exponential functions

e3

b e2.7

-R

a

c

e 0.8

d e −2

c

ln 0.9

d ln 0.15

c

5e ln 6

d e

c

C op

1 Use a calculator to evaluate correct to 3 significant figures:

e3 ln x = 64

d

e − ln x = 3

e x +1 = 8

d

e2 x − 3 = 16

c

e 2 x −1 = 6

d

e2

c

5  ×  e2 x + 3 , 1

ln 3

b ln1.4

op

y

a

Pr es s

-C

2 Use a calculator to evaluate correct to 3 significant figures:

e ln 2

b e  2

e ln x = 5

b

ln e x = 15

U

R

a

ni

ev ie

4 Solve.

 ln 9

y

1

a

ve rs ity

w

C

3 Without using a calculator, find the value of:

e2x = 25

w

b

c

b

ev

e x = 13

am

a

e3x = 7

-R

br

6 Solve, giving your answers in terms of natural logarithms.

1 2

ie

e x = 18

id

a

ge

5 Solve, giving your answers correct to 3 significant figures.

− ln

1

x+3

=4

es

b e5 x − 2 ø 35

e x . 10

op

Pr

y

a

s

-C

7 Solve, giving your answers in terms of natural logarithms.

43

ln x = 5

b

ln x = −4

w

2 ln( x + 2) − ln x = ln(2 x − 1)

d

ln(2 x + 1) = 2 ln x + ln 5

ln( x + 2) − ln x = 1

f

ln(2 + x 2 ) = 1 + 2 ln x

U

6e2x − 13e x − 5 = 0

w

ie

-R

ln( y + 2) − ln y = 1 + 2 ln x

b

e2x −  5e x + 6 = 0

s es

c

Pr

e2x +  2e x − 15 = 0

b

d  e x   − 21e − x = 4

ity

op y

12 Given that the function f is defined as f : x ֏ 5e x + 2 for x ∈ R, find an expression for f −1( x ).

ni ve rs

C w

ev

id

br

-C

am

2 ln( y + 1) − ln y = ln( x + y )

a

y

e

op

2 ln( x + 1 ) = ln ( 2 x + 3 )

11 Solve, giving your answers in exact form.

13 Solve 2 ln(3 − e2x ) = 1, giving your answer correct to 3 significant figures.

PS

14 Solve the simultaneous equations, giving your answers in exact form.

op

ev

ie

e2 x + y =  8e x + 6 y

-R s es

am

15 Solve ln(2 x + 1) ø ln( x + 4).

w

e3x + 4 y =  2e2 x − y

br

ln10 x − ln y = 2 + ln 2

b

C

U

e

2 ln x + ln y = 1 + ln 5

id g

a

PS

y

PS

-C

ie

b

d

C

ve

c

ni

ln(3 − x 2 ) = 2 ln x

ge

ie ev

R

a

a

ev

ln( x − 2) = 6

9 Solve, giving your answers correct to 3 significant figures.

10 Express y in terms of x for each of these equations.

R

ln(2 x + 1) = −2

c

rs

a

ity

C

8 Solve, giving your answers correct to 3 significant figures.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

w

ge

2.8 Transforming a relationship to linear form

Pr es s

-C

-R

am br id

ev ie

When we collect experimental data for two variables we often wish to find a mathematical relationship connecting the variables. When the data plots on a graph lie on a straight line the relationship can be easily found using the equation y = mx + c, where m is the gradient and c is the y-intercept. It is more common, however, for the data plots to lie on a curve.

y

Logarithms can be used to convert some curves into straight lines.

C

ve rs ity

op

This is the case for relationships of the form y = kx n and y = k ( a x ) where k, a and n are constants.

y C op

ni ge

U

WORKED EXAMPLE 2.15

R

ev ie

w

Logarithms to any base can be used but it is more usual to use those that are most commonly available on calculators, which are natural logarithms or logarithms to the base 10.

br am

y = aebx

-R

Answer

ev

id

ie

w

Convert y = aebx , where a  and b  are constants, into the form Y = mX + c.

-C

Taking natural logarithms of both sides gives:

ln y = ln a + bx

op

ity

ln y = bx + ln a

rs

ve

Now compare ln y = bx + ln a with Y = mX + c:

y op

R

ni

ev

   ln y = b     x    +  ln a

becomes the linear equation:

w

ge

The non-linear equation y = ae

C

U

  Y   =   m   X   +   c

bx

ni ve rs

(

)

w

(0.55, 0.84) (1.72, 0.26)

O

es

s

-R

br

ev

ie

id g

Find the value of a and the value of b correct to 2 decimal places.

am

ln y

C

2

e

U

The variables x and y satisfy the equation y = a e − bx , where a and b are constants. The graph of ln y against   x 2 is a straight line passing through the points (0.55, 0.84) and (1.72, 0.26) as shown in the diagram.

-C

w ie ev

R

WORKED EXAMPLE 2.16

y

ity

Pr

op y

es

s

The variables X and Y in Y = mX + c must contain only the original variables x and y. (They must not contain the unknown constants a and b.) The constants m and c must contain only the original unknown constants a and b. (They must not contain the variables x and y.)

C



-C

It is important to note: ●

ie

-R

am

br

ev

id

Y = mX + c, where Y = ln y, X = x,  m = b and c = ln a

op

w ie

es

Pr

y

ln y = ln a + ln ebx

C

44

s

ln y = ln( aebx )

Copyright Material - Review Only - Not for Redistribution

x2

ve rs ity y = a   ×  e − bx

ev ie

am br id

Answer

w

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C

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Chapter 2: Logarithmic and exponential functions

2

2

-C

ln y = ln( a   ×  e − bx ) 2

Pr es s

ln y = ln a + ln e − bx

op

y

ln y = ln a − bx 2

ve rs ity

ln y = −bx 2 + ln a

Now compare ln y = −b   x 2 + ln a with Y = mX + c:

w

C

-R

Taking natural logarithms of both sides gives:

y

ev ie

ln y = − b   x 2 + ln a

0.26 − 0.84 = −0.4957 … 1.72 − 0.55

U

id

∴ b = 0.50 to 2 decimal places

ev

rs

C 2593

1596

983

op

y

40

80

605

372

C

20

y

ve ni

10

U

5

ge

x

br

ev

id

The table shows experimental values of the variables x and y.

w

w ev

ie

WORKED EXAMPLE 2.17

R

-R

es

s

= −0.4957 …  ×  0.55 + c = 1.1126 … = 1.1126 … = e1.1126… to 2 decimal places.

ity

op

y

-C

am

0.84    c          ∴ ln a    a Hence a = 3.04  and b = 0.50

DID YOU KNOW? When one variable on a graph is a log but the other variable is not a log, it is called a semilog graph. When both variables on a graph are logs, it is called a log–log graph.

Pr

br

Using Y = mX + c, X = 0.55, Y = 0.84 and m = −0.4957 …

ie

w

ge

Gradient = m =

ie

R

ni

C op

  Y   =   m    X    +   c

-R

am

a By plotting a suitable straight-line graph, show that x and y are related by the equation y = k   ×   x n , where k and n are constants.

es

s

-C

b Use your graph to estimate the value of k and the value of n.

Pr

y = k  × xn

Take natural logarithms of both sides.

ity

n

Use the multiplication law.

ni ve rs

ln y = ln( k   ×   x ) ln y = n ln x + ln k

Use the power law.

y

ln y = ln k + ln x n

-R s es

-C

am

br

ev

ie

id g

w

e

C

U

op

a

R

ev

ie

w

C

op y

Answer

Copyright Material - Review Only - Not for Redistribution

45

ve rs ity

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ev ie

am br id

Now compare ln y = n ln x + ln k with Y = mX + c: ln y = n  ln x + ln k

-R

Y    = m    X   +     c

2.30

3.00

3.69

4.38

ln y

7.86

7.38

6.89

6.41

5.92

C op

1.61

y

ve rs ity

ln x

U

ln y

id

w

ge

10

ie

R

ni

ev ie

w

C

op

y

Pr es s

-C

Hence the graph of ln y against ln x needs to be drawn where: ● gradient = n ● intercept on vertical axis = ln k Table of values is

ev

br

8

-R

am

6

es

s

-C

4

O

ity

1

2

ln x 3

4

5

ev

ve

ie

The points form an approximate straight line, so x and y are related by the equation y = k   ×   x n .

ni

C

U

-R

am

br

ev

id

ie

w

ge

R

ln k ≈ 9       k ≈ e9

op

5.92 − 7.86 ≈ −0.70 to 2 significant figures. 4.38 − 1.61 ln k = intercept on vertical axis

b n = gradient ≈

y

w

rs

C

46

Pr

op

y

2

Pr

op y

es

s

-C

      k ≈ 8100 to 2 significant figures.

C

ity

DID YOU KNOW?

y

op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

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ev

ie

w

ni ve rs

The distributions of a diverse range of natural and man-made phenomena approximately follow the power law y = ax b . These include the size of craters on the moon, avalanches, species extinction, body mass, income and populations of cities.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

am br id

EXERCISE 2H

w

ge

C

U

ni

op

y

Chapter 2: Logarithmic and exponential functions

e

a = ex

y

+ by

y = 10 ax − b

f

x a yb = 8

c

y = a( x −b )

d

y = a(b x )

g

xa y = b

h

y = a (e − bx )

C op

id

ie

w

ge

U

R

(0.31, 4.02) (1.83, 3.22)

ln x O

ln y

(7, 4.33)

s

-C

-R

am

br

ev

3 The variables x and y satisfy the equation y = k   ×  e n ( x − 2), where k and n are constants. The graph of ln y against x is a straight line passing through the points (1, 1.84) and (7, 4.33) as shown in the diagram. Find the value of k and the value of n correct to 2 significant figures.

es

(1, 1.84)

Pr

y op

O

x

ity

4 Variables x and y are related so that, when log10 y is plotted on the vertical axis and x is plotted on the horizontal axis, a straight-line graph passing through the points (2, 5) and (6, 11) is obtained.

w

rs

C

ln y

y

ve rs ity

2 The variables x and y satisfy the equation y = a   ×   x n, where  a and n are constants. The graph of ln y against  ln x is a straight line passing through the points (0.31, 4.02) and (1.83, 3.22) as shown in the diagram. Find the value of a and the value of n correct to 2 significant figures.

ni

ev ie

w

C

op

2

b

Pr es s

y = e ax + b

-C a

-R

1 Given that a and b are constants, use logarithms to change each of these non-linear equations into the form Y = mX + c. State what the variables X and Y and the constants m and c represent. (Note: there might be more than one method to do this.)

y

ev

ve

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a Express log10 y in terms of x.

C

U

R

ni

op

b Express y in terms of x, giving your answer in the form y = a   ×  10bx .

id

ie

w

ge

5 Variables x and y are related so that, when ln y is plotted on the vertical axis and ln x is plotted on the horizontal axis, a straight-line graph passing through the points (2, 4) and (5, 13) is obtained.

br

ev

a Express ln y in terms of x.

-R

am

b Express y in terms of x.

Pr

ni ve rs

ity

7 The mass, m grams, of a radioactive substance is given by the formula m = m0 e − kt , where t is the time in days after the mass was first recorded and m0 and k are constants.

m

40.9

33.5

U

30

27.4

40

50

22.5

18.4

op

20

C

10

w

e

R

t

y

The table below shows experimental values of t and m.

ev

es

ie

s

-R

The half-life of a radioactive substance is the time it takes to decay to half of its original mass. Find the half-life of this radioactive substance.

-C

c

br

b Use your graph to estimate the value of m0 and k.

ev

id g

a Draw the graph of ln m against t.

am

ie

w

C

op y

es

s

-C

6 The variables x and y satisfy the equation 52 y = 32 x +1 . By taking natural logarithms, show that the graph of ln y against ln x is a straight line, and find the exact value of the gradient of this line and state the coordinates of the point at which the line cuts the y-axis.

Copyright Material - Review Only - Not for Redistribution

47

ve rs ity

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

w

4

6

8

10

63.3

57.7

52.8

48.7

45.2

ev ie

T

2

-R

t

am br id

ge

8 The temperature, T °C , of a hot drink, t minutes after it is made, can be modelled by the equation T = 25 + ke − nt , where k and n are constants. The table below shows experimental values of t and T .

Pr es s

-C

a Convert the equation to a form suitable for drawing a straight-line graph.

w ev ie

Estimate:

ve rs ity

c

C

op

y

b Draw the straight-line graph and use it to estimate the value of k and n.

i

the initial temperature of the drink

ii

the time taken for the temperature to reach 28 °C

y op y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

am

br

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es

s

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C op

y

iii the room temperature.

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ve rs ity am br id



log a a = 1, log a 1 = 0, log a a x = x , a log a x = x Product rule: log a (xy ) = log a x + log a y  x Division rule: log a   = log a x − log a y  y



Power rule:



1 Special case of power rule: log a   = − log a x  x

ve rs ity

Logarithms to the base of e are called natural logarithms.



e ≈ 2.718



ln x is used to represent log e x.



If y = e x then x = ln y.



All the rules of logarithms apply for natural logarithms.

U



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y=x

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1

x

1

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ity

Pr

y

Logarithms can be used to convert relationships of the form y = kx n and y = k ( a x ) , where k , a and n are constants, into straight lines of the form Y = mX + c , where X and Y are functions of x and of y.

y op y op -R s es

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id g

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ni ve rs

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Pr

op y

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id

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ve

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rs

C

op

y = ln x

s

Transforming a relationship to linear form ●

y = ex

C op

ni

y

y

log a (x ) m = m log a x

-R

ev ie

w

C

op

y ●

Natural logarithms

R

Pr es s

If y = a x then x = log a y.

-C



-R

The rules of logarithms



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ev ie

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Checklist of learning and understanding

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op

y

Chapter 2: Logarithmic and exponential functions

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

ev ie

w

END-OF-CHAPTER REVIEW EXERCISE 2

Solve the inequality 2 x . 7 , giving your answer in terms of logarithms.

[2]

2

Given that ln p = 2 ln q − ln(3 + q ) and that q . 0 , express p in terms of q not involving logarithms.

[3]

3

Solve the inequality 3  ×  23x + 2 , 8, giving your answer in terms of logarithms.

[4]

Pr es s

-C

Use logarithms to solve the equation 5 x + 3 = 7 x −1 ,

giving the answer correct to 3 significant figures.

[4]

ve rs ity

C

op

y

4

-R

1

Solve the equation 6(4x ) − 11(2 x ) + 4 = 0, giving your answers for x in terms of logarithms where appropriate. [5]

6

Solve the equation ln(5x + 4) = 2 ln x + ln 6 .

C op

y

5

ie

ln y

es

ln x

Pr

O

s

-C

(0, 2.0)

y

The variables x and y satisfy the equation y = Kx m , where K and m are constants. The graph of ln y against ln x is a straight line passing through the points (0, 2.0) and (6, 10.2), as shown in the diagram. Find the values of K and m, correct to 2 decimal places. [5]

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ie

w

rs

ity

op C

50

y

op

Given that y = 2 x , show that the equation

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8

U

ev

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q3 June 2011

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ge

2 x + 3(2 − x ) = 4

br

ev

id

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can be written in the form

s es

[3]

Pr

op y

ity

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q5 June 2010 x [3] 9 Given that (1.2)x = 6 y, use logarithms to find the value of    correct to 3 significant figures. y 10 The polynomial f( x ) is defined by

y

ni ve rs

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y

C w

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y

y

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[3]

-R

Cambridge International A Level Mathematics 9709 Paper 31 Q4 June 2011

s

am -C

[4]

y

Given that 12  ×  27 + 25  ×  9 − 4  ×  3 − 12 = 0, state the value of 3 and hence find y correct to 3 significant figures.

br

ii

Show that f( −2) = 0 and factorise f( x ) completely.

id g

i

op

f( x ) = 12 x 3 + 25x 2 − 4x − 12.

es

C w

2 x + 3(2 − x ) = 4,

giving the values of x correct to 3 significant figures where appropriate.

ev

ie

[3]

-R

Hence solve the equation

-C

ii

am

y2 − 4 y + 3 = 0.

R

(6, 10.2)

-R

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br

ev

id

7

[5]

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U

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R

ni

ev ie

w

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q1 November 2015

Copyright Material - Review Only - Not for Redistribution

ve rs ity am br id

ev ie

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Chapter 2: Logarithmic and exponential functions

11 Solve the equation 4 − 2 x = 10, giving your answer correct to 3 significant figures.

-R

Cambridge International A Level Mathematics 9709 Paper 31 Q1 June 2012

Pr es s

-C

12 Use logarithms to solve the equation e x = 3x − 2 , giving your answer correct to 3 decimal places.

[3]

y

Cambridge International A Level Mathematics 9709 Paper 31 Q1 November 2014

13 Using the substitution u = 3x , solve the equation 3x + 32 x = 33x giving your answer correct to 3 significant figures.

ve rs ity

op C w

[3]

[5]

Cambridge International A Level Mathematics 9709 Paper 31 Q2 November 2015

y

ev ie

14 The variables x and y satisfy the equation 5 y = 32 x − 4.

ge

U

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ni

C op

a By taking natural logarithms, show that the graph of y against x is a straight line and find the exact value of the gradient of this line. [3]

br

15

(3.1, 2.1)

ity

op

Pr

y

es

s

-C

-R

am

ln y

(2.3, 1.7)

51

x

O

rs

C w

[3]

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b This line intersects the x-axis at P and the y-axis at Q. Find the exact coordinates of the midpoint of PQ.

y

op

y op -R s es

-C

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id g

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ni ve rs

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ity

Pr

op y

es

s

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-R

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The variables x and y satisfy the equation y = K ( b x ) , where K and b are constants. The graph of ln  y against x is a straight line passing through the points (2.3, 1.7) and (3.1, 2.1), as shown in the diagram. Find the values of K and b, correct to 2 decimal places. [6]

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op

y

ve rs ity ni

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am br id

Pr es s

-C y

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C op

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ve rs ity

op C w ev ie

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Chapter 3 Trigonometry

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understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all six trigonometric functions for angles of any magnitude use trigonometrical identities for the simplification and exact evaluation of expressions and in the course of solving equations, and select an identity or identities appropriate to the context, showing familiarity in particular with the use of: sec 2 θ = 1 + tan2 θ and cosec 2 θ = 1 + cot 2 θ the expansion of sin( A ± B ), cos( A ± B ) and tan( A ± B ) the formulae for sin 2 A, cos 2 A and tan 2 A the expression of a sin θ + b cos θ in the forms R sin(θ ± α ) and R cos(θ ± α ).

Pr

op y

op

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ni ve rs

-R s

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• • • •

es

■ ■

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In this chapter you will learn how to:

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Chapter 3: Trigonometry

w

What you should be able to do

Pure Mathematics 1 Coursebook, Chapter 5

Sketch graphs of the sine, cosine and tangent functions.

y

ve rs ity

a b c

Find exact values of the sine, cosine and tangent of 30° , 45° and 60° and related angles.

C op

y

a sin120° b cos 225° c tan150°

U

3 Solve for 0° ø x ø 360° .

w

ge

a 5 sin x = 3 cos x b 2 cos2 x − sin x = 1

-R

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br

ev

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Solve trigonometric equations using sin θ the identities tan θ ≡ and cos θ sin2 θ + cos2 θ ≡ 1.

ie

R

Pure Mathematics 1 Coursebook, Chapter 5

y = 2 sin x + 1 y = tan( x − 90°) y = cos(2 x + 90°)

2 Find the exact values of:

ni

Pure Mathematics 1 Coursebook, Chapter 5

Check your skills 1 Sketch the graphs of these functions for 0° ø x ø 360°.

Pr es s

-C

-R

Where it comes from

op C w ev ie

ev ie

am br id

PREREQUISITE KNOWLEDGE

REWIND

es

s

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Why do we study trigonometry?

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As you know, scientists and engineers represent oscillations and waves using trigonometric functions. These functions also have further uses in navigation, engineering and physics.

-C

3.1 The cosecant, secant and cotangent ratios

Pr

ity

KEY POINT 3.1

ni ve rs

cot θ =

1  cos θ  =  tan θ  sin θ 

id g

Consider the right-angled triangle: y x y sin θ = cos θ = tan θ =   r r x r r x cosec θ = sec θ = cot θ =   y x y

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e

C

U

1 cos θ

ev

r

y

-R

br

θ

x

s

am

-C

op

sec θ =

ie

1 sin θ

R

cosec θ =

y

The three reciprocal trigonometric ratios are defined as:

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C

op y

es

s

There are a total of six trigonometric ratios. You have already learnt about the ratios sine, cosine and tangent. In this section you will learn about the other three ratios, which are cosecant (cosec), secant (sec) and cotangent (cot).

ev

This chapter builds on the work we covered in the Pure Mathematics 1 Coursebook, Chapter 5.

rs

We will also learn about the compound angle formulae, which form a basis for numerous important mathematical techniques. These techniques include solving equations, deriving further trigonometric identities, the addition of different sine and cosine functions and deriving rules for differentiating and integrating trigonometric functions, which we will cover later in this book.

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In addition to the three main trigonometrical functions (sine, cosine and tangent) there are three more commonly used trigonometrical functions (cosecant, secant and cotangent) that we will learn about. These are sometimes referred to as the reciprocal trigonometrical functions.

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WEB LINK Explore the Trigonometry: Compound angles station on the Underground Mathematics website.

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

2

Use

Divide both sides by y2 . 2

Use

C w

cot 2 θ + 1 ≡ cosec 2 θ

U

ni

C op

y

We will need these important identities to solve trigonometric equations later in this section.

w ev -R

am

br

cot 2 θ + 1 ≡ cosec 2 θ

id

1 + tan2 θ ≡ sec 2 θ

ge

KEY POINT 3.2

ie s

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The graphs of y = sin θ and y = cosec θ

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O

θ

90

–1

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–1

op

360

270

U

ni

180

y

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ie

90

1

rs

y = sin θ

O

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y

w

1

y = cosec θ

Pr

y op C

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y

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x r = cot θ and = cosec θ . y y

ve rs ity

y

op

2

 x  r  y  + 1 =  y 

Pr es s

-C

1 + tan2 θ ≡ sec 2 θ x 2 + y2 = r 2

w

r y = sec θ . = tan θ and x x

ev ie

y r 1+   =    x  x

Divide both sides by x 2 .

-R

2

am br id

x 2 + y2 = r 2

ge

The following identities can be found from this triangle:

-R

am

The function y = cosec θ is the reciprocal of the function y = sin θ .

y op -R s es

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The function y = sin θ is zero when θ = 0°, 180°, 360°, … , this means that y = cosec θ 1 is not defined at each of these points because is not defined. Hence vertical asymptotes 0 need to be drawn at each of these points.

Copyright Material - Review Only - Not for Redistribution

180

270

360 θ

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Chapter 3: Trigonometry

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ge

The graphs of y = cos θ and y = sec θ

O

O

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y = cot θ

360 θ

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am

1 sin 240° 1 = –sin 60° 1 = 3 − 2 2     = − 3

y

A

es

s

S

Pr

240°

ity

60° O

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y

C

-R s es

-C

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x

T

ni ve rs

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cosec 240° =

w

180

90

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R

WORKED EXAMPLE 3.1

Answer

ie

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O

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270

rs

180

Find the exact value of cosec 240° .

ev

y

Pr

y op C

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am -C

y = tan θ

90

360 θ

y

ni ge

U

R

We can find the graph of y = cot θ from the graph of y = tan θ .

O

270

–1

The graphs of y = tan θ and y = cot θ y

180

90

C op

C –1

1

360 θ

270

ve rs ity

180

90

ev ie

w

y = sec θ

Pr es s

-C

y = cos θ

op

y

y 1

y

-R

am br id

ev ie

We can find the graph of y = sec θ in a similar way from the graph of y = cos θ .

Copyright Material - Review Only - Not for Redistribution

270

360 θ

ve rs ity

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Use 1 + tan2 x ≡ sec 2 x.

Pr es s

sec 2 x − tan x − 3 = 0

Factorise.

U

ge

id

-R s g sec 150°

g

2π 3  π h sec  −  .  6 d cot

c

cosec x = −3

d

3sec x − 4 = 0

c

cot x = 2

d

2 cot x + 5 = 0

c

cot 2 x = 1

d

2 sec 2 x = 3

b

sec(2 x + 60°) = −1.5 for 0° ø x ø 180°

d

2 cosec(2 x − 1) = 3

s

sec x = −1

Pr

b

-R

cot x = 0.8

es

am

op y

a cosec x = 2

sec

op

y 7π 4

π 4 4π cot 3

c

ie

cosec

-C

b

h cot( −30°).

C

π 3

br

id

f

d sec 300°

ev

ni

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b cot

4 Solve each equation for 0 ø x ø 2π .

ity

5 Solve each equation for 0° ø x ø 180°.

sec 2 x = 4

b

y

ni ve rs

a cosec 2 x = 1.2

U

e

ie ev -R s es

am

br

c

cosec( x − 30°) = 2 for 0° ø x ø 360° π cot  x +  = 2 for 0 ø x ø 2 π  4

id g

a

op

6 Solve each equation for the given domains.

-C

C

y = –1

es cot 330°

ve

f

cot120°

c

rs

cosec135°

a sec x = 3

w

x

360

Pr ity

b cosec 45°

3 Solve each equation for 0° ø x ø 360°.

ie

270

ev

br

am -C

y op C ie ev

y

–1

180

C

R w

a sec 60°

2 Find the exact values of: π a cosec 6 5π e sec 6

R

63.4 90

O

–90

–3

1 Find the exact values of:

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C op

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1

–2

EXERCISE 3A

R

y=2

–45

The values of x are 63.4°, 135°, 243.4°, 315°.

e

y = tan x

2

tan x = 2   ⇒ x = 63.4° (calculator) or x = 63.4° + 180° = 243.4°

tan x = −1   ⇒ x = − 45° (not in required range)   or x = − 45° + 180° = 135° or x =  135° + 180° = 315°

56

y 3

w

w

C

ve rs ity

(tan x − 2)(tan x + 1) = 0 tan x = 2 or tan x = −1

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op

y

tan2 x − tan x − 2 = 0

w

Answer

-R

Solve sec 2 x − tan x − 3 = 0 for 0° ø x ø 360°.

ie

WORKED EXAMPLE 3.2

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Copyright Material - Review Only - Not for Redistribution

for − π ø x ø π

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Chapter 3: Trigonometry

sec 2 x = 9  

d sec x = cos x

e

cosec x = sec x

Pr es s

2 cot 2 θ − cosec θ = 13

e tan2 θ + 3sec θ = 0

b

3 cosec 2 θ = 13 − cot θ

d

cosec θ + cot θ = 2 sin θ

f

ve rs ity

y op

3 sec 2 θ = cosec θ

9 Solve each equation for 0° ø θ ø 180°.

2 sec 2 2θ = 3tan 2θ + 1

d

2 cot 2 2θ + 7 cosec 2θ = 2

U

sec 4 θ + 2 = 6 tan2 θ

b

w

tan2 θ + 3sec θ + 3 = 0

3 cot 2 θ + 5 cosec θ + 1 = 0

ev

id

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b

br

a

ge

10 Solve each equation for 0 ø θ ø 2 π .

y

c

ni

ev ie

a sec θ = 3 cos θ − tan θ

C op

C w

c

f

-R

-C

8 Solve each equation for 0° ø θ ø 360° . a 2 tan2 θ − 1 = sec θ

y = 1 + sec x

ii

-C

i

s

es

Pr

sin x + cos x cot x ≡ cosec x

c

sec x cosec x − cot x ≡ tan x

b

cosec x − sin x ≡ cos x cot x

d

(1 + sec x )(cosec x − cot x ) ≡ tan x

b

sec 2 x + sec x tan x ≡

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d

-R

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br

1  − cos2 x ≡ 1 − sin2 x  sec 2 x   − 1 sin x ≡ cosec x 1 − cos2 x 1 1 + ≡ 2 cosec 2 x 1 + cos x 1  − cos x

es

1 1 − sin x

1 +  tan2 x ≡ sec x cosec x tan x 1 + sin x 2 ≡ ( tan x + sec x ) 1  − sin x cos x cos x + ≡ 2 sec x 1 + sin x 1  − sin x

6 sec 3 θ − 5 sec 2 θ − 8 sec θ + 3 = 0

b

-R s es

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R

2 cot 3 θ + 3 cosec 2 θ − 8 cot θ = 0

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ni ve rs

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14 Solve each equation for 0° ø θ ø 180°.

ie

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C

PS

h

Pr

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g

f

s

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C

R

13 Prove each of these identities. 1 ≡ sin x cos x a tan x  + cot x c

y

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a

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y = 1 + cosec

ity

y op C

v

12 Prove each of these identities.

ev

y = cot 2 x

1  π x vi y = sec  2 x +  2  4 b Write down the equation of each of the asymptotes for your graph for part a vi. iv y = 1 − sec x

P

 π iii y = 2 cosec  x −   2

-R

am

11 a Sketch each of the following functions for the interval 0 ø x ø 2π .

P

1 x = 4   2 2 tan x = 3 cosec x

9 cot 2

c

ev ie

b

am br id

cosec 2 x = 4

a

R

w

ge

7 Solve each equation for −180° ø x ø 180°.

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57

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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3.2 Compound angle formulae

-R

EXPLORE 3.1

-C

1 Sunita says that: sin( A + B ) = sin A + sin B

Pr es s

Use a calculator with A = 20° and B = 50° to prove that Sunita is wrong.

op

y

2 The diagram shows ∆PQR .

A B

C

ve rs ity

RX is perpendicular to PQ.

R

w

∠A and ∠B are acute angles.

q

p

h

U

R

y

ni

a Write down an expression for h in terms of:

P

X

Q

p and B 1 b Using the formula area of triangle = ab sin C , we can write: 2 1 Area of ∆PQR = pq sin( A + B ). 2 ii

-R

am

br

ev

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i q and A

C op

ev ie

QR = p, PR = q and RX = h.

s

∆QRX

Use your expressions for the areas of triangles PQR, PRX and QRX to prove that: sin( A + B ) ≡ sin A cos B + cos A sin B

ity

3 In the identity sin( A + B ) ≡ sin A cos B + cos A sin B , replace B by − B to show that:

w

rs

C

58

ii

∆PRX

Pr

c

op

y

i

es

-C

Find an expression in terms of p, q, A and B for the area of:

y

ve

ie

sin( A − B ) ≡ sin A cos B − cos A sin B

op

ni

ev

4 In the identity sin( A − B ) ≡ sin A cos B − cos A sin B , replace A by (90° − A) to show that:

C

U

R

cos(A + B ) ≡ cos A cos B − sin A sin B

w

ge

5 In the identity cos( A + B ) ≡ cos A cos B − sin A sin B , replace B by − B to show that:

Pr ni ve rs

Summarising, the six compound angle formulae are:

y

sin( A + B ) ≡ sin A cos B + cos A sin B

op

sin( A − B ) ≡ sin A cos B − cos A sin B

-R s es

am

br

ev

ie

id g

w

e

cos( A − B ) ≡ cos A cos B + sin A sin B tan A + tan B tan( A + B ) ≡ 1 − tan A tan B tan A − tan B tan( A − B ) ≡ 1 + tan A tan B

C

U

cos( A + B ) ≡ cos A cos B − sin A sin B

-C

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C

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KEY POINT 3.3

R

ev

s

tan A − tan B tan A + tan B and tan( A − B ) ≡ 1+ tan A tan B 1− tan A tan B

op y

tan( A + B ) ≡

sin( A + B ) sin( A − B ) and tan( A − B ) as , show that: cos( A + B ) cos( A − B )

es

-C

am

6 By writing tan( A + B ) as

-R

br

id

ie

cos(A − B ) ≡ cos A cos B + sin A sin B

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y

Chapter 3: Trigonometry

ev ie

am br id

w

WORKED EXAMPLE 3.3

Find the exact value of:

-R

cos 42° cos12° + sin 42° sin12°

-C

Answer

b

sin105° = sin(60° + 45°)

y

y C op

ni

R

b

3 +1 2 2

cos 42° cos12° + sin 42° sin12°

Use cos ( A − B ) ≡ cos A cos B + sin A sin B.

U

ev ie

=

ve rs ity

w

C

 3   1   1  1  =  +    2   2   2   2 

ie ev

id br

3 2

s

-C

-R

am

=

w

ge

= cos(42° − 12°) = cos 30°

ity

ni A

s

C

es

sin B =

12 5 12 , tan B = , cos B = 5 13 13

op

y

ni ve rs

w ie

C

U

cos( A − B ) = cos A cos B + sin A sin B

ev

ie

id g

w

e

3 5 4 12  = −    +    5   13   5   13  33 = 65

s

-R

br am -C

ev

R

T

 4   5   3   12  =     + −     5   13   5   13  16 =− 65 b

x

5

Pr

sin( A + B ) = sin A cos B + cos A sin B

C

a

4 3 4 , cos A = −  , tan A = − 5 5 3

O

ity

op y

sin A =

B 12

w ie

C

-C

T

A

13

ev

x

br

O

am

–3

id

A

4

-R

5

S

ge

S

y

U

R

y

es

ev

Answer

y

ve

ie

w

a sin( A + B )

4 5 and cos B = , where A is obtuse and B is acute, find the value of: 5 13 b cos( A − B ) c tan( A − B )

rs

C

Given that sin A =

Pr

op

y

es

WORKED EXAMPLE 3.4

op

op

= sin 60° cos 45° + cos 60° sin 45°

C

a

Pr es s

a sin105°

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59

ve rs ity

tan A − tan B 1 + tan A tan B

w ev ie

am br id

tan( A − B ) =

ge

c

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ve rs ity

C w ev ie

-R

56 33

op

y

=

Pr es s

-C

 − 4  −  12   3  5  = 4 12  1+  −    3  5 

WORKED EXAMPLE 3.5

U

R

ni

C op

y

Solve the equation sin(60° − x ) = 2 sin x for 0° , x , 360° .

w

ge

Answer

sin(60° − x ) = 2 sin x

br

ev

id

ie

Expand sin(60° − x ). Use sin 60° =

-R

am

sin 60° cos x − cos 60° sin x = 2 sin x

es Pr rs

ve

ev

ie

w

3 = tan x 5

Add sin x to both sides. Rearrange to find tan x .

ity

3 cos x = 5 sin x

Multiply both sides by 2.

y

op

y

3 cos x − sin x = 4 sin x

C

60

s

-C

3 1 cos x − sin x = 2 sin x 2 2

3 1 and cos 60° = . 2 2

C

U

R

ni

op

x = 19.1°  or   x = 180° + 19.1°

-R

am

br

EXERCISE 3B

ev

id

ie

w

ge

The values of x are 19.1° and 199.1° .

s

-C

1 Expand and simplify cos( x + 30°).

es

e

tan 25° + tan 20° 1 − tan 25° tan 20°

Pr

cos 25° cos 35° − sin 25° sin 35°

b sin172° cos 37° − cos172° sin 37° d cos 99° cos 69° + sin 99° sin 69°

ity

c

tan 82° − tan 52° 1 + tan 82° tan 52°

b tan 75°

c

cos105°

19 π 12

g

cot

ni ve rs

f

C 7π 12

w

ie

cos

d tan( −15°) h sin

7π 12

-R

ev

4 and that 0° , x , 90° , without using a calculator find the exact value of cos( x − 60°). 5

s

am

br

4 Given that cos x =

f

es

π 12

e

sin

id g

e

U

a sin 75°

op

3 Without using a calculator, find the exact value of:

-C

R

ev

ie

w

C

a sin 20° cos 70° + cos 20° sin 70°

y

op y

2 Without using a calculator, find the exact value of each of the following:

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Chapter 3: Trigonometry

4 5 63 , where A and B are acute, show that sin( A + B ) = and sin B =  . 5 13 65 12 3 and sin B = , where A is obtuse and B is acute, find the value of: 6 Given that sin A = 13 5 b cos( A − B ) c tan( A + B ) a sin( A + B )

a sin( A + B )

w

ev ie

-R

4 8 and sin B = − and that A and B lie in the same quadrant, find the value of: 5 17 b cos( A + B ) c tan( A − B )

8 Given that tan A = t and that tan( A − B ) = 2, find tan B in terms of t.

ve rs ity

C

op

y

-C

7 Given that cos A = −

Pr es s

am br id

ge

5 Given that sin A =

ev ie

w

9 If cos( A − B ) = 3 cos( A + B ), find the exact value of tan A tan B.

ni

C op

y

10 a Given that 8 + cosec 2 θ = 6 cot θ , find the value of tan θ .

ge

U

R

b Hence find the exact value of tan(θ + 45°).

id

ie

w

11 a Given that x is acute and that 2 sec 2 x + 7 tan x = 17, find the exact value of tan x .

br

ev

b Hence find the exact value of tan(225° − x ).

-R

am

12 a Show that the equation sin( x + 30°) = 5 cos( x − 60°) can be written in the form cot x = − 3 .

c

sin(30° − x ) = 4 sin x

Pr

cos( x + 30°) = 2 sin x

ity

a

b

cos( x − 60° ) = 3 cos x

d

cos( x + 30°) = 2 sin( x + 60°)

b

2 tan(45° − x ) = 3tan x

w

rs

C

op

y

13 Solve each equation for 0° ø x ø 360°.

es

s

-C

b Hence solve the equation sin( x + 30°) = 5 cos( x − 60°) for −180° , x , 180°.

y

sin( x + 60°) = 2 cos( x + 45°)

d

sin( x − 45°) = 2 cos( x + 60°)

e

tan( x + 45°) = 6 tan x

f

cos( x + 225°) = 2 sin( x − 60°)

C

ie

w

U

op

c

ge

ve

2 tan(60° − x ) = tan x

ni

a

id

R

ev

ie

14 Solve each equation for 0° ø x ø 180°.

-R

am

br

ev

15 Solve the equation tan( x − 45°) + cot x = 2 for 0° ø x ø 180°.

16 Use the expansions of cos(5x + x ) and cos(5x − x ) to prove that cos 6x + cos 4x ≡ 2 cos 5x cos x.

PS

17 Given that sin x + sin y = p and cos x + cos y = q, find an expression for cos( x − y ), in terms of p and q.

Pr

op y

es

s

-C

P

C

ity

3.3 Double angle formulae

If B = A,

sin( A + A) ≡ sin A cos A + cos A sin A

U

op

sin( A + B ) ≡ sin A cos B + cos A sin B

y

ni ve rs

Consider

R

cos( A + B ) ≡ cos A cos B − sin A sin B

If B = A,

cos( A + A) ≡ cos A cos A − sin A sin A

-R

 ∴  cos 2 A ≡ cos2 A − sin2 A

s

-C

am

br

ev

ie

id g

Consider

w

e

C

 ∴  sin 2 A ≡ 2 sin A cos A

es

ev

ie

w

We can use the compound angle formulae to derive three new identities.

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61

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ge

tan A + tan B 1 − tan A tan B tan A + tan A tan( A + A) ≡ 1 − tan A tan A 2 tan A  ∴  tan 2 A ≡ 1 − tan2 A

Pr es s

-C

If B = A,

-R

am br id

ev ie

w

tan( A + B ) ≡

Consider

C

ve rs ity

op

y

These three formulae are known as the double angle formulae. We can use the identity sin2 A + cos2 A ≡ 1 to write the identity cos 2 A ≡ cos2 A − sin2 A in two alternative forms:

ni

C op

Summarising, the double angle formulae are:

U

ie

ity

C

rs

br

y

ie es Pr

op

y

ni ve rs

U

cos 2 x = cos2 x − sin2 x 2

2

-R s es

am

br

ev

ie

id g

w

e

4 3 7         =  −  −  −  =  5  5 25

-C

ie b

ity

sin 2 x = 2 sin x cos x  3   4  24  = 2  −   −  =  5   5  25

w

C

a

C

-C

C

op y

T

s

x

O 5

From the diagram: 3 3 4 sin x = − , cos x = − and tan x = 5 5 4

-R

am

–4 –3

ev

A

id

y S

w

ge

Answer

C

U

R

ni

op

a sin 2 x

3 and that 180° , x , 270°, find the exact value of: 5 x b cos 2 x c tan 2

ve

w ie

Given that sin x = −

ev

s Pr

y

es

-C

  ≡ 1 − 2 sin2 A

WORKED EXAMPLE 3.6

ev

2 tan A 1 − tan2 A

-R

  ≡ 2 cos2 A − 1

op

62

tan 2 A ≡

cos 2 A ≡ cos2 A − sin2 A

am

br

sin 2 A ≡ 2 sin A cos A

ev

id

w

ge

KEY POINT 3.4

R

y

Using cos2 A ≡ 1 − sin2 A gives: cos 2 A ≡ 1 − 2 sin2 A

R

ev ie

w

Using sin2 A ≡ 1 − cos2 A gives: cos 2 A ≡ 2 cos2 A − 1

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ve rs ity

w

ge

C

U

ni

op

y

Chapter 3: Trigonometry

am br id

ev ie

x 2 c tan x = 2 x 1 − tan 2 x 2 tan 3 2            = 4 1 − tan2 x 2 x x 2 x            3tan + 8 tan − 3 = 0 2 2     x x        3tan − 1   tan + 3  = 0    2 2 x 1 x          tan =   or  tan = −3 2 3 2 2 tan  

y

y

Solve.

x , 135°, which means 2

s

-C

-R

am

br

ev

id

ie

w

x is in the second quadrant. 2 x ∴ tan = −3 2 that

C op

ge

If 180° , x , 270° , then 90° ,

Rearrange.

Factorise.

ve rs ity

U

R

ni

ev ie

w

C

op

x

Pr es s

-C

-R

Use 2 A = x in the double angle formula.

op

Pr

y

es

WORKED EXAMPLE 3.7

63

C

ity

Solve the equation sin 2 x = sin x for 0° ø x ø 360°.

rs ve

y

ge

Rearrange.

2 sin x cos x − sin x = 0

id

ie

w

Factorise.

ev y

1 2

-R

am

br

sin x(2 cos x − 1) = 0

1

-C

sin x = 0  or  cos x =

s es Pr

O

60 90

180

–1

y op

U

es

s

-R

br

ev

ie

id g

w

e

C

When solving equations involving cos 2 x it is important that we choose the most suitable expansion of cos 2 x for solving that particular equation.

am

270 y = cos x

The values of x are 0°, 60°, 180°, 300°, 360°.

-C

R

y=

0.5

–0.5

ni ve rs

C w

or x = 360° − 60° = 300°

ev

ie

1    ⇒    x = 60° 2

ity

op y

sin x = 0   ⇒    x = 0°, 180°, 360° cos x =

op

U

    2 sin x cos x = sin x

R

Use sin 2 x = 2 sin x cos x.

ni

ev

sin 2 x = sin x

C

ie

w

Answer

Copyright Material - Review Only - Not for Redistribution

360 x

1 2

ve rs ity

ev ie

w

ge

am br id

-C

    cos 2 x + 3sin x = 2

Pr es s

Use cos 2 x = 1 − 2 sin2 x .

1 − 2 sin2 x + 3sin x = 2

y

ni

0.5 O

id

π 5π or x = π − = 6 6

–0.5

br

s es

π π 5π , , . 6 2 6

ity

op

op

ity

Pr

es

s

-C

op y

ni ve rs

5 Given that 3 cos 2 x + 17 sin x = 8, find the exact value of sin x.

2 cos 2θ + 3 = 4 cos θ

2 cos 2θ + 1 = sin θ

c

w ev

ie

id g

es

s

-R

br am

op

b

C

U

2 sin 2θ = cos θ

e

a

y

6 Solve each equation for 0° ø θ ø 360°.

-C

C w

C

3 and that 90° , x , 180°, find the exact value of: 5 1 b sin 4x c tan 2 x d tan x 2

a sin 2 x

ie

ev

4 Given that cos x = −

-R

527 and that 0° , x , 90°, find the exact value of: 625 b tan 2 x c cos x d tan x

a sin 2 x

ev

d tan 3x

am

br

id

ie

w

ge

U

R

ni

ev

1 Express each of the following as a single trigonometric ratio. 2 tan17° a 2 sin 28° cos 28° b 2 cos2 34° − 1 c 1 − tan2 17° 4 2 Given that tan x = , where 0° , x , 90°, find the exact value of: 3 a sin 2 x b cos 2 x c tan 2 x 3 Given that cos 2 x = −

R

y

ve

rs

EXERCISE 3C

ie

w

C

64

y= x 3π 2

2π y = sin x

-R

π 2

The values of x are

y

–1

π

π 2

Pr

am

-C

sin x = 1   ⇒    x =

π 6

w

1 π    ⇒    x = 2 6

y=1

1

U

R

sin x =

1   or  sin x = 1 2

y

C op

(2 sin x − 1)(sin x − 1) = 0 sin x =

Factorise.

ve rs ity

  2 sin2 x − 3sin x + 1 = 0

ge

ev ie

w

C

op

y

Rearrange.

ie

Answer

-R

Solve the equation cos 2 x + 3sin x = 2 for 0 ø x ø 2 π.

ev

WORKED EXAMPLE 3.8

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Copyright Material - Review Only - Not for Redistribution

1 2

ve rs ity

C

U

ni

op

y

Chapter 3: Trigonometry

w

2 cosec 2θ + 2 tan θ = 3sec θ

d

2 tan 2θ = 3 cot θ

tan 2θ = 4 cot θ

f

y

8 Express cos2 2 x in terms of cos 4x.

-R

ev ie

3 cos 2θ + cos θ = 2

-C

e

b

cot 2θ + cot θ = 3

Pr es s

c

2 sin 2θ tan θ = 1

am br id

a

ge

7 Solve each equation for 0° ø θ ø 180°.

b Express sin 3x in terms of sin x.

ve rs ity

w

C

op

9 a Use the expansions of cos(2 x + x ) to show that cos 3x ≡ 4 cos3 x − 3 cos x.

y

2 . sin 2θ

ge

b Hence find the exact value of tan

π π + cot . 12 12

w

U

R

ni

11 a Prove that tan θ + cot θ ≡

C op

ev ie

10 Solve tan 2θ + 2 tan θ = 3 cot θ for 0° ø θ ø 180°.

br

ev

id

ie

12 a Prove that 2 cosec 2θ tan θ ≡ sec 2 θ .

s

-C

13 a Prove that cos 4x + 4 cos 2 x ≡ 8 cos 4 x − 3.

-R

am

b Hence solve the equation cosec 2θ tan θ = 2 for −π , θ , π.

Pr

y

es

b Hence solve the equation 2 cos 4x + 8 cos 2 x = 3 for −π ø x ø π.

ity

b Express sin 3θ and cos 2θ in terms of sin θ . Hence show that sin18° is a root of the equation 4x 3 − 2 x 2 − 3x + 1 = 0.

ve

ie

c

y op

ni

15 Find the range of values of θ between 0 and 2 π for which cos 2θ . cos θ .

PS

16 Solve the inequality cos 2θ − 3sin θ − 2 ù 0 for 0° ø θ ø 360°.

PS

17

w

ge

C

U

PS

ie

es

3x a

s

b

x

Solving trigonometric inequalities is not in the Cambridge syllabus. You should, however, have the skills necessary to tackle these more challenging questions.

-R

A

am -C

C

Pr

Prove that a = 4b cos 2 x cos x.

ity

op y C

18 a Use the expansions of cos(2 x + x ) and cos(2 x − x ) , to prove that:

w

ni ve rs

PS

TIP

ev

br

id

y

ie

cos 3x + cos x ≡ 2 cos 2 x cos x

op

U

es

s

-R

br

ev

ie

id g

w

e

19 Solve the inequality cos 4θ + 3 cos 2θ + 1 , 0 for 0° ø θ ø 360°.

am

PS

-C

ev

b Solve cos 3x + cos 2 x + cos x . 0 for 0° , x , 360°.

C

R

ev

d Hence find the exact value of sin18°.

B

R

65

rs

w

C

op

14 a Show that θ = 18° is a root of the equation sin 3θ = cos 2θ .

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

w

ge

3.4 Further trigonometric identities

-R

am br id

ev ie

This section builds on our previous work by using the compound angle formulae and the double angle formulae to prove trigonometric identities.

op

y

Prove the identity tan 3x ≡

3tan x − tan3 x . 1 − 3tan2 x

y

U

ge

2 tan x + tan x 1 − tan2 x

w

Multiply numerator and denominator by (1 − tan2 x ).

ev

id

ie

 2 tan x  1 − tan x  2  1 − tan x 

-R

2 tan x + tan x(1 − tan2 x ) 1 − tan2 x − 2 tan2 x

Expand the brackets and simplify.

es

3tan x − tan3 x 1 − 3tan2 x

ity

≡ RHS

op

ni

ie

s

-C

2 sin x cos y 2 cos x sin y − 2 sin x sin y 2 sin x sin y cos y cos x ≡ − sin y sin x ≡ cot y − cot x ≡ RHS

es

Separate fractions. Simplify fractions.

-R s es

-C

am

br

ev

ie

id g

w

e

C

U

op

y

ni ve rs



Pr

2 sin x cos y − 2 cos x sin y 2 sin x sin y

Use compound angle formulae. Simplify denominator.

ity

op y C w ie ev

R

ev

am

2 sin( x − y ) cos( x − y ) − cos( x + y ) 2 sin x cos y − 2 cos x sin y ≡ (cos x cos y + sin x sin y ) − (cos x cos y − sin x sin y )

LHS ≡

-R

br

Answer



w

ge

2 sin( x − y ) ≡ cot y − cot x. cos( x − y ) − cos( x + y )

id

Prove the identity

C

U

R

WORKED EXAMPLE 3.10

y

ve

ev

ie

w

rs

C

66

Pr

op

y



s

-C



am

br



Use the double angle formula.

ni

tan 2 x + tan x 1 − tan 2 x tan x

R



Use the compound angle formula for tan.

C op

LHS ≡ tan(2 x + x )

ev ie

w

C

ve rs ity

Answer

Pr es s

-C

WORKED EXAMPLE 3.9

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

am br id

EXPLORE 3.2

w

ge

C

U

ni

op

y

Chapter 3: Trigonometry

-R

Pr es s

1 − cos 2x sin 2x

sin 2x + sin x cos 2x + cos x + 1

cosec 2x − cot 2x

y

-C

Odd one out

op

1

ve rs ity

C

cosec 2x + cot 2x

ev ie

w

sin 2x 1 + cos 2x

2 sec2 x sin 2x

U

R

ni

C op

y

cot x − 2 cot 2x

br

ev

id

ie

Create as many expressions of your own to match the ‘odd one out’.

w

ge

Find the trigonometric expression that does not match the other six expressions.

-R

am

(Your expressions must contain at least two different trigonometric ratios.)

es

s

-C

Compare your answers with your classmates.

1 − tan2 A ≡ cos 2 A 1 + tan2 A

d

cos 2 A + sin 2 A − 1 ≡ tan A cos 2 A − sin 2 A + 1

e

sin 3A + sin A ≡ cos A 2 sin 2 A

f

cos 3A − sin 3A ≡ cos A + sin A 1 − 2 sin 2 A

g

cos 2 A + 9 cos A + 5 ≡ 2 cos A + 1 4 + cos A

Pr

es

s

-R

am

ev

cos A + sin A ≡ sec 2 A + tan 2 A cos A − sin A

-C

cos3 A − sin3 A 2 + sin 2 A ≡ cos A − sin A 2

P

4 Prove the identity 8 sin2 x cos2 x ≡ 1 − cos 4x.

P

1 5 Prove the identity (2 sin A + cos A) ≡ (4 sin 2 A − 3 cos 2 A + 5). 2

P

6 Use the expansions of cos(3x − x ) and cos(3x + x ) to prove the identity:

ie

ev

es

s

-R

br

-C

am

cos 2 x − cos 4x ≡ 2 sin 3x sin x

C

w

2

id g

e

U

op

ni ve rs

3 Use the fact that 4 A = 2  ×  2 A to show that: sin 4 A ≡ 8 cos3 A − 4 cos A b cos 4 A + 4 cos 2 A ≡ 8 cos 4 A − 3 a sin A

y

h

ity

op y C w

h sin( A + B ) sin( A − B ) ≡ sin2 A − sin2 B

P

ie ev

cosec 2 A + cot 2 A ≡ cot A

b

c

R

y

ve

ni

id

2 Prove each of these identities. sin A cos A 2 cos( A − B ) a + ≡ cos B sin B sin 2 B

br

P

f

w

g

U

R

e

d (cos A + 3sin A)2 ≡ 5 − 4 cos 2 A + 3sin 2 A

op

tan 2 A − tan A ≡ tan A sec 2 A 1 cot 2 A + tan A ≡ cosec A sec A 2 sec 2A cosec 2 A ≡ 4 cosec 2 2 A

b 1 − tan2 A ≡ cos 2 A sec 2A

C

c

rs

tan A + cot A ≡ 2 cosec 2 A

ge

ie ev

a

ie

1 Prove each of these identities.

w

C

P

67

ity

op

Pr

y

EXERCISE 3D

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WEB LINK Try the Equation or identity?(II) resource on the Underground Mathematics website.

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

w

ge

3.5 Expressing a sin θ + b cos θ in the form R sin(θ θ ± α ) or R cos(θθ ± α )

am br id

ev ie

In this section you will learn how to solve equations of the form a sin θ + b cos θ = c.

-C

-R

EXPLORE 3.3

y

y

y = 3 sin θ + 4 cos θ

C

ve rs ity

op

5

Pr es s

Use graphing software to confirm that the graph of y = 3sin θ + 4 cos θ for −90° ø θ ø 360° is:

O

ev ie

w

−90

θ 180

90

270

360

U

R

ni

C op

y

−5

w

ge

1 What are the amplitude and period of this graph?

y = cos θ

-R

am

br

b

ie

y = sin θ

a

ev

id

2 Discuss with your classmates how this graph can be obtained by transforming the graph of:

b R sin(θ + β )

Pr

y

(You will need to use your graph to find the approximate values of α and β .)

ni

op

Let a sin θ + b cos θ = R sin(θ + α ) where R . 0 and 0° , α , 90°.

ge

C

U

R sin(θ + α ) = R(sin θ cos α + cos θ sin α )

w s

ity

Use cos2 α + sin2 α = 1.

U

a 2 + b2

op

= R2

y

ni ve rs

= R 2 (cos2 α + sin2 α )

a 2 + b2 and tan α =

ev

ie

id g

es

s

-R

br am

b a

w

e

C

∴ a sin θ + b cos θ can be written as R sin(θ + α ) where R =

-C

ie

w

C

a 2 + b2 = R 2 cos2 α + R 2 sin2 α

ev

es

Pr

op y

-C

am

(2)

ev

br

(1)

R sin α = b Equating coefficients of cos θ : sin α b b =    ⇒   tan α =   (2) ÷ (1): cos α a a Squaring equations (1) and (2) and then adding gives:

Hence, R =

ie

R cos α = a

Equating coefficients of sin θ :

-R

id

∴ a sin θ + b cos θ = R sin θ cos α + R cos θ sin α

R

y

ve

The formula for replacing a sin θ + b cos θ by R sin(θ + α ) is derived as follows.

R

ev

ie

w

rs

C

ity

op

es

y = R sin(θ + α )

a

68

s

-C

3 Use your knowledge of transformations of functions to write the function y = 3sin θ + 4 cos θ in the form:

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ve rs ity

ev ie

am br id

EXPLORE 3.4

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ni

op

y

Chapter 3: Trigonometry

Using a similar approach, show that:

a cos θ − b sin θ ≡ R cos(θ + α ) where R =

Pr es s

3

ve rs ity

y

a cos θ + b sin θ ≡ R cos(θ − α ) where R =

C op

ni U

R

KEY POINT 3.5

y

Summarising the results from Explore 3.4:

ev ie

w

C

op

2

b a b a 2 + b2 and tan α = a b a 2 + b2 and tan α = a

a 2 + b2 and tan α =

-R

a sin θ − b cos θ ≡ R sin(θ − α ) where R =

-C

1

w ie

b and 0° , α , 90°. a

ev

id

a 2 + b 2 , tan α =

-R

am

br

where R =

ge

a sin θ ± b cos θ ≡ R sin(θ ± α ) and a cos θ ± b sin θ ≡ R cos(θ ∓ α )

es

s

-C

WORKED EXAMPLE 3.11

ity

b Hence solve the equation 2 sin θ − 3 cos θ = 2 for 0° , θ , 360°.

rs

y

ve

a

2 sin θ − 3 cos θ = R sin(θ − α )

ni

ev

ie

Answer

op

w

C

op

Pr

y

a Express 2 sin θ − 3 cos θ in the form R sin(θ − α ), where R . 0 and 0° , α , 90°.  Give the value of α correct to 2 decimal places.

ge

C

U

R

∴ 2 sin θ − 3 cos θ = R sin θ cos α − R cos θ sin α Equating coefficients of sin θ : R cos α = 2

br

(2)

w

-R

s es Pr

ity

2 sin θ − 3 cos θ = 2

ni ve rs

b

32 + 22 = R 2    ⇒    R = 13

∴ 2 sin θ − 3 cos θ = 13 sin(θ − 56.31°)

  13 sin(θ − 56.31°) = 2

-R s es

-C

am

br

ev

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id g

w

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op

y

2 13 θ − 56.31° = 23.09°, 180° − 23.09° = 23.09°, 156.91° θ = 79.4°, 213.2°

sin(θ − 56.31°) =

R

ev

ie

w

C

op y

-C

am

3 (2) ÷ (1):  tan α =     ⇒   α = 56.31° 2 Squaring equations (1) and (2) and then adding gives:

ie

Equating coefficients of cos θ : R sin α = 3

ev

id

(1)

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

ev ie

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We can find the maximum and minimum values of the expression a sin θ + b cos θ by writing the expression in the form R sin(θ + α ).

-R

From the Pure Mathematics Coursebook 1, Chapter 5, Section 5.4, we know that −1 ø sin(θ + α ) ø 1, hence −R ø R sin(θ + α ) ø R .

Pr es s

-C

This can be summarised as:

ve rs ity

C op

y

The minimum value of a sin θ + b cos θ is − a 2 + b 2 and occurs when sin(θ + α ) = −1.

ge

U

WORKED EXAMPLE 3.12

R

a 2 + b 2 and occurs when sin(θ + α ) = 1.

The maximum value of a sin θ + b cos θ is

ni

ev ie

w

C

op

y

KEY POINT 3.6

id

ie

w

Express 2 cos θ − sin θ in the form R cos(θ + α ), where R . 0 and 0° , α , 90°.

-R

am

br

ev

Hence, find the maximum and minimum values of the expression 2 cos θ − sin θ and the values of θ in the interval 0° , θ , 360° for which these occur.

es

s

-C

Answer 2 cos θ − sin θ = R cos(θ + α )

Pr

op

y

∴ 2 cos θ − sin θ = R cos θ cos α − R sin θ sin α 70

(2)

rs

Equating coefficients of sin θ : R sin α = 1 1 ⇒ α = 26.57° (2) ÷ (1):  tan α = 2 Squaring equations (1) and (2) and then adding gives:

(1)

y op

ni

ev

ve

ie

w

C

ity

Equating coefficients of cos θ : R cos α = 2

C

U

R

22 + 12 = R 2 ⇒ R = 5

id

ie

w

ge

∴ 2 cos θ − sin θ = 5 cos ( θ + 26.57° )

s

es

Pr

ity

5 when θ = 333.4°, min value = − 5 when θ = 153.4°

y op

DID YOU KNOW?

-R s es

am

br

ev

ie

id g

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e

C

U

The French mathematician Joseph Fourier (1768−1830) found a way to represent any wave-like function as the sum (possibly infinite) of simple sine waves. The sum of sine functions is called a Fourier series. These series can be applied to vibration problems. Fourier is also credited with discovering the greenhouse effect.

-C

R

ev

ie

w

C

∴  Max value =

ni ve rs

op y

-C

-R

am

br

ev

22 + ( −1)2 and occurs when cos (θ + 26.57°) = 1 θ + 26.57° = 360° θ = 333.4° Minimum value is − 22 + ( −1)2 and occurs when cos(θ + 26.57°) = −1 θ + 26.57° = 180° θ = 153.4°

Maximum value is

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ev ie

am br id

EXERCISE 3E

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Chapter 3: Trigonometry

-R

1 a Express 15 sin θ − 8 cos θ in the form R sin(θ − α ), where R . 0 and 0° , α , 90°.  Give the value of α correct to 2 decimal places.

Pr es s

-C

b Hence solve the equation 15 sin θ − 8 cos θ = 10 for 0° , θ , 360°.

Give the exact value of R and the value of α correct to 2 decimal places.

ve rs ity

w

C

op

y

2 a Express 2 cos θ − 3sin θ in the form R cos(θ + α ), where R . 0 and 0° , α , 90°.  b Hence solve the equation 2 cos θ − 3sin θ = 1.3 for 0° , θ , 360°.

y

ev ie

3 a Express 15 sin θ − 8 cos θ in the form R sin(θ − α ), where R . 0 and 0° , α , 90°. 

U

R

ni

C op

Give the value of α correct to 2 decimal places.

w

ie

Find the greatest value of 30 sin θ − 16 cos θ as θ varies.

id

c

ge

b Hence solve the equation 15 sin θ − 8 cos θ = 3 for 0° , θ , 360°.

-R

am

br

ev

4 a Express 4 sin θ − 6 cos θ in the form R sin(θ − α ), where R . 0 and 0° , α , 90°. Give the exact value of R and the value of α correct to 2 decimal places.

s

es

Find the greatest and least values of (4 sin θ − 6 cos θ )2 − 3 as θ varies.

Pr

c

op

y

-C

b Hence solve the equation 4 sin θ − 6 cos θ = 3 for 0° , θ , 180°.

C

ity

5 a Express 3sin θ + 4 cos θ in the form R sin(θ + α ), where R . 0 and 0° , α , 90°.

w

rs

Give the value of α correct to 2 decimal places.

y

ve

op

Find the least value of 3sin θ + 4 cos θ + 3 as θ varies.

U

R

c

ni

ev

ie

b Hence solve the equation 3sin θ + 4 cos θ = 2 for 0° , θ , 360°.

w

-R

am

br

ev

id

Give the exact values of R and α .  π 1 1 ≡ sec 2  θ −  . b Hence prove that 2 ( cos θ + 3 sin θ ) 4  3 

π . 2

ie

ge

C

6 a Express cos θ + 3 sin θ in the form R cos (θ − α ), where R . 0 and 0 , α ,

-C

7 a Express 8 sin 2θ + 4 cos 2θ in the form R sin(2θ + α ), where R . 0 and 0° , α , 90°.

s

es

ni ve rs

ity

Pr

b Hence solve the equation 8 sin 2θ + 4 cos 2θ = 3 for 0° , θ , 360°. 10 as θ varies. c Find the least value of (8 sin 2θ + 4 cos 2θ )2

8 a Express cos θ − 2 sin θ in the form R cos(θ + α ) , where R . 0 and 0° , α , 90° .

op

y

Give the exact value of R and the value of α correct to 2 decimal places.

-R s es

am

br

ev

ie

id g

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e

C

U

b Hence solve the equation cos θ − 2 sin θ = −1 for 0° , θ , 360°. 1 c Find the least value of 2 as θ varies. ( 2 cos θ − 2 sin θ )

-C

R

ev

ie

w

C

op y

Give the exact value of R and the value of α correct to 2 decimal places.

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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9 a Express cos θ − sin θ in the form R cos(θ + α ), where R . 0 and 0 , α , Give the exact values of R and α .

-R

1 b Show that one solution of the equation cos θ − sin θ = 2 interval 0 < θ < 2 π.

19 π and find the other solution in the 12

6 is θ =

State the set of values of k for which the equation cos θ − sin θ = k has any solutions.

Pr es s

-C c

π . 2

Give the exact value of R and the value of α correct to 2 decimal places.

ve rs ity

C

op

y

10 a Express sin θ − 3 cos θ in the form R sin(θ − α ), where R > 0 and 0° , α , 90°. 

y

Find the greatest possible value of 1 + sin 2θ − 3 cos 2θ as θ varies and determine the smallest positive value of θ for which this greatest value occurs. 5 cos θ + 2 sin θ in the form R cos(θ − α ), where R . 0 and 0° , α , 90°.

ge

11 a Express

U

R

ni

ev ie

c

C op

w

b Hence solve the equation sin θ − 3 cos θ = −2 for 0° , θ , 360°.

id

ie

w

Give the value of α correct to 2 decimal places.

br

Find the smallest positive angle θ that satisfies the equation

-R

am

c

ev

b Find the smallest positive angle θ that satisfies the equation

5 cos θ + 2 sin θ = 3. 1 1 5 cos θ + 2 sin θ = −1. 2 2

es

s

-C

12 a Given that 3sec θ + 4 cosec θ = 2 cosec 2θ , show that 3sin θ + 4 cos θ = 1.

Pr

Give the value of α correct to 2 decimal places.

rs

Hence solve the equation 3sec θ + 4 cosec θ = 2 cosec 2θ for 0° , θ , 360°.

13 a Express sin(θ + 30°) + cos θ in the form R sin(θ + α ), where R . 0 and 0° , α , 90°. 

y

ve

PS

c

ev

Give the exact values of R and α .

U

R

ni

op

ie

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C

72

ity

op

y

b Express 3sin θ + 4 cos θ in the form R sin(θ + α ) , where R . 0 and 0° , α , 90°.

w

ge

ie

14 a Find the maximum and minimum values of 7 sin2 θ + 9 cos2 θ + 4 sin θ cos θ + 2.

id

PS

C

b Hence solve the equation sin(θ + 30°) + cos θ = 1 for 0° , θ , 360°.

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

am

br

ev

b Hence, or otherwise, solve the equation 7 sin2 θ + 9 cos2 θ + 4 sin θ cos θ = 10 for 0° , θ , 360°.

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ve rs ity ni

op

y

Chapter 3: Trigonometry

y

op

y

y = cosec θ

1

1

y = sec θ y = cot θ

360

270

ge

U

R

ni

−1

180

1 + cot 2 x ≡ cosec 2 x

-R

sin( A − B ) ≡ sin A cos B − cos A sin B

s



Pr



cos( A − B ) ≡ cos A cos B + sin A sin B

op

y

ve

tan( A + B ) ≡



cos 2 A ≡ cos2 A − sin2 A

ie

sin 2 A ≡ 2 sin A cos A

-R

am

br

ev

id



w

ge

C

U

ni

ie ev

R

tan A + tan B 1 − tan A tan B tan A − tan B tan( A − B ) ≡ 1 + tan A tan B

73

rs

cos( A + B ) ≡ cos A cos B − sin A sin B

ity

op C w



Double angle formulae

≡ 1 − 2 sin2 A

Pr

es

2 tan A 1 − tan2 A

where R =

a 2 + b 2 and tan α =

b a

-R s es

-C

am

br

ev

ie

id g

w

e

U

y

a cos θ ± b sin θ = R cos(θ ∓ α )

op



C

a sin θ ± b cos θ = R sin(θ ± α )

ni ve rs



ity

θ ± α ) or R cos (θθ ± α ) Expressing a sin θ + b cos θ in the form R sin(θ

R

ev

ie

w

C

op y

tan 2 A ≡

s

-C

≡ 2 cos2 A − 1



360

es

sin( A + B ) ≡ sin A cos B + cos A sin B

y





270

ev

br

am

-C

Compound angle formula



180

ie



id

1 + tan2 x ≡ sec 2 x

θ 90

w

Trigonometric identities ●

O

y

90

C op

360

270

1 tan θ

y

θ

ve rs ity

180

90

cot θ =

O

θ

ev ie

w

C

O −1

1 cos θ

-R

sec θ =

Pr es s

-C

y

1 sin θ

ev ie

am br id

Cosecant, secant and cotangent cosec θ =

C w

ge

U

Checklist of learning and understanding

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

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w

END-OF-CHAPTER REVIEW EXERCISE 3 Sketch the graph of y = 3sec(2 x − 90°) for 0° , x , 180°.

[3]

2

By expressing the equation cosec θ = 3sin θ + cot θ in terms of cos θ only, solve the equation for 0° , θ , 180°.

[5]

Pr es s

-C

-R

1

5

Solve the equation 2 cot 2 x + 5 cosec x = 10 for 0° , x , 360°.

6

a

Prove that sin( x + 60°) + cos( x + 30°) ≡ 3 cos x .

b

3 Hence solve the equation sin( x + 60°) + cos( x + 30°) = for 0° , x , 360°. 2

[3]

a

Prove that sin(60° − x ) + cos(30° − x ) ≡ 3 cos x .

[3]

b

2 Hence solve the equation sin(60° − x ) + cos(30° − x ) = sec x for 0° , x , 360°. 5

[3]

i

Show that the equation tan( x + 45°) = 6 tan x can be written in the form 6 tan2 x − 5 tan x + 1 = 0.

[3]

ii

Hence solve the equation tan( x + 45°) = 6 tan x , for 0° , x , 180°.

[3]

y

C op

ni

ie

ev

-R

s

Hence solve the equation tan( x + 45°) − tan(45° − x ) = 6 for 0° , x , 180°.

[3]

Express 3 cos θ + sin θ in the form R cos(θ − α ), where R . 0 and 0° , α , 90°, giving the exact value of R and the value of α correct to 2 decimal places.

[3]

ity

rs

y

ve

Hence solve the equation 3 cos 2 x + sin 2 x = 2, giving all solutions in the interval 0° ø x ø 360° .

U

R

ni

ii

op

C

es

b

Pr

[4]

ie

w

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q3 June 2010

Prove that tan( x + 45°) − tan(45° − x ) ≡ 2 tan 2 x.

10 i

ev

[3]

a

y op

9

74

w

ge

id

br

-C

8

am

7

[5]

ge

ie es

ity

[4]

2

[1]

y

ni ve rs

 2 sin 2θ − 3 cos 2θ + 3  Write down the greatest value of   . sin θ 

U

op

i Express 4 sin θ − 6 cos θ in the form R sin(θ − α ) , where R . 0 and 0° , α , 90° .

ie

id g

ii Solve the equation 4 sin θ − 6 cos θ = 3 for 0° ø θ ø 360°.

w

e

C

Give the exact value of R and the value of α correct to 2 decimal places.

ev

2

[3] [4] [2]

s

-R

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q8 June 2011

es

am

br

iii Find the greatest and least possible values of (4 sin θ − 6 cos θ ) + 8 as θ varies.

-C

13

[3]

Pr

op y ie

c

[3]

s

-C

1 ii solve the equation cos(60° − x ) + cos(300° − x ) = cosec x for 0° , x , 180°. 4 2 sin 2θ − 3 cos 2θ + 3 ≡ 4 cos θ + 6 sin θ . 12 a Prove the identity sin θ π b Express 4 cos θ + 6 sin θ in the form R cos(θ − α ), where R .0 and 0 , α , . 2 Give the exact value of R and the value of α correct to 2 decimal places.

w

C

[2]

-R

am

i find the exact value of cos15° + cos 255°

ev

ev

id

Hence

[3]

w

Prove that cos(60° − x ) + cos(300° − x ) ≡ cos x .

br

b

C

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q7 November 2013

11 a

R

[6]

U

R

ev ie

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C

op

3

ve rs ity

y 4

Cambridge International A Level Mathematics 9709 Paper 31 Q3 June 2016 1 [5] Given that cos A = , where 270° , A , 360°, find the exact value of sin 2 A. 4 2 [6] Solve the equation 2 tan x + sec x = 1 for 0° ø x ø 360°.

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Chapter 3: Trigonometry

y

ve rs ity

C op

Hence solve the equation cosec x − cot x = 16 − cot x for 0° , x , 180°. 4

4

ge

b

Prove the identity cosec 4 x − cot 4 x ≡ cosec 2 x + cot 2 x.

ni

R

16 a

U

ev ie

w

C

op

y

Pr es s

-C

-R

i By first expanding sin(2θ + θ ), show that sin 3θ = 3sin θ − 4 sin3 θ . [4] 2 sin θ 1 , the equation x 3 − x + 3 = 0 can be written in ii Show that, after making the substitution x = 6 3 3 [1] the form sin 3θ = . 4 1 iii Hence solve the equation x 3 − x + 3 = 0, giving your answers correct to 3 significant figures. [4] 6 Cambridge International A Level Mathematics 9709 Paper 31 Q8 November 2014 1 ≡ sec x.  [3] 15 a Prove the identity sin( x + 30°) + cos( x + 60°) 2 = 7 − tan2 x for 0° , x , 360°. b Hence solve the equation [6] sin( x + 30°) + cos( x + 60°) 14

[3] [6]

ie

id

ev

br

-R

Hence find all the roots of the equation 3 cos 2 x + 4 sin 2 x = 0 for 0° , x , 180°.

[3]

Pr

es

s

[3] 75

ity

y op

Give the exact value of R and the value of α correct to 2 decimal places.

y op y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

am

br

ev

id

ie

w

ge

C

U

R

ni

ev

ve

ie

w

rs

C

c

[3]

Express 3 cos 2 x + 4 sin 2 x in the form R sin(2 x + α ), where R . 0 and 0° , α , 90° .

-C

b

Show that the x-coordinates of the points where C1 and C2 intersect satisfy the equation 3 cos 2 x + 4 sin 2 x = 0.

am

a

w

17 The curves C1 and C2 have equations y = 1 + 4 cos 2 x and y = 2 cos2 x − 4 sin 2 x .

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ve rs ity

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ev ie

am br id

Find the set of values of x satisfying the inequality 3 x − 1 , 2 x + 1 .

[4]

Cambridge International A Level Mathematics 9709 Paper 31 Q1 November 2012

-R

1

Pr es s

-C

Solve the equation 2 3x − 1 = 3x , giving your answers correct to 3 significant figures.

Solve the equation 3x + 4 = 3x − 11 .

[3]

ii

Hence, using logarithms, solve the equation 3  ×  2 y + 4 = 3  ×  2 y − 11 , giving the answer correct to 3 significant figures.

[2]

y

ve rs ity

op

i

ev ie

w

C

3

[4]

Cambridge International A Level Mathematics 9709 Paper 31 Q2 November 2013

y

2

w

CROSS-TOPIC REVIEW EXERCISE 1

U

i

Solve the equation 2 x − 1 = 3 x .

ii

Hence solve the equation 2 5x − 1 = 3 5x , giving your answer correct to 3 significant figures.

w

ie

br

-R

am

es

s

-C

ln y

Pr

(5, 4.49)

ity

op

rs

C

76

[2]

Cambridge International A Level Mathematics 9709 Paper 31 Q1 June 2016

y

5

y

ev

ve

ie

w

(0, 2.14)

op

O

U

( )

C

R

ni

x

br

w

It is given that x satisfies the equation 32x = 5(3x ) + 14. Find the value of 3x and, using logarithms, find the value of x correct to 3 significant figures.

[4]

es

-R

am

i

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q2 June 2012

[1]

Pr

) + 14.

ity

Find the values of n and C .

ii

Explain why the graph of ln y against ln x is a straight line.

[5] [1]

w

Cambridge International A Level Mathematics 9709 Paper 31 Q3 June 2010

ie

id g

C

U

i

op

y

ni ve rs

The variables x and y satisfy the equation x n y = C , where n and C are constants. When x = 1.10 , y = 5.20 , and when x = 3.20, y = 1.05.

-R

br

ev

Given that 3e x + 8e − x = 14, find the possible values of e x and hence solve the equation 3e x + 8e − x = 14 correct to 3 significant figures.

[6]

s

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q3 June 2016

es

am

8

= 5(3

x

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q1 November 2016

e

R

ev

ie

7

2x

Hence state the values of x satisfying the equation 3

-C

C

op y

ii

s

-C

6

[5]

ev

id

ie

ge

The variables x and y satisfy the equation y = A b x , where A and b are constants. The graph of ln y against x is a straight line passing through the points (0, 2.14) and (5, 4.49), as shown in the diagram. Find the values of A and b, correct to 1 decimal place.

w

[3]

ev

id

4

ge

R

ni

C op

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q1 June 2015

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Cross-topic review exercise 1

The angles θ and φ lie between 0° and 180°, and are such that tan(θ − φ ) = 3 and tan θ + tan φ = 1.

9

Solve the equation 4x − 1 = x − 3 .

[3]

ii

Hence solve the equation 4 y +1 − 1 = 4 y − 3 correct to 3 significant figures.

[3]

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i

[3]

ii

When a has this value, solve the equation p( x ) = 0 .

[4]

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s

es

Find the values of a and b.

ii

When a and b have these values, find the quotient and remainder when p( x ) is divided by x 2 − 2 .

[5]

Pr

)

(

77

[3]

rs

ve

w

C

[5]

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id br

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am

[3]

By first using the factor theorem, factorise 16x − 24x − 15x − 2 completely. 2

s

3

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iii Hence solve the equation 2 ln(4x − 5) + ln( x + 1) = 3 ln 3.

[4]

Pr

[1]

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Cambridge International A Level Mathematics 9709 Paper 31 Q6 June 2014

Find the values of a and b.

ii

When a and b have these values, factorise p( x ) completely.

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i

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id g

[5] [3]

Cambridge International A Level Mathematics 9709 Paper 31 Q6 November 2015

es

s

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br am -C

C

op

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ni ve rs

The polynomial 8x 3 + ax 2 + bx − 1, where a and b are constants, is denoted by p( x ) . It is given that ( x + 1) is a factor of p( x ) and that when p( x ) is divided by ( 2 x + 1 ) the remainder is 1.

e

op y

[3]

Cambridge International A Level Mathematics 9709 Paper 31 Q6 November 2011

Show that 16x 3 − 24x 2 − 15x − 2 = 0.

ii

C

op

ni

Hence solve the equation cos 2θ + 3 sin 2θ = 2, for 0° , θ , 90°.

It is given that 2 ln(4x − 5) + ln( x + 1) = 3 ln 3. i

15

y

Express cos x + 3 sin x in the form R cos ( x − α ), where R . 0 and 0° , α , 90° , giving the exact value of R and the value of α correct to 2 decimal places.

i

U

ie

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13

w

w ie

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i

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y

-C

am

The polynomial 2 x 3 − 4x 2 + ax + b, where a and b are constants, is denoted by p( x ) . It is given that when p( x ) is divided by ( x + 1) the remainder is 4, and that when p( x ) is divided by ( x − 3) the remainder is 12.

C

op

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q5 November 2011

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q7 November 2012

14

ie

C op

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Find the value of a and hence verify that ( x − 3) is a factor of p( x ) .

U

i

ii

ev

y

The polynomial 4x 3 + ax 2 + 9x + 9 , where a is a constant, is denoted by p( x ) . It is given that when p( x ) is divided by (2 x − 1) the remainder is 10.

id

R

ev ie

11

ge

w

Cambridge International A Level Mathematics 9709 Paper 31 Q4 June 2013

12

R

-R

Cambridge International A Level Mathematics 9709 Paper 31 Q3 November 2015

C

op

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10

[6]

Pr es s

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Find the possible values of θ and φ .

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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The polynomial 3x 3 + 2 x 2 + ax + b, where a and b are constants, is denoted by p( x ) . It is given that ( x − 1) is a factor of p( x ) , and that when p( x ) is divided by ( x − 2) the remainder is 10. Find the values of a and b.

ii

When a and b have these values, solve the equation p( x ) = 0 .

Pr es s

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y

The polynomial x 3 + ax 2 + bx + 8, where a and b are constants, is denoted by p( x ) . It is given that when p( x ) is divided by ( x − 3) the remainder is 14, and that when p( x ) is divided by ( x + 2) the remainder is 24. Find the values of a and b.

ni

ii

ve rs ity

Cambridge International AS and A Level Mathematics 9709 Paper 21 Q7 November 2010 i

When a and b have these values, find the quotient when p( x ) is divided by x 2 + 2 x − 8 and hence solve the equation p( x ) = 0.

[5] [4]

w

ge

R

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U

17

ev ie

w

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op

y

i

C op

16

[4]

When a and b have these values, factorise 5x 3 + ax 2 + b completely, and hence solve the equation 53 y +1 + a   ×  52 y + b = 0, giving any answers correct to 3 significant figures.

[5]

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ii

Given that ( x + 2) and ( x + 3) are factors of 5x 3 + ax 2 + b, find the values of the constants a and b.

br

i

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18

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ie

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q4 November 2013

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q5 November 2014

Pr

y

[4]

ii

It is given that, when x 4 + x 3 + 3x 2 + px + q is divided by ( x 2 − x + 4) , the remainder is zero. Find the values of the constants p and q.

[2]

ity

Find the quotient and remainder when x 4 + x 3 + 3x 2 + 12 x + 6 is divided by ( x 2 − x + 4) .

y op

ni

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ve

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i

rs

op 19

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C

78

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iii When p and q have these values, show that there is exactly one real value of x satisfying the equation x 4 + x 3 + 3x 2 + px + q = 0 and state what that value is.

[3]

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id

br

ev

The angle α lies between 0° and 90° and is such that 2 tan2 α + sec 2 α = 5 − 4 tan α . Show that 3tan2 α + 4 tan α − 4 = 0 and hence find the exact value of tan α .

[4]

ii

It is given that the angle β is such that cot(α + β ) = 6. Without using a calculator, find the exact value of cot β .

[5]

-R

am

i

s

es

Pr

ity

Express 5 sin 2θ + 2 cos 2θ in the form R sin(2θ + α ), where R . 0 and 0° , α , 90°, giving the exact value of R and the value of α correct to 2 decimal places.

ni ve rs

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solve the equation 5 sin 2θ + 2 cos 2θ = 4, giving all solutions in the interval 0° ø θ ø 360°, 1 as θ varies. iii determine the least value of (10 sin 2θ + 4 cos 2θ )2

w

ie

[5] [2]

s

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ev

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q7 June 2013

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br

id g

e

C

U

ii

y

Hence

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i

am

21

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q7 November 2014

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R

ev

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C

op y

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20

w

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q6 November 2015

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ve rs ity -R

Find the values of a and b.

ii

When a and b have these values,

[5]

Pr es s

i

a show that the equation p( x ) = 0 has exactly one real root,

ve rs ity

y op C

ev ie

am br id

The polynomial p( x ) is defined by p( x ) = ax 3 + 3x 2 + bx + 12, where a and b are constants. It is given that ( x + 3) is a factor of p( x ). It is also given that the remainder is 18 when p( x ) is divided by ( x + 2).

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22

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Cross-topic review exercise 1

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b solve the equation p(sec y ) = 0 for −180° , y , 180°.

[3]

y C op

79

y op y op -R s es

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C

ity

Pr

op y

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s

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-R

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C

U

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ev

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rs

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op

Pr

y

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s

-C

-R

am

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ev

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U

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ni

ev ie

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q7 November 2016

Copyright Material - Review Only - Not for Redistribution

op

y

ve rs ity ni

C

U

ev ie

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-R

am br id

Pr es s

-C y

ni

C op

y

ve rs ity

op C w ev ie C

op

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rs

Chapter 4 Differentiation

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op

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R 80

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differentiate products and quotients use the derivatives of e x , ln x, sin x, cos x, tan x, together with constant multiples, sums, differences and composites find and use the first derivative of a function, which is defined parametrically or implicitly.

y op -R s es

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ev

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ni ve rs

C

ity

Pr

op y

■ ■ ■

ie

In this chapter you will learn how to:

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ve rs ity ge

C

U

ni

op

y

Chapter 4: Differentiation

w

Pure Mathematics 1 Coursebook, Chapter 7

Differentiate x n together with constant multiples, sums and differences.

1 Differentiate with respect to x. 3 a y = 5x 3 − 2 + 2 x x

Pr es s

-C

y

b

ve rs ity

op Pure Mathematics 1 Coursebook, Chapter 7

Differentiate composite functions using the chain rule.

a (3x − 5)4 b

ni

w ev

C

Pr

4 Find the stationary points on the curve y = x 3 − 3x 2 + 2 and determine their nature. 81

In this chapter we will learn how to differentiate the product and the quotient of two simple functions. We will also learn how to find and use the derivatives of exponential, logarithmic and trigonometric functions.

y

op

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w ie

3 Find the equation of the normal to the curve y = x 3 − 5x 2 + 2 x − 1 at the point (1, −3) .

ity

op

y

es

s

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Find stationary points on curves and determine their nature.

Why do we study differentiation?

4 1 − 2x

ie

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id

Find tangents and normals to curves.

br am

Pure Mathematics 1 Coursebook, Chapter 8

x 8 − 4x 5 + x 2 2x3

2 Differentiate with respect to x.

U

R

Pure Mathematics 1 Coursebook, Chapter 7

y=

C op

C

Check your skills

y

What you should be able to do

-R

Where it comes from

ev ie

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ev ie

am br id

PREREQUISITE KNOWLEDGE

id

ie

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C

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R

All of the functions that we have differentiated so far have been of the form y = f( x ) . In this chapter we will also learn how to differentiate functions that cannot be written in the form y = f( x ). For example, the function y2 + 2 xy = 4 .

br

-R

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Pr

op y

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Although these topics might seem like they are just for Pure Mathematics problems, differentiation is used for calculations to do with quantum mechanics and field theories in Physics.

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ni ve rs

The function y = ( x + 1)4 (3x − 2)3 can be considered as the product of two separate functions:

op

-R s es

am

br

ev

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To differentiate the product of two functions we can use the product rule.

C

U

y = uv where u = ( x + 1)4 and v = (3x − 2)3

-C

ev

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C

ity

4.1 The product rule

R

This chapter builds on the work from Pure Mathematics 1 Coursebook, Chapters 7 and 8 where we learnt how to find and use the derivative of x n .

ev

Lastly, we will learn how to find and use the derivative of a function where the variables x and y are given as a function of a third variable.

REWIND

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WEB LINK Explore the Calculus of trigonometry and logarithms and the Chain rule and integration by substitution stations on the Underground Mathematics website.

ve rs ity

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am br id

KEY POINT 4.1

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Some people find it easier to remember this rule as:

Pr es s

-C

-R

The product rule is: dv du d +v ( uv ) = u dx dx dx

op

y

‘(first function × derivative of second function) + (second function × derivative of first function)’ dy d d = ( x + 1)4 (3x − 2)3 + (3x − 2)3    ( x + 1)4   dx  dx dx         differentiate second + differentiate first first second

y

ev ie

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C

ve rs ity

So for y = ( x + 1)4 (3x − 2)3,

ni

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  = ( x + 1)4 × 9(3x − 2)2 + (3x − 2)3 × 4( x + 1)3

U

R

= ( x + 1)3 (3x − 2)2 [ 9( x + 1) + 4(3x − 2) ]

br

The product rule from first principles

-R

am

Consider the function y = uv where u and v are functions of x.

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ge

  = ( x + 1)3 (3x − 2)2 (21x + 1)

es

s

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A small increase δx in x leads to corresponding small increases δy, δu and δv iny, u and v.   y + δy = ( u + δu) ( v + δv)

Pr

y

Expand brackets and replace y with uv.

  uv + δy = uv + uδv + vδu + δuδv

op

Subtract uv from both sides.

δy = uδv + vδu + δuδv

ity

δy δv δu δv ---------------(1) =u +v + δu δx δx δx δx

op

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ve

ni

As δx → 0, then so do δy, δu and δv and

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C

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Divide both sides by δx.

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C

82

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dy δu du dv δy δv → , → and → . δx δx dx δx dx dx

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Equation (1) becomes:

dy dv du dv =u +v +0 dx dx dx dx dv du dy =u +v  ∴  dx dx dx

Pr

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s

-C

 

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1

y = (2 x − 1)(4x + 5) 2

y

dy when y = (2 x − 1) 4x + 5 . dx Answer

Find

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1

1 dy d d = (2 x − 1)  [(4x + 5) 2 ] + (4x + 5) 2   (2 x − 1)   dx  x d d  x     second differentiate first differentiate second + first

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WORKED EXAMPLE 4.1

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1    = (2 x − 1)  (4x + + 2 

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(

C

+

1  − 5) 2 (4)

differentiate first

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differentiate second

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first

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Chapter 4: Differentiation

4x + 5 (2)

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Chain rule.

2(2 x − 1)  = + 2 4x + 5 4x + 5 2(2 x − 1) + 2(4x + 5)  = 4x + 5 12 x + 8  = 4x + 5

y C op

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WORKED EXAMPLE 4.2

Simplify the numerator.

ve rs ity

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y

Pr es s

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Write as a single fraction.

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  y = (2 x − 3)2 ( x + 5)3

y

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dy d d [( x + 5)3 ] + ( x + 5)3   [(2 x − 3)2 ] = (2 x − 3)2     dx  d d x x        second first differentiate second + differentiate first

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Answer

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Find the x-coordinate of the points on the curve y = (2 x − 3)2 ( x + 5)3 where the gradient is 0.

Pr

Chain rule

83

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Chain rule

= 3(2 x − 3)2 ( x + 5)2 + 4(2 x − 3)( x + 5)3

Factorise.

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C

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   = (2 x − 3)2 [3( x + 5)2 (1)]  +( x + 5)3 [2(2 x − 3)1(2)]    

= (2 x − 3)( x + 5) [3(2 x − 3) + 4( x + 5)]

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ni

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Simplify.

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  = (2 x − 3)( x + 5) (10 x + 11) 2

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2

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dy = 0 when (2x − 3)( x + 5)2 (10x + 11) = 0 dx x+5 = 0 2x − 3 = 0 10 x + 11 = 0 3 x=   x = −5 x = −1.1   2

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EXERCISE 4A

Pr

g ( x − 3)2 ( x + 2)5

c

2x − 1

f

h (2 x − 1)5 (3x + 4)4

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x

3

y

ni ve rs

d ( x − 1) x + 5

b 5x(2 x + 1)3

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x( x − 2)5

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a

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2 Find the gradient of the curve y = x 2 x + 4 at the point ( −3, 9) .

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1 Use the product rule to differentiate each of the following with respect to x:

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x x+2 x ( x 2 + 2)3 (2 x − 5)(3x 2 + 1)2

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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3 Find the equation of the tangent to the curve y = (2 − x )3 ( x + 1)4 at the point where x = 1.

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4 Find the gradient of the tangent to the curve y = ( x + 2)( x − 1)3 at the point where the curve meets the y-axis.

Pr es s

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5 Find the x-coordinate of the points on the curve y = (3 − x )3 ( x + 1)2 where the gradient is zero.

y

6 Find the x-coordinate of the point on the curve y = ( x + 2) 1 − 2 x where the gradient is zero. 7 a Sketch the curve y = ( x − 1)2 (5 − 2 x ) + 3.

C op

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4.2 The quotient rule

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b The curve y = ( x − 1)2 (5 − 2 x ) + 3 has stationary points at A and B. The straight line through A and B cuts the axes at P and Q. Find the area of the triangle POQ.

ge

We can differentiate the function y =

x2 − 5 by writing the function in the form 2x + 1

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y = ( x 2 −  5)(2 x +  1)−1 and then by applying the product rule.

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br

ev

x2 − 5 as the division (quotient) of two separate Alternatively, we can consider y = 2x + 1 functions: u y= where  u = x 2 − 5 and v = 2 x + 1. v To differentiate the quotient of two functions we can use the quotient rule.

y

ve

du dv  − u  dx dx v2

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d  u   = dx  v 

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The quotient rule is:

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KEY POINT 4.2

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84

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Some people find it easier to remember this rule as (denominator  × derivative of numerator) − (numerator × derivative of denominator) (denominator)2

The quotient rule from first principles

u where u and v are functions of x. v A small increase δx in x leads to corresponding small increases δy, δu and δv in y, u and v.

y op -R s es

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Pr

op y

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Consider the function y =

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ve rs ity Pr es s

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ev -R s

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C

x2 − 5 . 2x + 1

rs br

denominator squared

-R

am

d (2 x + 1) dx

C

d ( x 2 − 5) − ( x 2 − 5)   × dx (2 x + 1)2   2

(2 x + 1)(2 x ) − ( x − 5)(2) (2 x + 1)2

s

-C

Pr

es

4x 2 + 2 x − 2 x 2 + 10 (2 x + 1)2

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2( x 2 + x + 5) (2 x + 1)2

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ni ve rs

 =

differentiate denominator  

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× 

numerator −   

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(2 x + 1)

differentiate numerator 

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C

op y

 =

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y

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denominator    

=

85

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x2 − 5 2x + 1

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Answer

dy = dx

Pr

Find the derivative of y =

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WORKED EXAMPLE 4.3

y=

y

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du dv  − u  x d dx v2

br



am

d  u = dx  v 

dy δu dv du δy δv → → → , . and dx δx dx dx δx δx

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Equation (1) becomes

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As δx → 0 , then so do δy, δu and δv and

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δy = δx

   

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    δy =

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    δy =

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    δy =

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    δy =

u + δu Subtract y from both sides. v + δv u u + δu Replace y with . −y v v + δv u + δu u − Combine fractions. v + δv v v ( u + δu) − u( v + δv ) Expand brackets and simplify. v ( v + δv ) vδu − uδv Divide both sides by δx.   v 2 + vδv δu δv v  − u δx δx   ---------------------(1) v 2 + vδv

am br id

y + δy =

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ni

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Chapter 4: Differentiation

Copyright Material - Review Only - Not for Redistribution

ve rs ity

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  d x −1 × [( x   +  2)3 ] dx

dy =  dx

numerator −   

differentiate denominator

− ( x   +  2)

3

×

ni

2 x − 1  ) ( 

  d  x − 1   dx 

y

differentiate numerator

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denominator 

C op

( x + 2)3 x −1

y

y=

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Answer

( x + 2)3 . x −1

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Find the derivative of y =

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WORKED EXAMPLE 4.4

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

U

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denominator squared

( x + 2)3 2  x − 1 

6( x   +  2)2 ( x − 1) −  ( x   +  2)3 2( x − 1) x − 1

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3

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br

-C

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EXPLORE 4.1

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2( x − 1) 2

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( x   +  2)2 (5x − 8)

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R

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2( x − 1) 2  =

Factorise the numerator.

rs

3

Multiply numerator and denominator by 2 x − 1.

ve

C

( x   +  2)2 [6( x − 1) − ( x   +  2)]

ie

w

 =

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s

x −1

y

op

 =

86

ev -R

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3( x + 2)2 x − 1  −    =

w

ge

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br

=

1  1 − x − 1 [3( x   +  2)2 (1)]  − ( x   +  2)3  ( x − 1) 2  (1)  2   x −1

id

(

ity

ni ve rs

Do you obtain the same answers?

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br

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Discuss with your classmates which method you prefer for functions such as these: the quotient rule or the product rule.

R

ev

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C

Pr

op y

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s

x2 − 5 and In Worked examples 4.3 and 4.4 we differentiated the functions y = 2x + 1 ( x + 2)3 y= using the quotient rule. Now write each of these functions as a product, x −1 such as y = ( x 2 − 5)(2 x + 1)−1, and then use the product rule to differentiate them.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

am br id

EXERCISE 4B

w

ge

C

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Chapter 4: Differentiation

f

5x 4 ( x 2 − 1)2

op C w

g

3x 2 − 7 x x2 + 2x + 5

d

3x + 1 2 − 5x

h

( x + 4)2 ( x 2 + 1)3

x−5 1 at the point  2,  −  .  x+4 2

ve rs ity

2 Find the gradient of the curve y =

Pr es s

-C

1 − 2x2 ( x + 4)2

y

e

-R

1 Use the quotient rule to differentiate each of the following with respect to x: 2x + 3 3x − 5 x2 − 3 b c a x−4 2−x 2x − 1

3 Find the coordinates of the points on the curve y =

( x − 1 )2

y

C op

ie ev

x −1 2x + 3

c

-R

am

br

id

6 Differentiate with respect to x: x b a 5x − 1

w

ge

U

R

ni

ev ie

where the tangent is parallel to the x-axis. 2x + 5 1 − 2x at which the gradient is 1. 4 Find the coordinates of the points on the curve y = x−5 x−4 at the point where the curve crosses the y-axis. 5 Find the equation of the tangent of the curve y = 2x + 1 3 − x2 x2 − 1

d

x +1 where the gradient is 0. x −1 x2 + 1 at the point ( −1, 2). 8 Find the equation of the normal to the curve y = x+2

5( x − 1)3 x+2

op

Pr

y

es

s

-C

7 Find the x-coordinate of the point on the curve y =

ity

9 The line 2 x − 2 y = 5 intersects the curve 2 x 2 y − x 2 − 26 y − 35 = 0 at three points.

C

PS

w

rs

a Find the x-coordinates of the points of intersection.

y op

R

ni

ev

ve

ie

b Find the gradient of the curve at each of the points of intersection.

w

ge

The derivative of e x

C

U

4.3 Derivatives of exponential functions

-C

-R

am

br

ev

id

ie

In Chapter 2, we learnt about the natural exponential function f( x ) = e x . This function has a very special property. If we use graphing software to draw the graph of f( x ) = e x together with the graph of its gradient function we find that the two curves are identical.

es

s

KEY POINT 4.3

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

Hence, we have the rule: d (e x ) = e x dx

Copyright Material - Review Only - Not for Redistribution

87

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

w

ge

An explanation of how this rule can be obtained is as follows.

ev ie

f(x) = ex

f(x + dx)

op

y

f(x)

C

y ie

ity

op

y

ve ni

C w

ge

ie -R

ev

id

br

s

d f( x ) [e ] = f ′( x ) × e f( x ) dx

Pr

es

am

= f ′( x ) × e f( x )

op y

-C

dy dy du = × dx du dx = e u × f ′( x )

C

ity

KEY POINT 4.4

In particular, d ax + b ] = a e ax + b [e dx

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ni ve rs

w

δx → 0 .

U

du = f ′( x ) dx

Using the chain rule:

f( x + δx) − f( x ) as ( x + δx ) − x

rs

w ie ev

dy = eu du

R

u = f( x )

where

f( x + δx) − f( x ) ( x + δx) − x

means the limit of

Pr

y

op

C

Consider the function y = e f( x ). y = eu

C op w

−1 → 1. δx

δx

-C

e

1.000 050

ev

1.000 500

The derivative of e f( x )

Let

lim

-R

1.005 017

br

1.051709

TIP

δx → 0

s

−1 δx

0.0001

es

δx

0.001

From the table, we can see that as δx → 0, dy ∴  = ex dx

ie

x

ni

U

0.01

ge

0.1

am

e

ev

x + dx

eδx − 1 for small values of δx. δx

id

w

Now consider

R

ev ie

x

ve rs ity

O

f(x + δx ) − f( x ) dy e x +δx − e x e x (eδx − 1) = lim = lim = lim δx → 0 δx → 0 δx δx dx δx → 0 (x + δx ) − x

δx

88

-R

y

Pr es s

-C

am br id

Consider the function f( x ) = e x and two points whose x-coordinates are x and x + δx where δx is a small increase in x.

Copyright Material - Review Only - Not for Redistribution

ve rs ity ge

C

U

ni

op

y

Chapter 4: Differentiation

ev ie

am br id

w

WORKED EXAMPLE 4.5

b xe −5 x

-C

a e2 x

differentiate index original function

d d d ( xe −5 x ) =   x   ×   (e −5 x )  +  e −5 x   ×   (x) dx dx dx =   x   ×  ( −5e −5 x )  +  e −5 x   ×  (1)

y

Product rule.

U

R

ni

C op

w ev ie

b

ge

=   −5xe −5 x   +  e −5 x

x 2   ×  3e3x −  e3x   ×  2 x x4 3x e (3x   −  2) = x3

Pr

es

s

Quotient rule.

89

op

ni

ev

WORKED EXAMPLE 4.6

y

ve

ie

w

rs

C

ity

op

y

=

-R

br

am

-C

c

d 3x d ( x2 ) (e ) −  e3x   ×   x2   ×    d x d x     =   x4

ev

id

ie

w

=  e −5 x (1 − 5x ) d  e 3x dx  x 2

e 3x x2

Pr es s

d 2x 2x 2x (e ) =  2   ×  e   = 2e   dx ւ ց

C

op

a

ve rs ity

y

Answer

c

-R

Differentiate with respect to x.

ie

br

ev

id

Answer

w

ge

C

U

R

The equation of a curve is y = e2 x − 9e x + 7 x . Find the exact value of the x-coordinate at each of the stationary points and determine the nature of each stationary point.

-R

am

y = e2 x − 9e x + 7 x

es Pr -R s es

-C

am

br

ev

ie

id g

w

e

C

U

op

ie ev

R

y

ni ve rs

dy = 0. dx

ity

op y

Stationary points occur when

w

C

Differentiate.

s

-C

dy = 2e2 x − 9e x + 7   dx d2 y = 4e2 x − 9e x dx 2

Copyright Material - Review Only - Not for Redistribution

ve rs ity

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

        2e2 x − 9e x + 7 = 0

am br id

ev ie

Factorise.

 (2e x − 7)(e x − 1) = 0

-R

(2e x − 7) = 0 or (e x − 1)  = 0 7  or e x = 1 2 7 x = ln    or  x = 0  2 d2 y When x = 0 , is negative ⇒ maximum point. dx 2 d2 y 7 When x = ln   , is positive ⇒ minimum point.  2  dx 2

y C op

U

w

ge

EXERCISE 4C

ie

R

ni

ev ie

w

C

ve rs ity

op

y

Pr es s

-C

ex =

d 3e −5 x

e

-C

s

−3

h 2 x + 3e

es

x

3e2 x + e −2 x 2

Pr

j

2

2(e3x − 1)

k

w

ve

ie

e2 x − 7

i

5 ex −

l

5(e x − 2 x )

1 e2 x

2

y

op

ni

3 The mass, m grams, of a radioactive substance remaining t years after a given time, is given by the formula m = 300e −0.00012t . Find the rate at which the mass is decreasing when t = 2000 .

w

ge

C

U

ev

R

M

ie Pr

-R

xe3x +

e6x 2

5xe −2 x

f

e −2 x x

i

x2ex − x ex + 2

ity

8 at the point where x = 0. 5 + e2 x

ni ve rs

5 Find the gradient of the curve y =

c

s

h

e6x x

es

ex − 1 ex + 2

-C

e

op y

y

6 Find the exact coordinates of the stationary point on the curve y = xe x .

e

C

U

op

7 The curve y = 2e2 x + e − x cuts the y-axis at the point P. Find the equation of the tangent to the curve at the point P and state the coordinates of the point where this tangent cuts the x-axis.

ie

id g

w

8 Find the exact coordinates of the stationary point on the curve y = (x − 4)e x and determine its nature.

-R

e2 x and determine its nature. x2

s es

am

br

ev

9 Find the exact coordinates of the stationary point on the curve y =

-C

C w

ev

id

br

b x 2 e 3x

am

xe x

d 2 x  e x g

ie

f

b Find the equation of the normal to the curve y = 1 − e2 − x at the point where y = 0.

a

ev

2e6 x

2 a Sketch the graph of the function y = 1 − e2 − x .

4 Differentiate with respect to x.

R

c

rs

C

90

ex

x 4e 2

ity

op

y

g

ev

b e −4 x

-R

e5 x

am

a

br

id

1 Differentiate with respect to x.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Chapter 4: Differentiation

w

ge

10 The equation of a curve is y = x 2 e − x .

am br id

ev ie

a Find the x-coordinates of the stationary points of the curve and determine the nature of these stationary points.

-R

b Show that the equation of the normal to the curve at the point where x = 1 is e2 x + ey = 1 + e2 .

e 2 x −1 . x

Pr es s

-C

11 Find the exact value of the x-coordinates of the points on the curve y = x 2 e −2 x at which

13 By writing 2 as e ln 2 , prove that

14 The equation of a curve is y = x(3x ) .

C op

y

Find the exact value of the gradient of the tangent to the curve at the point where x = 1.

ni

ev ie

w

PS

d (2 x ) = 2 x ln 2. dx

ve rs ity

C

op

P

y

12 Find the coordinates of the stationary point on the curve y =

ev

y

-R

am

br

id

ie

w

ge

U

R

4.4 Derivatives of natural logarithmic functions EXPLORE 4.2

4

s Pr

op

y

2

2

4

5

6

7

8

9 x

y op

ni

–2

U

–3

w

ge

–4

C

ev

3

rs

1

91

ve

ie

w

–1 O –1

ity

C

1

R

y = ln x

es

-C

3

br

ev

id

ie

Here is a sketch of the graph of y = ln x. Can you sketch a graph of its derivative (gradient function)? Discuss your sketch with your classmates.

Pr

op y

es

s

-C

-R

am

Now sketch the graphs using graphing software. Can you suggest a formula for the derivative of ln x ? (It might be helpful to look at the coordinates of some of the points on the graph of the derivative.)

The derivative of ln x

ity

In Chapter 2, you learnt that if y = ln x , then  e y = x. dy In Section 4.3, you learnt that if y = e x , then = ex . dx Using these two results we can find the rule for differentiating ln x .

op C

e

w

ev

ie

Rearrange and replace e y by x.

s

-R

br

id g

Differentiate both sides with respect to y.

es

-C

am

dx   e y = dy dy 1 = dx x

U

   y = ln x  e y = x

y

ni ve rs

C w ie ev

R

d2 y = 0. dx 2

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

am br id

KEY POINT 4.5

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

-C

-R

1 d (ln x ) = dx x

Pr es s

The derivative of ln f(x ) where

du = f ′( x ) dx

y

dy du dy = × dx du dx 1 = × f ′( x ) u f ′( x ) = f( x )

ni

Using the chain rule:

C op

dy 1 = du u

u = f( x )

br

ev

id

ie

w

ge

U

R

ev ie

w

C

Let y = ln u

ve rs ity

op

y

Consider the function y = ln f( x ).

s

-C

-R

am

d f ′( x ) [ln f( x )] = dx f( x )

Pr

y

es

KEY POINT 4.6

op

ni

C

U

R

WORKED EXAMPLE 4.7

y

ve

ev

ie

w

rs

C

ity

op

In particular, a d [ln( ax + b )] = dx ax + b

92

ie -R s

es Pr ity

‘Inside’

y -R s es

am

br

ev

ie

id g

w

e

C

U

op

ie ev

R

-C

ln 5 − x

‘Inside’ differentiated.

ni ve rs

C

d 4 [ln(4x − 3)] =   dx 4x − 3

w

b

c

ev

br

-C

d 2 (ln 2 x ) =   dx 2x 1 = x

op y

a

am

Answer

b ln(4x − 3)

id

a ln 2 x

w

ge

Differentiate with respect to x.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

w

ge

am br id

c Method 1:

C

U

ni

op

y

Chapter 4: Differentiation

1

Pr es s

‘Inside’

Method 2: using the rules of logarithms before differentiating.

ve rs ity

1  d d   ln 5 − x  =  ln(5 − x ) 2    dx dx   d 1          = ln(5 − x )  dx  2  1 d [ln(5 − x )] = × 2 dx 1 −1 = × 2 5−x 1 =− 2(5 − x )

C op

y

Use ln a m = m ln a

U

R

w

ge

es Pr

y

op

rs

b

ni

op

y

ln 2 x x3

d d d (2 x 4 ln 5x ) =  2 x 4   ×   (ln 5x )  + ln 5x   ×   (2 x 4 ) dx dx dx 5 =  2 x 4   ×   + ln 5x   ×  8x 3 5x =  2 x 3   + 8x 3 ln 5x

U

Product rule.

w ie ev Quotient rule.

es

Pr

ity

op

y

ni ve rs

C

U

w

e

ev

ie

id g

es

s

-R

br am -C

ev

ie

w

C

op y

b

s

-C

d d 3 3 d  ln 2 x  x   ×   dx ( ln 2 x ) − ln 2 x   ×   dx ( x )  =  dx  x 3  ( x 3 )2 2 x 3   ×     −  ln 2 x   ×  3x 2 2 x = x6 2 2 x − 3x ln 2 x = x6 1 − 3 ln 2 x = x4

-R

am

br

id

ge

C

R

ev

Answer

93

ity

a 2 x 4 ln 5x

ve

C w ie

Differentiate with respect to x.

a

‘Inside’

s

-C

-R

am

br

ev

id

ie

‘Inside’ differentiated.

WORKED EXAMPLE 4.8

R

‘Inside’ differentiated.

ni

ev ie

w

C

op

y

-C

-R

− 1 (5 − x ) 2 ( −1) d  ln 5 − x  = 2 dx  5−x 1 =− 2(5 − x )

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

am br id

EXERCISE 4D

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

a

ln 3x

b ln 7 x

d 5 + ln( x 2 + 1)

ln(ln x )

h

2 3x + ln    x

k

ln 2 − x

(

)2

ln(2 x + 1)

f

ln x − 3

i

2  5x + ln   1 − 2x 

l

ln(5x + ln x )

y C op

c

x ln(2 x + 1)

f

ln 5x x

i

ln(2 x + 1) 4x − 1

w

ln(3x − 2) x

ie

h

ev

x ln(ln x )

-R

2 ln x

am

g

e

br

d 3x ln 2 x

b 2 x 3 ln x

ge

x ln x

id

a

U

3 Differentiate with respect to x.

R

c

2 The answers to question 1 parts a and b are the same. Why is this the case? How many different ways can you justify this?

ni

ev ie

w

PS

j

ln( x + 3)5

ve rs ity

C

op

y

g

ln(2 x − 1)2

Pr es s

-C

e

-R

1 Differentiate with respect to x.

es

s

-C

4 a Sketch the graph of the function y = ln(2 x − 3).

op

Pr

y

b Find the gradient of the curve y = ln(2 x − 3) at the point where x = 5.

94

ity

6 A curve has equation y = x 2 ln 5x. dy d2 y and at the point where x = 2. Find the value of dx dx 2

y op

ni

ev

ve

ie

w

rs

C

5 Find the gradient of the curve y = e2 x − 5 ln(2 x + 1) at the point where x = 0.

w

ge

C

U

R

7 The equation of a curve is y = x 2 ln x. Find the exact coordinates of the stationary point on this curve and determine whether it is a maximum or a minimum point. ln x . Find the exact coordinates of the stationary point on this curve and x determine whether it is a maximum or a minimum point.

-R

am

br

ev

id

ie

8 The equation of a curve is y =

-C

9 Find the equation of the tangent to the curve y = ln(5x − 4) at the point where x = 1.

Pr

ity

 ( x + 2) (2 x − 1)  ln   x( x + 5)  

c

e y = ( x + 1)( x − 5)

ev

ie

w

C

U

e

id g

-R

br

am

TIP

i

y

  8 ln   2  ( x + 1) ( x − 2) 

dy , in terms of x, for each of the following. dx e y = 2x2 − 1 b e y = 3x 3 + 2 x

11 Find a

h

op

  3−x ln    ( x + 4) ( x − 1) 

ni ve rs

g

es

s

Take the natural logarithm of both sides of the equation before differentiating.

-C

R

ev

ie

w

C

op y

es

s

10 Use the laws of logarithms to help differentiate these expressions with respect to x. 1  a ln 5x − 1 b ln  c ln [ x( x + 1)5 ]  3x + 2  1 − 3x  2x + 3   x( x − 2)  d ln  e ln  f ln   x −1   x2   x + 4 

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C -R

am br id

dy when x = 1. dx

-C

Find the value of

1 y (2 x − 3) e + 4 . 5

w

12 A curve has equation x =

PS

ev ie

ge

U

ni

op

y

Chapter 4: Differentiation

y

Pr es s

4.5 Derivatives of trigonometric functions

ve rs ity

w

C

op

EXPLORE 4.3

Graphing software has been used to draw the graphs of y = sin x and y = cos x for

π 2

y = sin x –1

y π

2π x

3π 2

-R

–1

gradient function

w

O

2π x

3π 2

ie

π

y = cos x

1

gradient function

am

π 2

br

O

id

ge

1

y

ev

U

R

y

C op

ni

ev ie

0 ø x ø 2 π together with their gradient (derived) functions.

95

ity

Can you suggest how to complete the following formulae? d d (sin x ) = (cos x ) = dx dx

y C

U

The derivative of sin x

R

op

ni

ev

ve

ie

w

rs

C

op

Pr

y

es

s

-C

Discuss with your classmates why, for the function y = sin x , you would expect the graph of the gradient function to have this shape. Do the same for the graph of y = cos x .

br am

f(x) = sin x

s Pr

x

e

id g

δx → 0

y

-R s es

am

op

As δx → 0, cos δx → 1 and sin δx → δx.

br

= cos x

C

sin x cos δx + cos x sin δx − sin x δx

Expand sin( x + δx )

w

= lim

δx → 0

ie

sin( x + δx ) − sin x δx

U

= lim

ev

f( x + δx ) − f( x ) dy = lim dx δx → 0 ( x + δx ) − x

es

x + δx

ity

x

ni ve rs

O

-C

R

ev

ie

w

C

op y

-C

f(x)

-R

y f(x + δx)

ev

id

ie

w

ge

Consider the function f( x ) = sin x, where x is measured in radians, and two points whose x-coordinates are x and x + δx where δx is a small increase in x.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

am br id

KEY POINT 4.7

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

-R

d (sin x ) = cos x   dx Similarly, we can show that

op

y

Pr es s

-C

d (cos x ) = − sin x dx

w ie ev

cos2 x + sin2 x   cos2 x

-R

am

Use cos2 x + sin2 x = 1 .

-C



Pr

=  sec 2 x

y

ve

d (tan x ) = sec 2 x dx

ie

w

ge

id

WORKED EXAMPLE 4.9

C

U

R

ni

op

ie

w

rs

KEY POINT 4.8

ev

1 = sec x. cos x

s

es

Use

C

96

1  cos2 x

ity

op

y



2 sin x

x cos x

c

Pr

d d d ( x cos x ) =   x   ×   (cos x )  + cos x   ×   (x) dx dx dx = − x sin x + cos x

ni ve rs

-R s

-C

am

br

ev

ie

id g

w

e

C

U

op

y

Product rule.

es

C w ie ev

ity

op y

d d (2 sin x ) = 2 (sin x ) dx dx   = 2 cos x

R

b

d

es

Answer a

tan x x2

s

-C

b

-R

am

br

ev

Differentiate with respect to x. a

y

C op

U

ge

cos x   × cos x − sin x   × ( −sin x ) cos2 x

br



d d (sin x ) − sin x   ×   (cos x ) dx dx 2 (cos x )

id



cos x   ×  

Use the quotient rule.

ni

d d  sin x  (tan x ) = dx dx  cos x 

R

ev ie

w

C

ve rs ity

We can find the derivative of tan x using these two results together with the quotient rule.

Copyright Material - Review Only - Not for Redistribution

(3 + 2 sin x )5

ve rs ity d [(3 + 2 sin x )5 ] = 5(3 + 2 sin x )4   ×  2 cos x dx = 10 cos x(3 + 2 sin x )4

Chain rule.

ni

ie

ge

id

ev -R

du =a dx

es

s

dy dy du = × dx du dx = cos u   ×   a

op C w

ge

Similarly, it can be shown that:

s

-C

-R

am

br

ev

id

ie

d [cos( ax + b )] = − a sin( ax + b ) dx d   [tan( ax + b )] = a sec 2 ( ax + b ) dx

 

Pr

op y

es

WORKED EXAMPLE 4.10

c

π cos  2 x −   4 x2

y op w

e

ev

ie

id g

es

s

-R

br am

d

C

U

d d (2 sin 3x ) = 2 (sin 3x ) dx dx    = 2 × cos 3x × (3)    = 6 cos 3x

-C

ie ev

Answer a

4x tan 2 x

ni ve rs

b

ity

2 sin 3x

w

C

Differentiate with respect to x. a

It is important to remember that, in calculus, all angles are measured in radians unless a question tells you otherwise.

y

ve

ni

R

d [sin( ax + b )] = a cos( ax + b ) dx

TIP

U

ev

ie

KEY POINT 4.9  

97

rs

w

C

= a cos( ax + b )

Pr

op

y

Using the chain rule:

u = ax + b

ity

-C

am

dy = cos u du

where

br

Let   y = sin u

w

U

R

Derivatives of sin( ax + b ),   cos( ax + b )   and  tan( ax + b ) Consider the function y = sin( ax + b ) where x is measured in radians.

R

C op

y

ve rs ity

w

C

op

y

Pr es s

-C



d

Quotient rule.

-R

x 2   × sec 2 x − tan x   ×  2 x x4 x sec 2 x − 2 tan x            =   x3

ev ie

am br id

d d ( x2 ) x2   ×   (tan x ) −  tan x   ×   d   tan x  x x d d    =  dx  x 2  x4

c

ev ie

w

ge

C

U

ni

op

y

Chapter 4: Differentiation

Copyright Material - Review Only - Not for Redistribution

(3 − 2 cos 5x )4

ve rs ity

Pr es s

π π d    2  cos  2 x − 4     −  cos  2 x − 4    ×   dx [ x ] ( x 2 )2

ve rs ity

C

y

ev

id

br

-R

am

s

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op

Pr

y

es

EXERCISE 4E

ity

a

2 + sin x

d

3sin 2x

g

tan( 3x + 2 )

ve

rs

b 2 sin x + 3 cos x e

4 tan 5x

f

π 3 cos 4 x + 2 tan2  2 x −   4

s es

x cos x 1 sin3 2 x

x 2 tan x

c

d x cos3 2 x

tan x x 3x sin 2 x

g k

ni ve rs

j

sin2 x − 2 cos x

h l

sin x 2 + cos x sin x + cos x sin x − cos x

e x sin 2 x

g

e x (2 cos x − sin x )

h x 3e cos x

j

x ln(sin x )

k

cos 2 x e2 x+1

l

es

s

-R

br

ev

5 Find the gradient of the curve y = 3 sin 2 x − 5 tan x at the point where x = 0.

am

d e( sin x − cos x )

f

C

ln(cos x )

etan 3x

w

i

c

ie

e x cos x

e

e

id g

esin x

U

b e cos 2 x

a

y

4 Differentiate with respect to x.

-C

C w ie ev

i

f

5x cos 3x

Pr

5 cos 3x sin x 3x − 1

e

b

ity

op y

-C

x sin x

2 cos 3x − sin 2 x

C -R

am

br

e

3 Differentiate with respect to x. a

f

c

w

( 3 − cos x )4

5 cos2 3x π 2 sin3  2 x +   6

ie

d

b

id

sin3 x

2 cos x − tan x π 2 cos  3x −   6

ev

ge

2 Differentiate with respect to x. a

c

i

op

ni

π h sin  2 x +   3

U

R

ev

ie

w

C

1 Differentiate with respect to x.

op

98

Chain rule.

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4 3 d  3 − 2 cos 5x )    = 4 ( 3 − 2 cos 5x ) × 10 sin 5x (  dx     = 40 sin 5x(3 − 2 cos 5x )3

d

R

C op w

ge

U

R

ni

ev ie

w

π π −2 x 2 sin  2 x −  − 2 x cos  2 x −    4 4 =  x4 π π −2 x sin  2 x −  − 2 cos  2 x −       4 4     =   3 x

Quotient rule.

y

op

y

c

Product rule.

-R

am br id

-C

= 8x sec 2 2 x   + 4 tan 2 x

d π   cos  2 x −   x 2   ×     x d d  4   =  2 x dx    

w

d d d (4x tan 2 x ) = 4x   ×   (tan 2 x ) + tan 2 x   ×   (4x ) dx dx dx = 4x   ×  sec 2 2 x  (2) + tan 2 x   ×  (4)

b

ev ie

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Copyright Material - Review Only - Not for Redistribution

x sin 2 x e2 x

ve rs ity

C

U

ni

op

y

Chapter 4: Differentiation

am br id

ev ie

w

ge

π 6 Find the exact value of the gradient of the curve y = 2 sin 3x − 4 cos x at the point  ,  −2  . 3 

-R

Pr es s

-C

C op

y

ve rs ity

ni

π π 10 Prove that the normal to the curve y = x sin x at the point P  ,   intersects the x-axis at the point ( π, 0).  2 2

ge

P

U

R

ev ie

w

C

op

y

P

7 Given that y = sin2 x for 0 ø x ø π , find the exact values of the x-coordinates of the points on the curve 3 . where the gradient is 2 5 8 Prove that the gradient of the curve y = is always positive. 2 − tan x d 1 , find (sec x ). 9 a By writing sec x as x d cos x d 1 , find b By writing cosec x as (cosec x ). dx sin x d cos x , find (cot x ). c By writing cot x as dx sin x

am

br

ev

id

ie

w

11 The equation of a curve is y = 5 sin 3x − 2 cos x . Find the equation of the tangent to the curve at the point  π ,  −1  . Give the answer in the form y = mx + c , where the values of m and c are correct to 3 significant figures. 3 

s

-C

-R

12 A curve has equation y = 3 cos 2 x + 4 sin 2 x + 1 for 0 ø x ø π . Find the x-coordinates of the stationary points of the curve, giving your answer correct to 3 significant figures.

es

π . Find the exact value of the x-coordinate of the stationary 2 point of the curve and determine the nature of this stationary point.

Pr

ity

ve

point of this curve.

ie

sin 2 x π for 0 ø x ø . Find the exact value for the x-coordinate of the stationary 2x e 2

rs

14 A curve has equation y =

w

C

op

y

13 A curve has equation y = e x cos x for 0 ø x ø

y

ev

π e 3x for 0 , x , . Find the exact value for the x-coordinate of the stationary sin 3x 2 point of this curve and determine the nature of this point.

w

ge

C

U

R

ni

op

15 A curve has equation y =

16 A curve has equation y = sin 2 x − x for 0 ø x ø 2π . Find the x-coordinates of the stationary points of the curve, and determine the nature of these stationary points.

PS

17 A curve has equation y = tan x cos 2 x for 0 ø x ,

es

op y

s

-C

-R

am

π . Find the x-coordinate of the stationary point on the 2 curve giving your answer correct to 3 significant figures.

ity

All the functions that we have differentiated so far have been of the form y = f( x ) . These are called explicit functions as y is given explicitly in terms of x. However, many functions cannot be expressed in this form, for example x 3 + 5xy + y3 = 16 .

ni ve rs

C

br

ev

ie

id g

w

e

C

U

op

y

When a function is given as an equation connecting x and y, where y is not the subject, it is called an implicit function. dv du   dy = dy × du  and the product rule  d ( uv ) = u +v The chain rule   are used   dx du dx  dx dx dx  extensively to differentiate implicit functions.

-R s es

am

This is illustrated in the following Worked examples.

-C

w ie ev

Pr

4.6 Implicit differentiation

R

ev

br

id

ie

PS

Copyright Material - Review Only - Not for Redistribution

99

ve rs ity

ev ie

w

ge

Differentiate each of these expressions with respect to x.

c x 3 + 5xy

-R

b 4x 2 y5

-C

a y3

Pr es s

Answer

d d dy ( y3 ) = ( y3 ) × dx dy dx dy = 3 y2 dx

Chain rule.

ve rs ity

y

d d d (4x 2 y5 ) = 4x 2 ( y5 ) + y5 (4x 2 ) dx dx dx d  = 4x 2 ( y5 ) + 8xy5   dx dy  = 20 x 2 y 4 + 8xy5   dx

Product rule. Chain rule:

ie -R

Use the product rule for

es

Pr

ity rs

y

ve

op

dy for the curve x 3 + y3 = 4xy . Hence find the gradient of the curve at the point (2, 2). dx

w

ge

Answer

C

U

ni

ev

R

Find

x 3 + y3 = 4xy

d (4xy ) dx d d ( y) + y (4x ) = 4x dx dx dy = 4x + 4y dx

ev

id

ie

Differentiate with respect to x.

s es Pr

y C w -R s es

-C

am

br

ev

ie

id g

When x = 2 and y = 2,

op

4 y − 3x 2 3 y 2 − 4x

Rearrange.

U

=

Factorise.

ni ve rs

= 4 y − 3x 2

Rearrange terms.

ity

= 4 y − 3x 2

-R

=

e

op y

-C

am

br

d d ( x3 ) + ( y3 ) dx dx d dy 3x 2 + ( y3 ) × dy dx dy 3x 2 + 3 y2 dx dy 2 dy − 4x 3y dx dx dy (3 y2 − 4x ) dx dy dx

C w ie ev

R

d (5xy ). dx

s

am

-C y op

WORKED EXAMPLE 4.12

ie

w

C

100

ev

id

br

d d d ( x 3 + 5xy ) = ( x3 ) + (5xy ) dx dx dx d d ( y) + y (5x ) = 3x 2 +  5x dx dx dy = 3x 2 +  5x + 5y dx

c

dy d ( y5 ) = 5 y 4 . dx dx

w

ge

U

R

ni

b

C op

ev ie

w

C

op

a

y

am br id

WORKED EXAMPLE 4.11

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Copyright Material - Review Only - Not for Redistribution

ve rs ity

w

ge

C

U

ni

op

y

Chapter 4: Differentiation

am br id

ev ie

dy 4(2) − 3(2)2 = = −1 dx 3(2)2 − 4(2)

Pr es s

-C

-R

Gradient of the curve at the point (2, 2) is −1.

op

y

WORKED EXAMPLE 4.13

ve rs ity

w

C

The equation of a curve is 3x 2 + 2 xy + y2 = 6. Find the coordinates of the two stationary points on the curve. Answer

y

d d d (3x 2 ) + (2 xy ) + y2 dx dx dx d d d 6x + 2 x ( y) + y y2 ( 2x ) + dx dx dx dy dy 6x + 2 x + 2y + 2y dx dx dy dy x +y dx dx dy (x + y) dx dy dx

Differentiate with respect to x.

( ) = ddx (6)

Use product rule for

( )=0

Use chain rule for

w

ie

=0

ev

Divide by 2 and rearrange.

s

Pr

− y − 3x x+y

Factorise. Rearrange.

es

= − y − 3x

-R

= − y − 3x

101

ity

=

dy = 0. dx − y − 3x = 0 y = −3x

op

y

ve

ni

ev

C ie ev

br

id

6x 2 = 6 x = ±1

w

ge

U

R

Substituting y = −3x into the equation of the curve gives: 3x 2 + 2 x( −3x ) + ( −3x )2 = 6

es

s

-C

-R

am

The stationary points are ( −1, 3) and (1,  −3).

Pr

op y

EXERCISE 4F

ity

c

5x 2 + ln y

6x 2 y 3

f

y2 + xy

h x sin y + y cos x

i

x 3 ln y

k 5 y + e x sin y

l

2 xe cos y

x 3 − 7 xy + y3

j

x cos 2 y

-R s es

am

br

ev

ie

id g

w

e

U

g

op

e

C

d 2 + sin y

y

b x 3 + 2 y2

ni ve rs

y5

-C

R

ev

ie

w

C

1 Differentiate each expression with respect to x. a

d (2 xy ). dx

d ( y2 ). dx

rs

Stationary points occur when

ie

w

C

op

y

-C

am

br

id

ge

U

R

ni

C op

ev ie

3x 2 + 2 xy + y2 = 6

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

b

x 2 y + y 2 = 5x

c

2 x 2 + 5xy + y2 = 8

d

x ln y = 2 x + 5

e

2e x y + e2 x y3 = 10

f

ln( xy ) = 4 − y2

g

xy3 = 2 ln y

h

ln x − 2 ln y + 5xy = 3

ev ie

Pr es s

-R

am br id

-C

w

ge

a

dy for each of these functions. dx x 3 + 2 xy + y3 = 10

2 Find

op

y

3 Find the gradient of the curve x 2 + 3xy − 5 y + y3 = 22 at the point (1, 3).

ve rs ity

5 A curve has equation 2 x 2 + 3 y2 − 2 x + 4 y = 4 . Find the equation of the tangent to the curve at the point (1,  −2) .

y

ev ie

w

C

4 Find the gradient of the curve 2 x 3 − 4xy + y3 = 16 at the point where the curve crosses the x-axis.

ge

U

R

ni

C op

6 The equation of a curve is 4x 2 y + 8 ln x + 2 ln y = 4. Find the equation of the normal to the curve at the point (1, 1).

id

ie

w

7 The equation of a curve is 5x 2 + 2 xy + 2 y2 = 45.

es

8 The equation of a curve is y2 − 4xy − x 2 = 20.

a Find the coordinates of the two points on the curve where x = 4.

Pr

y

Find the equation of the tangent to the curve at the other point.

ve

c

ity

b Show that at one of these points the tangent to the curve is parallel to the x-axis.

rs

op

ie

w

C

102

s

-C

b Hence find the coordinates of the two points.

-R

am

br

ev

a Given that there are two points on the curve where the tangent is parallel to the x-axis, show by differentiation that, at these points, y = −5x.

y op

ni

ev

9 The equation of a curve is y3 − 12 xy + 16 = 0.

C

U

R

a Show that the curve has no stationary points.

w

ge

b Find the coordinates of the point on the curve where the tangent is parallel to the y-axis.

br

ev

id

ie

10 Find the gradient of the curve 5e x y2 + 2e x y = 88 at the point (0, 4).

s

-C

-R

am

11 The equation of a curve is x 2 − 4x + 6 y + 2 y2 = 12 . Find the coordinates of the two points on the curve at 4 which the gradient is . 3

Pr

ity

ni ve rs

ev

ie

id g

w

e

C

U

op

y

d2 y determine the nature of each of 14 Find the stationary points on the curve x 2 − xy + y2 = 48. By finding dx 2 these stationary points.

es

s

-R

br

PS

13 The equation of a curve is y = x x . Find the exact value of the x-coordinate of the stationary point on this curve.

am

R

ev

ie

w

PS

-C

C

op y

es

12 The equation of a curve is 2 x + y ln x = 4 y . Find the equation of the tangent to the curve at the point with 1 coordinates  1,   .  2

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Chapter 4: Differentiation

w

ge

4.7 Parametric differentiation

am br id

ev ie

Sometimes variables x and y are given as a function of a third variable t.

-R

For example, x = 1 + 4 sin t and y = 2 cos t.

Pr es s

-C

The variable t is called a parameter and the two equations are called the parametric equations of the curve.

x

1

3.83

5

y

2

1.41

0

3 π 4

π

5 π 4

3 π 2

7 π 4



3.83

1

−1.83

−3

−1.83

1

−1.41

−2

−1.41

0

1.41

ie 3

4

5

6 x

103

rs

ity

–2 –3

2

s

1

es

–1 O –1

Pr

op

y

–2

C

y

ve ni

op

w ie

-R

am -C

1

EXPLORE 4.4

ev

ev

br

id

y 3 2

–3

2

w

ge

Plotting the coordinates on a grid gives the following graph.

–4

y

1 π 2

C op

1 π 4

ve rs ity

0

ni

t

U

R

ev ie

w

C

op

y

We can find the curve given by the parametric equations x = 1 + 4 sin t, y = 2 cos t for 0 ø t ø 2π by finding the values of x and y for particular values of t.

w

ge

a x = t 3,

id

ie

y = t2

-R

am

br

ev

y = t3 − t 1 1 x=t+ , y=t− t t

b x = t2 , c

C

U

R

1 Use tables of values for t, x and y to sketch each of the following curves. Use graphing software to check your answers.

y = sin 2t

Pr

op y

a x = sin t ,

es

s

-C

2 Use graphing software to draw the curves given by these parametric equations.

ity

x = sin 4t, y = sin 3t

ni ve rs

c

C

U

op

y

3 The curves in question 2 are called Lissajous curves. Use the internet to find out more about these curves. (A Lissajous curve is shown at the start of this chapter.)

-R s es

am

br

ev

ie

id g

w

e

When a curve is given in parametric form, in terms of the parameter t, we can use the dy chain rule to find in terms of t. dx

-C

R

ev

ie

w

C

b x = sin 2t , y = sin 3t

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

w

ge

am br id

WORKED EXAMPLE 4.14

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ve rs ity

C op

ni

w ie ev -R

y w

ge

C

U

WORKED EXAMPLE 4.15

R

op

ni

ev

ve

ie

w

rs

C

104

ity

op

Pr

y

es

-C

1 b When t = 2, gradient = − ,  x = 4 and y = 1. 2 Using y − y1 = m( x − x1 ) 1   y − 1 = − ( x − 4) 2 1    y = − x + 3 2

s

am

br

 =

U

 =

Chain rule.

ge

dy = dx

⇒     

id

ev ie

y=

R

dx = 2t dt dy = −2 5 − 2t    ⇒      dt dy dt  ×  dt dx 1 −2  ×   2t 1 − t

x = t2  

a

w

C

op

Answer

y

y

Pr es s

-C

-R

The parametric equations of a curve are x = t 2 , y = 5 − 2t. dy in terms of the parameter t. a Find dx b Find the equation of the tangent to the curve at the point where t = 2.

Pr

dx = 4 sin θ cos θ dθ dy 4 ⇒      = 4 sec 2 θ = dθ cos2 θ

dy dy dθ =  ×  dx dθ dx 1 4 = 2  ×  4 sin θ cos θ cos θ 1      = sin θ cos3 θ

-R s es

am

br

ev

ie

id g

w

e

C

U

op

Chain rule.

y

y = 4 tan θ   

ity

x = 1 + 2 sin2 θ   ⇒     

-C

R

ev

ie

w

C

a

ni ve rs

op y

Answer

es

s

-C

-R

am

br

ev

id

ie

π 3π The parametric equations of a curve are x = 1 + 2 sin2 θ , y = 4 tan θ for , θ , . 2 2 dy a Find in terms of the parameter θ . dx b Find the coordinates of the point on the curve where the tangent is parallel to the y-axis.

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ve rs ity ge

C

U

ni

op

y

Chapter 4: Differentiation

am br id

ev ie

w

b Tangent is parallel to the y-axis when:   sin θ cos3 θ = 0

 sin θ = 0   ⇒   θ = π    ⇒   (1, 0)

Pr es s

EXERCISE 4G

y

ve rs ity

w ev ie

π 3π . ,θ , 2 2

∴ The point where the tangent is parallel to the y-axis is (1, 0).

C

op

y

-C

There are no solutions to cos θ = 0 in the range

-R

 sin θ = 0  or  cos θ = 0 

b

x = 2 + sin 2θ , y = 4θ + 2 cos 2θ

c

x = 2θ − sin 2θ , y = 2 − cos 2θ

d

x = 3tan θ , y = 2 sin 2θ

x = 1 + tan θ , y = cos θ

f

x = cos 2θ − cos θ , y = sin2 θ

x = 1 + 2 sin2 θ , y = 4 tan θ

h

ev

s

es

x = e 2t , y = t 3 et + 1 5 x = ln(1 − t ), y = t

x = 2 ln(t + 3) , y = 4et

j

Pr

k

x = 2 + e − t , y = et − e − t

l

x = 1 + t , y = 2 ln t

105

ity

op

y

i

-R

br

am

-C

g

ie

x = 2t 3, y = t 2 − 5

id

a

e

2 The parametric equations of a curve are x = 3t, y = t 3 + 4t 2 − 3t. Find the two values of t for which the curve has gradient 0.

y

ve

ie

w

rs

C

w

ge

U

R

ni

C op

dy 1 For each of the following parametric equations, find in terms of the given dx parameter.

op

-R

am

br

ev

id

ie

w

ge

C

U

R

ni

ev

3 The parametric equations of a curve are x = 2 sin θ , y = 1 − 3 cos 2θ . Find the π exact gradient of the curve at the point where θ = . 3 4 4 The parametric equations of a curve are x = 2 + ln(t − 1), y = t +   for t . 1. t Find the coordinates of the only point on the curve at which the gradient is equal to 0.

s

es

op y

-C

5 The parametric equations of a curve are x = e2t , y = 1 + 2tet .  Find the equation of the normal to the curve at the point where t = 0.

Pr

ity

dy t (2 − t ) = . dx 2e3t

ni ve rs

a Show that

-R s es

am

br

ev

ie

id g

w

e

C

U

op

y

b Show that the tangent to the curve at the point (1,  −1) is parallel to the x-axis and find the exact coordinates of the other point on the curve at which the tangent is parallel to the x-axis.

-C

R

ev

ie

w

C

6 The parametric equations of a curve are x = e2t , y = t 2 e −t − 1.

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ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

dy = 4 cos2 θ (2 cos2 θ − 1). dx b Hence, find the coordinates of the stationary point.

Pr es s

-C

-R

a Show that

ev ie

am br id

w

ge

7 A curve is defined by the parametric equations x = tan θ , y = 2 sin 2θ , for π 0 ,θ , . 2

9 , for t . 0. t

dy t2 − 9 = 2 . dx t + 4t b The curve has one stationary point. Find the y-coordinate of this point and determine whether it is a maximum or a minimum point. a Show that

ve rs ity

ev ie

w

C

op

y

8 The parametric equations of a curve are x = t + 4 ln t, y = t +

ge

U

11 The parametric equations of a curve are x = ln(tan t ), y = 2 sin 2t, for π 0,t , . 2

ni

WEB LINK

y

ev

ve

ie

w

PS

rs

C

106

ity

op

Pr

y

es

s

-C

-R

am

br

ev

id

ie

w

10 The parametric equations of a curve are x = 2 sin θ + cos 2θ , y = 1 + cos 2θ , for π 0 øθ ø . 2 2 sin θ dy . a Show that = dx 2 sin θ − 1 b Find the coordinates of the point on the curve where the tangent is parallel to the x-axis. 3 3 c Show that the tangent to the curve at the point  ,   is parallel to the  2 2 y-axis.

op

R

ni

C op

y

9 The parametric equations of a curve are x = 1 + 2 sin2 θ , y = 1 + 2 tan θ .  Find π the equation of the normal to the curve at the point where θ = . 4

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

am

br

ev

id

ie

w

ge

C

U

R

dy = sin 4t . dx b Hence show that at the point where x = 0 the tangent is parallel to the x-axis.

a Show that

Copyright Material - Review Only - Not for Redistribution

Try the Parametric points resource on the Underground Mathematics website.

ve rs ity am br id

Product rule

-R

dv du d +v ( uv ) = u dx dx dx

op

du dv  −  u  dx dx v2

d (e x ) = e x dx

ev ie

w



ve rs ity

C

Exponential functions d  e ax + b  = ae ax + b dx 

ni

1 d (ln x ) = dx x

a d [ln( ax + b )] = dx ax + b

f ′( x ) d [ln(f( x ))] = f( x ) dx

w

ge



U

R

Logarithmic functions

d f( x ) [e ] = f ′( x ) × e f( x ) dx

y



y



d  u   = dx  v 

Pr es s

-C

Quotient rule

d (cos x ) = − sin x dx

d [cos( ax + b )] = − a sin( ax + b ) dx

d (tan x ) = sec 2 x dx

d [tan( ax + b )] = a sec 2 ( ax + b ) dx

ie



107

y op y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

am

br

ev

id

ie

w

ge

C

U

R

ni

ev

ve

ie

w

rs

C

ity

op

Pr

y

es

s

-C

-R

br

d (sin x ) = cos x dx

am

d [sin( ax + b )] = a cos( ax + b ) dx



ev

id

Trigonometric functions



C op



w

ev ie

ge

Checklist of learning and understanding

C

U

ni

op

y

Chapter 4: Differentiation

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

The parametric equations of a curve are

-R

9 x = 1 + ln(t − 2), y = t + , for t . 2. t 2 dy (t − 9)(t − 2) . = Show that dx t2 Find the coordinates of the only point on the curve at which the gradient is equal to 0.

i

y

[2]

iii Find the exact coordinates of the other point on the curve at which the tangent is parallel to the x-axis.

[2]

ie

w

The parametric equations of a curve are x = e3t , y = t 2 et + 3. dy t (t + 2) i Show that = . dx 3e2t ii Show that the tangent to the curve at the point (1, 3) is parallel to the x-axis.

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q7 November 2011

[3] [3]

y

ev

ie

ii

y = 3sin x + tan 2 x 6 y= 1 + e2 x

rs

w

i

ity

Find the gradient of each of the following curves at the point for which x = 0.

ve

C

4

Pr

y op

108

The equation of a curve is 2 x 2 + 3xy + y2 = 3.

C

U

w

Find the equation of the tangent to the curve at the point (2,  −1), giving your answer in the form ax + by + c = 0, where a, b and c are integers.

br

es

s

-C

The equation of a curve is y3 + 4xy = 16 . dy 4y =− 2 . i Show that dx 3 y + 4x ii Show that the curve has no stationary points.

Pr

[2]

ity

C

[4]

ni ve rs

iii Find the coordinates of the point on the curve where the tangent is parallel to the y-axis.

w

[4]

op

The equation of a curve is y = 6 sin x − 2 cos 2 x.

w

ie

[5]

ev

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q3 June 2015

es

s

-R

br

id g

e

C

U

1 Find the equation of the tangent to the curve at the point  π, 2  . Give the answer in the 6  form y = mx + c , where the values of m and c are correct to 3 significant figures.

am

7

y

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q7 June 2015

-C

ie ev

[4]

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q7 June 2014

op y

6

[6]

-R

Show that the curve has no stationary points.

am

ii

ev

id

ie

5

ge

R

ni

op

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q2 June 2014

i

R

[4]

es

s

-C

-R

am

br

ev

id

3

[3]

C op

ge

U

R

ni

ev ie

w

2

[3]

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q4 November 2010 dy when x = 4 in each of the following cases: Find the value of dx [4] i y = x ln( x − 3), x −1 . [3] ii y = x +1 Cambridge International AS & A Level Mathematics 9709 Paper 21 Q5 June 2011

ve rs ity

op

y

ii

Pr es s

-C

1

C

ev ie

w

END-OF-CHAPTER REVIEW EXERCISE 4

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ve rs ity y

-R

A

D

B x

C

ve rs ity

op

y

O

w ev ie

Pr es s

-C

8

ev ie

am br id

w

ge

C

U

ni

op

y

Chapter 4: Differentiation

C

Pr

es

The equation of a curve is 3x 2 + 4xy + y2 = 24. Find the equation of the normal to the curve at the point (1, 3), giving your answer in the form ax + by + c = 0 where a, b and c are integers.

ity

y

rs

C

10

op

y

ve ni

ev

id

ie

w

ge

C

U

R

Pr

1− x dy , obtain an expression for in terms of x. Hence show that the gradient of 1+ x dx

ity

By first differentiating

the normal to the curve at the point ( x,  y ) is (1 + x )  (1 − x 2 ) .

[5]

op C

w ev

ie

id g

es

s

-R

br am -C

[4]

Cambridge International A Level Mathematics 9709 Paper 31 Q9 June 2010

e

U

y

The gradient of the normal to the curve has its maximum value at the point P shown in the diagram. Find, by differentiation, the x-coordinate of P.

R

ii

ni ve rs

ie

w

C

op y

i

 1 − x   .  1+ x 

1 x

-R

O

es

-C

–1

s

am

br

ev

P

The diagram shows the curve y =

ev

[8]

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q6 November 2016

ie

w

ie

s

-C

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q7 November 2015

op

y

9

[3]

-R

am

br

iii Find the value of the gradient of the curve at B.

ev

id

w

ge

U

R

ni

C op

y

The parametric equations of a curve are x = 6 sin2 t, y = 2 sin 2t + 3 cos 2t, for 0 ø t , π. The curve crosses the x-axis at the points B and D and the stationary points are A and C , as shown in the diagram. dy 2 i Show that = cot 2t − 1. [5] dx 3 [3] ii Find the values of t at A and C , giving each answer correct to 3 decimal places.

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109

ve rs ity

am br id

ev ie

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

11 The parametric equations of a curve are x = 3(1 + sin2 t ), y = 2 cos3 t.

-R

dy in terms of t, simplifying your answer as far as possible. [5] dx Cambridge International A Level Mathematics 9709 Paper 31 Q2 November 2011

Pr es s

-C

Find

ve rs ity

ev ie

w

C

op

y

12 The equation of a curve is ln( xy ) − y3 = 1. dy y = . i Show that dx x(3 y3 − 1) ii Find the coordinates of the point where the tangent to the curve is parallel to the y-axis, giving each coordinate correct to 3 significant figures.

[4] [4]

ni

C op

y

Cambridge International A Level Mathematics 9709 Paper 31 Q7 November 2012

ge

U

R

13 The curve with equation 6e2 x + ke y + e2 y = c , where k and   c are constants, passes through the point P with coordinates (ln 3, ln 2). Show that 58 + 2 k = c .

ii

Given also that the gradient of the curve at P is −6, find the values of k and c.

w

i

ie

ev

id

br

-R

am

es

s

-C

Pr

y

ity

op

rs

w

ni

op

y

ve

ie ev

y -R s es

am

br

ev

ie

id g

w

e

C

U

op

ie

w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

am

br

ev

id

ie

w

ge

C

U

R ev

R

-C

[5]

Cambridge International A Level Mathematics 9709 Paper 31 Q5 June 2011

C

110

[2]

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op

y

ve rs ity ni

C

U

ev ie

w

ge

-R

am br id

Pr es s

-C y

ni

C op

y

ve rs ity

op C w ev ie

rs

w

ge

C

U

R

ni

op

y

ve

ie

w

C

ity

op

Pr

y

es

s

-C

-R

am

br

ev

id

ie

w

ge

U

R ev

Chapter 5 Integration

111

id

es

s

-C

-R

am

br

ev

1 extend the idea of ‘reverse differentiation’ to include the integration of e ax + b , , ax + b 2 sin( ax + b ) , cos( ax + b ) and sec ( ax + b ) use trigonometrical relationships in carrying out integration understand and use the trapezium rule to estimate the value of a definite integral.

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

P2

■ ■ ■

ie

In this chapter you will learn how to:

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

ev ie

w

PREREQUISITE KNOWLEDGE

What you should be able to do

Chapter 4

Differentiate standard functions.

a

dy when: dx y = 3sin 2 x − 5 cos x

b

y = e5 x − 2

c

y = ln(2 x + 1)

1 Find

Pr es s

-C y

ve rs ity

op

2 Prove that cos 4x + 4 cos 2 x ≡ 8 cos 4 x − 3.

y

Use standard trigonometric relationships to prove trigonometric identities.

U

ni

C op

C w

Chapter 4

ev ie

R

Check your skills

-R

Where it comes from

w

ge

Why do we study integration?

br

ev

id

ie

In this chapter we will learn how to integrate exponential, logarithmic and trigonometrical functions and how to apply integration to solve problems.

f( x ) d x using a numerical method. This

y

P2 Lastly, we will learn how to find an estimate for

WEB LINK

rs

w

ve

5.1 Integration of exponential functions

op

y

Since integration is the reverse process of differentiation, the rules for integrating exponential functions are:

dx =

1 ax + b e +c a

∫e

c

∫e

id g

1 −2 x +c e −2 1 = − e −2 x + c 2

dx    =

5x − 4

dx =

1 5x − 4 e +c 5

-R s es

am

br

−2 x

dx

op

∫e

5x − 4

C

b

dx

w

1 3x e +c 3

−2 x

ie

dx =

U

3x

e

∫e

∫e

ev

b

y

Pr

es c

d (e x ) = e x dx d (e ax + b ) = ae ax + b dx

ity

dx

Answer a

In Chapter 4, we learnt the following rules for differentiating exponential functions:

ni ve rs

3x

-C

R

ev

ie

w

C

Find:

∫e

s

-C

op y

WORKED EXAMPLE 5.1

a

REWIND

-R

am

ax + b

ie

∫e

dx = e x + c

ev

id x

br

∫e

Explore the Calculus of trigonometry and logarithms station on the Underground Mathematics website.

w

ge

C

U

ni

ie ev

R

Pr

a

KEY POINT 5.1

This chapter builds on the work we did in Pure Mathematics 1 Coursebook, Chapter 9, where we learnt how to integrate ( ax + b ) n , where  n ≠ −1.

f( x ) d x using an algebraic method.

ity

op

a b

C

112

s

∫ is used when we are not able to find the value of ∫

b

es

-C

-R

am

It is important that you have a good grasp of the work covered in Chapter 3 on trigonometry, as you will be expected to know and use the trigonometrical relationships when solving integration problems.

REWIND

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ve rs ity ge

C

U

ni

op

y

Chapter 5: Integration

ev ie

am br id

Evaluate

w

WORKED EXAMPLE 5.2



2

9e3x −1 d x.

Pr es s

2

9 9e3x −1 d x =  e3x −1  3 1

y

ni

ev ie

w

  = 3e2 (e3 − 1)

WORKED EXAMPLE 5.3

ge

U

R

ve rs ity

C

= (3e5 ) − (3e2 )

Substitute limits.

y

op

1

C op



2

-C

Answer

-R

1

y

-R

x

2

s

1

es

( x 2 + e −2 x ) d x

y

ni

C

U

Substitute limits.

ge

R

op

2

 x 3 1 −2 x  = − e  2  3 1

w

8 1 1 1 =  − e −4  −  − e −2  3 2  3 2  7 1 1 = + e −2 − e −4 3 2 2 ≈ 2.39 units2

op y

es

s

-C

-R

am

br

ev

id

ie

Simplify.

−4 x

dx

−x

dx

2 − 3x

dx

ev -R s es

am -C

c

ie

h

∫e ∫ 2e ∫ 6e

y

ity

Pr 3 x −1

e

U

1 x 2

b

br

g

2x

e

d

∫ e dx ∫ 4e dx ∫ e dx

id g

a

ni ve rs

1 Find:

R

ev

ie

w

C

EXERCISE 5A

op

ev

1

C



2

rs

ie

Area =

ity

Answer

113

ve

w

C

op

Pr

y

Find the area of the shaded region.

w

O

-C

am

br

ev

id

ie

w

y = x2 + e–2x

Copyright Material - Review Only - Not for Redistribution

f i

∫ 6e ∫e ∫ 2e

3x

dx

2x + 4

dx

8x − 3

dx

ve rs ity

(e x + e2 x )2 d x

1

0



h

1

0

U

c

2x

− 1)2  dx

(e x − 2)2 dx e2 x

∫ ∫

ln 2



1



2

5e −2 x d x

0

8

 e

∫ (e

2 x −1 d x

f

(e x + 1)2  d x

0

2

 2e x − 5  d x  ex 

i

0

(e x + 1)2 dx e2 x

ev

ie

w

dy = 6e2 x + 2e − x . Given that the curve passes through the point (0, 2), find the equation dx

y

es

-C

5 point  1,  2  , find the equation of the curve.  e 

-R

d2 y dy −2 x .  Given that = −8 when x = 0 and that the curve passes through the 2 = 20e dx dx

s

5 A curve is such that

Pr

6 Find the exact area of the region bounded by the curve y = 1 + e2 x − 5 , the x-axis and the lines x = 1 and x = 3.

rs

ity

op

1x

x

id

3

ie

O

w

ge

C

U

R

ni

ev

y

y

ie

y = 2e 2 – 2x + 3

op

7

ve

C

w

e 4x d x

ge

id

am

br

of the curve.

w

f

y



e

ni

ev ie

1 2



ve rs ity

e1+ 2 x  d x

4 A curve is such that

114

c

0

0

R

) dx

C op



1

8 − ex dx 2e2 x



b

0

g

3x

Pr es s

y op C



3

e 3x d x

0

d

w



2

x

e

3 Evaluate: a

∫ 5e (2 + e

b

ev ie



4 + e2 x dx e2 x

(1 + e x ) dx

-R

−x

-C

d

∫e

am br id

a

ge

2 Find:

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

-R

y = xex

Pr ity x

y

3

op

ni ve rs

C w ie

O

ev

es

y

op y

b

d ( xe x − e x ) = xe x . dx

s

Show that

-C

8 a

am

br

ev

Find the exact area of the shaded region.

w

ev

0

(4e −2 x + 5e − x ) d x .

-R





s

-C

am

b Hence find the value of

ie

(4e −2 x + 5e − x ) d x, where a is a positive constant.

br

0

es



a

e

Find

id g

9 a

C

U

R

Use your result from part a to evaluate the area of the shaded region.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C ev ie -R

M

Pr es s

x

-C

O

w

ge

y

10

am br id

PS

U

ni

op

y

Chapter 5: Integration

ev ie

3

R

ln 3

x

U

ln 2

O

ni

2

y

y = ex

C op

y

11

w

C

P PS

Find the area of the shaded region.

ve rs ity

op

y

The diagram shows the curve y = 2e x + 8e − x − 7 and its minimum point M.

w

ie

e x d x.

b Hence show that



3

2

s

1 ax + b

ity

y

1 1 dx = ln( ax + b ) + c,   ax + b . 0 a ax + b

C

U ge

id

ie

w

WORKED EXAMPLE 5.4

1

Pr

1 = 4   ln(2 x + 1) + c  2

op

C w ie

e

-R s

-C

am

br

id g

1   2

ev

U

R

= 2 ln(2 x + 1) + c,   x . −

Valid for 2 x + 1 . 0 .

y

1

ity

4

∫ 2x + 1 dx  = 4∫ 2x + 1 dx

es

w

C

  = 2 ln x + c,   x . 0

ev

ie

es

2

∫ x dx = 2 ∫ x dx

ni ve rs

op y

Answer

b



s

-R



am

-C

ev

br

Find each of these integrals and state the values of x for which the integral is valid. 2 4 6 dx a b dx c dx x 2x + 1 2 − 3x

a

In Chapter 4, we learnt the following rules for differentiating logarithmic functions: d 1 (ln x ) = , x . 0 x dx

op



ni

ie ev

R

ve

1



REWIND

rs

KEY POINT 5.2

w

C

op

Pr

Since integration is the reverse process of differentiation, the rules for integration are:

∫ x dx = ln x + c,  x . 0

27  . ln y dy = ln  4e 

-R

ln 2

y

-C

5.2 Integration of



ln 3

es

am

br

a Find the value of

ev

id

ge

The diagram shows the graph of y = e x. The points (ln 2, 2) and (ln 3, 3) lie on the curve.

Copyright Material - Review Only - Not for Redistribution

+

d [ ln( ax + b ) ] = ax dx a = , ax + b . 0 ax + b It is important to remember that ln x is only defined for x . 0.

115

ve rs ity

1

 1 = 6 ln(2 − 3x ) + c  −3 

op

∫x

n

-R

1 x n +1 + c to obtain her answer. n +1

dx =

ge

R

y

She tries to use the formula

∫ (2x + 3) dx. 1 ( ax + b ) She tries to use the formula ( ax + b ) dx = ∫ a ( n + 1) −1

br

id

Fiona is also asked to find

n +1

+ c to obtain her

-R

am

-C

answer.

n

w

dx.

ie

−1

ni

∫x

Fiona is asked to find

U

ev ie

w

C

ve rs ity

EXPLORE 5.1

Pr es s

2 3

y

-C

= −2 ln(2 − 3x ) + c,   x ,

Valid for 2 − 3x . 0.

C op

am br id

ev ie

∫ 2 − 3x dx  = 6∫ 2 − 3x dx

c

ev

6

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

op

Pr

y

es

s

Discuss with your classmates why Fiona’s methods do not work.

rs

y op C ie

1 dx 3x + 1

ev

am

br

∫ 2(3x1+ 1) dx = 21 ∫

w

1 ln(6x + 2) + c 6

Pr

op y

es

-C

1 1 =     ln(3x + 1) + c  2  3 1 = ln(3x + 1) + c 6

-R

id

=

Fausat writes:

1 dx 6x + 2

ni

∫ 2(3x1+ 1) dx = ∫

ge

R

Rafiu writes:

1

∫ 2(3x + 1) dx.

ve

Rafiu and Fausat are asked to find

U

ev

ie

w

C

ity

EXPLORE 5.2

s

116



−2

−3

1 d x.  x

1 we can see that the shaded areas x that represent the two integrals are equal in magnitude. However, one of the areas is −2 3 1 1 dx = − below the x-axis, which suggests that d x. −3 x 2 x

y=

1 x

2

3 x

1

w

–3

s

-R

ev

ie



es

am

br

id g

e

C

U

From the symmetry properties of the graph of y =



y 2

y

2

1 d x and x

op



3

ni ve rs

Consider the integrals

-C

R

ev

ie

w

C

ity

Decide who is correct and discuss the reasons for your decision with your classmates.

Copyright Material - Review Only - Not for Redistribution

–2

–1 O –1 –2

1

ve rs ity

C

3

1 d x gives: x

w

am br id

2

ev ie



ge

Evaluating

U

ni

op

y

Chapter 5: Integration



-C

2

This implies that

−2

1 3 2 d x = − ln = ln . 2 3 x

y

−3



−2

−3



WEB LINK

−2

−3

−2 1 2 d x =  ln x  −3 = ln( −2) − ln( −3) = ln 3 x

U

R

ni

C op

y

There is, however, a problem with this calculation in that ln x is only defined for x . 0 so ln( −2) and ln( −3) do not actually exist. 1 dx = ln x + c . x In conclusion, the integration formulae are written as:



br

ev

id

ie

w

ge

Hence for x , 0, we say that

ity

y C

U

ity e

2

0

y op

f

C

ev

1

1 dx 2x + 1

3 dx 2x − 7

s

2 dx 4x + 1

∫ ∫

4

-R

b

c

w

U

e

e

1

∫ 2x dx 5 ∫ 2 − 3x dx

es

2

b

br

4

am

∫ ∫

id g

2 Evaluate: 5 1 dx a 0 x +2 d

ni ve rs

6

∫ x dx 6 ∫ 2x − 5 dx

-C

C w ie ev

1 Find:

d

ev

s Pr

op y

es

-C

am

Simplify.

-R

br

= −2 ln 7 + 2 ln 4 4 = 2 ln 7

EXERCISE 5B

a

w ie

Substitute limits.

id

2

ie



3

 6   6 dx =    ln 2 − 3x   2 − 3x  −3 2 = ( −2 ln 7) − ( −2 ln 4)

ge

R

Answer 3

op

6 d x. 2 − 3x

ni

ev

2

ve

3

rs

C w ie



-R

Pr

y op

WORKED EXAMPLE 5.5

Find the value of

It is normal practice to only include the modulus sign when finding definite integrals.

1 1 dx = ln ax + b + c ax + b a

s



TIP

es

1 dx = ln x + c x

-C



am

KEY POINT 5.3

R

A more in-depth explanation for this formula can be found in the Two for one resource on the Underground Mathematics website.

1 d x we obtain: x

ve rs ity

ev ie

w

C

op

If we try using integration to find the value of

-R

3 1 3 d x =  ln x 2 = ln 3 − ln 2 = ln 2 x

Pr es s



3

Copyright Material - Review Only - Not for Redistribution

c f

1

∫ 3x + 1 dx 3 ∫ 2(5x − 1) dx 4

3 dx −1 2 x + 5

∫ ∫

3

2

4 dx 3 − 2x

117

ve rs ity

 2 + 5  dx  3x − 2 

-R

3



b Hence show that

5

4x d x = 8 + ln 9. 2x − 1

U

R

1

y

ev ie

4x A ≡2+ , find the value of the constant A. 2x − 1 2x − 1

ni

4 a Given that

ve rs ity

 2x − 1 + 4  dx 2x + 3  −1 



w

C

op

y

c

Pr es s

-C

1

 3 − 4  dx  x 2x − 1 

C op



3

4

b

w

10

ev ie



am br id

a

ge

3 Evaluate:

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

id

0

br

dy 3 = 2x + . dx x+e

w

6x 2 + 5x 23 5 − 25 ln  . dx =  3 2x − 5 2

-R

am

-C

6 A curve is such that

1

ie



b Hence show that

ev

ge

5 a Find the quotient and remainder when 6x 2 + 5x is divided by 2 x − 5.

es

y

y op

x

2 . Given that the shaded region has area 4, find the value of k. x+3

id

w

ge

The diagram shows part of the curve y =

C

k

ie

1

U

R

O

ni

ev

ve

ie

w

rs

C

ity

op

7

Pr

y

PS

118

s

Given that the curve passes through the point (e, e2 ), find the equation of the curve.

dy 2 = 3 −  . The tangents to the curve at the points dx x P and Q intersect at the point R. Find the coordinates of R.

s

-C

-R

am

br

ev

8 The points P (1,  −2) and Q(2,  k ) lie on the curve for which

PS

op y

es

5.3 Integration of sin( ax + b ), cos( ax + b ) and sec 2 ( ax + b )

Pr

-R s es

am

y

ie

w

C

op

dx d [cos( ax + b )] = − a sin( ax + b ) dx dx d [tan( ax + b )] = a sec 2 ( ax + b ) dx

br

id g

e

U

d (cos x ) = − sin x dx

ev

ni ve rs

d

d (tan x ) = sec 2 x dx

d [sin( ax + b )] = a cos( ax + b ) dx

ity

d (sin x ) = cos x dx

-C

R

ev

ie

w

C

In Chapter 4, we learnt how to differentiate some trigonometric functions:

Copyright Material - Review Only - Not for Redistribution

ve rs ity

TIP

ev ie



am br id

KEY POINT 5.4

w

ge

C

U

ni

op

y

Chapter 5: Integration

Since integration is the reverse process of differentiation, the rules for integrating are: 1

∫ cos(∫ax + b) dx = a sin(ax + b) + c

-C

-R

∫ cos x dx = sin x + c

Pr es s

y

2

C op

U ∫ cos 3x dx

b

∫ cos 3x dx = 3 sin 3x + c

c

∫ 4 sec

c

∫ 4 sec

1

1

rs

C

C

(3 − 2 sin 2 x ) d x.

w

ge



π 4

U

R

ni

op

y

ve

w ie ev

WORKED EXAMPLE 5.7

ev

id

-R

Substitute limits.

es

s

br am

3π π    =  + cos  − ( 0 + cos 0 )  4 2 3π    =  + 0  − (0 + 1)  4  3π    = −1 4

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

0

π

4 2 (3 − 2 sin 2 x ) d x =  3x + cos 2 x  2  0

-C



π 4

ie

0

Answer

x  dx 2

2

x  dx = 4 2

x  dx 2 x = 4 ×  2 tan  + c  2

∫ sec

  = 8 tan

ity

op

Pr

y

es

s

-C

∫ sin 2x dx = − 2 cos 2x + c

Find the exact value of

2

ev

br am

Answer a

w

b

ie

∫ sin 2x dx

id

a

ge

Find:

-R

R

ni

WORKED EXAMPLE 5.6

y

ve rs ity

op w ev ie

1

∫ sec (ax + b) dx = a tan(ax + b) + c

x dx = tan x + c

C

2

1

+ b ) dx = − cos(ax + b ) + c ∫ sin(ax a ∫

∫ sin x dx = − cos x + c ∫ sec

It is important to remember that the formulae for differentiating and integrating these trigonometric functions only apply when x is measured in radians.

Copyright Material - Review Only - Not for Redistribution

2

x +c 2

119

ve rs ity

1 Find:

b

Pr es s

∫ 3sin(2x + 1) dx

i

∫ 2 sec (5x − 2) dx

(sin 2 x + cos x ) d x

e



1 π 6

1 sin  x  d x 2  (cos 2 x + sin x ) d x

0

id

0



2 π 3

0

U

R

h

b

ge



1 π 2

∫ sec

br

x cos x d x.



1 π 6

sec 2 2 x d x

0

f

1 π 4 1 − π 4



(5 − 2 sec 2 x ) d x

ity

dy = 1 − 3sin 2 x. dx

Pr

y op C

2

s



1 π 3

0

4 A curve is such that

2 x  dx

es

b Hence find the exact value of

120

c

2

-R

d ( x sin x + cos x ). dx

am

Find

-C

3 a

y

cos 4x d x

0

d

f

C op



∫ 5 cos 3x  dx

ie

a

1 π 6

ni

ev ie

2 Evaluate:

∫ sin 2 dx

e

w

∫ 2 cos(1 − 5x ) dx

x

c

ev

w

∫ 3sin 2x  dx

ve rs ity

g

C

op

y

d

∫ cos 4x dx

-R

∫ sin 3x dx

-C

a

ev ie

am br id

EXERCISE 5C

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

y op

ni

C

dy π = 4 when x = 0 and that the curve passes through the point  ,  −3  , find the equation of 2  dx

w

ie

am

br

the curve.

ev

id

Given that

d2 y = −12 sin 2 x − 2 cos x. dx 2

ge

R

5 A curve is such that

U

ev

ve

ie

w

rs

π Given that the curve passes through the point  , 0  , find the equation of the curve. 4 

es

s

-C

-R

π π dy = 4 sin  2 x −  . 6 The point  , 5    lies on the curve for which  2  dx 2 a Find the equation of the curve.

y op

ie

x

C

U

O

-R s es

am

br

ev

ie

id g

w

e

The diagram shows part of the curve y = 1 + 3 sin 2 x + cos 2 x. Find the exact value of the area of the shaded region.

-C

ev

R

π . 3

ity

y

ni ve rs

7

w

C

PS

Pr

op y

b Find the equation of the normal to the curve at the point where x =

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C w ev ie π

x

Pr es s

-C

O

-R

M

y

ge

8

am br id

PS

U

ni

op

y

Chapter 5: Integration

y

ni U

w

π 6

π 3

br

id

O

ge

R

1 2

y

y = sin x

x

ev

ev ie

w

√3 2

C op

9

ie

C

P PS

Find the exact area of the shaded region.

ve rs ity

op

y

The diagram shows part of the curve y = 3sin 2 x + 6 sin x and its maximum point M.

-R

s

Pr (

)

π 2 3 −1 − 12

ity

(sin −1 y ) d y =

121

3 −1 . 2

rs

1 2

y

y op

y = cos x

ie ev

π 4

π 3

x

-C

O

-R

am

br

id

2

w

ge

2 1

C

U

√2

R



3 2

sin x d x.

ve

10

b Hence show that



π 3 π 6

ni

P PS

ev

ie

w

C

op

y

a Find the exact value of

es

-C

am

π π 1 3   lie on the curve. The diagram shows part of the graph of y = sin x. The points  ,   and  ,   6 2  3 2 

s

es

Pr

cos x d x.

(cos −1 y ) d y =

π (3 2 − 4) + 24

3− 2 . 2

y

1 2

w

id g

e

5.4 Further integration of trigonometric functions

C

U



2 2

op

ni ve rs

w

-R

s

-C

am

br

ev

ie

Sometimes, when it is not immediately obvious how to integrate a trigonometric function, we can use trigonometric identities to rewrite the function in a form that we can then integrate.

es

ie

R

ev

b Hence show that



π 3 π 4

ity

a Find the exact value of

C

op y

π π 1 2 and  ,     lie on the curve. The diagram shows part of the graph of y = cos x. The points  ,   3 2  4 2 

Copyright Material - Review Only - Not for Redistribution

ve rs ity

am br id

ev ie

cos 2 x ≡ 1 − 2 sin2 x and cos 2 x ≡ 2 cos2 x − 1

op

y

WORKED EXAMPLE 5.8

x dx =

ni

∫ (1 − cos 2x ) dx

U

Use sin2 x ≡

 x − 1 sin 2 x  + c   2 1 x − sin 2 x + c 4

s es

rs

w

op

y

ve

ie

x dx .

w

ge

4

am

br

Answer

ev

id

∫ sin

C

U

R

ni

WORKED EXAMPLE 5.9

Find

Use sin2 x ≡

es

Use cos2 2 x ≡

1 (1 + cos 4x ). 2

op C

e

ev

ie

id g

w

1 3 1 sin 4x + c x − sin 2 x + 4 32 8

-R s es

-C

am

br

      =

U



y

ni ve rs

ity

Pr

-C

1 (1 − cos 2 x ). 2

s

2

1 =  (1 − cos 2 x )  2  1 = (1 − 2 cos 2 x + cos2 2 x ) 4 1 1 1 =  1 − 2 cos 2 x + + cos 4x   4 2 2 3 1 1 = − cos 2 x + cos 4x 8 2 8 3 1 1 sin 4 x dx =  − cos 2 x + cos 4x  dx 8 2  8

op y C w ie ev



-R

sin 4 x = (sin2 x )2

R

1 (1 + cos 4x ). 2

Pr

op

 x + 1 sin 4x  + c   4 3 x + sin 4x + c 8

Use cos2 2 x ≡

ity

3 2 3 = 2

=

ie ev

-R

br am

-C

2

C

122

3

∫ 3 cos 2x dx = 2 ∫ (1 + cos 4x ) dx

y

b

1 (1 − cos 2 x ). 2

ev

id

1 2 1  = 2 1  = 2

2

ge

R

∫ sin

a

∫ 3 cos 2x dx

y

2

Answer

b

C op

x dx

w

2

ie

w ev ie

ve rs ity

C

Find:

∫ sin

-R

Pr es s

-C

Rearranging these two identities gives: 1 1 sin2 x ≡ (1 − cos 2 x ) and cos2 x ≡ (1 + cos 2 x ) 2 2

a

w

ge

To integrate sin2 x and cos2 x we must use the double angle identities:

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Chapter 5: Integration

w

ge

To integrate tan2 x we use the identity 1 + tan2 x ≡ sec 2 x.

am br id

ev ie

Rearranging this identity gives tan2 x ≡ sec 2 x − 1.

y

op π 4

5  tan x d x = 2

0



π 4

ve rs ity

C w

5 tan2 x d x.

0

Answer



ev ie

-R



Find the exact value of

π 4

Pr es s

-C

WORKED EXAMPLE 5.10

(5 sec 2 x − 5) d x

0

Use tan2 x ≡ sec 2 x − 1.

y

π

Substitute limits.

U

R

ni

C op

  =  5 tan x − 5x  4 0

-R

am

br

ev

id

ie

w

ge

5π  − (0 − 0)  =  5 −  4  5π  = 5 − 4

es

s

-C

Worked example 5.11 uses the principle that integration is the reverse process of differentiation.

ity

a Show that if y = sec x then

123

dy = tan x sec x. dx

0

w

Use the quotient rule.

ie

1 cos x

id

y =

a

ge

Answer

C

U

(cos x ) (0) − (1) ( − sin x ) dy = dx cos2 x sin x  = cos2 x   = tan x sec x (cos 2 x + 5 tan x sec x ) d x

0 π

s es Use

ni ve rs

1

Pr



1 π 4

ie ev

Substitute limits.

-R s es

-C

am

br

ev

ie

id g

w

e

C

U

op

1 =  + 5 2  − (0 + 5) 2  9 =5 2− 2

R

∫ tan x sec x dx = sec x.

y

4 1 =  sin 2 x + 5 sec x   2  0

w

C

b

ity

op y

-C

-R

am

br

ev

R

(cos 2 x + 5 tan x sec x ) d x.

y



ni

ev

b Hence find the exact value of

op

1 π 4

ve

ie

w

rs

C

op

Pr

y

WORKED EXAMPLE 5.11

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

w

ge ∫

(5 + cosec 2 x tan x ) d x.

Pr es s

-C

b Hence find the exact value of

1 π 3 1 − π 3

-R

a Prove that 2 cosec 2 x tan x ≡ sec 2 x.

2 tan x sin 2 x

2 cosec 2 x tan x ≡

C

a

sin x    and cos x sin 2 x = 2 sin x cos x.

Use tan x =

ve rs ity

op

y

Answer

2 sin x (2 sin x cos x ) cos x 1             ≡ cos2 x

C op

Simplify.

U

R

ni

ev ie

w

   ≡

-C

w ie

Simplify.

es

3

ev

am

π

Use part a.

-R

 5 + 1 sec 2 x  d x   2 1

   =

Pr

10 π + 3 3

w

rs

ity

op

y

 5π 3   5π 3 = + − − − 2   3 2   3

op C

U

R

ni

WORKED EXAMPLE 5.13

y

ve

ie

w

ge

a Use the expansions of cos(3x − x ) and cos(3x + x ) to show that:

0

1 2



1 π 4

ev

(cos 2 x − cos 4x ) d x 1

4 1 1 1 =  sin 2 x − sin 4x  2 2 4 0

π

Simplify.

C

U

R

Use part a.

0

op

sin 3x sin x d x =

ity

≡ 2 sin 3x sin x 1 π 4

ni ve rs

1  1  − 0  − (0 − 0)   2   2  1 = 4

ie ev -R s

-C

am

br

id g

w

e

=

es

C w ie



Pr

≡ (cos 3x cos x + sin 3x sin x ) − (cos 3x cos x − sin 3x sin x )

ev

b

-R

cos 2 x − cos 4x ≡ cos(3x − x ) − cos(3x + x )

op y

a

0

1 sin 3x sin x d x = . 4

-C

Answer



s

am

b Hence show that

1 π 4

es

br

id

ie

cos 2 x − cos 4x ≡ 2 sin 3x sin x

y

ev



3 1 =  5x + tan x   − 1 π 2 

C

124

1 π 3 1 − π 3

s

id

(5 + cosec 2 x tan x ) d x =

br



b

ge

   ≡ sec 2 x 1 π 3 1 − π 3

y

am br id

WORKED EXAMPLE 5.12

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Copyright Material - Review Only - Not for Redistribution

ve rs ity 1 Find:

e

op

ve rs ity



sin x d x

0 1 π 8



sin2 3x dx

∫ 6 tan (3x ) dx

f



cos 4 x dx

cos2 2 x d x

0

b

e



1 π 3 1 π 6



1 π 6

2

2

3tan x d x

4 sin2 x d x

0

ie

0

y op C w ie ev -R

s es Pr

2 sin 5x cos 2 x d x =

ni ve rs



ity

1 π 3 1 π 6

11 − 3 3 . 42

op

U

5 . 12

w

e

0

2 sin3 x d x =

C



y

Show that sin 3x ≡ 3sin x − 4 sin3 x. 1 π 3

ie

17 . 6

ev

0

(4 cos3 x + 2 cos x ) d x =

-R



1 π 6

s

-C

am

b Hence show that

es

br

id g

Show that cos 3x ≡ 4 cos3 x − 3 cos x.



1 π 4



1 π 6

0

Pr

ity

ni

U

ge id

br

f

8 d x = 1. tan x + cot x

am

-C op y

1 π 2 1 π 3



b Hence show that 8 a

1 + cos 4 x dx cos2 x

2 . sin 2 x

sin 7 x + sin 3x ≡ 2 sin 5x cos 2 x

b Hence show that

7 a

1 π 6

Use the expansions of sin(5x + 2 x ) and sin(5x − 2 x ) to show that:

C w ie ev

rs

ie ev

R

6 a



1 π 6

tan2 2 x d x

(2 sin x + cos x )2 d x

0

3 . 6

cosec 2 2 x d x =

Show that tan x + cot x ≡

b Hence show that

R

1 π 4 1 π 6

ve

op w



b Hence, show that

5 a



0

cos x dy , show that if y = cot x then = − cosec 2 x. d x sin x

By differentiating

C

4 a

cos2 x d x

0

ev

e

y

0

f

c

-R

(cos x − 3sin x )2 d x

(cos2 x + sin 2 x ) d x

es

1 π 3



1 π 6

s

id

b

br

0

 1   dx  cos2 x − cos2 x  

am ∫

-C

d



1 π 6



w

3 Find the value of: a

c

1 π 3

0

ge

R

d



ni

ev ie

w

a

2

U

C

2 Find the value of: 1 π 3

c

2

y

x dx

x   dx 2

∫ 4 cos

C op

2

b

-R

∫ 2 tan

x dx

y

d

2

Pr es s

∫ 3 cos

-C

a

ev ie

am br id

EXERCISE 5D

w

ge

C

U

ni

op

y

Chapter 5: Integration

Copyright Material - Review Only - Not for Redistribution

5 dx 1 + cos 4x 125

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

9 The diagram shows part of the curve y = cos x + 2 sin x. Find the exact volume of the solid formed when the shaded region is rotated through 360 ° about the x-axis.

ev ie

Pr es s

4 d x = 2(1 + 3 ). 1 − sin 2 x

ni

0

U

5.5 The trapezium rule b

ev

br

a

-R

a

s

-C

cannot be integrated algebraically.

op

Pr

y

es

In these situations an approximate answer can be found using a numerical method called the trapezium rule. 126

rs

y

ve U id

h

h

h

y5

C

y4

y6

h

w

h

h

a

x

b

am

br

O

y = f(x) y3

y2

ie

ge

y1

ev

R

ni

ev

ie

y

op

w

C

ity

This numerical method involves splitting the area under the curve y = f( x ) between x = a and x = b into equal width strips.

y0

es

s

-C

-R

The area of each trapezium-shaped strip can be found using the formula: 1 area = (sum of parallel sides) × width 2

C



b

a

Pr

curve and hence an approximate value for

f( x ) d x.

ity

op y

The sum of the areas of all the strips gives an approximate value for the area under the

b−a and the 6 length of the vertical edges (ordinates) of the strips are y0 ,   y1,   y2 ,   y3 ,   y4 ,   y5   and   y6 .

y

op C

U

The sum of the areas of the trapezium-shaped strips is:

-R s es

am

br

ev

ie

id g

w

e

h h h h h h ( y0 + y1 ) + ( y1 + y2 ) + ( y2 + y3 ) + ( y3 + y4 ) + ( y4 + y5 ) + ( y5 + y6 ) 2 2 2 2 2 2 h = [ y0 + y6 + 2( y1 + y2 + y3 +   y4 + y5 )] 2

-C

ev

ie

w

ni ve rs

For the previous diagram, there are 6 strips, each of width h, where h =

R

A

f( x ) d x by the integration methods that we have covered so far or maybe the function

am



b

y = f(x)

f( x ) d x. Sometimes we might not be able to find the value of

ie

id



y

w

ge

We already know that the area, A, under the curve y = f( x ) between x = a and x = b can be found by evaluating

x

y



1 π 6

C op

Hence show that

ve rs ity

op C w ev ie

R

P2

1π 2

dy = sec x tan x. dx

b Show that if y = sec x then

c

O

1 Show that sec x + sec x tan x ≡ . 1 − sin x 2

y

10 a

-R

am br id

-C

PS

y

w

ge

PS

Copyright Material - Review Only - Not for Redistribution

O

a

b

x

ve rs ity

C

U

ni

op

y

Chapter 5: Integration

am br id

ev ie

w

ge

When there are n strips and the ordinates are y0 ,   y1,   y2 ,   y3 ,  … ,   yn −1,   yn then the sum of the areas of the strips is:



b

f ( x ) d x using: a

op

y

KEY POINT 5.5

TIP • The width of the strip is the interval along the x-axis.

b−a h [ y0 + yn + 2( y1 + y2 + y3 +  … + yn −1 )] where h = . n 2 a An easy way to remember this rule in terms of the ordinates is: f ( x ) d x ≈ half width of strip × (first + last + twice the sum of all the others).

a

w

ge ity

127

Answer

0.5

op

y

ve

y

C

U

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w

ge

ev

id

x

1.8863

0.6137

0.9945

y0

y1

y2

es

3.5

ity

2

s

3.5

Pr

2

-R

br am

0.5

0.5

h [ y0 + y2 + 2 y1 ] 2 1.5 ≈ [1.8863 + 0.9945 + 2 × 0.6137] 2 ≈ 3.08115 …

y C

U

op

ev

R

ni ve rs

Area ≈

ie

id g

w

e

≈ 3.08 (to 2 decimal places)

-R s

-C

am

br

ev

ie

It can be seen from the diagram that this is an over-estimate since the top edges of the strips all lie above the curve.

es

w

C

op y

-C

O

x

3.5

ni

ev

a = 0.5,  b = 3.5 and  h = 1.5

R

O

rs

ie

w

C

op

Pr

y

es

s

-C

Use the trapezium rule with 2 intervals to estimate the shaded area, giving your answer correct to 2 decimal places. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value.

-R

am

y

ev

id

br

The diagram shows the curve y = x − 2 ln x.

y

• You should not use a numerical method when an algebraic method is available to you, unless you are specifically asked to do so.

ie

WORKED EXAMPLE 5.14

x

• The number of ordinates is one more than the number of strips.

C op

b

U

R



y

ve rs ity

f (x ) dx ≈

ni

C



b

ev ie

w

Pr es s

-C

Hence we can find an approximate value for

-R

h h h h h ( y0 + y1 ) + ( y1 + y2 ) + ( y2 + y3 ) +  …  + ( yn − 2 + yn −1 ) + ( yn −1 + yn ) 2 2 2 2 2

Copyright Material - Review Only - Not for Redistribution

ve rs ity

y

-R

The diagram shows the curve y = 2 4x − x 2 .  Use the trapezium rule with 4 intervals to estimate the



4

-C

2 4x − x 2 d x giving your answer correct

Pr es s

value of

ev ie

w

ge

am br id

WORKED EXAMPLE 5.15

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

0

O

4x

0

2 3

4

2 3

0

y0

y1

y2

y3

y4

id br

[ y0 + y4 + 2( y1 + y2 + y3 )] 0 + 0 + 2 2 3 + 4 + 2 3   

)

3+4

4x

ity

op C

It can be seen from the diagram that this is an under-estimate since the top edges of the strips all lie below the curve.

rs

w

3

≈ 10.93 (to 2 decimal places)

ni

op

y

ve

ie ev

2

es

(

1

Pr

y

0

128

h 2 1 ≈ 2 ≈4

2 4x − x 2   d x ≈

am

4

O

w

y

ie

U

4

ev

3

-R

2

s

1

ge

0

-C

R

x



Worked example 5.14 involves a ‘concave’ curve and the diagram shows that the trapezium rule gives an over-estimate for the area.

w

ge

br

ev

id

ie

Worked example 5.15 involves a ‘convex’ curve and the diagram shows that the trapezium rule gives an under-estimate for the area.

es

s

-C

-R

am

If a curve is partially convex and partially concave over the required interval it is not so easy to predict whether the trapezium rule will give an under-estimate or an over-estimate for the area.

Pr

op y

The more strips that are used, the more accurate the estimate will be.

ni ve rs 0

f

2 dx 3 + ex

w

1 dx 2 + tan x



12

ie



1 π 3

0

ev

e

4

op



c

C

e x − 12 d x

2

log10 x d x

s

-R

id g

am

1

2 dx ln( x + 5)

br

d



2

0

e

2

b

es



7

x2 − 2 dx

U



4

y

1 Use the trapezium rule with 2 intervals to estimate the value of each of these definite integrals. Give your answers correct to 2 decimal places.

-C

R

ev

ie

w

C

ity

EXERCISE 5E

a

DID YOU KNOW?

C

U

R

y

ni

a = 0,  b = 4 and  h = 1

y

Answer

C op

ev ie

w

C

ve rs ity

op

y

to 2 decimal places. State, with a reason, whether the trapezium rule gives an under-estimate or an overestimate of the true value.

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If you double the number of strips in a trapezium rule approximation, the error reduces to approximately one quarter the previous error.

ve rs ity

C w ev ie

x

-R

O

ge

y

am br id

2

U

ni

op

y

Chapter 5: Integration

x 2 e − x d x , giving your answer correct to

ve rs ity

2 decimal places.

1 cosec  x  d x, giving your answer correct to 2 decimal places. 2 

C op



7 π 4 1 π 4

y

Use the trapezium rule with 6 intervals to estimate the value of

ni

3 a

ev ie



5

1

w

C

op

y

4 intervals to estimate the value of

Pr es s

-C

The diagram shows part of the curve y = x 2 e − x .  Use the trapezium rule with

y

w

s es Pr

y

129

ity

op



y

C

ge

rule with 3 intervals to estimate the value of

1 π 2

1 π.  Use the trapezium 2

e x cos x d x, giving your answer

w

U

R

ni

The diagram shows the curve y = e x cos x for 0 ø x ø

op

ve

ie

x

rs

C w

1 π 2

O

ev

ie

1 cosec  x  d x. 2 

ev

7 π 4 1 π 4

-C

4



am

br

true value of

-R

id

ge

U

R

1 b Use a sketch of the graph of y = cosec  x  for 0 ø x ø 2 π, to explain 2  whether the trapezium rule gives an under-estimate or an over-estimate of the

0

br

ev

id

ie

correct to 2 decimal places. State, with a reason, whether the trapezium rule

-C

-R

am

gives an under-estimate or an over-estimate of the true value of

es Pr 2

3

x

ity

1

ni ve rs

O

w

e

C

U

op

y

ex . Use the trapezium rule with 4 The diagram shows part of the curve y = 2x intervals to estimate the area of the shaded region, giving your answer correct to 2 decimal places.

-R s es

am

br

ev

ie

id g

State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the shaded area.

-C

C w ie ev

R

e x cos x d x.

0

s

y

op y

5



1 π 2

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WEB LINK Try the When does the trapezium rule give the exact answer here? resource on the Underground Mathematics website.

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ax + b

y

Pr es s

1 dx = ln x + c x 1 1 cos x dx = sin x + c

1

2

The trapezium rule can be used to find an approximate value for

id



1

∫ sec (ax + b) dx = a tan(ax + b) + c

U

The trapezium rule

ni

x dx = tan x + c



h

b−a . n

y op y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

am

br

ev

id

ie

w

ge

C

U

R

ni

ev

ve

ie

w

rs

C

ity

op

Pr

y

es

s

a

130

ev

+ yn + 2( y1 + y2 + y3 +  … + yn −1 )] where h =

-C

0

f( x ) d x. If the region under

-R

br

am

∫ f(x ) dx ≈ 2 [ y

b

a

the curve is divided into n strips each of width h, then: b

y

∫ sec

1

∫ sin(∫ax + b) dx = − a cos(ax + b) + c

C op



ve rs ity

∫ sin x dx = − cos x + c

ge

P2



2

+c

∫ cos(ax + b) dx = a sin(ax + b) + c



op

R

ev ie

w

C



ax + b

w



1

∫ e ∫ dx = a e + c 1 1 ∫ ax +∫ b dx = a ln ax + b

dx = e x + c

ie



x

-R

∫e

-C



ev ie

am br id



Integration formulae

w

ge

Checklist of learning and understanding

Copyright Material - Review Only - Not for Redistribution

ve rs ity ge

C

U

ni

op

y

Chapter 5: Integration



Find the exact value of

−2

−4

3

-R

[4]

 4 − 2 x  d x , giving your answer in the form a + ln b  , where a,  b and c  1 − 2x 

c are integers.

[4]

ve rs ity

y

ev ie

w

C

P2

4 3 π − . 3 6

(1 + 2 tan2 x ) d x =

Pr es s

op

y

2

ev ie

am br id ∫

Show that

1 π 3 1 π 6

-C

1

w

END-OF-CHAPTER REVIEW EXERCISE 5

U

R

ni

C op

y

R

O

x

w

ge

1

(3e x − 2)2 d x =

(5e −2 x + 2e −3x ) d x , where k is a positive constant.



4

ii

y ev

br

Hence show that

ie



-R

s

[5]

es

Pr

1 π 3 12 sin2 1 π 4

[3]

x cos2 x d x =

π 3 3 . + 16 8

[3]

x

2x

[3]

C 1 (8 + π ) . 2

w

(3 + 2 tan2 θ ) dθ =

ie



[4]

s

-R

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q6 June 2011

es

am

) dx.

ev

1 π 4 1 − π 4

id g

Show that

br

b

op

∫ 4e (3 + e

U

Find

e

10 a

y

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q3 June 2013

-C

ev

R



ity

i

3

1

w

C

op y

9

[2]

11 8x d x = 8 − 10 ln . 7 2x + 5 3 Show that 12 sin2 x cos2 x ≡ (1 − cos 4x ) . 2

Hence show that

ni ve rs

b

[1] [5]

8x A ≡ 4+ . 2x + 5 2x + 5

Find the value of the constant A such that

am

a

-C

8

op

(5e −2 x + 2e −3x ) d x.

0

3 3 21 d x = ln . 5x + 1 5 11

2

[4]

C

ni

Show that





id

7

Hence find the exact value of

U

R

b

ge

ev

0

w



k

[5]

ie

Find

131

9 2 23 e − 12e + . 2 2

rs

a

[5]

Pr

1

ve

ie



Show that

0

6

(cos 3x − sin 2 x ) d x.

0

ity

op C w

5

1 π 6

[4]

s



Find the exact value of

y

4

es

-C

-R

am

br

ev

id

ie

The diagram shows part of the curve y = 2 − x 2 ln( x + 1) . The shaded region R is bounded by the curve and by the lines x =   0, x = 1 and y = 0 . Use the trapezium rule with 4 intervals to estimate the area of R giving your answer correct to 2 decimal places. State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the area of R.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

y By differentiating

[4]

[3]

cos x dy , show that if y = cot x then = − cosec 2 x. sin x dx

By expressing cot 2 x in terms of cosec 2 x and using the result of part i, show

y

1 π. 4

1 + cos 4 2 x dx. cos2 2 x

w

ie

[3]

Pr

es

s

Without using a calculator, find the exact value of b



14

4

 2 +   6  d x, giving your answer in the form  3x − 2 

ln( ae ), where a and b are integers.

ity

op

Find

ii

[3]

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q8 June 2010

y

-C

1 dx. 1 − cos 2 x

13 i

[5]

w

rs

C

Q

ev

br

id

ie

w

ge

U

R

op

y

ni

ev

14

y

ve

ie

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q7 June 2016

C

132



1 1 can be expressed as cosec 2 x. 1 − cos 2 x 2

-R

am

br

Hence, using the result of part i, find

[4]

ev

id

ge

iii Express cos 2 x in terms of sin2 x and hence show that



C op

cot 2 x d x = 1 −

ni



1 π 2 1 π 4

U

that

R

(2 sin x + cos x )2 d x .

ve rs ity

op ev ie

w

ii

0

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q7 June 2012

C

12 i



1 π 4

[5]

Pr es s

Hence find the exact value of

5 3 + 2 sin 2 x − cos 2 x. 2 2

-R

Show that (2 sin x + cos x )2 can be written in the form

-C

ii

am br id

11 i

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

π

-R

am

O

x

es

ii

Find

0

ev

ity



1 π 2

x sin x d x.

[3]

y

iii Hence evaluate

[5] [2]

ni ve rs

C

d (sin x − x cos x ). dx

Pr

Show that the normal to the curve at Q passes through the point ( π, 0).

op y

i

ie

op

[3]

w ie ev

1 3. 8

-R

sin 3x sin x d x =

[3]

Cambridge International A Level Mathematics 9709 Paper 31 Q4 June 2010

s

-C



1 π 3 1 π 6

1 (cos 2 x −  cos 4x ) ≡  sin 3x sin x. 2

C

U

e

Hence show that

am

ii

Using the expansion of cos(3x − x ) and cos(3x + x ), prove that

id g

15 i

br

R

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q8 November 2010

es

w

s

-C

1 1 The diagram shows the curve y = x sin x, for 0 ø x ø π . The point Q  π, π  lies on the curve. 2 2 

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op

y

ve rs ity ni

C

U

ev ie

w

ge

-R

am br id

Pr es s

-C y

ni

C op

y

ve rs ity

op C w ev ie

Chapter 6 Numerical solutions of equations y

op

w

ge

C

U

R

ni

ev

ve

rs

w

C

ity

op

Pr

y

es

s

-C

-R

am

br

ev

id

ie

w

ge

U

R ie

133

id

es

s

-C

-R

am

br

ev

locate approximately a root of an equation, by means of graphical considerations and/or searching for a sign change understand the idea of, and use the notation for, a sequence of approximations that converges to a root of an equation understand how a given simple iterative formula of the form xn +1 = F( xn ) relates to the equation being solved, and use a given iteration, or an iteration based on a given rearrangement of an equation, to determine a root to a prescribed degree of accuracy.

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

■ ■ ■

ie

In this chapter you will learn how to:

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ve rs ity

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

ev ie

w

PREREQUISITE KNOWLEDGE

What you should be able to do

Check your skills

IGCSE / O Level Mathematics

Substitute numbers for letters in complicated formulae and algebraic expressions, including those involving logarithmic functions, exponential functions and major and minor trigonometric functions.

1 Evaluate each of the following expressions when x = 1.5. 2 1 a x+ 2 3 x 

U

R

y

b

ie ev -R

es

s

Understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations.

c

Pr

y = x 2 − 6x + 10 y=x

ity

op

y

4 Find the roots of the equation x 2 − 7 x + 10 = 0.

C w

ie

ev

-R

Sketch graphs of quadratic or cubic functions or the trigonometric functions sine, cosine, tangent.

ity

Pr

es

s

5 Sketch the graph, y = f( x ), of each function. a

f( x ) = x 3 + 1

b

f( x ) = cos 2 x − 3

c

f( x ) = 4x 2 − 16x

op

y

6 Sketch the graph, y = g( x ), of each function. ex a g( x ) = 2 b g( x ) = ln( x − 1)

ev

ie

id g

w

C

ni ve rs

Sketch graphs of exponential or logarithmic functions or the trigonometric functions cosecant, secant, cotangent.

c

g( x ) = cosec x

es

s

-R

br am

y − 3x + 7 w y 1 + 3= 3 5 x 0=

3 Show that the x-coordinates of the solutions to the simultaneous equations:

rs ve

Know that the values of x that make both sides of an equation equal are called the roots of the equation.

e

U

R

ev

Chapters 2 and 3

-C

y = 4x 2 + 13, given x > 0

also satisfy the equation x 2 − 7 x + 10 = 0.

ni

U ge id br am

-C

op y C w

ln x + 5

Also, the roots of f( x ) = 0 are the x-intercepts on the graph of y = f( x ).

IGCSE / O Level Mathematics Pure Mathematics 1 Coursebook, Chapter 5

ie

d

2 Rearrange each formula so that x is the subject.

w

ge id br am

-C

y op C w ie ev

Pure Mathematics 1 Coursebook, Chapter 1

R

e−x

a

Pure Mathematics 1 Coursebook, Chapters 1 and 2

134

c

C op

Rearrange complicated formulae and equations.

ni

IGCSE / O Level Mathematics

b 100 sin x − 99x, (x in radians)

ve rs ity

ev ie

w

C

op

y

Pr es s

-C

-R

Where it comes from

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Chapter 6: Numerical solutions of equations

w

ge

Why solve equations numerically?

am br id

ev ie

You are used to solving equations using direct, algebraic methods such as factorising or using the quadratic formula.

-C

-R

You might be surprised to learn that not all equations can be solved using a direct, algebraic process.

op

y

Pr es s

Numerical methods are ways of calculating approximate solutions to equations. They are extremely powerful problem-solving tools and many are available. They are widely used in engineering, computing, finance and many other applications.

C op

w

id



ie



ni



understand when the results are likely to be reasonable understand how to use available software correctly select an appropriate method when choices are available write our own programs when we need to do so, if we have the programming skills!

U

R



y

Knowing how a numerical method works helps us to:

ge

ev ie

w

C

ve rs ity

A well-programmed computer is quickly able to solve a complicated equation using a numerical method. So why do we need to understand how the method works?

s

-C

-R

am

br

ev

To be a powerful problem-solving tool, the method must be used correctly. This might mean the method needs to work quickly or work to obtain a particular level of accuracy. When used incorrectly, numerical methods might be very slow or even fail to work, as we will see later in this chapter.

op

Pr

y

es

In this chapter we will focus on one numerical method, the numerical solution of an equation using an iterative formula.

As we will see throughout this chapter, a good starting point is very important.

rs

ie

w

C

ity

6.1 Finding a starting point

ni

op

y

ve

We need a method to find an approximate solution to an equation first.

ev

135

C

U

R

We might find this using, for example, a graphical approach or the change of sign approach.

br

ev

id

ie

w

ge

For suitable functions, each of these approaches will produce two values: one above and one below the solution that we are finding.

-R

am

WORKED EXAMPLE 6.1

op y

es

s

-C

By sketching graphs of y = x 3 and y = 4 − x, show that the equation x 3 + x − 4 = 0 has a root α between 1 and 2.

Pr

Answer

ity

4

y = x3

1

4

w ev

ie

id g

es

s

-R

br am -C

x

op

y 2

C

O

TIP

e

U

R

ev

ie

w

y = 4–x

ni ve rs

C

y

Copyright Material - Review Only - Not for Redistribution

Make sure your sketch shows all the key features of each graph, such as the coordinates of turning points and the coordinates of any intercepts.

ve rs ity

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

ev ie

There is one point of intersection and its x-coordinate is between 1 and 2. At this point x 3 = 4 − x .

-R

This means that the equation x 3 = 4 − x has one root between 1 and 2.

Pr es s

-C

Rearranging this equation gives x 3 + x − 4 = 0.

C

ve rs ity

op

y

Hence x 3 + x − 4 = 0 has a root, α , between 1 and 2.

ev ie

w

WORKED EXAMPLE 6.2

U

R

ni

C op

y

Show, by calculation, that the equation f( x ) = x5 + x − 1 = 0 has a root α between 0 and 1.

ge

Answer

Find the value of α such that f(α ) = 0.

am

br

ev

id

ie

w

Why? Think about it and check the explanation that follows.

f(0) = 05 + 0 − 1 = −1

-R

As α is between 0 and 1, then f(0) might have the opposite sign to f(1).

ity

op C

rs

To see why the change of sign method works in this case, look at the graph of y = f( x ) .

ve

ie

w

This conclusion is important as it completes the argument.

Pr

y

The change of sign indicates the presence of a root, so 0 , α , 1.

136

es

s

-C

f(1) = 15 + 1 − 1 = 1

op

y

f(1) is positive

x

-R

am

br

1

ie

O

ev

id

w

ge

C

U

R

ni

ev

y

es

s

-C

There are no breaks f(0) is negative in the graph: it is continuous.

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

Since it starts below the x-axis at x = 0 and finishes above the x-axis at x = 1, it must cross the x-axis somewhere in between, say at x = α , where 0 , α , 1. So α is a root of the equation, that is, f(α ) = 0.

Copyright Material - Review Only - Not for Redistribution

ve rs ity ge

C

U

ni

op

y

Chapter 6: Numerical solutions of equations

am br id

ev ie

w

WORKED EXAMPLE 6.3

Pr es s

-C

-R

a By sketching a suitable pair of graphs, show that the equation cos x = 2 x − 1 (where x is in radians) has 1 only one root for 0 ø x ø π . 2 b Verify by calculation that this root lies between x = 0.8 and x = 0.9.

op

y

Answer

Why are these graphs suitable?

ve rs ity

C

a Draw the graphs of y = cos x and y = 2 x − 1.

w

y

π 2

y C op x

s

Pr

es

The graphs intersect once only and so the equation 1 cos x = 2 x − 1 has only one root for 0 ø x ø π . 2

ity

137

b cos x = 2 x − 1 so cos x − 2 x + 1 = 0

w

rs

C

op

y

-C

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am

br

ev

id

0.5

w

O

–1

y = cos x

ge

U

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ni

1

ie

ev ie

y = 2x – 1

y

ev

ve

ie

Let f( x ) = cos x − 2 x + 1 and so f( x ) = 0 then

U

R

ni

op

f(0.8) = cos 0.8 − 2(0.8) + 1 = 0.0967…

ge

C

f(0.9) = cos 0.9 − 2(0.9) + 1 = −0.1783…

am

br

ev

id

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w

Change of sign indicates the presence of a root.

-C

-R

EXERCISE 6A

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es

Pr

b Using your answer to part a, explain why the equation x 2 − 1 + x = 0 has two roots.

ity

Show, by calculation, that the smaller of these two roots lies between x = −1 and x = 0 .

ni ve rs

c

op

y

2 a By sketching graphs of y = x 3 + 5x 2 and y = 5 − 2 x, determine the number of real roots of the equation x 3 + 5x 2 + 2 x − 5 = 0.

br

ev

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3 The curve y = x 3 + 5x − 1 cuts the x-axis at the point (α , 0).

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id g

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C

U

b Verify by calculation that the largest root of x 3 + 5x 2 + 2 x − 5 = 0 lies between x = 0 and x = 2 .

es

s

b Using calculations, show that 0.1 , α , 0.5.

-R

am

a By sketching the graph of y = x 3 and one other suitable graph, deduce that this is the only point where the curve y = x 3 + 5x − 1 cuts the x-axis.

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1 a On the same axes, sketch the graphs of y = 1 + x and y = x 2.

Copyright Material - Review Only - Not for Redistribution

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

w

ge

4 a On the same axes, sketch the graphs of y = ln( x + 1) and y = 3x − 4.

am br id

ev ie

b Hence deduce the number of roots of the equation ln( x + 1) − 3x + 4 = 0 .

-R

5 a By sketching graphs of y = e x and y = x + 6, determine the number of roots of the equation e x − x − 6 = 0.

Pr es s

-C

b Show, by calculation, that e x − x − 6 = 0 has a root that lies between x = 2.0 and   x = 2.1.

b Show, by sketching the graph of y = e5 x and one other suitable graph, that this is the only root of this equation.

w ev ie

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C

op

y

6 a Show, by calculation, that ( x + 2)e5 x = 1 has a root between x = 0 and x = −0.2.

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C op

y

7 a Show by calculation that the equation cos−1 2 x = 1 − x has a root α , where 0.4 ø α ø 0.5.

w ie -R

-C

am

3π 4

s es

y

π 2

Pr ity

op

π 4

O

–0.5

0.5

1

x

y op

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w

–1

rs

C

138

y π

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br

id

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b The diagram shows the graph of y = cos−1 2 x where −0.5 ø x ø 0.5.

Pr

op y

es

s

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br

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C

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By drawing a suitable graph on a copy of the diagram, show that α is the only root of cos −1 2 x = 1 − x for −0.5 ø x ø 0.5. sin x π 8 The curve with equation y = where x , − intersects the line y = 1 at 2 2x + 3 the point (α , 1). π radians and one a By sketching the graph of y = sin x for −2 π , x , − 2 other suitable graph, deduce that this is the only point where the curve sin x y= intersects the line y = 1. 2x + 3

ity

9 a By sketching a suitable pair of graphs, show that the equation x 3 + 4x = 7 x + 4 has only one root for 0 ø x ø 5 .

op

10 a Show graphically that the equation 2 x = x + 4 has exactly two roots.

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b Verify by calculation that this root lies between x = 2 and x = 3.

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11 a By sketching two suitable graphs on the same diagram, show that the π equation cot x = x 2 has one root between 0 and radians. 2 b Show, by calculation, that this root, α , lies between 0.8 and 1.

am

PS

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b Show, by calculation, that the larger of the two roots is between 2.7 and 2.8.

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PS

ni ve rs

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b Using calculations, show that −2 , α , − 1.9.

Copyright Material - Review Only - Not for Redistribution

REWIND You might need to look back at Chapter 3 before attempting Question 11.

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12 a By sketching a suitable pair of graphs, deduce the number of roots of the equation x = tan 2 x for 0 , x , 2 π.

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Chapter 6: Numerical solutions of equations

b Verify, by calculation, that one of these roots, α , lies between 2.1 and 2.2.

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13 a Show graphically that the equation cosec x = sin x has exactly two roots for 0 , x , 2 π.

14 f( x ) = 20 x 3 + 8x 2 − 7 x − 3

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C

PS

Pr es s

b Use an algebraic method to find the value of the larger root correct to 3 significant figures.

op

y

-C

PS

ev ie

w

a The equation 20 x 3 + 8x 2 − 7 x − 3 = 0 has exactly two roots.

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C op

y

Without factorising the cubic equation, show, by calculation, that one of these roots is between 0.5 and 1.

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ge

b Show, by sketching the graph of y = f( x ) for −1 ø x ø 1, that the smaller root is between −1 and 0.

15 A guitar tuning peg is in the shape of a cylinder with a hemisphere at one end.

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PS

am

br

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Explain why it is not necessary to use a numerical method to find the two solutions of this equation.

c

r mm

Pr

20 mm

a Show that πr 3 + 30 πr 2 − 1200 = 0.

139

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b Show that the value of r is between 3 mm and 4 mm.

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ni

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EXPLORE 6.1

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ie

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rs

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y

The base radius of the cylinder is r mm .

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s

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The cylinder is 20 mm long and the whole peg is made from 800 mm 3 of plastic.

ie

id

He states:

w

ge

Anil is trying to find the roots of f( x ) = tan x = 0 between 0 and 2π radians.

-R

am

br

ev

f(3.1) . 0 and f(3.2) . 0. There is no change of sign so there is no root of tan x = 0 between 3.1 and 3.2.

-C

Mika states:

Pr

op y

es

s

y = tan x and it meets the x-axis where x = π radians so Anil has made a calculation error.

y

ni ve rs

DID YOU KNOW?

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am

br

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id g

w

e

C

U

op

Interestingly, cubic and quartic equations can be solved algebraically. This has been known since the 16th century and was shown by the Italian mathematicians Niccolò Fontana (Tartaglia) and Lodovico Ferrari. Quintic equations cannot be solved algebraically in general. This was shown by the Norwegian mathematician Niels Abel and the French mathematician Évariste Galois in the early 19th century.

-C

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C

ity

Discuss Anil and Mika’s statements.

Copyright Material - Review Only - Not for Redistribution

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6.2 Improving your solution

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Once we have an approximate location for a solution, we can improve it.

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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One common way to do this is to use an iterative process, finding values that are closer and closer to α , the root we are looking to find.

op

y

Pr es s

-C

For example, we have shown, by calculation, that the equation f( x ) = x5 + x − 1 = 0 has a root α between 0 and 1. This is a reasonable starting point but probably not accurate enough if we wanted a solution to this equation in the real world. What if we needed to know the value correct to 1 significant figure? How could we find it?

ni

C op

y

The root we are trying to find is a value of x, so rearranging x5 + x − 1 = 0 into the form x = F( x ) seems reasonable (in other words, x = some function of x). One possible rearrangement of this form is x = 5 1 − x .

ge

As we already know that 0 , α , 1, a good starting point is x = 0.5.

br

0.5 = 0.870 55...

Pr

No

rs

0.870 55 …

0.5

ity

op C

y

ve

w

Yes – then use this value as the estimate for the root. No – then find a more accurate value.

-R es

s

-C

y

ni ve rs

w -R

x

es

s

1

ev

y=51–x

br am -C

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e

y=x

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The graphs show that the values x and F( x ) are not very close together as the vertical distance between the line and curve is quite large.

op

ity

Pr

op y C w

0.5

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C

ev

id am

br

y 1

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Look at the graphs of y = x and y = 5 1 − x on the same diagram.

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ge

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op

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Are x and F( x ) the same correct to 1 significant figure?

Are the values the same to the required number of significant figures or decimal places?

ev



F( x ) = 5 1 − x

es

Compare it with x. x

140

-R

5

y



0.5 =

1 − x for the starting value.

s

51−

am

Work out the value of

-C



5

Remember from Section 6.1 that α is just the special name given to the solution that you are looking for.

ev

id

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Now we can test our value to see how accurate it is. How?

REWIND

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U

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ev ie

w

C

ve rs ity

Let us consider a rearrangement of f( x ) = x5 + x − 1 = 0.

Copyright Material - Review Only - Not for Redistribution

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C

U

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op

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Chapter 6: Numerical solutions of equations

w

ge

How do we improve the accuracy of the estimate for the root?

am br id

ev ie

We can see in the following diagram that 0.870 55... is a better approximation for the root α than 0.5.

U

R

ni

The vertical distance between the line and curve is now smaller, so the values are converging.

-R

am

br

ev

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ge ity

rs

y

ve

ni

C

0.664 38...

No

w

0.870 55...

x = 5 1 − 0.870 55... = 0.664 38...

br

ev

id

No

ie

0.870 55...

ge

0.5

141

Are x and F( x ) the same correct to 1 significant figure?

U

ie

w

51−

F( x ) = 5 1 − x

x

ev

es Pr

x

1

O

C

op

y

y=51–x

Testing x = 0.870 55... gives

R

s

-C

y=x

op

0.5

y

ev ie

w

C

ve rs ity

op

1

C op

y

Pr es s

-C

y

-R

Is 0.870 55... close enough?

s F( x ) = 5 1 − x

Pr

Are x and F( x ) the same correct to 1 significant figure?

0.762 27...

0.762 27...

0.750 26...

y

0.742 62...

No

op

0.742 62...

No

C

0.77413...

TIP

No

Yes

w

0.77413...

ity

0.72197...

No

ni ve rs

0.72197...

U

0.80383...

No

e

0.80383...

ie

id g

0.664 38...

-R

am

br

ev

Each test is called an iteration. Each iteration gives a value of x that is closer to the actual value of the root than the previous value of x.

es

s

It seems as if the value of α is 0.8 correct to 1 significant figure.

-C

R

ev

ie

w

C

op y

x

es

-C

When we continue testing and comparing values, we find:

-R

am

We can see that 0.664 38... is a better approximation of the root α than 0.870 55... because the values are closer together. Is 0.664 38... close enough?

Copyright Material - Review Only - Not for Redistribution

Using a table is helpful, but still very repetitive and a lot of work!

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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y C op es

s

-C

-R

am

br

ev

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start

id

0.5

w

ge

U

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ni

ev ie

w

C

ve rs ity

op

y

Pr es s

-C

1

am br id

y

ge

This (enlarged) diagram shows what is happening with each new value.

ity

op

Pr

y

y=x

y = 51 – x

y op x

1

br

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C

142

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am

Each value that comes from F( x ) is used as the next value for x. The spiral is often called the cobweb pattern.

es

s

Summary

Pr

op y

When trying to solve f( x ) = 0, rearrange the equation to the form x = F( x ).

U

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id g

w

C

e

The first approximation of the root is called x1 (sometimes it is called x0). The next approximation of the root is called x2 , and so on.

y

op

In order to avoid unnecessary repetition and to make sure the process is recorded accurately, the method outlined is usually described using subscript notation.

s es

am

xn +1 = F( xn ) = 5 1 − xn

-R

br

ev

A relationship such as x = F( x ) where F( x ) = 5 1 − x is the basis for an iterative formula:

-C

R

ev

ie

w

ni ve rs

Subscript notation and formulae

ity

C

When α is the value of x such that x = F( x ) then α is also a solution of f( x ) = 0.

Copyright Material - Review Only - Not for Redistribution

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C

U

ni

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Chapter 6: Numerical solutions of equations

x2 = 0.870 55...

No

x3 = 0.664 38...

No No No

ev ie

x8 = 0.762 27...

No No Yes

U

R

ni

x9 = 0.750 26...

No

y

C w

x7 = 0.742 62...

ve rs ity

op

x5 = 0.72197...

C op

y

x4 = 0.80383... x6 = 0.77413...

ev ie

Are they the same correct to 1 significant figure?

Pr es s

-C

am br id

xn +1 = F( xn ) = 5 1 − xn

-R

x1 = 0.5

w

ge

Looking at the values we already have we can see how much easier this is to record:

-R

am

br

TIP

ev

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w

ge

In this format we test the accuracy of the values by comparing each value of xn in the sequence with the previous value.

Does your calculator have an Ans key?

s

Pr

y

• Type the value for x1 into your calculator and press = .

es

-C

Using the Ans key reduces the number of key strokes needed to use an iterative process.

op

e.g. 0.5  =

143

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Ans

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51−

rs

e.g.

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C

• Type the formula into your calculator using Ans instead of xn .

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Try this now and check that you generate the values that are given in the table.

am

The equation x 2 + x − 3 = 0 has a negative root, α .

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WORKED EXAMPLE 6.4

es

3 − 1 with a starting value of x1 = −2.5 xn to find α correct to 2 decimal places. Give the result of each iteration to 4 decimal places, where appropriate.

Pr

ity

ni ve rs

Answer x2 + x − 3 = 0

op ev

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w

Isolate x.

-R



es

s

3 −1 x

br am

x=

C

U e

id g

x x 3 0 + − = x x x x 3 x +1− = 0 x

y

Divide through by x.

2

-C

ev

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C

op y

b Use the iterative formula xn +1 =

a

3 − 1. x

s

-C

a Show that this equation can be rearranged as x =

R

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ni

ev

Now each time you press = the calculator gives you the next value of xn generated (so x2 , x3 , x4 , …).

Copyright Material - Review Only - Not for Redistribution

ve rs ity

3 −1 xn

Are these the same correct to 2 decimal places?

ev ie

3 − 1 = −2.3636 −2.2

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C

x4 = −2.2692

w

x5 = −2.3220

ev ie

x6 = −2.2920

No No No

U

R

x8 = −2.2993

No No

ni

x7 = −2.3089

No

x9 = −2.3047

w

ge

Yes

y

2

op

ev

ity

w

rs

C

Change of sign indicates presence of root, therefore α = −2.30, to 2 decimal places.

op C

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ni

WORKED EXAMPLE 6.5

y

ve

ie ev

The change of sign test is an important test. If the root passes the change of sign test it must be correct.

Pr

f( −2.295) = ( −2.295) + ( −2.295) − 3 = −0.0279...

s

f( −2.305) = ( −2.305)2 + ( −2.305) − 3 = 0.00802...

es

-C

When α = −2.30 then −2.305 , α , − 2.295 and so

Any value between −2.305 and −2.295 must round to −2.30 to 2 decimal places.

-R

br

am

We can prove this, using a change of sign test:

TIP

ie

id

The root seems to be α = −2.30 to 2 decimal places.

144

y

op

-R

x3 =

No

Pr es s

-C

3 − 1 = −2.2 −2.5

y

x2 =

C op

xn +1 =

w

ge

x1 = −2.5

am br id

b

C

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ni

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ie

w

ge

The equation e x − 1 = 2x has a root α = 0.

br

ev

id

a Show by calculation that this equation also has a root, β , such that 1 , β , 2 .

-R

am

b Show that this equation can be rearranged as x = ln(2 x + 1) .

es

s

-C

c Use an iteration process based on the equation in part b, with a suitable starting value, to find β correct to 3 significant figures.

Pr

op y

Give the result of each step of the process to 5 significant figures.

ity

ni ve rs

a Let f( x ) = e x − 1 − 2 x = 0 and look for a change of sign.

y

f(1) = e − 1 − 2 = −0.2817…

C

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op

f(2) = e2 − 1 − 2(2) = 2.38905…

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br

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w

e

Change of sign indicates presence of root.

-C

R

ev

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w

C

Answer

Copyright Material - Review Only - Not for Redistribution

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Isolate e.

ev ie

ex − 1 = 2x

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b

C

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Chapter 6: Numerical solutions of equations

-R

ex = 2x + 1

y

ve rs ity

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c A suitable iterative formula is xn +1 = ln(2 xn + 1) with a starting value of x1 = 1.5. (Any value between 1 and 2 would be an acceptable starting value.)

ev ie

x1 = 1.5

xn +1 = = ln(2 xn + 1)

Are these the same correct to 3 significant figures?

y

w

C



Pr es s

-C

x = ln(2 x + 1)

Take logs to base e.

ni

C op

x2 = ln(2 × 1.5 + 1) = 1.38629… = 1.3863

U

R

x3 = ln(2 × 1.38629… + 1) = 1.32776… = 1.3278

w

ge

x4 = 1.29623… = 1.2962

id br

No

-R

am

x8 = 1.26051… = 1.2605

Yes

es

s

-C

Pr

The root seems to be β = 1.26 correct to 3 significant figures. 145

We can prove this, using a change of sign test:

ity

y op C

x7 = 1.26362… = 1.2636

No

No

ev

x6 = 1.26910… = 1.2691

No

No

ie

x5 = 1.27884… = 1.2788

No

w

rs

When β = 1.26 then 1.255 , β , 1.265 and so:

y

ev

ve

ie

f(1.255) = e1.255 − 1 − 2(1.255) = −0.0021616…

U

R

ni

op

f(1.265) = e1.265 − 1 − 2(1.265) = 0.01309…

w

ge

C

Change of sign indicates presence of root, therefore β = 1.26 correct to 3 significant figures.

-R

am

br

ev

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You might notice that the pattern of convergence is different in this example. This is clear when you look at the graphs of y = x and y = ln(2 x + 1) on the same grid. The type of convergence shown in the graph on the right is often called a staircase pattern.

s es y = ln(2x + 1)

y op w

2 x

ie

1

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-C

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br

O

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e

y=x

C

U

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ni ve rs

1

w

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C

Pr

op y

-C

y

2

Copyright Material - Review Only - Not for Redistribution

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am br id

EXERCISE 6B

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Pr es s

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1 The equation x 3 + 5x − 7 = 0 has a root α between 1 and 1.2. 7 − xn 3 can be used to find the value of α . The iterative formula xn +1 = 5 a Using a starting value of x1 = 1.1, find and write down, correct to 4 decimal places, the value of each of x2 , x3 , x4 , x5 , x6.

ve rs ity

2 a Show that the equation ln( x + 1) + 2 x − 4 = 0 has a root α between x = 1 and x = 2 .

U

R

ni

correct to 3 decimal places.

4 − ln( xn + 1) with an initial value of x1 = 1.5 to find the value of α 2

y

ev ie

b Use the iterative formula xn +1 =

C op

w

C

op

y

b Write the value of x6 correct to 2 decimal places and prove that this is the value of α correct to 2 decimal places.

w

ge

3 The equation ( x − 0.5) e3x = 1 has a root α .

ev

br

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a Show, by sketching the graph of y = e3x and one other suitable graph, that α is the only root of this equation. b Use the iterative formula xn +1 = e −3x + 0.5 with x1 = 0.5 to find the value of α correct to 2 decimal places. Give the value of each of your iterations to 4 decimal places.

s

-C

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am

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es

Pr

5 − xn 3 , with a suitable value for x1, to find the value of this root correct 9 to 4 decimal places. Give the result of each iteration to 6 decimal places.

b Use the iterative formula xn +1 =

w

rs

C

146

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op

y

4 a By sketching a suitable pair of graphs, show that the equation x 3 + 10 x = x + 5 has only one root that lies between 0 and 1.

y

ve

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5 The equation cos−1 3x = 1 − x has a root α .

ev

π π and . 15 12 1 b Show that the given equation can be rearranged into the form x = cos (1 − x ). 3 1 c Using the iterative formula xn+1 = cos (1 − xn ) with a suitable starting value, x1, find the value of α 3 correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

br

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C

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a Show, by calculation, that α is between

-R

5 − 2 xn − xn 3 5

s

x n +1 =

es

-C

am

6 The terms of a sequence with first term x1 = 1 are defined by the iterative formula:

Pr

ni ve rs

ity

a Use this formula to find the value of α correct to 2 decimal places. Give the value of each term you calculate to 4 decimal places.

C

U

7 The equation x 4 − 1 − x = 0 has a root, α , between  x = 1 and x = 2 .

w

e

a Show that α also satisfies the equation x = 4 1 + x .

id g

op

y

b The value α is the root of an equation of the form ax 3 + bx 2 + cx + d = 0 , where a, b, c and d are integers. Find this equation.

ev

br

es

s

-R

Use your iterative formula, with a starting value of x1 = 1.5, to find α correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

am

c

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b Write down an iterative formula based on the equation in part a.

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ev

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w

C

op y

The terms converge to the value α .

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ve rs ity

w

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8 The equation cosec x = x 2 has a root, α , between 1 and 2.

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Chapter 6: Numerical solutions of equations

-R

am br id

ev ie

 1  1 The equation can be rearranged either as x = sin−1  2  or x = . x  sin x a Write down two possible iterative formulae, one based on each given rearrangement.

-C

Use the starting value 1.5.

ve rs ity

C

Show that the other formula from part a converges to α and find the value of α correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

9 The terms of a sequence, defined by the iterative formula xn +1 = ln( xn 2 + 4), converge to the value α . The first term of the sequence is 2.

ev ie

w

Pr es s

c

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y

b Show that one of the formulae from part a fails to converge.

ni

C op

y

a Find the value of α correct to 2 decimal places. Give each term of the sequence you find to 4 decimal places.

U

R

b The value α is a root of an equation of the form x 2 = f( x ). Find this equation.

x

es

s

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3 11 The graphs of y =   − 1 and y = x intersect at the points O(0, 0) and A.  2 a Sketch these graphs on the same diagram.

Pr

b Using logarithms, find a suitable iterative formula that can be used to find the coordinates of the point A.

y

op

12 The diagram shows a container in the shape of a cone with a cylinder on top.

ni

PS

ve

w ie ev

Calculate the length of the line OA, giving your answer correct to 2 significant figures. Give the value of any iterations you calculate to a suitable number of significant figures.

rs

c

ity

C

op

y

PS

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am

br

ev

id

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w

ge

sin ( x − 1) + 1 = 0 has a root, α , between   x = 1 and x = 1.4. 2x − 3 3 − sin ( x − 1) a Show that α also satisfies the equation x = . 2 b Using an iterative formula based on the equation from part a, with a suitable starting value, find the value of α correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

10 The equation

C

U

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The height of the cylinder is 3 times its base radius, r.

ie

w

ge

The volume of the container must be 5500 cm 3.

br

ev

id

The base of the cone has a radius of r cm.

-R

am

a Write down an expression for the height of the cone in terms of r. 3r 33 cm

s

The equation in part b has a root, α .

r

5500 − 11πr 2 . 2π d Using an iterative formula based on the equation in part c, and a starting value of 8, find the value of α correct to 3 decimal places. Give the value of each of your iterations to an appropriate degree of accuracy. 3

ni ve rs

ity

Pr

Show that α is also a root of the equation r =

op

id g

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e

C

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13 The equation x 3 − 7 x 2 + 1 = 0 has two positive roots, α and β , which are such that α lies between 0 and 1 and β lies between 6 and 7.

es

s

-R

br

ev

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By deriving two suitable iterative formulae from the given equation, carry out suitable iterations to find the value of α and of β , giving each correct to 2 decimal places. Give the value of each of your iterations to 4 decimal places.

am

PS

y

e Explain what information is given by your answer to part d, in the context of the question.

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c

es

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b Show that 2 πr 3 + 11πr 2 − 5500 = 0 .

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147

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14 a Show graphically that the equation cot x = sin x has a root, α , which is such that 0 , α ,

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PS

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

Using an iterative formula based on the equation in part b, with an initial value of 0.9, find the value of α correct to 2 decimal places. Give the value of each iteration to an appropriate number of decimal places.

-R

c

ev ie

b Show that the equation in part a can be rearranged as x = sin−1 cos x .

π . 2

Pr es s

ni

C op

w ev ie

EXPLORE 6.2

y

ve rs ity

Find an example of a formula, based on a different rearrangement of the equation given in part a, that also converges to the value of α , with an initial value of 0.9, in fewer iterations than the formula used in part c.

C

op

y

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d The equation given in part a can be rearranged in other ways.

U

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You are going to explore why using rearrangements of equations into the form x = F( x ) sometimes fails.

w

ge

Possible causes of failure to consider are the starting value and the formula itself.

br

ev

id

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Discuss why formulae based on: ●

x = 51− x



x = ln(2 x + 1) (the staircase pattern)

-C

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am

(the cobweb pattern)

es

s

each converged to a root.

ity

y

C

148

Pr

op

y

Look at the equation e x = 3x + 1 and some related graphs. y

y=

y op C

y = ex

ie O

x

x

-C

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am

O

ev

id br

y=x

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ni

ev

ve

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w

rs

y = 3x + 1

ex – 1 3

s

ex − 1 . 3 Does this give a convergent iterative formula? Explain your answer.

ni ve rs

op

y

Using one of the equations in questions 1 to 5 or 7 to 14, draw the graph of y = x and the graph of a function F( x ) , where x = F( x ) is a rearrangement of the equation given and for which x = F( x ) will fail to converge.

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br

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e

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U

Explain any similarities or differences between your example and the rearrangement x =

-C

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ev

ie

w

C

Look at the equations in Exercise 6B.

ity

Pr

op y

es

Discuss the rearrangement x =

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ex − 1 . 3

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Chapter 6: Numerical solutions of equations

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ev ie

am br id

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6.3 Using iterative processes to solve problems involving other areas of mathematics

2

2

y

Pr es s

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1  1 − cos x   1 + cos x  + = cot 2 x + . a Show that    2  2 sin x   2 sin x  2 2  1 − cos x   1 + cos x  + b Hence, given that α is a root of the equation   2 sin x  = x, show that α is also a root of  2 sin x  π 2 the equation x = tan−1 for 0 , x , . 2 2x − 1 π 2 for 0 , x , . Verify by calculation c It is given that α is the only root of the equation x = tan−1 2 2x − 1 that the value of α lies between 0.9 and 1.0.

ni

C op

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ve rs ity

op C w ev ie

-R

WORKED EXAMPLE 6.6

ie ev

2

am

1 + 2 cos x + cos2 x 1 − 2 cos x + cos2 x + 4 sin2 x 4 sin2 x

=

2 + 2 cos2 x 4 sin2 x

=

1 + cos2 x 2 sin2 x

=

cos2 x 1 + 2 sin2 x 2 sin2 x

es

Pr ity rs ve

op

Simplify.

w

ge

C

Write in terms of cotx.

ev

id

ie

Collect terms.

br

Pr

2x − 1 2

tan x =

2 2x − 1

C

U

Take the reciprocal of each side.

e

ie

w

Make x the subject.

ev

2 2x − 1

es

s

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id g

br

y

1 = tan x

Left-hand side: write in terms of tan x. Right-hand side: write as a single fraction.

op

x−

am

Square root.

ity

1 2

cot x =

x = tan−1

-C

Rearrange.

ni ve rs

w

1 2

ie ev

-R s es

1 =x 2

cot 2 x = x −

R

149

Write as separate fractions.

am

-C

cot 2 x +

C

op y

b

Simplify.

U

R

cosec 2 x cot 2 x + 2 2 2 1 + cot x cot 2 x = + 2 2 1 = cot 2 x + 2 =

Collect terms.

ni

ev

ie

w

C

op

y

-C =

Square.

y

 1 − cos x   1 + cos x   2 sin x  +  2 sin x 

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2

a

s

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id

Answer

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d Using an iterative formula based on the equation in part c, find the value of α correct to 3 significant figures.

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ve rs ity

ge

C

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

2 = 0.44683… 2(1) − 1

2 2 xn − 1

xn +1 = tan −1

ev ie

2 = 0.979 92… 2(0.95) − 1

No

y

ev ie

2 = 0.964 94… 2(0.97992…) − 1

x5 = 0.968 66… x6 = 0.970 48…

-C

w No

ev

x4 = 0.972 34…

No No Yes

es

s

x7 = 0.969 58…

y

Pr

α seems to be 0.970 correct to 3 significant figures. We can prove this, using a change of sign test:

ity

op

When α = 0.970 then  0.9695 , α , 0.9705 and so 2 = −0.000571... 2(0.9695) − 1

y op

2 = 0.000924... 2(0.9705) − 1

C

ge

f(0.9705) = 0.9705 − tan−1

w

U

R

ni

ev

f(0.9695) = 0.9695 − tan−1

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ie

w

rs

C

150

No

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am

br

id

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x3 = tan−1

ie

U

R

ni

x2 = tan−1

Are they the same correct to 3 significant figures?

C op

x1 = 0.95

ve rs ity

op

y

Change of sign indicates the presence of a root.

w

C

-R

-C

f(1) = 1 − tan −1

2 = −0.10685… 2(0.9) − 1

Pr es s

am br id

f(0.9) = 0.9 − tan −1

w

2 and so f( x ) = 0 then: 2x − 1

Let f( x ) = x − tan −1

c

s

-C

-R

am

Therefore  α = 0.970 correct to 3 significant figures.

ev

br

id

ie

Change of sign indicates presence of root.

Pr

op y

es

WORKED EXAMPLE 6.7

The diagram shows the design of a company logo, ABC .

ity

ni ve rs

op

br

ev

ie

id g

w

b Showing all your working, use an iterative formula based on the equation in part a, with an initial value of 1.85, to find θ correct to 3 significant figures.

es

s

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c Hence find the length of AB, given that AC is 8 cm.

am

C

θ

C

e

a Show that θ = 2 sin θ .

U

Angle ACB is θ radians.

y

The area of the unshaded segment is to be the same as the area of the shaded triangle.

-C

R

ev

ie

w

C

AB is an arc of a circle with centre C.

A

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B

ve rs ity

ev ie

w

ge am br id

-R

Divide through by

Pr es s

θ = 2 sin θ

ve rs ity

θ n + 1 = 2 sin θ n

ev ie

Add sin θ . ✓

Are they the same correct to 3 significant figures?

θ 2 = 2 sin1.85 = 1.922 55…

No No

U

θ 3 = 2 sin1.922 55… = 1.877 53…

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ni

No

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w

θ 4 = 1.906 64…

-R

θ 6 = 1.900 05…

ity

No

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w

θ10 = 1.896 25… θ11 = 1.895 00…

Yes

op

ni

ev

151

No

rs

C

θ 9 = 1.894 30…

No

No

Pr

op

y

θ8 = 1.897 35…

No

No

s

θ 7 = 1.892 56…

es

-C

am

br

ev

θ5 = 1.888 26…

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op C

θ 1 = 1.85

w

b

θ seems to be 1.90 correct to 3 significant figures.

ge

C

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1 2 r . 2

y

sin θ = θ − sin θ

y

-C

1 2 1 r sin θ = r 2 (θ − sin θ ) 2 2

C op

Answer a

C

U

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op

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Chapter 6: Numerical solutions of equations

id

ie

w

We can prove this, using a change of sign test with f(θ ) = θ − 2 sin θ = 0:

am

f(1.895) = 1.895 − 2 sin1.895 = −0.000809...

s

-C

f(1.905) = 1.905 − 2 sin1.905 = 0.01565...

-R

br

ev

When θ = 1.90 then  1.895 < θ < 1.905 and so

y op -R s es

-C

am

br

ev

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e

C

U

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ev

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ni ve rs

C

ity

Pr

op y

es

Change of sign indicates presence of root, therefore θ = 1.90 correct to 3 significant figures.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ge

C

U

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

ev ie

w

c Using trigonometry:

Pr es s

-C 0.95

ev ie

w

B

0.5 AB 8 16 sin 0.95 = AB

Multiply by 16.

ni U

ie

w

ge

R

sin 0.95 =

y

C

ve rs ity

op

y

C

C op

8 cm

-R

A

-R s

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EXERCISE 6C

am

br

ev

id

AB = 13.014..... = 13 cm (correct to 2 significant figures)

es

e2 x + x has a stationary point with x-coordinate lying between 1 and 2. x3 3 x a Show that the x-coordinate of this stationary point satisfies the equation x = + 2 x . 2 e b Using a suitable iterative process based on the equation given in part a, find the value of the x-coordinate of this stationary point correct to 4 decimal places. Give each iteration to 6 decimal places.

Pr

ni

y

ev

2 The parametric equations of a curve are x = t 2 + 6, y = t 4 − t 3 − 5t.

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C

152

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1 The curve with equation y =

ge

C

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The curve has a stationary point for a value of t that lies between 1 and 2.

ev

id

ie

3t 2 + 5 4

br

3

am

t =

w

a Show that the value of t at this stationary point satisfies the equation

es

Hence find the coordinates of the stationary point, giving each coordinate correct to 1 significant figure.

Pr

3 In the diagram, triangle ABC is right-angled and angle BAC is θ radians. The point O is the mid point of AC and OC = r. Angle BOC is 2θ radians and BOC is a sector of the circle with centre O. The area of triangle ABC is 2 times the area of the shaded segment.

op

y

ni ve rs

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B

U

a Show that θ satisfies the equation sin 2θ = θ .

es

C

w ie

s

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br

am



θ

A

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id g

e

π b This equation has one root in the interval 0 , θ , . 2 Use the iterative formula θ n+1 = sin 2θ n to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

-C

R

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op y

c

s

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b Use an iterative process based on the equation in part a to find the value of t correct to 3 decimal places. Show the result of each iteration to 6 decimal places.

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O

r

C

ve rs ity

y

w ev ie

Pr es s

ve rs ity

π 8

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U

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EXPLORE 6.3

s

-C

The golden ratio

153 1

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C

φ–1

1

ity

1

1

es Pr

op

y

φ

y

op

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br

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C

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ni

ev

When a rectangle with sides in the ratio 1: φ (phi) can be cut into a square of side 1 and a smaller rectangle, also with sides in the ratio 1: φ , the original rectangle is called a golden rectangle and the ratio 1: φ is called the golden ratio. φ −1 1 = . This means that φ 1 By using a suitable iterative formula based on this ratio, with a starting value of 1, find the value of φ correct to 5 significant figures.

-R

Explore the link between the golden ratio and the sequence of Fibonacci numbers:

Pr

op y

es

s

-C

1, 1, 2, 3, 5, 8, ..., ...

ni ve rs

The relationship between the Fibonacci numbers and the golden ratio produces the golden spiral.

w ie ev

-R s es

am

br

id g

e

From a tight centre, petals flow outward in a spiral, growing wider and larger as they do so. Nature has made the best use of an incredible structure. This perfect arrangement of petals is why the small, compact rose bud grows into such a beautiful flower.

C

U

op

y

This can be seen in many natural settings, one of which is the spiralling pattern of the petals in a rose flower.

-C

ev

ie

w

C

ity

DID YOU KNOW?

R

x

y

O

C op

ev ie

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C

op

y

-C

b Show also that the x-coordinate of P satisfies  1  1 the equation x = tan−1  .  2x  4 c Using an iterative formula based on the equation in part b with initial value x1 = 0.3 find the x− - coordinate of P correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

d Use integration by parts twice to find the exact π area enclosed between the curve and the x-axis from 0 to . 8

P3

P

-R

am br id

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4 The diagram shows the curve y = x 2 cos 4x for π 0 ø x ø . The point P is a maximum point. 8 a Show that the x-coordinate of P satisfies the equation 4x 2 tan 4x = 2 x.

C

U

ni

op

y

Chapter 6: Numerical solutions of equations

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

Each time you generate a value of x you carry out an iteration.



When the point of intersection is at x = α and your iterations are values that are getting closer and closer to α , you are converging to α . Is it accurate enough?

Yes → Stop, No →



Let x2 = F( x1 )

work out F( x2 ).

Is it accurate enough?

Yes → Stop, No →



Let x3 = F( x2 )

work out F( x3 ).

Is it accurate enough?

Yes → Stop, No →



Let xn+1 = F( xn )



The relationship xn +1 = F( xn ) is called an iterative formula.



The sequence of values of x given by this formula are called x1, x2 , x3 , ..., xn.



When the value found is accurate enough, then it is usually given a particular name such as α .



The success and speed of the iterative process depends on the function chosen for F( xn ) and the value chosen for x1.

y op y op -R s es

-C

am

br

ev

ie

id g

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e

C

U

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ev

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w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

am

br

ev

id

ie

w

ge

C

U

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ni

ev

ve

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rs

C

ity

op

Pr

y

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s

-C

-R

am

br

ev

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U

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Yes → Stop.

C op



work out F( xn+1). Is it accurate enough?

y

work out F( x1 ).

ve rs ity

Start with x1

op



y

Pr es s

-C



C w ev ie

R 154

w

The process of finding a sequence of values that become closer and closer to a point of intersection of y = x and y = F( x ) is called an iterative process.

-R



ev ie

ge

Checklist of learning and understanding

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ve rs ity ge

C

U

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Chapter 6: Numerical solutions of equations:

ev ie

am br id

6 The terms of the sequence generated by the iterative formula xn +1 = 7 converge to α .

Pr es s

State an equation satisfied by α , and hence find the exact value of α . x The equation x 3 + − 8 = 0 has one real root, denoted by α . 2 a Find, by calculation, the pair of consecutive integers between which α lies.

b

C op

[2]

U

Show by calculation that this root lies between 0.5 and 1.

w

ie

ev

-R

ln(14 − x 3 ) − 1 Show that this root also satisfies the equation x = . 2 Use an iteration process based on the equation in part c, with a suitable starting value, to find the root correct to 4 decimal places. Give the result of each step of the process to 6 decimal places.

s

ity

The sequence of values given by the iterative formula xn+1 = value x1 = 1, converges to α .

y

op

ni

ev

ie

ev

id

br

[1]

Find the value of p and the value of q.

[2]

d

Hence find an approximate value of y that satisfies the equation 5 y − 2 = y ln 5.

[2]

a

By sketching a suitable pair of graphs, show that the equation 7 − x 6 = x 2 − 1 has exactly two real roots, α and β , where α is a positive constant.

Pr

ity

y

8 − x. x Write down an iterative formula based on the equation in part a. Use this formula, with a suitable starting value, to find the value of α correct to 3 significant figures. Give the result of each iteration to 5 significant figures. 5

op

Show that α satisfies the equation x =

-R s es

-C

[3] [1]

C

ev

br

Hence write down the value of β correct to 3 significant figures.

am

d

ie

id g

w

e

c

ni ve rs

b

U

op y C w ie ev

-R

[3]

es

Use the iterative formula based on the equation in part a with starting value x = p, to find the value of α correct to 3 decimal places.

c

6

s

-C

b

[2]

Show that α also satisfies the equation x = ln x + 2.

am

a

[3]

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ge

C

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R

5

R

xn (1 + sec xn ) − tan xn with initial sec 2 xn − 1

Use this formula to find α correct to 2 decimal places, showing the result of each iteration to 4 decimal places. π b Show that α also satisfies the equation tan x − 2 x = 0 where 1 ø x , . 2 ln x 2 − 6 = x − 5 has a root α , such that p , α , q, where p and q are consecutive The equation 2 integers.

a

[3]

2

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rs

C

4

[1]

es

Pr

d

op

y

-C

c

[2]

[3]

am

b

[2]

By sketching a suitable pair of graphs, show that the equation e2 x +1 = 14 − x 3 has exactly one real root.

ge

a

[2]

[3]

br

3

[3]

Use this iterative formula to determine α correct to 1 decimal place. Give the result of each iteration to 4 decimal places.

id

R

c

8 1 − converges, then xn 2

y

Show that, if a sequence of values given by the iterative formula xn +1 = it converges to α .

ni

b

ve rs ity

y op ev ie

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C

2

 x + 1  with initial value x = 1.5, 1  n x 3  n

Use this formula to find α correct to 2 decimal places. Give the result of each iteration to an appropriate number of decimal places.

-C

a

-R

1

w

END-OF-CHAPTER REVIEW EXERCISE 6

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[3] [1]

155

ve rs ity

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a

ve rs ity

C op

ni

ev ie

[3]

y

A curve has parametric equations 1 x = , y = t +1 t

8

w

C

op

y

b

[4]

Pr es s

-C

a

ev ie

am br id

 1 + 2e6 x  dx = 0.6 , where the constant a > 0.  0  3x + 5 1  5  Show that a = ln  2.8 + ln  .  3a + 5   6  Use an iterative formula based on the equation in part a, with a starting value of a = 0.2, to find the value of a correct to 3 decimal places. Give the result of each iteration to an appropriate number of decimal places.

It is given that

7

w

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C

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

U

R

The point P on the curve has parameter p. It is given that the gradient of the curve at P is −0.4.

ie

es

s

A

Pr

ity

rs

c

Use the iterative formula θ n+1 = cos−1

θ r

O

C

[3] B

[2]

ve

Verify by calculation that θ lies between 0.8 and 1.2 radians.

op

y

3π to find the value of θ correct to 3 decimal places. 32θ n Give the result of each iteration to 5 decimal places.

[3]

w

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C

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y

b

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156

[3]

ev

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In the diagram, A, B and C are points on the circumference of the circle with centre O and radius r. The shaded region, ABC , is a sector of the circle with centre C. Angle OCA is equal to θ radians. 3 The area of the shaded region is equal to of the area of the circle. 8 3π . a Show that θ = cos−1 32θ

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9

[3]

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b

1

2 5 ( p + 1) 4 . 5 Use an iterative process based on the equation in part a to find the value of p correct to 3 decimal places. Use a starting value of 1 and show the result of each iteration to 5 decimal places. Show that p =

id

a

y

br

ev

id

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10 The functions f and g are defined, for 0 ø x ø π , by f( x ) = e x − 2 and g( x ) = 5 − cos x.

y = 5 – cos x

-R

am

The diagram shows the graph of y = f( x ) and the graph of y = g( x ).

es

Pr

Given that p , q , verify by calculation that p is 0.16 correct to 2 decimal places.

ity

y = ex–2

b

Show that q satisfies the equation q = 2 + ln(sin q ) .

c

Given also that 1.5 , q , 2.5 , use the iterative formula qn+1 = 2 + ln(sin qn ) to calculate q correct to 2 decimal places, showing the result of each iteration to 4 decimal places.

[1] [3]

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[3]

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[4] x

O

11 The equation x = cos x + sin x has a root α that lies between 1 and 1.4.  π a Show that α is also a root of the equation x = 2 cos  x −  .  4  π b Using the iterative formula xn+1 = 2 cos  xn −  , find the value of α , giving your answer  4 correct to 2 decimal places.

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a

s

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The gradients of the curves are equal both when x = p and when x = q.

Copyright Material - Review Only - Not for Redistribution

[3]

ve rs ity

ev ie

am br id

π π π π By sketching each of the graphs of y = sec x and y =  − x   + x  for − , x ø on the 2 4  2 2 π  π  same diagram, show that the equation sec x =  − x   + x  has exactly two real roots in 2  4  π π the interval − , x ø . 2 2

-R

Pr es s

y

Verify by calculation that the smaller root, α , is −0.21 correct to 2 decimal places.

[2]

y

ve rs ity

[1]

w

ev

id

ie

Using an iterative formula based on the equation given in part b, with an initial value of 1, find the value of β correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

[3]

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am

br

d

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U

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c

[3]

π  π  Show that the equation sec x =  − x   + x  can be written in the form 2  4  2 2 πx + π − 8 sec x . x= 8 π  π  π π The two real roots of the equation sec x =  − x   + x  in the interval − , x ø are denoted 2  4  2 2 α and β .

ni

op ev ie

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b

C op

12 a

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P3

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Chapter 6: Numerical solutions of equations:

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FAST FORWARD

13

157

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y

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P3

Pr

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You need to have studied the skills in Chapter 8 before attempting questions 13 and 14.

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1

y = cos3 x

α

π 6

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s

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  0.9 Use the iterative formula α n+1 = sin−1   with α1 = 0.2, to find the value of α correct 2  3 − sin α n  to 3 significant figures. Give the result of each iteration to 5 significant figures.

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ev

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am -C

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C

4+a . ln a Use an iterative formula based on the equation in part a to find the value of a correct to 3 decimal places. Use an initial value of 5 and give the result of each iteration to 5 decimal places. Show that a =

br

b

ln x d x = 5, where a is a constant greater than 1.

1

id g

a



a

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14 It is given that

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x

The diagram shows part of the curve y = cos3 x, where x is in radians. The shaded region between the curve, the axes and the line x = α is denoted by R. The area of R is equal to 0.3. α 0.9 a Using the substitution u = sin x, find cos3 x d x . Hence show that sin α = . 3 − sin2 α 0

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π 3

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Copyright Material - Review Only - Not for Redistribution

[5] [3]

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ev ie

am br id

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1+ x 1 The equation of a curve is y = for x . − . Show that the gradient of the curve is 1 + 2x 2 always negative.

Pr es s

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ve rs ity

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y

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Pr

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y

1 1 tan 2 x for 0 ø x ø π . Find the x-coordinates 2 2 of the points on this part of the curve at which the gradient is 4. The diagram shows the part of the curve y =

rs

2 The parametric equations of a curve are x = ln(1 − 2t ), y = , for t , 0. t dy 1 − 2t i Show that = 2 . dx t ii Find the exact coordinates of the only point on the curve at which the gradient is 3.

[3]

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Cambridge International AS & A Level Mathematics 9709 Paper 21 Q4 November 2012

[4]

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s

-C

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am

The equation of a curve is x 2 y + y2 = 6x. dy 6 − 2 xy i Show that = . dx x 2 + 2 y ii Find the equation of the tangent to the curve at the point with coordinates (1, 2), giving your answer in the form ax + by + c = 0.

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0

1 d x , giving 6 + 2e x

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U





1

y

Use the trapezium rule with two intervals to estimate the value of your answer correct to 2 decimal places. (e x − 2)2 ii Find dx . e2 x

i

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Cambridge International AS & A Level Mathematics 9709 Paper 21 Q6 June 2010

[3] [4]

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Cambridge International AS & A Level Mathematics 9709 Paper 21 Q6 November 2012

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C w ie

[3]

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3

P2 5

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[5]

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q3 November 2011

4

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x

1π 2

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1π 4

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Cambridge International A Level Mathematics 9709 Paper 31 Q1 November 2013

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1

2

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CROSS-TOPIC REVIEW EXERCISE 2

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ev ie

[2]

Verify by calculation that 4.5 , α , 5.0.

[2]

Pr es s

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Cambridge International AS & A Level Mathematics 9709 Paper 21 Q4 November 2016

rs

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U

ni

ev

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Explain why the gradient of the curve is never negative.

w ev

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1 1 a find the exact value of tan π + cot π, 8 8

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1 π 2

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6 dθ . tan θ + cot θ

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2 . sin 2θ

Hence

b evaluate

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Prove that tan θ + cot θ ≡

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i

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Cambridge International A Level Mathematics 9709 Paper 31 Q5 November 2015

am

10

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ii

op

C

U

The equation of a curve is xy( x − 6 y ) = 9a 3, where a is a non-zero constant. Show that there is only one point on the curve at which the tangent is parallel to the x-axis, and find the coordinates of this point.

w

ie

[7]

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Cambridge International A Level Mathematics 9709 Paper 31 Q4 November 2016

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11

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Cambridge International AS & A Level Mathematics 9709 Paper 21 Q5 June 2014

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id br



a

It is given that the positive constant a is such that (4e2 x + 5) d x = 100 . −a 1 i Show that a = ln(50 + e −2 a − 5a ) . 2 1 ii Use the iterative formula an +1 = ln(50 + e −2 a − 5an ) to find α correct to 3 decimal places. 2 Give the result of each iteration to 5 decimal places.

iii Find the value of x for which the gradient is least.

R

[3]

Cambridge International A Level Mathematics 9709 Paper 31 Q4 November 2014

1 The equation of a curve is y = e −2 x tan x, for 0 ø x , π. 2 dy and show that it can be written in the form e −2 x ( a + b tan x )2, i Obtain an expression for dx where a and b are constants.

9

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8

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Cambridge International AS & A Level Mathematics 9709 Paper 21 Q4 November 2015 1 1 , y = tan3 t, where 0 ø t , π. The parametric equations of a curve are x = 2 cos3 t dy = sin t. i Show that dx ii Hence show that the equation of the tangent to the curve at the point with parameter t is y = x sin t − tan t.

7

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iii Use an iterative formula xn +1 = 8 − 2 ln xn to find α correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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ii

1 By sketching a suitable pair of graphs, show that the equation ln x = 4 − x has exactly 2 one real root, α .

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6

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Cross-topic review exercise 2

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

6 intersects the line y = x + 1 at the point P. x2 Verify by calculation that the x-coordinate of P lies between 1.4 and 1.6.

-C ii

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Cambridge International AS & A Level Mathematics 9709 Paper 21 Q4 November 2014

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Cambridge International A Level Mathematics 9709 Paper 31 Q5 November 2012

1 A curve is defined by the parametric equations x = 2 tan θ , y = 3 sin 2θ ,  for 0 ø θ , π. 2 dy = 6 cos 4 θ − 3 cos2 θ . i Show that dx ii Find the coordinates of the stationary point. 3 3 . iii Find the gradient of the curve at the point  2 3,    2

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[4] [3] [2]

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q5 June 2016

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Pr

op y C w

(

s

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0

1 1 8 2−π . 2 dx = 4 (sec x − tan x )

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1 π 4

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1 ≡ 2 sec 2 x − 1 + 2 sec x tan x. (sec x − tan x )2

[1]

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1 ≡ sec x + tan x. sec x − tan x

iv Hence show that

y

Show that

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ii

1 dy ,  show that if y = sec x then = sec x tan x. cos x dx

ve

By differentiating

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br am

ii

1 y = 3 cos 2 x − 5 sin x at  π,  −1  , 6  3 3 x + 6xy + y = 21 at (1, 2).

iii Deduce that

16

[4]

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q5 June 2013

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i

ie

[3]

For each of the following curves, find the exact gradient at the point indicated:

14

15

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ii

dy 2(t + 1) = . dx et Find the equation of the normal to the curve at the point where t = 0 .

Show that

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The parametric equations of a curve are x = e2t, y = 4tet. i

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 6   x + 1  , with initial value x1 = 1.5, to determine n the x-coordinate of P correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

iii Use the iterative formula xn +1 =

[2]

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q6 November 2010 13

160

 6 .  x + 1

Show that the x-coordinate of P satisfies the equation x =

Pr es s

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The curve with equation y =

12

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y

y C op

M

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O 1

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Pr es s

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17

ev ie

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Cross-topic review exercise 2

x

id

Find the exact value of the area of the region enclosed by the curve and the lines x = 0 , x = 2 and y = 0 .

[4]

br

ev

[5]

-R

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s

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ii

Show that the x-coordinate of M can be written in the form ln a, where the value of a is to be stated.

am

i

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The diagram shows the curve y = 4e 2 − 6x + 3 and its minimum point M.

Pr

[4]

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y

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am

s

es

y

x dx.

C

6 d x , giving ln( x + 2)

1

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2

[3]

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q6 November 2013

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s

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br am

[3]

op

2

[3]

Use the trapezium rule with 2 intervals to estimate the value of your answer correct to 2 decimal places.

-C

[3]

Pr ity

∫ ∫ 3 cos

U

ev

b

e2 x + 6 dx , e2 x

ni ve rs

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w

a

ii

[6]

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q6 June 2012

Find

C

i

ev

id

br

1

1 Show that p = ( p + 2) 6 − . 2 Use an iterative process based on the equation in part i to find the value of p correct to 3 decimal places. Use a starting value of 0.7 and show the result of each iteration to 5 decimal places.

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op

1 , y = (t + 2 ) (2t + 1)2 The point P on the curve has parameter p and it is given that the gradient of the curve at P is −1.

ii

P2

Cambridge International A Level Mathematics 9709 Paper 31 Q7 June 2016

A curve has parametric equations x =

i

20

[5]

C

U

R

ni

Find the coordinates of the points on the curve where the tangent is parallel to the x-axis.

w

w ie ev

ii

19

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161

The equation of a curve is x 3 − 3x 2 y + y3 = 3. x 2 − 2 xy dy = 2 . i Show that dx x − y2

ity

18

rs

C

op

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Cambridge International AS & A Level Mathematics 9709 Paper 21 Q5 June 2012

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

By sketching a suitable pair of graphs, show that the equation e2 x = 14 − x 2 has exactly two real roots.

-R

i

am br id

21

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Show by calculation that the positive root lies between 1.2 and 1.3. 1 iii Show that this root also satisfies the equation x = ln(14 − x 2 ). 2 iv Use an iteration process based on the equation in part iii, with a suitable starting value, to find the root correct to 2 decimal places. Give the result of each step of the process to 4 decimal places.

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Pr es s

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ii

[2] [1]

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b Deduce the value of β correct to 2 decimal places.

[1]

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Pr

Cambridge International A Level Mathematics 9709 Paper 31 Q8 June 2014

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y O x

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am

P

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23

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op C

[3]

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By sketching each of the graphs y = cosec x and y = x( π − x ) for 0 , x , π, show that the equation cosec x = x( π − x ) has exactly two real roots in the interval for 0 , x , π. 1 + x 2 sin x ii Show that the equation cosec x = x( π − x ) can be written in the form x = . π sin x iii The two real roots of the equation cosec x = x( π − x ) in the interval 0 , x , π are denoted by α and β , where α , β . 1 + xn 2 sin xn to find α correct to 2 decimal places. a Use the iterative formula xn +1 = π sin xn Give the result of each iteration to 4 decimal places.

i

ni

ev ie

w

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q7 June 2011

22

162

[3]

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Pr

x = 2 ln(t + 2), y = t 3 + 2t + 3.

i

Find the gradient of the curve at the origin.

ii

At the point P on the curve, the value of the parameter is p. It is given that the gradient 1 of the curve at P is . 2 1 a Show that p = − 2. 2 3p + 2 b By first using an iterative formula based on the equation in part a, determine the coordinates of the point P. Give the result of each iteration to 5 decimal places and each coordinate of P correct to 2 decimal places.

[1]

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[5]

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id g

[4]

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-R

Cambridge International A Level Mathematics 9709 Paper 31 Q10 June 2015

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br am -C

R

ev

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w

C

op y

The diagram shows part of the curve with parametric equations

Copyright Material - Review Only - Not for Redistribution

ve rs ity ii

ev ie

Prove that 2 cosec 2θ tan θ ≡ sec 2 θ . Hence

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i

am br id

24

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Cross-topic review exercise 2

op

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1

b find the exact value of

π

2 cosec 4x tan 2 x d x.

C

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q6 June 2015

Hence

ie cos 4 θ dθ .

-R

s es

By first expanding cos(2 x + x ) , show that cos 3x = 4 cos3 x − 3 cos x.



Pr

163

rs

(2 cos3 x − cos x ) d x =

5 . 12

[5]

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ie

w

0

ity

Hence show that

1 π 6

[5]

y

op

ni

C

y

T1

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U

27

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am

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id

T3

O

x

s

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T4

Pr

T2

ity

The diagram shows the curve y = shown.

1 − x 10e 2

sin 4x for x ù 0. The stationary points are labeled T1, T2 , T3 , … as

ni ve rs

ii

It is given that the x-coordinate of Tn is greater than 25. Find the least possible value of n.

op

w ev

ie

id g

es

s

-R

br am -C

[6] [4]

Cambridge International A Level Mathematics 9709 Paper 31 Q10 June 2014

e

U

y

Find the x-coordinates of T1 and T2 , giving each x-coordinate correct to 3 decimal places.

R

i

C

op y C w

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ev

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q8 November 2011

R ie

[3]

Cambridge International A Level Mathematics 9709 Paper 31 Q9 June 2011

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y C

ii

[3]

ev



1 π 4

0

op

i

1 1 π ø θ ø π. 2 2

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am

b find the exact value of

26

y

ii

[4]

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Prove the identity cos 4θ + 4 cos 2θ ≡ 8 cos 4 θ − 3.

ni

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25

a solve the equation cos 4θ + 4 cos 2θ = 1 for −

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0

w ev ie



6

[3]

Pr es s

-C

a solve the equation 2 cosec 2θ tan θ = 5 for 0 , x , π.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

-C

the gradient of the curve is

1 . 2

ev ie

 48 p − 16   p2 + 8  .

i

Show that p =

ii

Show by calculation that 2 , p , 3.

[5] [2]

ve rs ity

y op C

3x 2 . At the point on the curve with positive x-coordinate p, x2 + 4

-R

The equation of a curve is y =

28

Pr es s

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ev ie

w

iii Use an iterative formula based on the equation in part i to find the value of p correct to 4 significant figures. Give the result of each iteration to 6 significant figures.

[3]

1 dy , show that if y = sec θ then = tan θ sec θ . cos θ dθ

U

By differentiating

ii

Hence show that

w

ge

i

d2 y = a sec 3 θ + b sec θ , giving the values of a and b. dθ 2



ev

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iii Find the exact value of

1 π 4

(1 + tan2 θ − 3 sec θ tan θ ) dθ .

[5]

s

-C

0

ity

op

Pr

y

es

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q8 November 2012

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-C

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id g

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e

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ni ve rs

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Pr

op y

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164

[3] [4]

-R

am

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29

id

R

ni

C op

y

Cambridge International AS & A Level Mathematics 9709 Paper 21 Q6 June 2016

Copyright Material - Review Only - Not for Redistribution

op

y

ve rs ity ni

C

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ev ie

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am br id

Pr es s

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C op

y

ve rs ity

op C w ev ie

Chapter 7 Further algebra

y op w

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165

id

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P3 This chapter is for Pure Mathematics 3 students only.

br

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• • •

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recall an appropriate form for expressing rational functions in partial fractions, and carry out the decomposition, in cases where the denominator is no more complicated than: ( ax + b )( cx + d )( ex + f ) ( ax + b )( cx + d )2 ( ax + b )( cx 2 + d ) use the expansion of (1 + x ) n , where n is a rational number and x , 1.

op y



ev

In this chapter you will learn how to:

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ve rs ity

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

w

What you should be able to do

Pure Mathematics 1 Coursebook, Chapter 6

Equate coefficients of polynomials.

Pr es s

-C

y op

Expand ( a + b ) n where n is a positive integer.

Pure Mathematics 1 Coursebook, Chapter 6

a

Ax 2 − 3x + C ≡ 6x 2 + Bx − 9

2 Find the first 3 terms, in ascending powers of x, in the expansion of: a

(1 + 2 x )7

ni

C op

y

b (3 − 2 x )5

Divide polynomials.

3 Find the quotient and remainder when x 2 − 8x + 4 is divided by x − 3.

id

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U

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Chapter 1

1 Find the value of A, B and C for: b (2 − A)x 2 + 5x + 2C ≡ 3x 2 −  3Bx + 8

ve rs ity

C

Check your skills

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Where it comes from

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ev ie

am br id

PREREQUISITE KNOWLEDGE

br

ev

Why do we study algebra?

FAST FORWARD

Pr

y

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s

-C

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am

At IGCSE / O Level we learnt how to add and subtract algebraic fractions. In this chapter we will learn how to do the ‘reverse process’. This reverse process is often referred to as splitting a fraction into its partial fractions. In Mathematics it is often easier to deal with two or more simple fractions than it is to deal with one complicated fraction.

ity

op

In the Pure Mathematics 1 Coursebook, Chapter 6, you learnt how to find the binomial expansion of ( a + b ) n for positive integer values of n. After working through this chapter you will be able to expand expressions of the form (1 + x ) n for values of n that are not positive integers (providing x , 1).

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Combining your partial fraction and binomial expansion skills will enable you to obtain 2x − 1 series expansions of complicated expressions such as . 2 x 2 + 3x − 20

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166

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7.1 Improper algebraic fractions

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A numerical improper fraction is defined as a fraction where the numerator ù the 11 denominator. For example, is an improper fraction. This fraction can be expressed as 5 1 2 + , which is the sum of a positive integer and a proper fraction. 5 So how do we define an algebraic improper fraction?

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P( x ) , where P( x ) and Q( x ) are polynomials in x, is said to be an algebraic Q( x ) improper fraction if the degree of P( x ) ù the degree of Q( x ) .

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x 3 − 3x 2 + 7 is an improper algebraic fraction because the x−2 degree of the numerator (3) is greater than the degree of the denominator (1). For example, the fraction

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KEY POINT 7.1

The algebraic fraction

In Chapter 8 you will be shown another use for partial fractions: how to integrate rational expressions 2x − 1 such as 2 x 2 + 3x − 20 by first splitting the expression into partial fractions.

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WEB LINK Explore the Polynomials and rational functions station on the Underground Mathematics website.

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x 3 − 3x 2 + 7 as the sum of a polynomial and x−2

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We can use long division to write the fraction

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Chapter 7: Further algebra

a proper algebraic fraction.

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− x2 + 2x

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− 2x + 7 − 2x + 4 3

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3 x − 3x + 7 = x2 − x − 2 + x−2 x−2

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EXPLORE 7.1

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3

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y

− x2 + 0x

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Pr es s

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x3 − 2x2

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x2 − x − 2 x − 2 x − 3x 2 + 0 x + 7 3

2x − 3 ( x + 2)( x − 1)

x 3 + 2x2 − 7 − ( x + 2)( x + 1)

es

3x x−5

Pr

op

6x 3 − 2 x + 1 2x 2 − 1 167

4x 5 − 1 3x 7 + 2

2x 4 − 8 2 x − 2x − 1

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x 2 − 4x 5 − x2

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1 2x + 1

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1 Discuss with your classmates which of the following are improper algebraic fractions.

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EXERCISE 7A

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2 Write each improper fraction in question 1 as the sum of a polynomial and a proper fraction.

b

x 3 + 4x 2 + 3x − 1 x2 + 2x + 3

e

4x 3 − 3 2x + 1

f

x 4 + 2x2 − 5 x2 + 1

s

c

Pr

es

7 x 3 + 2 x 2 − 5x + 1 x2 − 5

x3 + x2 − 7 D ≡ Ax 2 + Bx + C + , find the values of A, B, C and D. x−3 x−3

3 Given that

x 4 + 5x 2 − 1 E ≡ Ax 3 + Bx 2 + Cx + D + , find the values of A, B, C , D and E. x +1 x +1

4 Given that

2 x 4 + 3x 3 + 4x 2 + 5x + 6 Cx + D ≡ Ax + B + 3 , find the values of A, B, C and D. x3 + 2x x + 2x

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2 Given that

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d

6x + 1 3x + 2

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8x 2x − 5

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1 Express each of the following improper fractions as the sum of a polynomial and a proper fraction.

Copyright Material - Review Only - Not for Redistribution

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7.2 Partial fractions

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

5 4 + x+3 x−2 5( x − 2) + 4( x + 3)    = ( x + 3)( x − 2) 9x + 12    = ( x + 3)( x − 2)

Case 3

2 3 + x − 1 x2 + 5 2( x 2 + 5) + 3( x − 1)    = ( x − 1)( x 2 + 5)  =

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3 2 + x + 1 ( x + 1)2 3( x + 1) + 2    = ( x + 1)2 3x + 5    = ( x + 1)2

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Denominator in answer has: one linear factor and one quadratic factor.

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Denominator in answer has: one linear repeated factor.

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Denominator in answer has: two distinct linear factors.

2 x 2 + 3x + 7 ( x − 1)( x 2 + 5)

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Case 2

Pr es s

Case 1

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At IGCSE / O Level, we learnt how to add and subtract algebraic fractions. The following table shows three different cases for addition of algebraic fractions.

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In this section we will learn how to do the reverse of this process. We will learn how to split (or decompose) a single algebraic fraction into two or more partial fractions.

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Case 1: Proper fraction where the denominator has distinct linear factors

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If the denominator of a proper algebraic fraction has two distinct linear factors, the fraction can be split into partial fractions.

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Pr

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KEY POINT 7.2

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px + q A B ≡ + ( ax + b )( cx + d ) ax + b cx + d

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A B C + + ax + b cx + d ex + f



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px + q

( ax + b ) ( cx + d ) ( ex + f )

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KEY POINT 7.3

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This rule can be extended for 3 or more linear factors.

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168

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Multiply throughout by (2 x + 1)( x − 3). (1)

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A B 2 x − 13 ≡ + (2 x + 1)( x − 3) 2 x + 1 x − 3 2 x − 13 ≡ A( x − 3) + B (2 x + 1)

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Answer

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Pr

2 x − 13 in partial fractions. (2 x + 1)( x − 3)

C

Express

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WORKED EXAMPLE 7.1

Copyright Material - Review Only - Not for Redistribution

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Chapter 7: Further algebra

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Method 1: Choose appropriate values for x. Let x = 3 in equation (1):

1 in equation (1): 2

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1 1  2  −  − 13 = A  − − 3  + B ( −1 + 1)  2   2 7 −14 = − A 2 A=4 2 x − 13 4 1 ≡ − ∴ (2 x + 1)( x − 3) 2 x + 1 x − 3

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Let x = −

Pr es s

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 2(3) − 13 = A(3 − 3) + B (6 + 1) −7 = 7 B B = −1

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Method 2: Equating coefficients.

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2 x − 13 ≡ A( x − 3) + B (2 x + 1)

es Pr

y

(2) A + 2B = 2 (3) −3A + B = −13 A = 4,  B = −1 2 x − 13 4 1 ≡ − ∴ (2 x + 1)( x − 3) 2 x + 1 x − 3

169

Solve equations (2) and (3) simultaneously.

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Equate the coefficients of x and the constants.

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2 x − 13 ≡ ( A + 2 B )x − 3A + B

Expand brackets and collect like terms.

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Case 2: Proper fraction with repeated linear factor in the denominator

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If the denominator has a repeated linear factor, the fraction can be split into partial fractions using the rule:

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KEY POINT 7.4

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Multiply throughout by ( x − 4)2 .

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(1)

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2 x − 11 A B ≡ + x − 4 ( x − 4)2 ( x − 4)2 2 x − 11 ≡ A( x − 4) + B

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Answer

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2 x − 11 in partial fractions. ( x − 4)2

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Express

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WORKED EXAMPLE 7.2

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px + q A B ≡ + ax + b ( ax + b )2 ( ax + b )2

Copyright Material - Review Only - Not for Redistribution

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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Let x = 4 in equation (1):

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2(4) − 11 = A(4 − 4) + B B = −3

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1 − 9x − 8x 2 in partial fractions. ( x − 2)(2 x + 3)2

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1 − 9x − 8x 2 A B C ≡ + + x − 2 2 x + 3 (2 x + 3)2 ( x − 2)(2 x + 3)2

Multiply throughout by ( x − 2)(2 x + 3)2 .

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Answer

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Express

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WORKED EXAMPLE 7.3

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Pr

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Let x = 2 in equation (1):

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1 − 9(2) −  8(2)2 = A(4 + 3)2 + B (2 − 2)(4 + 3) + C (2 − 2) −49 = 49 A A = −1

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3 in equation (1): 2

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Let x = −

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170

(1)

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1 − 9x − 8x 2 ≡ A(2 x + 3)2 + B ( x − 2)(2 x + 3) + C ( x − 2)

2

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C

3 3 3 3 1 − 9  −  −  8  −  = A( −3 + 3)2 + B  − − 2  ( −3 + 3) + C  − − 2   2  2  2   2  7 7  − = − C 2 2 C =1

es

s

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Let x = 0 in equation (1): 1 = 9 A − 6B − 2C

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op -R s es

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C w

1 − 9x − 8x 2 1 2 1 ≡− − + x − 2 2 x + 3 (2 x + 3)2 ( x − 2)(2 x + 3)2

ev

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Substitute A = −1 and C = 1 in equation (2).

Pr

op y

(2)

B = −2



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C

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Pr es s

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Let x = 0 in equation (1): 2(0) − 11 = A(0 − 4) − 3  A = 2 2 x − 11 2 3 ≡ − ∴ x − 4 ( x − 4)2 ( x − 4)2

Copyright Material - Review Only - Not for Redistribution

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Chapter 7: Further algebra

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Case 3: Proper fraction with a quadratic factor in the denominator that cannot be factorised

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In this syllabus you will only be asked questions where the quadratic factor is of the form cx 2 + d.

KEY POINT 7.5

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Pr es s

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If the denominator of a proper algebraic fraction has a quadratic factor of the form cx 2 + d that cannot be factorised, the fraction can be split into partial fractions using the following rule.

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px + q A Bx + C ≡ + ( ax + b )( cx 2 + d ) ax + b cx 2 + d

w

Answer

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8 + 2x − x2 in partial fractions. ( x − 1)( x 2 + 2)

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Express

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WORKED EXAMPLE 7.4

es

8 + 2 x − x 2 ≡ A( x 2 + 2) + Bx( x − 1) + C ( x − 1)

(1)

Pr

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Let x = 1 in equation (1):

171

8 + 2(1) − (1)2 = A(12 + 2) + B (1)(1 − 1) + C (1 − 1) 9 = 3A A=3

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Let x = 0 in equation (1): 8 = 6−C C = −2

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Multiply throughout by ( x − 1)( x 2 + 2).

s

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8 + 2x − x2 A Bx + C ≡ + ( x − 1)( x 2 + 2) x − 1 x 2 + 2

-C

es

s

−4x − 2 4x + 2 8 + 2x − x2 3 3 ≡ + 2 ≡ − x − 1 x2 + 2 ( x − 1)( x 2 + 2) x − 1 x +2

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Let x = −1 in equation (1): 5 = 9 + 2B + 4 B = −4

ni ve rs

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If an algebraic fraction is improper we must first express it as the sum of a polynomial and a proper fraction and then split the proper fraction into partial fractions.

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Case 4: Improper fractions

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Pr es s

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Answer

2 x 2 + 5x − 11 in partial fractions. x2 + 2x − 3

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Express

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WORKED EXAMPLE 7.5

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

y

The fraction is improper. Hence it must first be written as the sum of a polynomial and a proper fraction.

x−5 2 x + 5x − 11 x−5 ≡2+ 2 x2 + 2x − 3 x + 2x − 3

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2

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2 x 2 + 4x − 6

C op

)

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2 x 2 + 2 x − 3  2 x 2 + 5x − 11

x−5 into partial fractions: x2 + 2x − 3

id

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x−5 A B ≡ + ( x + 3)( x − 1) x + 3 x − 1 x − 5 ≡ A( x − 1) + B ( x + 3)

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Split the proper fraction

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Multiply throughout by ( x + 3 )( x − 1 ).

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Pr

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Let x = 1 in equation (1): −4 = 4B B = −1

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(1)

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Let x = −3 in equation (1): −8 = −4 A A=2

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2 1 2 x 2 + 5x − 11 ≡2+ − 2 x +3 x −1 x + 2x − 3

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x−5 2 1 ≡ − ( x + 3)( x − 1) x + 3 x − 1

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172

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Application of partial fractions

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Partial fractions can be used to help evaluate the sum of some sequences.

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WORKED EXAMPLE 7.6

1 1 1 can be written in partial fractions as − , find the sum of the first x( x + 1) x x +1 1 1 1 1 n terms of the series + + + + ……… . Hence, state the sum to infinity. 1× 2 2 × 3 3 × 4 4 × 5

op C

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1 1 1 1 1 1 + + + + …  + + 1× 2 2 × 3 3 × 4 4 × 5 ( n − 1) n n( n + 1)

e

Sn =

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Answer

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1  1 1  1 1  1 1  1 1  1 1  1 =  −  +  −  +  −  +  −  +  … +  − + − 1 2  2 3  3 4  4 5  n − 1 n   n n + 1  1 = 1− n +1

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Given that the algebraic fraction

Copyright Material - Review Only - Not for Redistribution

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1 →0 n +1

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As n → ∞,  

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Chapter 7: Further algebra

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∴ S∞ = 1

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DID YOU KNOW?

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EXERCISE 7B

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ve rs ity

op

The series in Worked example 7.6 is called a telescoping series, because its partial sums only have a fixed number of terms after cancellation.

e

6x 2 + 5x − 2 x( x − 1)(2 x + 1)

c

15x + 13 ( x − 1)(3x + 1)

f

11x + 12 (2 x + 3)( x + 2)( x − 3)

c

x2 − 2 x( x − 1)2

f

3x + 4 ( x + 2)( x − 1)2

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7 x + 12 2 x( x − 4)

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x −1 (3x − 5)( x − 3)

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d

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6x − 2 ( x − 2)( x + 3)

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1 Express the following proper fractions as partial fractions.

d

36x 2 + 2 x − 4 (2 x − 3)(2 x + 1)2

e

11x 2 + 14x + 5 (2 x + 1)( x + 1)2

es

3 ( x + 2)( x − 2)2

Pr

y op

b

3x 2 + 4x + 17 (2 x + 1)( x 2 + 5)

c

2 x 2 − 6x − 9 (3x + 5)(2 x 2 + 1)

br

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am

x 2 − 13x − 5 in partial fractions. 2 x − 3x 2 − 3x + 2 3

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4x 3 + x 2 − 16x + 7 2 x 3 − 4x 2 + 2 x

4x 3 − 9x 2 + 11x − 4 B C D ≡ A+ + 2 + .  x x 2x − 1 x 2 (2 x − 1)

6 a Factorise 2 x 3 − 3x 2 − 3x + 2 completely. b Hence express

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5 Find the values of A, B, C and D such that

22 − 17 x + 21x 2 − 4x 3 ( x − 4)( x 2 + 1)

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x2 + 3 x2 − 4

6x 2 − 21x + 50 (3x − 5)(2 x 2 + 5)

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b

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2 x 2 + 3x + 4 ( x − 1)( x + 2)

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4 Express the following improper fractions as partial fractions. a

d −

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2 x 2 − 3x + 2 x( x 2 + 1)

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3 Express the following proper fractions as partial fractions.

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b

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2x ( x + 2)2

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2 Express the following proper fractions as partial fractions.

Pr

24 − x in partial fractions. 2 x − 11x 2 + 12 x + 9 3

ity

b Hence express

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ni ve rs

8 The polynomial p( x ) = 2 x 3 + 5x 2 + ax + b is exactly divisible by 2 x + 1 and leaves a remainder of 9 when divided by x + 2.

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a Find the value of a and the value of b.

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120 in partial fractions. p( x )

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Express

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b Factorise p( x ) completely.

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7 a Factorise 2 x 3 − 11x 2 + 12 x + 9 completely.

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173

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1 1 1 1 + + + + … is a telescoping series. If it is a 1× 2 × 3 2 × 3 × 4 3 × 4 × 5 4 × 5 × 6 telescoping series, find an expression for the sum of the first n terms and the sum to infinity.

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EXPLORE 7.2

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10 Investigate whether the series

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Pr es s

PS

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Find the sum to infinity of this series.

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9 a Express

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2 in partial fractions. x( x + 2) b Find an expression for the sum of the first n terms of the series: 2 2 2 2 2 +  … + + + + 1× 3 2 × 4 3 × 5 4 × 6 5 × 7

PS

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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There is another method that can be used to decompose a proper algebraic fraction where the denominator contains only distinct linear factors. This alternative method is called Heaviside’s cover-up method. Find out more about this method and use it to 4x 2 + 15x + 29 in partial fractions. express ( x − 2)( x + 1)( x + 3)

op

Pr

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7.3 Binomial expansion of (1 + x ) n for values of n that are not positive integers 174

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REWIND

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In the Pure Mathematics 1 Coursebook, Chapter 6, we learnt that the binomial expansion of (1 + x ) n, where n is a positive integer is:

y op

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 n  n  n  n  n (1 + x ) n =   +   x +   x 2 +   x 3 + … +   x n 0 3 1 2          n

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n( n − 1) 2 n( n − 1)( n − 2) 3 x + x + … + x n. 2! 3!

am

br

(1 + x ) n = 1 + nx +

C

and that this terminating series can also be written in the form:

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The general binomial theorem can be used for any rational value of n . It states that:

n( n − 1) 2 n( n − 1)( n − 2) 3 x + x + … where n is rational and x , 1 2! 3!

op ie

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The proof of the general binomial theorem is beyond the scope of this course.

C



the series is infinite the expansion is only valid for x , 1 .

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There are two very important differences when n is not a positive integer. These are:

s es

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 n = −3 and n = 1  .  2

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Consider, however, the following two examples where n is not a positive integer

-C

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(1 + x ) n = 1 + nx +

Pr

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KEY POINT 7.6

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Chapter 7: Further algebra

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( −3)( −4) 2 ( −3)( −4)( −5) 3 ( −3)( −4)( −5)( −6) 4 x + x + x +… 2! 3! 4!

am br id

(1 + x ) −3 = 1 + ( −3)x +

= 1 − 3x + 6x 2 − 10 x 3 + 15x 4 + …

-R

 1 − 1 − 3 − 5  1 − 1 − 3  1 − 1  2  2 2  2  2  2 3  2  2  2  2 4 1  x+ x + x + x +… = 1+  2 2! 3! 4! 1 1 3 5 4 1 x − x +  … = 1 + x − x2 + 2 8 16 128

Pr es s

op

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(1 +

1 x)2

C op

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WORKED EXAMPLE 7.7

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When x , 1, the binomial series converges and has a sum to infinity (in a similar way to a geometric series converging for r , 1, where r is the common ratio).

ni

ev ie

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C

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We can see that in each case the sequence of coefficients forms an infinite sequence.

br

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Answer

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Expand (1 − 3x )−2 in ascending powers of x up to and including the term in x 3, and state the range of values of x for which the expansion is valid.

s

es

( −2)( −3) ( −2)( −3)( −4) ( −3x )2 + ( −3x )3 + … 2! 3!

rs

Expansion is valid for −3x , 1 1 ∴ x , . 3

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= 1 + 6x + 27 x 2 + 108x 3 + …

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WORKED EXAMPLE 7.8

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(1 − 3x ) −2 = 1 + ( −2)( −3x ) +

Pr

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Use n = −2 and replace x by −3x in the binomial formula:

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Answer

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ev

id

Find the first four terms in the expansion of (1 + 2 x 2 )−4 and state the range of values of x for which the expansion is valid.

Pr

ity

ni ve rs

Expansion is valid for 2 x 2 , 1

y

1 2

-R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

∴ x ,

op

ie

w

C

op y

es

Use n = −4 and replace x by 2 x 2 in the binomial formula: ( −4)( −5) ( −4)( −5)( −6) (1 + 2 x 2 ) −4 = 1 + ( −4)(2 x 2 ) + (2 x 2 )2 + (2 x 2 )3 + … 2! 3! = 1 − 8x 2 + 40 x 4 − 160 x 6 + …

Copyright Material - Review Only - Not for Redistribution

DID YOU KNOW? Isaac Newton is credited with the generalised binomial theorem for any rational power. A poet once wrote that ‘Newton’s Binomial is as beautiful as the Venus de Milo. The truth is that few people notice it.’

175

ve rs ity

x−5 . 1 + 2x

C

ve rs ity

op

y

Pr es s

-C

Answer x−5 = ( x − 5)(1 + 2 x )−1 1 + 2x Use n = −1 and replace x by 2 x in the binomial formula: ( −1)( −2) ( −1)( −2)( −3) (1 + 2 x ) −1 = 1 + ( −1)(2 x ) + (2 x )2 + (2 x )3 + … 2! 3! = 1 − 2 x + 4x 2 − 8x 3 + …

ev ie

w

-R

Find the coefficient of x 3 in the expansion of

ev ie

w

ge

am br id

WORKED EXAMPLE 7.9

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ni

C op

y

( x − 5)(1 + 2 x )−1 = ( x − 5)(1 − 2 x + 4x 2 − 8x 3 + …)

w

ge

U

R

= x(1 − 2 x + 4x 2 − 8x 3 ) − 5(1 − 2 x + 4x 2 − 8x 3 + …)

id

ie

Term in x 3 is x(4x 2 ) + ( −5)( −8x 3 ) = 44x 3.

-C

-R

am

br

ev

∴ The coefficient of x 3 is 44.

y

es

s

EXERCISE 7C

ve

e

1 + 2x

f

3

h

1 − 3x (1 + 2 x )3

ni U

1 − 3x 1+ x 1 − 2x

i

w

R

(1 − 2 x )−4

y

2 (1 − 4x )2

c

op

g

−3

(1 + 3x )−1

C

1+ x   2

rs

d

b

ge

C w ie

(1 + x )−2

ev

a

ity

op

Pr

1 Expand the following, in ascending powers of x, up to and including the term in x 3. State the range of values of x for which each expansion is valid.

176

b

1 − 2x2

c

-R

3

ev

(1 + x 2 )−3

am

a

br

id

ie

2 Find the first 3 terms, in ascending powers of x, in the expansion of each of the following expressions. State the range of values of x for which each expansion is valid. 1 − 4x 2

)

3

2 + 3x . 1 − 5x 2

3 Find the first 5 terms, in ascending powers of x, in the expansion of

es

s

-C

(

Pr

(3x − 1)−2

b

2x − 1

Give reasons for your answers.

ni ve rs

5 Expand (2 x − 1)−3 in ascending powers of x, up to and including the term in x 3.

y

ie

w

C

a

ity

op y

4 Can the binomial expansion be used to expand each of these expressions?

ev

1

op C

U

R

6 Show that the expansion of (1 + x ) 2 is not valid when x = 3. 3

7 Given that (1 − 3x )−4 − (1 + 2 x ) 2 ≈ 9x + kx 2 for small values of x, find the value of the constant k.

PS

8 Given that

ie

id g

w

e

PS

-R s es

-C

am

br

ev

b a + ≈ −3 + 12 x for small values of x, find the value of a and the value of b. 1 − x 1 + 2x

Copyright Material - Review Only - Not for Redistribution

ve rs ity

9 When (1 + ax )−3, where a is a positive constant, is expanded the coefficients of x and x 2 are equal.

w

ge

PS

C

U

ni

op

y

Chapter 7: Further algebra

ev ie

am br id

a Find the value of a.

b When a has this value, obtain the first 5 terms in the expansion.

-R

PS

11 The first 3 terms in the expansion of (1 + ax ) n are 1 − 24x + 384x 2.

Pr es s

a Find the value of a and the value of n.

ve rs ity

b Hence find the term in x 3.

R

ni

C op

7.4 Binomial expansion of ( a + x ) n for values of n that are not positive integers

y

op

y

-C

10 When (3 − 2 x )(1 + ax ) 3 is expanded the coefficient of x 2 is −15. Find the two possible values of a.

C w ev ie

2

PS

ge

U

To expand ( a + x ) n , where n is not a positive integer, we take a outside of the brackets:

ie

w

n

br

ev

id

x ( a + x )n = a n  1 +   a

-R

am

Hence the expression is:

es

s

-C

KEY POINT 7.7

Pr

ity rs

y

ve ni

ev

ie

WORKED EXAMPLE 7.10

op

w

C

op

y

2 3   x x n( n − 1)  x  n( n − 1)( n − 2)  x  + ⋯  for   + ( a + x ) n = a n  1 + n   + ,1      a  a  a 2! 3! a  

ie

−3

-R

am

br

x (2 + x )−3 = (2)−3  1 +   2

ev

id

Answer

w

ge

C

U

R

Expand (2 + x )−3 in ascending powers of x, up to and including the term in x 3 and state the range of values of x for which the expansion is valid.

s

es

1 8

-R s es

-C

am

br

ev

ie

id g

w

e

C

U

op

ev

R

y

ni ve rs

ity

Pr

3 3 2 5 3    1 − 2 x + 2 x − 4 x + …  1 3 3 2 5 3 = − x+ x − x + … valid for x , 2 8 16 16 32 =

ie

w

C

op y

-C

2 3   x ( −3)( −4)  x  ( −3)( −4)( −5)  x  x = (2)−3  1 + ( −3)   + ,1 + + …  for  2     2! 2 3! 2 2  

Copyright Material - Review Only - Not for Redistribution

177

ve rs ity

1



1 2

Simplify ( 4 )

w

-R

s es Pr

ity

op C

178

1 to . 2

ev

id

br

am

-C

1 3 3 1 45 × − × =− 2 128 2 8 256

y

Coefficient of x 2 is

1 2

ie

ge

U

R

ni

ev ie

  − 1 − 3 2   1 1  x  2  2  x  − + … = (1 − 3x )  1 + − − +  4  2  4 2 2!     1 1 3 2 x + … = (1 − 3x )  1 + x + 2 8 128   1 1 3 2 3 1 3 2 x + …  − x 1 + x + x + … = 1 + x + 2  8 128 8 128   2 



1 2.

y

1− x   4



C op

w

C

= (1 − 3x )(4)

1 2

Factor the 4 out of ( 4 − x )

ve rs ity

op

y

− 1 − 3x = (1 − 3x )(4 − x ) 2 4−x −

-R

-C

Answer

1 − 3x . 4−x

Pr es s

Find the coefficient of x 2 in the expansion of

ev ie

w

ge

am br id

WORKED EXAMPLE 7.11

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

w

rs

EXERCISE 7D

op

y

e

C

ge

3x + 8

(5 − 2 x )−1 4 (3 − x )3

c f

w

3

b

ie

d

ni

(2 + x )−2

U

a

id

R

ev

ve

ie

1 Expand the following, in ascending powers of x, up to and including the term in x 3 . State the range of values of x for which each expansion is valid. 9−x 1 + 2x (2 x − 5)3

b

8 − 3x 2

c

(

3 − x2

)

5

s

3

-R

(2 + x 2 )−2

-C

a

am

br

ev

2 Find the first 3 terms, in ascending powers of x, in the expansion of each of the following expressions. State the range of values of x for which each expansion is valid.

Pr

op y

es

3 Expand (1 − 2 x ) 3 8 − 2 x in ascending powers of x, up to and including the term in x 3 .

ni ve rs

ity

b Use your answers to part a to find the first 3 terms in the expansion of (2 −   x )−1(1 +  3x )−2, stating the range of values of x for which this expansion is valid. 1 5 + x + bx 2 . 4 4 Find the value of a, the value of b and the term in x 3 .

5 The first 3 terms in the expansion of ( a − 5x )−2 are

PS

6 In the expansion of 3 + ax where a ≠ 0 , the coefficient of the term in x 2 is 3 times the coefficient of the term in x 3. Find the value of a.

op C

U

w

e

ev

ie

id g

es

s

-R

br am -C

y

PS

R

ev

ie

w

C

4 a Expand (2 − x )−1 and (1 + 3x )−2 giving the first 3 terms in each expansion.

Copyright Material - Review Only - Not for Redistribution

ve rs ity −1

−1

−1



x x x ≡ 1+  . x+2 2  2

-R

2 b Show that  1 +   x

ev ie

w

ge

2 2 7 a Find the first 4 terms in the expansion of  1 +  , where , 1.  x x

am br id

PS

C

U

ni

op

y

Chapter 7: Further algebra

−1

x x x 1 +  , where , 1. 2 2  2 d Explain why your expansions are different. Find the first 4 terms in the expansion of

ve rs ity

EXPLORE 7.3

ev ie

w

C

op

y

Pr es s

-C

c

ie

f(x)

w

(1 + x)−1 (1 + 2x)−1

am

br

[1 + (3x + 2x2)]−1

s

-C

-R

2(1 + 2x)−1 − (1 + x)−1

ev

id

ge

U

R

ni

C op

y

1  , discuss why f( x ) can be written in each of (1 + x )(1 + 2 x ) the following ways:

1 Given that f( x ) =

Pr

179

ity

3 Discuss with your classmates which method you would prefer if you were asked for:

w

rs

C

op

y

es

2 Use the binomial theorem to expand each of these three expressions in ascending powers of x, up to the term in x 3.

y

ev

ve

ie

a a linear approximation for f( x )

C

a cubic approximation for f( x ).

ie

w

ge

c

U

R

ni

op

b a quadratic approximation for f( x )

br

ev

id

7.5 Partial fractions and binomial expansions

s

-C

-R

am

One of the most common reasons for splitting a fraction into partial fractions is so that binomial expansions can be applied.

Pr

op y

es

WORKED EXAMPLE 7.12

10 x + 1 ,  express f( x ) in partial fractions and hence obtain the expansion of f( x ) in (1 − 2 x )(1 + x ) ascending powers of x, up to and including the term in x 3. State the range of values of x for which the expansion is valid.

ity

op

y

ni ve rs

Multiply throughout by (1 − 2 x )(1 + x ).

w ev

ie

(1)

-R s es

am

br

id g

e

10 x + 1 A B ≡ + (1 − 2 x )(1 + x ) 1 − 2 x 1 + x 10 x + 1 ≡ A(1 + x ) + B (1 − 2 x )

C

U

Answer

-C

R

ev

ie

w

C

Given that f( x ) =

Copyright Material - Review Only - Not for Redistribution

ve rs ity

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

-R

am br id

−9 = 3B B = −3

ev ie

Let x = −1 in equation (1):

Pr es s

-C

1 in equation (1): 2

y

Let x =

3 A 2 A=4

ni U

R

Expanding (1 − 2 x )−1 gives:

y

10 x + 1 4 3 ≡ − ≡ 4(1 − 2 x )−1 − 3(1 + x )−1 (1 − 2 x )(1 + x ) 1 − 2 x 1 + x

( −1)( −2) ( −1)( −2)( −3) ( −2 x )2 + ( −2 x )3 + … 2! 3! 1 1 = 1 + 2 x + 4x 2 + 8x 3 + … valid for −2 x , 1  i.e. − , x , 2 2

br

ev

id

ie

w

ge

(1 − 2 x ) −1 = 1 + ( −1)( −2 x ) +

-C

-R

am

Expanding (1 + x )−1 gives:

s

( −1)( −2) 2 ( −1)( −2)( −3) 3 x + x 2! 3! = 1 − x + x2 − x3 + … valid for x , 1  i.e. − 1 , x , 1

es

10 x + 1 = 4(1 + 2 x + 4x 2 + 8x 3 + …) − 3(1 − x + x 2 − x 3 + …) (1 − 2 x )(1 + x )

ve

ie

w

rs

ity

C



Pr

op

y

(1 + x )−1 = 1 + ( −1)x + 180

y w 1

ie

1 2

-R

1 1 ,x, . 2 2

s es Pr

op y

ity

7x − 1 in partial fractions. (1 − x )(1 + 2 x )

ni ve rs

1 a Express

y

C w

ie

5x 2 + x in ascending powers of x, up to and including the term in x 3. (1 − x )2 (1 − 3x )

s

-R

ev

id g

am

br

b Hence obtain the expansion of  

-C

op

5x 2 + x in partial fractions. (1 − x )2 (1 − 3x )

e

2 a Express

7x − 1 in ascending powers of x, up to and including the term in x 3. (1 − x )(1 + 2 x )

es

ev

b Hence obtain the expansion of  

U

C w

0

1 2

-C

am



ev

ge br

id

−1

EXERCISE 7E

ie

1 1 , x , and −1 , x , 1 2 2

C

U

R

The expansion is valid for the range of values of x satisfying both −

op

ni

ev

= 1 + 11x + 13x 2 + 35x 3 + …

Hence, the expansion is valid for −

R

C op



ve rs ity

ev ie

w

C

op

6=

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

21 in partial fractions. ( x − 4)( x + 3)

ni

24 in ascending powers of x, up to and including the term in x 2. ( x − 4)( x + 3)

y

b Hence obtain the expansion of

U

R

19 − 7 x − 6x 2 . (2 x + 1)(2 − 3x )

C op

w

Pr es s

b Hence obtain the coefficient of x 2 in the expansion of  

5 a Express

ev ie

19 − 7 x − 6x 2 in partial fractions. (2 x + 1)(2 − 3x )

ve rs ity

C

op

y

-C

4 a Express

7 x 2 + 4x + 4 in ascending powers of x, up to and including the term in x 3 . (1 − x )(2 x 2 + 1)

-R

b Hence obtain the expansion of  

w

ge

7 x 2 + 4x + 4 in partial fractions. (1 − x )(2 x 2 + 1)

am br id

3 a Express

C

U

ni

op

y

Chapter 7: Further algebra

ie

ev

6x 2 − 24x + 15 in ascending powers of x, up to and including the term in x 2. ( x + 1)( x − 2)2

op

Pr

y

es

s

-C

-R

am

br

b Hence obtain the expansion of

w

ge

6x 2 − 24x + 15 in partial fractions. ( x + 1)( x − 2)2

id

6 a Express

181

rs ve

ev

ie

Partial fractions

y

w

C

ity

Checklist of learning and understanding

ni

s

-C

es

Case 4: If an algebraic fraction is improper we must first express it as the sum of a polynomial and a proper fraction and then split the proper fraction into partial fractions.

Pr

op y



-R

am

br

ev

id

ie

w

ge

C

U

R

op

Case 1: If the denominator of a proper algebraic fraction has two distinct linear factors, the px + q A B fraction can be split into partial fractions using the rule ( ax + b )( cx + d ) ≡ ax + b + cx + d ● Case 2: If the denominator of a proper algebraic fraction has a repeated linear factor, the px + q A B ≡ + fraction can be split into partial fractions using the rule ax + b ( ax + b )2 ( ax + b )2 ● Case 3: If the denominator of a proper algebraic fraction has a quadratic factor of the form cx 2 + d that cannot be factorised, the fraction can be split into partial fractions using the rule px + q A Bx + C ≡ + ( ax + b )( cx 2 + d ) ax + b cx 2 + d



ity

(1 + x ) n = 1 + nx +

n( n − 1) 2 n( n − 1)( n − 2) 3 x + x + … for x , 1. 2! 3!

ni ve rs



es

s

-R

br

ev

ie

id g

w

e

C

U

op

2 3   x x n( n − 1)  x  n( n − 1)( n − 2)  x  + …  for + ( a + x ) n = a n  1 + n   + , 1.      a  a  a 2! 3! a  

am



y

Binomial expansion of (a + x ) n for values of n that are not positive integers

-C

R

ev

ie

w

C

Binomial expansion of (1 + x ) n for values of n that are not positive integers

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ev ie

am br id

Expand (1 − 2 x )−4 in ascending powers of x, up to and including the term in x 3, simplifying the coefficients. [4]

2

Expand

Cambridge International A Level Mathematics 9709 Paper 31 Q1 June 2011 3 − 2x ) 2

op

ve rs ity

y

Expand (2 − x )(1 + in ascending powers of x, up to and including the term in x 2, simplifying the coefficients.

C 5

Expand

6

Expand

7

Expand

1 + 3x in ascending powers of x up to and including the term in x 2, simplifying the coefficients. [4] (1 + 2 x ) Cambridge International A Level Mathematics 9709 Paper 31 Q2 June 2013

y

Expand

U

ev

id

ie

10    in ascending powers of x, up to and including the term in x 2, simplifying the coefficients. [4] (2 − x )2

-R

s

Pr

es

[4]

8x 2 + 4x + 21 Bx + C A in the form + 2 . x+2 ( x + 2)( x 2 + 5) x +5

[4]

y

op

ni

C

ge

w

4x 2 − 5x + 3 in partial fractions. ( x + 2)(2 x − 1)

[5]

ie 2

s

es

i

4x + 12 . ( x + 1)( x − 3)2 Express f( x ) in partial fractions.

ii

Hence obtain the expansion of f( x ) in ascending powers of x, up to and including the term in x 2 .

Pr

2

ity

[5]

ie

ni ve rs

op y

[6]

Cambridge International A Level Mathematics 9709 Paper 31 Q3 June 2015

14 Let f( x ) =

[5]

y

op -R s

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

Cambridge International A Level Mathematics 9709 Paper 31 Q8 June 2016

es

C

-R

-C

am

13 Show that for small values of x 2 , (1 − 2 x 2 )−2 − (1 + 6x 2 ) 3 ≈ kx 4 , where the value of the constant k is to be determined.

w

[5]

ev

br

12 Express

9x 3 − 11x 2 + 8x − 4 B C D ≡ A+ + 2 + , find the values of A, B, C and D. 2 x x 3x − 2 x (3x − 2)

id

R

11 Given that

U

ev

ve

ie

7 x 2 − 3x + 2 in partial fractions. [5] x( x 2 + 1) Cambridge International A Level Mathematics 9709 Paper 31 Q3 June 2013

rs

10 Express

w

C A B 12 in the form + + . x x2 2x − 3 x 2 (2 x − 3)

ity

Express

C

9

1 in ascending powers of x, up to and including the term in x 2, simplifying the coefficients. [4] 4 − 5x

am

-C

op

y

Express

[4]

w

ge

1 in ascending powers of x, up to and including the term in x 3 , simplifying the coefficients. 1 + 3x

br

R

ni

4

8 182

[4]

Cambridge International A Level Mathematics 9709 Paper 31 Q2 November 2016

C op

w

Pr es s

(6x ) in ascending powers of x up to and including the term in x 3, simplifying the coefficients. [4]

-C

31−

-R

1

3

ev ie

w

END-OF-CHAPTER REVIEW EXERCISE 7

Copyright Material - Review Only - Not for Redistribution

ve rs ity am br id

ev ie

w

ge

C

U

ni

op

y

Chapter 7: Further algebra

2x2 − 7x − 1 . ( x − 2)( x 2 + 3) Express f( x ) in partial fractions.

ii

Hence obtain the expansion of f( x ) in ascending powers of x, up to and including the term in x 2 .

y op

i

3x . (1 + x )(1 + 2 x 2 ) Express f( x ) in partial fractions.

ii

Hence obtain the expansion of f( x ) in ascending powers of x, up to and including the term in x 3.

Cambridge International A Level Mathematics 9709 Paper 31 Q8 November 2010

w ie ev

id br

-R

am

es

s

-C

Pr

y

183

ity

op

rs

C w

ni

op

y

ve

ie ev

y -R s es

am

br

ev

ie

id g

w

e

C

U

op

ie

w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

am

br

ev

id

ie

w

ge

C

U

R ev

R

-C

[5]

C op

y

[5]

ge

U

R

ni

ve rs ity

C w

[5]

Cambridge International A Level Mathematics 9709 Paper 31 Q7 November 2013

16 Let f( x ) =

ev ie

[5]

Pr es s

-C

i

-R

15 Let f( x ) =

Copyright Material - Review Only - Not for Redistribution

op

y

ve rs ity ni

C

U

ev ie

w

ge

-R

am br id

Pr es s

-C y

ni

C op

y

ve rs ity

op C w ev ie C

ity

op

Pr

y

es

s

-C

-R

am

br

ev

id

ie

w

ge

U

R 184

op

y

ve

ni

w

ge

C

U

R

ev

ie

w

rs

Chapter 8 Further calculus id

ie

P3 This chapter is for Pure Mathematics 3 students only.

br

es

s

-C

-R

am

use the derivative of tan −1 x 1 extend the ideas of ‘reverse differentiation’ to include the integration of 2 x + a2 kf ′ ( x ) , and integrate such functions recognise an integrand of the form f( x ) use a given substitution to simplify and evaluate either a definite or an indefinite integral integrate rational functions by means of decomposition into partial fractions recognise when an integrand can usefully be regarded as a product, and use integration by parts.

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

■■ ■ ■ ■ ■

ev

In this chapter you will learn how to:

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ve rs ity ge

C

U

ni

op

y

Chapter 8: Further calculus

w

What you should be able to do

Chapter 4

Differentiate e x , ln x, sin x, cos x and tan x.

1 Differentiate with respect to x. a sin 3x b ex

ve rs ity

c

1 , sin( ax + b ), ax + b cos( ax + b ) and sec 2 ( ax + b ) and use trigonometrical relationships in carrying out integration.

ni

w

b

ie

ge

ev

id br

-R

am -C

s

es Pr

y

ity

op

rs

C w

5 x +1

c d

2

2

3 Express in partial fractions. 2x a ( x − 1)( x + 3) 4x 2 − 6 b x( x 2 + 2) 5 c x( x − 1)2 3x 2 − 3x − 3 d x2 − x − 2

ni

op

y

ve

ie

∫ e dx ∫ sin 2x dx 5 ∫ 3x − 2 dx ∫ sec (3x ) dx

C op

a

U

R ev

ln(5x − 3)

2 Find each of these integrals.

Express rational functions as partial fractions.

w

ge

C

U

R

+1

d tan 2 x − 5 cos x

Integrate e ax + b ,

Chapter 7

2

y

y op C w

Check your skills

Pr es s

-C

-R

Where it comes from

Chapter 5

ev ie

ev ie

am br id

PREREQUISITE KNOWLEDGE

br

ev

id

ie

Why do we study calculus?

-R

am

This chapter will further extend your knowledge and skills in differentiation and integration.

op

y

ni ve rs

The skills that we learn in this chapter will be very important when studying Chapter 10. Chapter 10 is about solving differential equations, many of which model real-life situations.

-R s es

am

br

ev

ie

id g

w

e

C

U

Integration by parts plays a major role in engineering and its application is often found in problems including electric circuits, heat transfer, vibrations, structures, fluid mechanics, air pollution, electromagnetics and digital signal processing.

-C

R

ev

ie

w

C

ity

Pr

op y

es

s

-C

In particular, we will learn about integration by substitution and integration by parts, which can be considered as the ‘reverse process’ of the chain rule and the product rule for differentiation that we learnt about earlier in this course. These new rules will enable you to integrate many more functions than you could previously.

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WEB LINK Explore the Chain rule and integration by substitution and the Product rule and integration by parts stations on the Underground Mathematics website.

185

ve rs ity

w

ge

8.1 Derivative of tan −1 x

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

tan y = x

ev ie

am br id

The function y = tan −1 x can be written as:

-C

y

ni

dy 1 = dx 1 + x 2

U

w

ge

-R

am

br

ev

id

1 d (tan −1 x ) = 2 dx x +1

ie

w ev ie

Use tan y = x .

ve rs ity

C

dy 1 = dx 1 + tan2 y

C op

op

y

Use sec 2 y   = 1 + tan2 y.

KEY POINT 8.1

R

Pr es s

-R

Differentiate both sides with respect to x.

dy sec 2 y   =1 dx dy 1 = dx sec 2 y  

s

-C

WORKED EXAMPLE 8.1

y

es

Differentiate with respect to x.

b

tan −1 x

c

x  tan −1   x−2

ity rs

op

ni U

C

d 1   ×  3 (tan −1 3x ) = dx (3x )2 + 1 3 = 9x 2 + 1

Chain rule.

(

x

w ie ev

1

1

)2 + 1

 × 

1 1 × x +1 2 x 1 = 2 x ( x + 1)

1 −2 x 2

-R

)

es Pr

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

ity

op y

=

C

Chain rule.

s

am

(

d tan −1 x = dx

-C

b

br

id

a

ge

R

Answer

y

ve

ev

Pr

tan −1 3x

op

a

ie

w

C

186

Copyright Material - Review Only - Not for Redistribution

ve rs ity +1

 × 

d  x  dx  x − 2 

( x − 2)2 ( x − 2)(1) − ( x )(1) × ( x − 2)2 x + ( x − 2)2 −2 = 2 x 2 − 4x + 4 −1 = 2 x − 2x + 2 2

y

Simplify.

ni

C op

y

ve rs ity

op C EXERCISE 8A

U

R

Chain rule and quotient rule.

Pr es s

-C

=

w ev ie

C

( x x− 2 )

2

w

1

ev ie

am br id

x   −1   tan  x − 2   =

-R

d dx

c

ge

U

ni

op

y

Chapter 8: Further calculus

tan −1( x − 1)

br

e

tan −1 x 2

-R

am

tan −1 5x

w

d

b

2 Differentiate with respect to x.

x 3

c

tan −1

f

2x  tan −1   x + 1

ie

tan −1 2 x

id

a

ev

ge

1 Differentiate with respect to x.

x tan −1 x

op

Pr

y

es

b

ity

y op ie -R es ity

Pr

op y C

op C w ev

ie

id g

1 1 x dx = tan −1 + c 5 5 x 2 + 25

-R s es

am

br



e

Answer

y

ni ve rs

1 dx. x 2 + 25

U



-C

R

ev

ie

w

WORKED EXAMPLE 8.2

Find

In Section 8.1 we learnt that 1 d (tan −1 x ) = 2 . dx x +1

s

This can be extended to: 1 1 x dx = tan −1 + c a a x2 + a2



REWIND

ev

id

1 dx = tan −1 x + c x2 + 1

-C



am

br

KEY POINT 8.2

w

ge

Since integration is the reverse process of differentiation we can say that:

C

1 + a2

ni

x2

U

8.2 Integration of

ve

rs

C w ie ev

−1

4 Show that the tangent to the curve y = tan x at the point where x = −1 is perpendicular to the normal to the curve at the point where x = 1. Find the x-coordinate of the point where this tangent and normal intersect.

PS

R

s

-C

tan −1 2 x c e x tan −1 x x x 3 Find the equation of the tangent to the curve y = tan −1   at the point where x = 2.  2 a

Copyright Material - Review Only - Not for Redistribution

187

ve rs ity

op

=

1 × 2

=

1 2

w

C



ev ie

w

ge

1 x2 +

-R 3 2

dx

2 tan −1 3

 x   23  + c  

 2 tan −1  3 

2  x +c 3 

am

w ie s es Pr

op

1 1    =  tan −1 1  −  tan −1 0  2  2  1 π 1 =  ×  −  × 0 2 4 2  π = 8

y op ie

br

ev

id

EXERCISE 8B

w

ge

C

U

R

ni

ev

ve

ie

w

rs

C

188

2

1 1 x d x =  tan −1  2 2 0 x +4  2

y

0

1 d x. x +4 2

ity



2

0

-C

Answer

2

ev



Find the exact value of

-R

br

id

WORKED EXAMPLE 8.4

ge

U

R

ev ie

1 2

y

dx =

y

+3

C op

1 2

dx.

Pr es s

-C

Answer

∫ 2x

+3

ve rs ity

1 2

ni

∫ 2x

am br id

WORKED EXAMPLE 8.3

Find

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

op y

d

2

1 dx + 16

e



1 dx 4x 2 + 1

dx

c



1 dx 4x 2 + 3

f

∫ 2 + 3x

-R

∫ 16 + x

2

s

∫ 9x

1

b

Pr

-C



es

am

1 Find the following integrals. 1 dx a x2 + 9

1

2

dx



1 2

0

2

2 4x + 1 2

c

dx

y

3 The diagram shows part of the curve y =

U

− 2

1 dx   3x + 2 2

y

w

e

ev

ie

id g

–1

es

s

-R

br am



2

. Find the exact value

x2 + 1 of the volume of the solid formed when the shaded region is rotated completely about the x-axis.

-C

ev

R

b

op

1 dx 2 x +9

C

0

ity



3

ni ve rs

a

ie

w

C

2 Find the exact value of each of these integrals.

Copyright Material - Review Only - Not for Redistribution

O

1

x

ve rs ity

ev ie

w

ge

k f ′(x) f(x)

am br id

8.3 Integration of

C

U

ni

op

y

Chapter 8: Further calculus

-R

Since integration is the reverse process of differentiation and ln x only exists for x . 0, we can say that:

f ′( x ) d x = ln f( x ) + c f( x )

y

ve rs ity



U

w ie

id

ev

Answer

2x dx. +3

2

br

∫x

am

Find

ge

WORKED EXAMPLE 8.5

-R

R

ni

ev ie

w

C

This can be extended to: k f ′( x ) dx = k ln f( x ) + c f( x )

In Chapter 4, we learnt that d [ ln f( x ) ] = ff(′(xx)) . dx

C op

op

y



REWIND

Pr es s

-C

KEY POINT 8.3

ity

op

y

ve ni

C

U

w ie

sin 3x dx cos 3x

-R

am

br

∫ tan 3x dx = ∫

id

Answer

ev

R

∫ tan 3x dx.

ge

ev

ie

WORKED EXAMPLE 8.6

Find

189

rs

w

C

op

es

2x dx = ln x 2 + 3 + c x +3 2

Pr



y



s

-C

If f( x ) = x 2 + 3 then f ′( x ) = 2 x .

s es

op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

y

ni ve rs

C

1 ln cos 3x + c 3

ev

ie

w

=−

−3sin 3x dx cos 3x

Pr

1

∫ tan 3x dx = − 3 ∫

ity

op y

-C

If f( x ) = cos 3x then f ′( x ) = −3sin 3x.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

w

ge

am br id

WORKED EXAMPLE 8.7

-C

Answer

-R

3x 2 + 5x dx. 3 + 5x 2

∫ 2x

y

If f( x ) = 2 x 3 + 5x 2 then f ′( x ) = 6x 2 + 10 x. 3x 2 + 5x 1 6x 2 + 10 x x = d dx 2 x 3 + 5x 2 2 2 x 3 + 5x 2 1 = ln 2 x 3 + 5x 2 + c 2



C op

U

w ie ev

am

Answer

-R

0

ge



2e x dx 1 + ex

id

1

br

R

WORKED EXAMPLE 8.8

Find

y

ve rs ity

ni

ev ie

w

C

op



Pr es s

Find

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

0

s es



ex dx 1 + ex 1

rs

1+ e  2 

Simplify the logarithms.

ie ∫ 1 + sin x dx



e

x dx 2 − x2

s

cot x dx





4x − 10 dx x 2 − 5x + 1

f



sec 2 x dx 1 + tan x



ln 2

c



1

c

Pr

op y

d

-R

b

es



cos x

am

6x 2 dx x3 − 1

-C

a

br

1 Find the following integrals.

ev

id

EXERCISE 8C

w

ge

C

U

R

ni

op

y

ve

w ev

ie

= 2 ln 

Substitute limits.

ity

=  2 ln 1 + e x  0 = 2 ln(1 + e) − 2 ln 2

C

190

1

Pr

op

0

2e x dx = 2 1 + ex

y



1

-C

If f( x ) = 1 + e x then f ′( x ) = e x.

1π 6

cot x d x

w

1π 4

sin x cos x 1 d x = − ln 2. 1 − 2 cos 2 x 8

-R s

1π 3

f

C

ex + e− x dx = ln(1 − e2 x ) − x + c. ex − e− x

es

am

0

ie





1π 4

br

4 Show that



0

3x 2 dx x3 + 2

ev

3 Show that

e

2

y

x +1 dx x + 2x − 1 2



op

ity ni ve rs

1

b

U

2

tan x d x

e



1π 6

-C

ie ev

R

d



1π 4

id g

a

w

C

2 Find the exact value of each of these integrals.

Copyright Material - Review Only - Not for Redistribution

0

e2 x dx 1 + e2 x

2x + 3 dx ( x + 1)( x + 2)

ve rs ity

C ev ie x

op

y

The diagram shows part of the curve y =

4x . Find the exact value of p for which the shaded region has x2 + 1

ve rs ity

an area of 4.

C w ev ie

-R

p

Pr es s

-C

O

w

ge

y

5

am br id

PS

U

ni

op

y

Chapter 8: Further calculus

8.4 Integration by substitution

REWIND

U

R

ni

C op

y

Integration by substitution can be considered as the reverse process of differentiation by the chain rule.

br

-R

am

s

-C

Pr

y op

U

C

du du = 2  ⇒  dx = dx 2

w ie ev

Rewrite the integral in terms of u and simplify.

1  3  2 2  u − u  du  

=

2 2 2 2 u − u +c 5 3

=

2 2 (2 x + 1) 2 − (2 x + 1) 2 + c 5 3

3

3

Rewrite in terms of x.

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

ity

5

Integrate with respect to u.

s



es

op y

du 2

=

5

C

u

Pr

am -C

 u − 1 2 

∫4

br

2 x + 1 dx =

-R

id

ge

ev

u = 2 x + 1  ⇒  

ve

u −1   2

u = 2 x + 1  ⇒ x =

ni

ie

Answer

R



191

4x 2 x + 1 dx.

ity

Use the substitution u = 2 x + 1 to find

rs

w

C

op

y

es

WORKED EXAMPLE 8.9

∫ 4x

dy dy du = × dx du dx

ev

id

ie

w

ge

This method is used when a simple substitution can be applied that will transform a difficult integral into an easier integral. The integral must be completely rewritten in terms of the new variable.

In the Pure Mathematics 1 Coursebook Chapter 7, we learnt the chain rule for differentiation:

Copyright Material - Review Only - Not for Redistribution

ve rs ity



16 sec 2 u du 4 sec 2 u

=

∫ 4 du

U

ie

w

ge

Simplify.

ev

id

Integrate with respect to u.

br

R

y

=

Use tan2 u + 1 ≡ sec 2 u.

ni



16 sec 2 u du 4 tan2 u + 4

=

Rewrite the integral in terms of u and simplify.

C op



ve rs ity



ev ie

= 4u + c

-R

am

Rewrite in terms of x.

Pr

y

es

s

-C

x = 4 tan −1   + c  2

op

y

ve

2

U

ni

Answer

R

∫ sin 2x cos x dx.

rs

C w ev

ie

Use the substitution u = sin x to find

ity

op

WORKED EXAMPLE 8.11

192

ge

C

u = sin x   ⇒ sin2 2 x = (2 sin x cos x )2

id

ie

w

= 4 sin2 x cos2 x

br

s

Replace dx by

es

ev

Integrate with respect to u. Rewrite in terms of x.

ev

ie

id g

es

s

-R

br am -C

TIP

w

e

U

Replace sin2 2 x by 4u2 − 4u 4.

op

4 3 4 5 u − u +c 3 5 4 4 = sin3 x − sin5 x + c 3 5 =

R

Pr

ni ve rs

4

ity

C

2

du and simplify. cos x

C

-C

op y

2

2

ie

w

du

∫ sin 2x cos x dx = ∫ sin 2x cos x cos x = sin 2 x du ∫ = (4u − 4u ) du ∫ 2

-R

du du = cos x ⇒  dx = dx cos x

am

u = sin x ⇒

ev

= 4u2 (1 − u )2

y

w

C

op

y

x = 2 tan u   ⇒ x 2 = 4 tan2 u   dx x = 2 tan u   ⇒   = 2 sec 2 u   ⇒ dx = 2 sec 2 u du du   8 8  2 sec 2 u du  dx =  2 2 x +4  4 tan u + 4 

-R

-C

Answer

8 dx. x +4 2

Pr es s



Use the substitution x = 2 tan u to find

ev ie

w

ge

am br id

WORKED EXAMPLE 8.10

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Copyright Material - Review Only - Not for Redistribution

When using substitution in definite integrals you must also remember to convert the limits to limits for u.

ve rs ity ge

C

U

ni

op

y

Chapter 8: Further calculus

3

0

x d x. x +1

y

ve rs ity

C op

4

id

1

1  1 −  2 2  u − u  du  

Integrate with respect to u.

w



Write the integral in terms of u and simplify.

ie

=

ni



U

4

ev

Evaluate the function at the limits.

s es Pr

rs

C

y

ve

 dx, use the substitution u = x 2 − 3 to show that I =

C

id

es

5x

Pr

∫ 5x + 1 dx, u = 5x + 1 1

∫x

1 − 2 x 2 dx, u = 1 − 2 x 2

d

∫e

e x + 2 dx, u = e x + 2

f

∫x

s

5

x

3x − 1 dx, u = 3x − 1

-C

y 0

w ie ev

)

d

x

x d x, u = 3 − x 3− x



0

π

 x2   x2  2 x cos  d x , u = cos    4   4 

s

(



4

-R

3 d x, u = x 2− x

am

1

id g



2

br

c

b

e

0

sin2 θ dθ .

0

C

cos3 x d x, u = sin x

es



1π 2

U

4 Find the exact value of each of these integrals. a



1π 2

op



ity

x2 d x, use the substitution x = sin θ to show that I = 0 1 − x2 Hence find the exact value of I .

3 Given that I =

1  du. u

b

ni ve rs

op y C w ie ev

R

e

-R

br

am

∫ cos x sin x dx, u = sin x

-C

c



w

2 Find these integrals, using the given substitution. x a dx, u = x + 2 ( x + 2)4



1 2

op

ni

x −3 2

ie

Hence find I .

x



ge

R

1 Given that I =

U

w ie

193

ity

op

y

-C

16 2 = − 4 −  − 2  3  3  8 = 3

EXERCISE 8D

ev

-R

am

br

1  2 3 =  u 2 − 2u 2  1 3

ev

R



ge

ev ie

w

C

op

y

u = x + 1 ⇒   x = u − 1  du u = x +1⇒ = 1  ⇒ dx = du dx Find the new limits for u: x = 3 ⇒ u = 3+1= 4 x = 0⇒ u = 0+1=1 x=3 u=4 x u −1 dx = du x u + 1 x=0 u =1

Pr es s

-C

Answer



ev ie

Use the substitution u = x + 1 to find

-R

am br id

w

WORKED EXAMPLE 8.12

Copyright Material - Review Only - Not for Redistribution

ve rs ity

f

1

0

y

Pr es s

5 Use the substitution u = x − 2 to find the exact value of



2

1

h

2

0

4 d x. 1 + ( x − 2)2

ve rs ity

op

y

U

R

ln x . Using the substitution x u = ln x, find the exact value of the volume of the solid formed when the shaded region is rotated completely about the x-axis.

cos x

-R

am

br

ev

id

ie

w

ge

y

O

es

Pr

y

−π

O

ity

op

rs

C w

y

ve

ie

op

ni

id

ie

w

ge

C

U

R EXPLORE 8.1

br

s

es

op y

-C

-R

am

2

ev

2 dx. x −1 Try using some of the integration techniques that you have learnt so far to find this integral and discuss what happens with your classmates.



2 dx, the denominator x 2 − 1 can be factorised to ( x − 1)( x + 1). x2 − 1 2 Hence the fraction 2 can be split into partial fractions. x −1

ity

Pr



op C

(1)

es

s

-R

br

ev

ie

id g

w

Let x = 1 in equation (1): 2 = 2A A=1

am

y

Multiply throughout by ( x − 1)( x + 1).

e

U

ni ve rs

A B 2 ≡ + x −1 x −1 x +1 2 ≡ A( x + 1) + B ( x − 1) 2

-C

R

ev

ie

w

C

In the integral

x

y

8.5 The use of partial fractions in integration

Consider

e

s

-C

sin x . The curve 8 The diagram shows part of the curve y = 3 e has rotational symmetry of order 2 about the origin. Using the substitution u = cos x, find the exact value of the total area bounded by the curve and the x-axis between x = −π and x = π.

194

π x 2

C op

ni

ev ie

y

O

7 The diagram shows part of the curve y =

ev

2

2

6 The diagram shows part of the curve y = 2 sin2 x cos3 x. Using the substitution u = sin x, find the exact area of the shaded region.

w

C

−2 2

x2 , x = 2 cos θ 4 − x2

-R



5

∫ ( x + 4 ) dx, x = 2 tan θ ∫ x(2 − x ) dx, x = 2 sin θ

w

sec 2 x tan3 x d x , u = tan x

ev ie

π

2

ge

1 6

π

-C

g



1 3

am br id

e

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Copyright Material - Review Only - Not for Redistribution

π x

ve rs ity

C

U

ni

op

y

Chapter 8: Further calculus

C op

ni U

dx.

+ 3)

w ie

br

ev

id

Answer

2

ge

ev ie

R

3−x

∫ (x + 1)(x

y

ve rs ity

x −1 +c x +1

WORKED EXAMPLE 8.13

Find

ev ie

Pr es s

y

op w

C

= ln

-R

am br id ∫

-C



w

ge

Let x = −1 in equation (1): 2 = −2 B B = −1 2 1 1 ∴ 2 ≡ − x −1 x −1 x +1 1 1  2 − dx =  Hence dx 2  x − 1 x + 1 x −1 = ln x − 1 − ln x + 1 + c

-R

am

First split into partial fractions.

Let x = 0

3 = 3A + C

⇒  C = 0

Let x = 1

2 = 4 A + 2 B + 2C

⇒ B = −1

rs

y

ve 1

x



= ln x + 1 −

C ie

1 ln x 2 + 3 + c 2

ev

id

2

am

br

w

2

s

-C

es Pr dx.

Answer

y

ni ve rs

2

ity

op y

op

First split into partial fractions.

U

9 A B C ≡ + + x + 2 x − 1 ( x − 1)2 ( x + 2)( x − 1)2

id g

w

e

C

Multiply throughout by ( x + 2)( x − 1)2 .

-R s es

am

br

ev

ie

9 ≡ A( x − 1)2 + B ( x + 2)( x − 1) + C ( x + 2)

-C

C w ie ev

R

9

∫ (x + 2)(x − 1)

op



∫  x + 1 − x + 3  dx 1 1 2x = ∫ x + 1 dx − 2 ∫ x + 3 dx

dx =

WORKED EXAMPLE 8.14

Find

195

-R

+ 3)

ge

2

ni

3−x

∫ (x + 1)(x

U

C w ie ev

es

⇒ A=1

ity

4 = 4A

op

Let x = −1



R

Pr

y

3 − x ≡ A( x 2 + 3) + Bx( x + 1) + C ( x + 1)

Multiply throughout by ( x + 1)( x 2 + 3) .

s

-C

3−x A Bx + C ≡ + ( x + 1)( x 2 + 3) x + 1 x 2 + 3

Copyright Material - Review Only - Not for Redistribution

ve rs ity

y

=



C

1

1

x2 + 2x − 5 dx ( x − 3)( x 2 + 1)

y w

ie

ev

f

es ∫

1



1

4x + 5 dx (2 x + 1)( x + 2)

c

x +x+2 dx ( x + 1)(1 + x 2 )

f

Pr



2

2(4x − 9)

∫ (2x + 1)(x − 5) dx 2 x 2 + 4x − 21 dx ( x − 1)( x + 4)



0

h

0



0



2



2

−2

1 − 2x dx (2 x + 1)( x + 1)2

y

e



2

1

2

op

1 − x − 2x2 dx x 2 (1 − x )



U

R

2

rs



3

ve



2

6x 2 − 19x + 20 dx ( x + 1)( x − 2)2

0

ity

3x + x dx 2 −1 ( x − 2)( x + 1)

b

ni

y op C w ie ev



1

c

s

-C



e

2x − 5

∫ (x + 2)(x − 1) dx

-R

am

br

b

2 Evaluate each of these integrals. 2 5x + 13 dx a x ( + 2)( x + 3) 1

g

C op

ni U ge

id



d

 dx 

−2

ve rs ity

op C

1 Find each of these integrals. 1 a dx x(2 − x )

196

2

3 x+2 − +c x −1 x −1

= ln

d

  dx

3

= ln x + 2 − ln x − 1 − 3( x − 1)−1 + c

w ev ie

1

∫  x + 2 − x − 1 + 3(x − 1)

EXERCISE 8E

R

1

ev ie



∫  x + 2 − x − 1 + (x − 1)

-R

dx =

2

⇒  C = 3 ⇒ A = 1  ⇒ B = −1

Pr es s

9

∫ (x + 2)(x − 1)

-C



9 = 3C 9 = 9A 9 = A − 2 B + 2C

am br id

Let x = 1 Let x = −2 Let x = 0

w

ge

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

i

1

4 − 3x dx (2 x − 1)( x + 2) 8 dx (2 − x )(4 + x 2 ) 8x 2 − 3x − 2 dx 4x 3 − 3x + 1

1

π

0

6 Show that



ln 2

1 e2 x 27 d x = ln . 2 20 (1 + e )(2e x + 1) x

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

0

cos x 1 d x = ln 2. 9 − sin2 x 6

s



1 2

x+3 2 d x = ln 3. x2 − 2

-R

 

2 2

1

es

-C

op y

5 Show that

3 2

c

Pr



am

4 Show that

C

2 x 2 + 5x + 1 dx ( x + 1)( x + 2)

w



3

ie

b

br

2

x2 + 2x + 3 dx ( x − 1)( x + 2)

ev



3

id

a

ge

3 By first dividing the numerator by the denominator, evaluate each of these integrals.

Copyright Material - Review Only - Not for Redistribution

4x 2 + 2 x − 5 dx ( x + 1)(2 x − 1)

ve rs ity

C

U

ni

op

y

Chapter 8: Further calculus

w

2x

REWIND

ev ie

∫xe

am br id

Consider

ge

8.6 Integration by parts

dx . The function to be integrated is clearly the product of two simpler 2x

-R

functions x and e . If we try using a substitution to simplify the integral we will find that it does not help. To integrate this function we require a new technique.

Pr es s

dv

du

∫ u dx dx + ∫ v dx dx

C op

ni

dv

U

R

KEY POINT 8.4

y

Rearranging gives the formula for integration by parts:

ev ie

w

C

uv =

ve rs ity

op

y

-C

Integrating both sides of the equation for the product rule for differentiation with respect to x and using the fact that integration is the reverse process of differentiation we can say that:

du

-R

am

es Pr rs

y op

1 1 x e2 x − e2 x + c 2 4

ity



ni ve rs

C

-R s es

am

br

ev

ie

id g

w

e

C

U

op

y

easy but we do not know how to integrate ln x to find v. Hence we must let u = ln x and dv = x. This is demonstrated in Worked example 8.16. dx

-C

w

Consider trying to find

es

op y

dv carefully. dx du dv x ln x dx. If we let u = x and = ln x , then finding is dx dx

It is important to choose our values for u and

ie

dx

s

-C

2x

ie

ge

id

am

br

2x

w

du

2x

=

C

dv

∫ dx dx = uv − ∫ v dx dx: 1 1 ∫ x e dx = 2 xe − ∫ 2 e

ev

1 2x e 2

-R

v=

ni

R

dv = e2 x ⇒ dx

ve

du =1 dx

U



Substitute into u

ev

197

Pr

op C w ie

dx .

Answer u=x

ev

2x

ity

y

∫xe

s

-C

WORKED EXAMPLE 8.15

Find

ev

br

id

ie

w

ge

∫ u dx dx = uv − ∫ v dx dx

The following Worked examples show how to use this formula.

R

In Chapter 4, we learnt the product rule for differentiation:   dv du d +v . ( uv ) = u dx dx dx

Copyright Material - Review Only - Not for Redistribution

ve rs ity

y op

w ev ie -R du

2

U

R

2

y

ev ie

dv

∫ u dx dx = uv − ∫ v dx dx: 1 1 1 ∫ x ln x dx = 2 x ln x − ∫ 2 x ⋅ x dx 1 1 = x ln x − ∫ 2 x dx 2

C op

C

dv =x ⇒ dx

ve rs ity

du 1 = dx x 1 2 v= x 2

u = ln x ⇒

Substitute into

Pr es s

-C

Answer

w

ge

∫ x ln x dx .

ni

Find

am br id

WORKED EXAMPLE 8.16

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ev

ie

1 2 1 x ln x − x 2 + c 2 4

-R

am

br

id

=

y

es

s

-C

WORKED EXAMPLE 8.17

Pr

∫ ln x dx.

C

ity

op

dv

rs

op

y

ve

C du

∫ u dx dx = uv − ∫ v dx dx: 1 ∫ ln x dx = x ln x − ∫ x ⋅ x dx = x ln x − 1 dx ∫

-R s

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

= x ln x − x + c

es

op y

-C

am

Substitute into

w

v=x

ie

dv =1 ⇒ dx

U

du 1 =   dx x

id

u = ln x ⇒

ni

∫ ln x dx as ∫ 1 ⋅ ln x dx.

ge

First write

br

R

ev

ie

w

Answer

ev

Find

198

w

ge

2

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Chapter 8: Further calculus

am br id

ev ie

w

ge

Sometimes we might need to use integration by parts more than once to integrate a function.

sin x d x .

dv = sin x dx

Pr es s

⇒ v = − cos x dv

du

y

ni 2

2

(1)

es

s

du =2 dx

dv

du

199

ity

Substitute into

Pr

⇒ v = sin x

∫ u dx dx = uv − ∫ v dx dx

ve

ie

w

C

op

y

dv = cos x dx



∫ 2x cos x dx

rs

-C

u = 2x

am

Use integration by parts a second time to find

-R

br

id

ge

U

2

R

C op

ev ie

∫ u dx dx = uv − ∫ v dx dx ∫ x sin x dx = −x cos x − ∫ − 2x cos x dx = − x cos x + 2 x cos x dx ∫

Substitute into

w

C w

du = 2x dx



op

u = x2

ve rs ity

y

Answer

ie

2

ev

∫x

-C

Find

-R

WORKED EXAMPLE 8.18

y op

ni

ev

∫ 2x cos x dx = 2x sin x − ∫ 2 sin x dx

C

U

R

= 2 x sin x + 2 cos x + c

w ie

id

ev

sin x dx = − x 2 cos x + 2 x sin x + 2 cos x + c

br

2

-R

am

∫x

ge

Substituting in (1) gives:

s es

ln x 1 dx, use integration by parts to show that I = (ln x )2 + c. x 2

ity



U

⇒ v = ln x

y

1 du =   dx x

op

1 dv = dx x



-R s es

am

br

ev

ie

id g

w

e

C

u = ln x

ni ve rs

Answer

-C

R

ev

ie

w

C

Given that I =

Pr

op y

-C

WORKED EXAMPLE 8.19

Copyright Material - Review Only - Not for Redistribution

ve rs ity du

-R

Rearrange.

Pr es s

Divide both sides by 2.

C op

ge



1π 2

U

R

ni

WORKED EXAMPLE 8.20

y

ve rs ity

4x sin 2 x d x.

w

C

op

y

-C

2 I = (ln x )2 + k 1 k I = (ln x )2 + 2 2 1 ∴  I = (ln x )2 + c 2

Find the exact value of

ev ie

2

w ev ie

w

ge

dv

∫ u dx dx = uv − ∫ v dx dx ln x I = (ln x ) − ∫ x dx

am br id

Substitute into

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3



du =4 dx

-C

-R

am

u = 4x

br

Answer

ev

id

ie

0

s es

dv

du

∫ u dx dx = uv − ∫ v dx dx ∫ 4x sin 2x dx = −2x cos 2x + ∫ 2 cos 2x dx

op

200

Pr

y

dv 1 = sin 2 x ⇒ v = − cos 2 x dx 2

ve

ie

w

rs

C

ity

Substitute into

y op



4x sin 2 x d x =  −2 x cos 2 x + sin 2 x  02

C

0

ni



1π 2

U

-R s

-C

EXERCISE 8F

am

br

ev

id

ie

w

= ( −π cos π + sin π ) − (0 + sin 0) =π

∫ x sin 2x dx

∫ x cos x dx

c

∫ x ln 2x dx

e

∫x

f





2

c



1

b



1



0

b

ity

dx

3

ln x dx

am

1

op

y e

C

(2 − x )e −3x d x

ie

ln x d x

ln x dx x

0

xe2 x d x

0

f

-R

id g



3

br

d

1

e

0

x ln 2 x d x

ev

x cos 3x d x

s



1π 6

es

a

U

2 Evaluate each of these integrals.

-C

w ie ev

R

es

d

x

Pr

∫ 3xe

C

a

ni ve rs

op y

1 Find each of these integrals.

w

∴ 

ge

R

ev

= −2 x cos 2 x + sin 2 x + c

Copyright Material - Review Only - Not for Redistribution

−1 5

(1 − 3xe −5 x ) d x

ve rs ity

C

U

ni

op

y

Chapter 8: Further calculus

0

1

d

x(ln x )2 d x



e

0

−∞

0

w

x 2 cos 2 x d x

c

ev ie



e



b

(ln x ) d x

1π 4



π



π

x 2 sin x d x

0

x2ex dx

f

-R



2

am br id

a

2

ge

3 Use integration by parts twice to find the exact value of each of these integrals.

e x sin x d x

0

Pr es s

a

y

op

b

y

y = x2 ln x

e

w

d

ln x x

y

ie

y=

x

y = x2 e–x

e

x

x

Pr

y op

y

201

y x

C

U

R

O

op

ni

ev

ve

ie

w

rs

C

ity

5

4

O

es

O

s

-C

-R

am

ev

ge id

y

br

c

π

O

U

R

ni

O

y

ve rs ity

C w ev ie

y = x sin x

C op

y

-C

4 Find the exact value of the area for each of these shaded regions.

br

ev

id

ie

w

ge

The diagram shows the curve y = e − x x + 2 . Find the exact value of the volume of the solid formed when the shaded region is rotated completely about the x-axis.

-R

am

8.7 Further integration

s

-C

Some integrals can be found using different integration techniques.

Pr

ni ve rs

ity

EXPLORE 8.2

∫ x(x − 2) dx.

1 1 ( x − 2)9 + ( x − 2)8 + c 9 4

w

e

U

Nadia uses the substitution u = x − 2 and her answer is:

op

y

7

C

1 Nadia and Richard are asked to find

id g

1 1 x( x − 2)8 − ( x − 2)9 + c 8 72

br

ev

ie

Richard uses integration by parts and his answer is:

-R

s es

am

Use each of their methods to show that they are both correct and then use algebra to explain why their answers are equivalent.

-C

R

ev

ie

w

C

op y

es

The activities in Explore 8.2 look at why our answers to an indefinite integral might ‘appear’ to be different when using different integration techniques when they are actually equivalent.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

partial fractions.

ev ie

ii

∫ 2x

w

ge

i

4x + 1 dx using: 2 + x −1 the substitution u = 2 x 2 + x − 1

-R

-C

Find

am br id

2 a

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

4x + 1 dx. 2 + x −1 Use algebra to show that your two answers to part a are equivalent.

Pr es s

∫ 2x

C

op

c

ve rs ity

y

b Suggest another method for finding

ni

C op

y

This activity expects you to be able to recognise the appropriate substitution to use when integrating by substitution.

U

R

ev ie

w

Explore 8.3 is designed to make you think about the different integration techniques that might be available to you for a particular integral.

id

ie

w

ge

EXPLORE 8.3

∫ ln 5x dx

∫x

2 x + 1 dx 5

es

y C w

s es Pr ity

op

id g

w

e

C

U

Some rational functions can be split into partial fractions that can then be integrated.

Integration by parts dv

du

-R s es

am

br

ev

ie

∫ u dx dx = uv − ∫ v dx dx

-C



y

ni ve rs

C w ie ev

Substitutions can sometimes be used to simplify the form of a function so that its integral can be easily recognised. When using a substitution, we must ensure that the integral is completely rewritten in terms of the new variable before integrating.

Integrating rational functions ●

ie ev

-R

-C

kf ′( x ) dx = k ln f( x ) + c f( x )

Integration by substitution





3x − 2 dx



op

ni U ge

id

br

am

k f ′( x ) f (x)

op y



dx

∫ x sin 4x dx

x sin x d x

ve

ie ev

R ●

2x dx x+5

s ∫ cos

Pr

2x + 3 dx + 3x − 10

Find each integral.

Integrating



dx

5

Checklist of learning and understanding

R

x2

3

ity

2

∫ x  e

∫ (2x + 1)

rs

y op

∫x

w

C

202

-R



3x 2 dx x3 + 1

2x + 3 dx x + 3x − 10 2

-C



am

br

ev

Discuss with your classmates which integration methods can be used to find each of these integrals.

Copyright Material - Review Only - Not for Redistribution

sin x dx 3 + cos x

ve rs ity ge

C

U

ni

op

y

Chapter 8: Further calculus

1

w ev ie -R

i      Prove that cot θ + tan θ ≡ 2 cosec 2θ .

1 ln 3. [4] 2 Cambridge International A Level Mathematics 9709 Paper 31 Q5 November 2013

ie

w

cosec 2θ dθ =

x2  d x . (4 − x 2 )

ev

1

0

Using the substitution x = 2 sin θ , show that I =

s 4t 3 ln(t 2 + 1) dt.

0

Use the substitution x = t + 1 to show that I =

ni

2

U

C w

ie -R

2

s

1

7 4 3 dx = − ln . 12 2 ( x + 1)2 ( x + 3)2

[5]

ity

0

[2]

es



Pr

iii Hence show that

[5]

[2]

  2 1 1 1 1 Using your answer to part i, show that  . ≡ + +  2 − x + 1 x + 3 ( x + 3)2 ( x + 1)  ( x + 1)( x + 3) 

-C op y

ev

id

2 in partial fractions. ( x + 1)( x + 3)

br

Express

[3]

Cambridge International A Level Mathematics 9709 Paper 31 Q7 June 2011

C

Cambridge International A Level Mathematics 9709 Paper 31 Q8 June 2010

∫ (4 + tan 2x ) dx.

y op C w

1π 4

sin ( x + 61 π ) d x. sin x

-R s es

am -C

[5]

Cambridge International A Level Mathematics 9709 Paper 31 Q5 June 2015

ie

U



1π 2

br

id g

e

b Find the exact value of

[3]

ev

a Find

2

ev

8

ni ve rs

w

(2 x − 2) ln x d x.

1

Hence find the exact value of I .

am

i



5

op

ve

rs



2

ge

ii

ie

[4]

ity

C w ie ev

R

[3]

es

Hence find the exact value of I .

The integral I is defined by I =

ii

R

4 sin2 θ dθ .

0

Cambridge International A Level Mathematics 9709 Paper 31 Q5 November 2010

i

7



1π 6

Pr

ii

op

y

-C

i

am

Let I =

-R

br



1π 6

id

ge

Hence show that

[3]

C op

ni



1π 3

U

R

ii



y

ve rs ity

4

6

1π 4

(1 + 3 tan x ) [5] dx cos2 x 0 Cambridge International A Level Mathematics 9709 Paper 31 Q2 June 2014

Use the substitution u = 1 + 3tan x to find the exact value of

y

op C

ln x d x. [5] x Cambridge International A Level Mathematics 9709 Paper 31 Q3 November 2013

Pr es s



Find the exact value of

4

3

5

[5]

Cambridge International A Level Mathematics 9709 Paper 31 Q2 June 2016

w ev ie

xe −2 x d x.

0

y

2

1 2



Find the exact value of

-C

1

am br id

END-OF-CHAPTER REVIEW EXERCISE 8

Copyright Material - Review Only - Not for Redistribution

203

ve rs ity



Hence show that

w

1π 6

0

Pr es s

y

w

C

ve rs ity

op

M

y x

C op

1

U

R

x

2

ni

ev ie

O

The diagram shows part of the curve y = (2 x − x 2 )e 2

and its maximum point M .

Find the exact x-coordinate of M .

ii

Find the exact value of the area of the shaded region bounded by the curve and the positive x-axis.

br

-R

am

0

4x 2 + 5x + 3 d x = 8 − ln 9. 2 x 2 + 5x + 2

[10]

es

s

-C



4

Pr

op

y

Cambridge International A Level Mathematics 9709 Paper 31 Q9 June 2012 y

y op 2

O

x

w

ge

M

C

U

R

ni

ev

ve

ie

w

rs

C

ity

12

ie

ev

id

br

am

-R

[5]

Cambridge International A Level Mathematics 9709 Paper 31 Q9 November 2010

s

es

y

Pr

op y

O

R 1

p

x

2

y

ie

w

ni ve rs

C

ity

M

w

e

ev

-R

[6]

Cambridge International A Level Mathematics 9709 Paper 31 Q10 November 2015

s

am -C

[4]

ie

id g

Find the exact value of the x-coordinate of M . Calculate the value of p for which the area of R is equal to 1. Give your answer correct to 3 significant figures.

br

i ii

C

U

op

x The diagram shows the curve y = for x ù 0, and its maximum point M . The shaded region R is 1 + x3 enclosed by the curve, the x-axis and the lines x = 1 and x = p.

es

13

[5]

Find the exact area of the shaded region bounded by the curve, the x-axis and the line x = 2.

-C

ii

ev

[5]

Cambridge International A Level Mathematics 9709 Paper 31 Q7 November 2016

The diagram shows the curve y = x 3 ln x and its minimum point M . i Find the exact coordinates of M.

R

[4]

ev

id

ie

w

ge

i

2 11 By first expressing 4x + 5x + 3 in partial fractions, show that 2 x 2 + 5x + 2

204

[4]

1 3 [4] ln . 2 2 Cambridge International A Level Mathematics 9709 Paper 31 Q5 November 2016

tan θ sec 2θ dθ =

y

10

Prove the identity tan 2θ − tan θ ≡ tan θ sec 2θ .

-C

ii

ev ie

i

-R

9

am br id

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Copyright Material - Review Only - Not for Redistribution

ve rs ity ge

C

U

ni

op

y

Cross-topic review exercise 3

ev ie

am br id

w

CROSS-TOPIC REVIEW EXERCISE 3 P3 This exercise is for Pure Mathematics 3 students only.

1

1

0

3x + 1 2

d x.

[3]

32 in ascending powers of x, up to and including the term in x 3. ( x + 2)3

[4]

4

Expand

1 in ascending powers of x, up to and including the term in x 2 . 4 − 2x

[4]

5

Evaluate

w

ge

ie

5 7 . − 4 4e2

s

[5]

1 in ascending powers of x, up to and including the term in x 2 , simplifying ( 1 − 4x ) the coefficients. 1 + 2x . Hence find the coefficient of x 2 in the expansion of ( 4 − 16x )

Pr

es

Expand

ity

ii

[3] [2]

ve

y

w ie

When (1 + ax ) −2, where a is a positive constant, is expanded in ascending powers of x, the coefficients of x and x 3 are equal.

ge

C

U

R

ni

9

op

ev

Cambridge International A Level Mathematics 9709 Paper 31 Q2 June 2012

i

Find the exact value of a.

ii

When a has this value, obtain the expansion up to and including the term in x 2 , simplifying the coefficients.

[3]

-R

am

br

ev

id

ie

w

[4]

α

O

x

y

ie

w

ni ve rs

C

ity

Pr

op y

s

y

es

-C

Cambridge International A Level Mathematics 9709 Paper 31 Q4 November 2012

10

ie

ev

s

-R

Cambridge International A Level Mathematics 9709 Paper 31 Q5 June 2012

es

am

br

id g

w

e

C

U

op

1 1 The diagram shows the curve y = 8 sin x − tan x for 0 ø x , π. The x-coordinate of the maximum 2 2 point is α and the shaded region is enclosed by the curve and the lines x = α and y = 0 . 2 [3] i Show that α = π. 3 ii Find the exact value of the area of the shaded region. [4]

-C

ev

[5]

rs

C

op

y

i

[5]

ev

(x + 2)e −2 x d x =

0

-C

am



1

-R

Show that

id

7

1 + 2x in ascending powers of x, up to and including the term in x 3. 1− x

br

Expand

y

x 2 sin x d x .

0

6

C op

ni

1 π 2

U

w ev ie

R



ve rs ity

Expand

C

3

8

R

[3]

op



Pr es s

-C

Evaluate

y

2

-R

Expand (1 − 3x ) −5 in ascending powers of x, up to and including the term in x 3.

1

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205

ve rs ity

5

0

a

Show that a =

b

Use the iterative formula an +1 =

[4]

y ie

ev

id

br

Pr

y

es

s

-C

M

ity

rs

and its maximum point M.

Show that the x-coordinate of M is 2.

[3]

x 2 e2 − x d x .

op



2

y

ve 0

0

ln(2 x ) d x = 1, where a . 1.

C

U

w

ie

ev

id g

[3]

Cambridge International A Level Mathematics 9709 Paper 31 Q6 November 2014

es

s

-R

br am

[6]

op

y

1 ln 2  x exp  1 +  , where exp( x ) denotes e .  a  2  1 ln 2  to determine the value of a correct to Use the iterative formula an +1 = exp  1 + an  2  2 decimal places. Give the result of each iteration to 4 decimal places. Show that a =

e

ii

-R

Pr

1

ity



a

ni ve rs

It is given that   i

[5]

Cambridge International A Level Mathematics 9709 Paper 31 Q8 November 2011

-C

C w ie ev

R

16

w

25  f( x ) d x = ln .  2 

s



1

[4]

es

Show that

op y

ii

Bx + C A + . 2−x 4 + x2

ev

id

Express f( x ) in the form

-C

i

ie

12 + 8x − x 2 . (2 − x )(4 + x 2 )

br

Let f( x ) =

am

15

[6]

Cambridge International A Level Mathematics 9709 Paper 31 Q9 June 2015

ge

U

R

Find the exact value of

C

ie

x

ni

w

The diagram shows the curve y = x e

ii

[3]

-R

am

y

O

i

[5]

w

ge

119 to determine the value of a correct to 2 ln an − 1 2 decimal places. Give the result of each iteration correct to 4 decimal places.

2 2−x

ev

[3]

119 . 2 ln a − 1

a

op C

206

4 sin2 θ dθ .

0

x ln x d x = 30 , where a . 1.

1

14

π 6

C op



It is given that

R



Using the substitution x = sin θ , show that I = 3 π Hence show that I = − . 3 2

ve rs ity

b

4x 2 d x. 1 − x2

ni

ev ie

13

a



[7]

U

w

C

op

y

Let I =

1

2

Pr es s

-C

0

4 5 − 3x d x = 4 ln . 3 ( x + 1)(3x + 1)

-R



Show that

ev ie

am br id

11

12

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Copyright Material - Review Only - Not for Redistribution

ve rs ity Hence obtain the expansion of

Pr es s

y

ve rs ity

Let f( x ) =

x 2 − 8x + 9 . (1 − x )(2 − x )2

Express f( x ) in partial fractions.

ii

Hence obtain the expansion of f( x ) in ascending powers of x, up to and including the term in x 2 .

w

ie ev -R

am

[5]

[5]

b

Hence obtain the expansion of f( x ) in ascending powers of x, up to and including the term in x 3.

[5]

a

Given that

b

Hence show that

Express f( x ) in partial fractions.

es

s

-C

Pr

B C D x3 − 2 ≡ A+ + 2 + , find the values of the constants A, B, C and D. x x 2x − 1 x (2 x − 1) 3 9 1 x3 − 2  d x = − 2 ln + ln 3 . 2 4 4 x (2 x − 1)

ve

2

21

ni

y

O

e

x

s

-C

M

-R

am

br

ev

id

ie

w

ge

C

U

R

[5]

y

2

[5]

op



ity

2

rs

y op C w ie

C op

ni

U

ge id

5x − 2 . ( x − 1)(2 x 2 − 1)

br

Let f( x ) =

[5]

Cambridge International A Level Mathematics 9709 Paper 31 Q9 November 2014

1

ev

y

i

a

20

[5]

Cambridge International A Level Mathematics 9709 Paper 31 Q9 June 2014

op C w ev ie

R

19

[5]

4 + 12 x + x 2 in ascending powers of x, up to and (3 − x )(1 + 2 x )2

including the term in x 2 .

18

ev ie

4 + 12 x + x 2 in partial fractions. (3 − x )(1 + 2 x )2

-C

ii

Express

-R

i

am br id

17

w

ge

C

U

ni

op

y

Cross-topic review exercise 3

es

Find the exact values of the coordinates of M.

[5]

ii

Find the exact value of the area of the shaded region bounded by the curve, the x-axis and the line x = e.

[5]

ni ve rs

ity

Pr

i

w

C

op y

The diagram shows the curve y = x 2 ln x and its minimum point M.



4

y op

Show that

4x ln x d x = 56 ln 2 − 12.

e

2

1

w

π

ie

cos3 4x d x.

[5] [5]

0

ev

Cambridge International A Level Mathematics 9709 Paper 31 Q8 June 2013

s es

am



24

-R

id g

Use the substitution u = sin 4x to find the exact value of   

br

ii

C

i

U

22

-C

R

ev

ie

Cambridge International A Level Mathematics 9709 Paper 31 Q9 November 2011

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207

ve rs ity

ev ie

Express 4 cos θ + 3 sin θ in the form R cos(θ − α ), where R . 0 and 0 , α ,

Hence

a solve the equation 4 cos θ + 3 sin θ = 2 for 0 , θ , 2 π, 50

∫ (4 cos θ + 3 sin θ )  dθ . 2

[4] [3]

ve rs ity

b find

C

op

y

-C

ii

y C op

ni

ev ie

w

Cambridge International A Level Mathematics 9709 Paper 31 Q9 June 2013

y op y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

am

br

ev

id

ie

w

ge

C

U

R

ni

ev

ve

ie

w

rs

C

ity

op

Pr

y

es

s

-C

-R

am

br

ev

id

ie

w

ge

U

R 208

1 π. Give 2 [3]

Pr es s

the value of α correct to 4 decimal places.

-R

i

am br id

23

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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op

y

ve rs ity ni

C

U

ev ie

w

ge

-R

am br id

Pr es s

-C y

ni

C op

y

ve rs ity

op C w ev ie

rs

op

y

ve

w

ge

C

U

ni

ie

w

C

ity

op

Pr

y

es

s

-C

-R

am

br

ev

id

ie

w

ge

U

R ev

R

Chapter 9 Vectors

209

id

ie

P3 This chapter is for Pure Mathematics 3 students only.

br

-R

am

es

s

-C

use standard notations for vectors, in two dimensions and three dimensions add and subtract vectors, multiply a vector by a scalar and interpret these operations geometrically calculate the magnitude of a vector and find and use unit vectors use displacement vectors and position vectors find the vector equation of a line find whether two lines are parallel, intersect or are skew find the common point of two intersecting lines find and use the scalar product of two vectors.

Pr

ity

op

y

ni ve rs

-R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

C

op y

■ ■ ■ ■ ■ ■ ■ ■

ev

In this chapter you will learn how to:

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ve rs ity

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

w

IGCSE / O Level Mathematics

Recall and apply: Pythagoras’ theorem simple trigonometry

B

to solve problems in two and three dimensions.

C

H

Find the angle between AG and AH.

w

2 Three points A, B and C are such that A(1, 7) ,

ie

ev

-R

a Find the equation of the perpendicular bisector of the line AB. Give your answer in the form y = mx + c.

b Write down the equation of the line parallel to AB that passes through the point C .

rs

ity

Pr

es

s

Use and apply the general equation of a straight line in gradient/intercept form. Calculate the coordinates of the midpoint of a line. Find and interpret the gradients of parallel and perpendicular lines.

B (7, 3) and C (0,  −6) .

ve

y

3 The diagram shows a cuboid with CD = 5 cm, DH = 6 cm and EH = 3 cm.

C

U

ni

op

Divide a quantity in a given ratio.

w ie ev

es

C

G A

E

D

6 cm

H

ity ni ve rs

The point N is the midpoint of EF .

y

a Write down the length of MH .

U

op

b Find the exact length of NH . 4 The point P has coordinates (3, 9). The vector

e

C

Translate points in 2-dimensional space using column vectors.

ev

ie

id g



w

 −1  PQ =  .  3  Write down the coordinates of the point Q.

es

s

-R

br am -C

F

The point M lies on DH such that DM : MH = 1:2.

Pr

op y C w ie IGCSE / O Level Mathematics

B

3 cm

s

-C

-R

am

br

id

ge

Recall and apply properties of prisms, other solids and plane shapes, such as trapezia and parallelograms.

5 cm

w ie

R

ev

IGCSE / O Level Mathematics

ev

R

6 cm

C op

ni U op



C

210

E 3 cm

ge

id

br

am

-C y



A

D

Show familiarity with the following areas of coordinate geometry in two dimensions: ●

G

5 cm

ve rs ity

C w ev ie

R

IGCSE / O Level Mathematics Pure Mathematics 1 Coursebook, Chapter 3

F

y

op

y



1 The diagram shows a cuboid with CD = 5cm, DH = 6 cm and EH = 3 cm.

Pr es s



Check your skills

-R

What you should be able to do

-C

Where it comes from

ev ie

am br id

PREREQUISITE KNOWLEDGE

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Chapter 9: Vectors

w

ge

Why are vectors important?

ev ie

w

C

ve rs ity

op

y

Pr es s

-C

-R

am br id

ev ie

Vectors are used to model many situations in the world around us. They are used by engineers to ensure that buildings are designed safely. They are used in navigation to ensure that planes and boats, for example, arrive at their destinations in differing winds or water currents. There are countless examples of the use of vectors to model the movement of objects in mathematics and physics. Displacement, velocity and acceleration are all examples of vector quantities. Vectors are essential to describing the position and location of objects. A boat lost at sea can be found if its bearing and distance from each of two fixed points is known. These are just a few examples. Our world is 3-dimensional, as far as we know, and in this chapter we will be studying vectors in two and three dimensions. As the mathematics of vectors can be applied to all vectors, the study of vector geometry and algebra in general is important.

id

ie

w

ge

U

R

ni

C op

y

It is helpful if you have a good understanding of simple coordinate geometry before you start working with vectors because you will be able to make comparisons between the coordinate geometry that you have previously studied and the vector geometry introduced in this chapter.

br

ev

Recall that a vector has both magnitude (size) and direction.

op

es

Pr

y

9.1 Displacement or translation vectors

rs

Examples of displacement or translation vectors

op

y

Sami walks 100 m in a straight line on a bearing of 045° from place A to place B. Jan is also standing at place A. Jan walks directly to Sami. Jan therefore also walks for 100 m on a bearing of 045° from place A. Jan follows the same displacement vector as Sami. The diagram shows two arrows.

ev

ie

y

-R

3 M

2

Q

es

s

-C

1

Pr

op y

w

ge

am

br

id

1

x

–3

P

op

ev

–2 L

R

3

ni ve rs

ie

w

–1

2

y

O

-R s es

-C

am

br

ev

ie

id g

w

e

C

–1

ity

C

–2

U



C

U

R

ni

ev

ve

ie

w

C

ity

A movement through a certain distance and in a given direction is called a displacement or a translation. The magnitude or length of a displacement represents the distance moved.



Explore the Vector geometry station on the Underground Mathematics website.

s

-C

-R

am

You should have some knowledge of how to work with vectors in 2-dimensional space from your previous studies. You will be reminded of this here, before being introduced to vectors in 3-dimensional space.

WEB LINK

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211

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

ev ie

w

ge

The arrow from L to M is the same length and has been drawn in the same direction as the arrow from P to Q.

-R

This means that the points L and P have been displaced or translated by the same vector to M and Q, respectively.

y

Notation using upper case

Pr es s

-C

The arrow LM and the arrow PQ represent the same displacement.

    The order of the letters is important as ML would represent a displacement from M to L.

C op

ni

U

w

ge

ie

id

 

es

Pr

B

ity x

y

2

ni

1

ve

–1 O

–2

rs

1

op

y op C w ie

–3

3 is the x-component and 1 is the y-component of the displacement. As 3 and 1 are both positive, the movement is to the right and up, respectively.

3 2

A

When you write your vectors using the correct notation, you are less likely to make errors.

s

-C



y

212

TIP

-R

am

br

ev

Any displacement can be represented by the displacement in the x-direction followed by the displacement in the y-direction. These are called the components of the displacement.  3 In the following example, the displacement vector AB =   .  1

ev

In some books LM is used for LM .

y

  

In the previous diagram, LM = PQ . Even though the arrows are drawn on different parts of the page, the displacement shown by each is the same.

Components of a displacement in two dimensions

Components of a displacement in three dimensions

C

U

R

TIP

The arrow above the upper case letters always goes from left to right. Reverse the letters to show the opposite direction.

w ev ie

R

ve rs ity

C

op

A displacement from L to M is written as LM.

s

z P(x, y, z)

es Pr

op y

y

P(x, y, z)

y

x

y

x

ni ve rs

w

C

ity

P(x, y, z)

ie

C

U

op

x

z

id g

w

e

A displacement vector in a 3-dimensional system needs to have three components.

ev -R s es

am



br

 4 vector PQ =  0  .    1

ie

For example, the point P (1, 3, 7) translates to the point Q(5, 3, 8) by the displacement

-C

ev

R

-R

z

-C

y

am

br

ev

id

ie

w

ge

In a 3-dimensional system of coordinates, a point P is located using three coordinates, ( x, y, z ). The axis for the third coordinate, the z-coordinate, intersects with the x- and y-axes at the origin and the three axes are perpendicular to each other. The axes may be rotated, as shown.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Chapter 9: Vectors

w

ge

Notation using lower case

am br id

ev ie

Sometimes lower case letters are used for vectors. In text, these are in bold type. For example, a or b.

C

ve rs ity

op

y

This diagram shows three vectors, a, b and c .

a

b

y

c

ni

C op

w ev ie

Pr es s

-C

-R

When written with a pen or pencil, they should be underlined with a bar or a short, wavy line instead. For example, a or b. Sometimes the bar or wavy line is written over the top of the ɶ lower case letter.

b

y

es

a

Pr

3 of the length of vector b, but it has opposite direction. 4 3 In other words c = − b. 4 A quantity (usually a number) used to scale a vector is called a scalar. So in ‘2a’ the vector a is being multiplied by the scalar 2.

y

C

U

Parallel vectors

R

op

ni

ev

ve

rs

w

C

ity

op

Vector c is

ie

-R

-C

a

s

am

br

ev

id

ie

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ge

U

R

The vectors a and b have the same direction but not the same magnitude or length. Vectors a and b are parallel but not equal. In fact, a is half the length of b. When the length of a is doubled, the vectors are equal. In other words b = 2a . When a is scaled so that it is the same length as b, then the vectors are equal.

-R

ev

ie

w

ge

id

br

am

This may also be written, for example, as 4c = −3b. Can you write an expression for b in terms of c ?

Two vectors are therefore parallel provided that one is a scalar multiple of the other. For example, when a = λ b, where λ is some scalar, then a and b are parallel.

Combining displacements (adding and subtracting)

213

TIP

 4 Starting at point A, make the displacement AB =   followed by the displacement  2  3 BC =   .  7

es

y op A

es

s

-R

br

ev

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id g

w

C

ni ve rs

U

 

e

 

am



  . This is usually called the

The x-component of the resultant vector is easily found by adding the x-components of AB and BC. The y-component of the resultant vector is easily found by adding the y-components of AB and BC.

-C

R

ev

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w

C



ity

 7 The overall displacement shown by AC, is   9 resultant vector.

Pr

What is the result?



C

s

-C

op y





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B

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

ev ie

w

ge

This operation is called vector addition. KEY POINT 9.1

-R

The formula for vector addition:

  

Pr es s

-C

AB + BC = AC

     0  , as there is no overall displacement. In other words AC + CA =

y

  .

ni



C op

7  −7  0  −7  =   , so CA =    +   −9  −9  0 9

U

R

ev ie

w

C

 0  

ve rs ity

op

y

The displacement AC followed by the displacement CA must return you to point A.

br

w

 0  0  =   . In other words AC − AC =   0  0

 

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id

7 7 Alternatively, we may write   −  9 9

ev

ge

As −7 and −9 are both negative, the movement is to the left and down, respectively.

 

-R

-C

am

It is reasonable, therefore, to state that CA = − AC .

y

es

s

KEY POINT 9.2

 

C

ity

op

Pr

CA = − AC is always true.

214

  .



y

ve



ni

op

Vectors AC and CA have opposite direction.

As a − b is the same as a + − b, vector subtraction is defined as the addition of the negative of a vector. The displacement a − b is therefore the displacement a followed by the displacement −b.

es

s

-C

a, b and c are displacement vectors such that c = a + b.

-R

br

am

EXPLORE 9.1

ev

id

ie

w

ge

C

U

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ev

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w

rs

Taking the negative of a vector changes the direction of the vector and does not change its length.

C

ity

Pr

op y

Using diagrams to help you, show that a = c − b and that a + b = b + a.

ni ve rs

op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

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y

a + b = b + a means that the addition of vectors is commutative.

ev

ie

w

KEY POINT 9.3

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ve rs ity ge

C

U

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op

y

Chapter 9: Vectors

Q

M

Pr es s

S

b

y

-C

P

op

PQRS is a quadrilateral. M is the midpoint of RS. 1 PQ = a, PS = b and PM = (2a + 5b ). 4 a Find an expression in terms of a and b for SM. Give your answer in its simplest form.

 

y

 

ni



C op



ve rs ity

C w

-R

R

a

ev ie

ev ie

am br id

w

WORKED EXAMPLE 9.1



ge

U

R

b Find an expression for QR.

ie -R

s

es

rs

  

y op

ni

Substitute known vectors in a and b.

C

Expand and simplify.

w ie ev

-R

s es

op y

Pr ity

The magnitude of a vector is the length of the line segment representing the vector. This is found using the components of the displacement and Pythagoras’ theorem.

ni ve rs

C

y

op

-R s

-C

am

br

ev

ie

id g

w

e

C

U

R

Notation: The magnitude of a vector p is written as p . This is also known as the modulus of the vector.

es

w ie

 

Use SR = 2SM and QP = − PQ.

ge

id

br

am

-C

2 PQRS is a trapezium.

Magnitude of a vector

ev

215

ity

Pr

Expand and simplify.

U

R

c

 

Substitute SP = − PS . Substitute given vectors in a and b.

ve

y op C w ev

ie

b

ev

id am

             SM = − PS + PM or PM − PS   1 SM = (2a + 5b ) − b 4   1 5 1 1 SM = a + b − b = a + b 2 4 2 4      QR = QP + PS + SR      QR = − PQ + PS + 2SM  1 1 QR = −a + b + 2  a + b  2 4   3 QR = b 2  3  QR = PS so the lines QR and PS are parallel and SM = SP + PM

-C

a

br

Answer

w

c Describe quadrilateral PQRS, justifying your answer.

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TIP A diagram can often be useful in vector questions.

ve rs ity

ev ie

w

ge

am br id

WORKED EXAMPLE 9.2

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

w ie

id

4

ge

U

R

ni

3

Draw a diagram showing the x- and y-components.

y

ev ie

a

ve rs ity

a

w

C

Answer

C op

op

y

Pr es s

-C

-R

 4  3 a Find the magnitude of each of the vectors a =   , b =  −  and a – b. 3    4  −1   5 b p =  0  and q =  8  . Find the magnitude of the vector p + q.      −3   8

42 + 32 = 25 = 5

ev

Apply Pythagoras’ theorem.

-R

Again, draw a diagram to show the x- and y-components.

s

-C

am

br

|a| =

es

b

rs

y

ve

( −3)2 + 42 =

25 = 5

Apply Pythagoras’ theorem.

C

U

R

ev

b =

ni

ie

3

op

w

C

216

ity

op

Pr

y

4

w

ie

3

br

a

–b

ev

id

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The diagram has been given to help you to see the components of the vector a − b.

-R

am

–4

Pr

49 + 1 =

50 = 5 2

ni ve rs

y

C

op

It is not a good idea to try to draw a diagram here.

w

id g

42 + 82 + 52 = 105

ev

ie

Apply Pythagoras’ theorem in 3 dimensions.

es

s

-R

br

am

 4  8 =    5

  and  

Apply Pythagoras’ theorem. (Notice that a − b is not the same as a − b .)

e

U

 5   −1   4 p+q = 0 + 8 = 8       8   −3   5

-C

C w ie ev

R

72 + ( −1)2 =

ity

op y

3

 7 = a−b =   −1 

b

es

s

-C

4

Copyright Material - Review Only - Not for Redistribution

ve rs ity

w

ge

Unit vectors

C

U

ni

op

y

Chapter 9: Vectors

am br id

ev ie

A unit vector is any vector of magnitude or length 1.

-R

Notation: aˆ (read as a hat) is the unit vector in the direction of a.

    p1     This means that, for a vectorp p =  p2  , the corresponding unit vector is pˆ =    p3    

ie ev

5  . 2

3

op

y

ve ni

C

2

9 =3

Apply Pythagoras’ theorem to find the magnitude of the given vector.

w

5 ) + 22 =

ie

(

ev -R s

Pr id g

op





es

s

-R

br

ev

ie

Find the unit displacement vector in the direction AC.

am

y

  11   and BC =  −2  .     −1  

C

e

U



w

 3 Points A, B and C are such that AB =  4   1

-C

ie ev

R

Divide each component by the magnitude.

ity ni ve rs

WORKED EXAMPLE 9.4

w

C

op y

-C

 Unit vector in the direction  

 5    5  3 .  is  2  2   3 

es

am

br

id

5   = 2

U

ev

R

√5

ge

w ie

rs

2 3

√5 3

  

ity

C

2

1

The diagram shows that, geometrically, we are finding lengths of similar triangles when finding components of unit vectors.

Pr

op

y

es

s

-C

am

br

 Find the unit vector in the direction  

-R

id

w

ge

WORKED EXAMPLE 9.3

Answer

p1  p   p2  . p  p3   p 

C op

U

R

ni

ev ie

w

C

ve rs ity

op

y



y

find the magnitude of the displacement then divide the components of the vector by the magnitude.

Pr es s



-C

To find a unit displacement vector in a particular direction:

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217

ve rs ity

ev ie

 AC is the resultant vector formed from   AB + BC .

-R

C

op

y

Pr es s

-C

y

200 = 10 2

Now find the magnitude by applying Pythagoras’ theorem.

ge

R



Substitute column vectors and sum components.

C op

AC = 142 + 22 + 02 =

   

ni



ve rs ity

 3   11   14 AC =  4  +  −2  =  2       1   −1   0

U

ev ie

w

C

   AC = AB + BC

w

B

w

ge am br id

Answer

A

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3



Divide each component by the magnitude of AC .

id

br

ev



es

s

-C

-R

am

 14  1  2 .  10 2   0 

ie

Unit displacement vector in the direction of AC is

op

Pr

y

EXERCISE 9A



A

ity

1 The diagram shows the vectors AB and BC on 1 centimetre squared paper.

rs



C

ve

ie

w

C

218

y op



ni

ev

Using 1cm as 1 unit of displacement:



ge

C

U

R

a write AB and BC as column vectors b write down the column vector AC .

w

ie

E

y op

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

F

D

ev

    show that DF − DE = EF .

ie

Using 1cm as 1 unit of displacement:

-R s es

-C

am

br

a find the column vector EF b

B

ev

id

br

am

2

   The diagram shows the vectors DE and DF on 1 centimetre squared paper.

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ve rs ity Q

ev ie

am br id

w

ge

R

-R

3

C

U

ni

op

y

Chapter 9: Vectors

  

Pr es s

-C

P

4 In triangle ABC , X and Y are the midpoints of AB and AC, respectively.

ev ie

A

ve rs ity

  AX = a and AY = b.

w

C

op

P

y

Justify the statement QR = PR − PQ.

X

ni

B

U

R



C op



XY = k BC

b

ie ev

-R

G

H

s

E

C

219

B

ve

x

y

ie

w

rs

C

ity

A

M

Pr

op

y

es

D

O

ev

N

F

z

-C

am

br

id

5 The diagram shows a cuboid, ABCDEFGH.

y

C

w

ge

Write down the value of the constant k. PS

Y

y

a Use a vector method to prove that BC is parallel to XY .

ie

id

      ABCDEFGH is a regular octagon. AB = p, AH = q, HG = r and HC = s.

-R

-C

A

ev

br

b AN.

am

a 6

  AM

w

ge



C

U

R

ni

op

M is the midpoint of BC and the point N is on FG such that FN : NG is 1:3.  12  Given that AG =  4  , find the displacement vector:    2 

H

C

s

r

Pr

 EB.

D

G

ity

b Explain why s = kp and find the exact value of k.

ni ve rs

E

y

F

op

w ie

-R s es

am

br

ev

ie

id g

w

e

C

U

7 Use a vector method to show that, when the diagonals of a quadrilateral bisect one another, then the opposite sides are parallel and equal in length.

-C

ev

R

P

s

es

ii

C

op y

 BC

B

q

a Using the vectors p, q, r and s, write down expressions for: i

p

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ve rs ity

C w

ge

9.2 Position vectors

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

-R

am br id

ev ie

A position vector is simply a means of locating a point in space relative to an origin, usually O. This means a position vector is fixed in space. Position vectors are sometimes called location vectors.

y

Pr es s

-C

Up to this point, the displacement vectors we have been working with have been free vectors. This means they are not fixed to any particular starting point or origin. Many free vectors can represent the same displacement.

C

ve rs ity

op

Position vectors are different because they start at a given point or origin. This fixed starting point means there is only one arrow that represents each vector.

 

U

ni

C op

The fixed starting point is often called O, as in the origin of a system of Cartesian coordinates.

y

w ev ie

R

Look back at vectors LM and PQ at the start of this chapter.

Since we can apply the same mathematics to all vectors, the results derived for displacement vectors (free vectors) also apply to position vectors.

ev

id

br

-R

am

Position vector of any point in a plane

s Pr

y op

ity rs ve

b





w

ge

This is because OP is some scalar multiple of a plus some scalar multiple of b:

C

ni

op

y

Given that a and b are position vectors relative to the origin O and that P is any other point in the same plane as a and b, then the position vector of P can be written using a and b.

U

ie

w

C

a O

R

ev

es

-C

Consider this case in 2-dimensional space.

220



ie

w

ge

 1 So a point with coordinates A(1, 2, 3) can be represented by the position vector OA =  2  .    3

P

br

ev

id

ie

OP = ma + nb, where m and n are scalars.

-C

-R

am

All the other vectors in this plane can be written using combinations of the vectors a and b.

es

s

Cartesian components

Pr

op y

When the vectors a and b are

y

C

U

op

the plane looks like the usual set of x- and y-axes (called the Oxy plane) we use for Cartesian coordinates.

-R s es

am

br

ev

ie

id g

w

e

In this case, the unit vector in the x-direction is called i and the unit vector in the y-direction is called j.

-C

R

ev

ie

w

ni ve rs

C

ity

perpendicular to each other and ● unit vectors ●

REWIND

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ve rs ity

w ev ie

P(x, y)

-R

am br id

ge

y

C

U

ni

op

y

Chapter 9: Vectors

y op

  In this case, OP is some scalar multiple of i plus some scalar multiple of j:

y

  , which can also be written as

ge

U

R

ni

 4 The point P with coordinates (4, 5) has position vector   5 4 i + 5 j.

C op

ve rs ity

OP = xi + yj, where x and y are scalars.

C w ev ie

x

i

O

Pr es s

-C

j

id

ie

w

This can be extended to three dimensions.

-R

am

br

ev

The unit vector in the z-direction is called k. z

es

s

-C

Q(x, y, z)

C

Pr

k

j x

i

ve

ie

w

rs

O

221

ity

op

y

y

 

y

op

C

U

R

ni

ev

In this case, OQ is some scalar multiple of i plus some scalar multiple of j plus some scalar multiple of k:

id

ie

w

ge

OQ = xi + yj + zk, where x, y and z are scalars.

s

-C

-R

am

br

ev

 4 The point Q with coordinates (4, 2, 3) has position vector  2  , which can also be written   as 4 i + 2 j + 3k.  3

es Pr

op y

WORKED EXAMPLE 9.5

 .  

ity

 −4   and c =  2    24 

op

y

ni ve rs

 2  0 a =  5 , b =  4     −3   6

C

U

a Write a, b and c in terms of i, j and k.

w

 ii AC

ie

 AB

id g

i

e

b Giving your answers in terms of i, j and k, find the vectors representing:

-R s es

am

br

ev

c Describe the relationship between the points A, B and C , justifying your answer.

-C

R

ev

ie

w

C

The position vectors of the points A, B and C are given by a, b and c , respectively, where

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ve rs ity

a = 2 i + 5 j − 3k, b = 4 j + 6 k and c = −4 i + 2 j + 24 k.

b

i

B A

O

C op

y

ve rs ity

     AB = AO + OB = OB − OA Substitute the Cartesian form.  AB = b − a = 4 j + 6 k − (2 i + 5 j − 3k ) Expand and simplify.  AB = −2 i − j + 9 k       Substitute the Cartesian form. ii AC = AO + OC = OC − OA  AC = c − a = −4 i + 2 j + 24 k − (2 i + 5 j − 3k ) Expand and simplify.  AC = −6 i − 3 j + 27 k   AC = 3AB and so these vectors are parallel. Since the vectors also have point A in common, then A, B and

br

c

s

-C

-R

am

C must be collinear (so they lie on the same straight line).

op

Pr

y

es

WORKED EXAMPLE 9.6

ity

The diagram shows a cube OABCDEFG with sides of length 4 cm. Unit vectors i, j and k are parallel to OC , OA and OD, respectively. The point M lies on FB such that FM = 1 cm, N is the midpoint of OC and P is the centre of the square face DEFG .

rs

 



ve

ie

w

C

222

w

br

-R

C

s F

E P G

M

w ie ev

C

N

es

s

-R

br

i

id g

B

P′

e

A

am

O

j

-C

k

C

U

R

D

y

ni ve rs

ie

w

C

ity

 NP = 2 j + 4 k

es

Using the displacement of P from N and M :

Pr

op y

Answer

op

N



ev

id i



P is directly above this point P ′ and so the second component P ′P is parallel to the z-direction and is +4k.

B

-C

O

A

j

am

k

y C

M

ie

G

The cube is of side 4 cm and P ′ is the centre of the square AOCB. NP is parallel to the y-direction and so the first part of the vector NP is 2j.

U

R

F

D

op

ni

E

ge

ev

Express each of the vectors NP and MP in terms of i, j and k. P

ev

ev

id

ie

w

ge

U

R

ni

ev ie

w

C

op

y

Pr es s

-C

A simple sketch should help you to calculate the answer correctly.

ev ie

a

-R

am br id

Answer

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Copyright Material - Review Only - Not for Redistribution

ve rs ity

w ev ie -R

F

P

M

Pr es s

-C

ev ie

O

A

B

j

i

C

N

ni

R

Or we can use position vectors:

In the z-direction, we must move a distance equal to MF . This is 1 unit and so the third component is +k.

ve rs ity

op w

C

k

In the y-direction, again we are moving in the negative direction, so the second component is −2j.

y

G

y

D

 

U

   NP = OP − ON

br

ev

id

ie

w

ge

OP is 2 units in x-direction, 2 units in y-direction and 4 units in the z-direction and ON is 2 units in x-direction.

     NP = OP − ON = 2 i + 2 j + 4 k − 2 i  NP = 2 j + 4 k      MP = OP − OM OP = 2 i + 2 j + 4 k and ON = 2 i

Simplify.

 

OM is 4 units in x-direction, 4 units in y-direction and 3 units in z-direction.

Pr

y

es

s

-C

-R

am

Subtract vectors.

op

ity

   OP = 2 i + 2 j + 4 k and OM = 4 i + 4 j + 3k      MP = OP − OM = 2 i + 2 j + 4 k − (4 i + 4 j + 3k )   MP = −2 i − 2 j + k

223

Subtract vectors.

op

y

Expand and simplify.

ie

br

ev

id

EXERCISE 9B

w

ge

C

U

R

ni

ev

ve

ie

w

rs

C

Starting with the x-direction, move 2i left. This is the opposite to the positive direction indicated by the red arrow, so the first component is −2i .

C op

am br id

E

ge

 

MP = −2 i − 2 j + k

C

U

ni

op

y

Chapter 9: Vectors



-R

am

1 Relative to an origin O, the position vectors of the points A and B are given by OA = 5 i + 3 j and OB = 5 i + λ j + µ k .





s

es

Pr

b Given that triangle OAB is right-angled with area 5 34 and µ = 10, find the value of λ .

2 The diagram shows a cuboid OABCDEFG with a horizontal base OABC . The cuboid has a length OA of 8 cm, a width AB of 4 cm and a height BF of d cm. The point M is the midpoint of CE and the point N is the point on GF such that GN = 6 cm. The unit vectors i, j and k are parallel to OA, OC and OD, respectively. It is given that OM = 4 i + 2 j + k.

ity

C

O

y i

ev s

es

B

M

op

k j

-R

br

-C

F d cm

w

id g

 

Find the unit vector in the direction MN.

am

c

N

E

ie

 b Find ON.

e

a Write down the height of the cuboid.

6 cm

D

C

U

R

 

PS

G

ni ve rs

ev

ie

w

C

op y

-C

a In the case where λ = 0 and µ = 7, find the unit vector in the direction of AB.

Copyright Material - Review Only - Not for Redistribution

4 cm 8 cm

A

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

  



Find AD , BC , AB and DC .



w

ii

ev ie

w

the point M , the midpoint of the line AB 1 the point P such that BP = BD . 3

ve rs ity

C

op

b Find the coordinates of i

ev ie

Deduce that ABCD is a parallelogram.

y

ii

 .  

Pr es s

-C

a i

am br id

 5  −6  13   2       a = 2 , b = 5 , c = −3 and d =  −6         4  0  0  4 

-R

ge

3 The position vectors of the points A, B, C and D are given by a, b, c and d, respectively, where



ni

C op

y

4 Relative to an origin O, the position vector of A is i − 2 j + 5 k and the position vector of B is 3i + 4 j + k.



U

R

a Find the magnitude of AB.

w

ie ev

id

5 The position vectors of the points A, B and C are given by

-R

am

PS

Find the exact area of triangle OAB.

br

c

ge

b Use Pythagoras’ theorem to show that OAB is a right-angled triangle.

es

s

-C

a = 5 i + j + 2 k, b = 7 i + 4 j − k and c = i + j + qk.

op

Pr

y

Given that the length of AB is equal to the length of AC, find the exact possible values of the constant q.

6 The diagram shows a triangular prism, OABCDE . The uniform cross-section of the prism, OAE, is a right-angled triangle with base 24 cm and height 7 cm. The length, AB, of the prism is 20 cm. The unit vectors i, j and k are parallel to OA, OC and OE, respectively. The point N divides the length of the line DB in the ratio 2:3.

ity

i

24 cm

A

ie

id

   Given that the magnitude of OP is equal to the magnitude of OQ, find the value of the constant a.

ev



7 Three points O, P and Q are such that OP = 2 i + 5 j + ak and OQ = i + (1 + a ) j − 3k.

PS

8 Relative to an origin O, the position vectors of the points P and Q are given by

Pr

 2 k + 13    and OQ =  −8  .      −32 k 



ity

−6 k −2   8(1 + k )

ni ve rs

C

op y

  OP = 

es

s

-C

-R

am

br

PS

Given that OPQ is a straight line:

op

y

a find the value of the constant k







w

e

find the magnitude of the vector PQ.

-R s es

am

br

ev

ie

id g

c

C

U

b write each of OP and OQ in the form xi + yj + zk

-C

w

B 20 cm

w

ge

C

op

j

C

U

R



y

k

O



ie

N 7 cm

ni



a Find the magnitude of DB.

ev

E

ve

w ie ev

 

b Find ON.

R

D

rs

C

224

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ve rs ity

C

ev ie

w

ge

9 An ant has an in-built vector navigation system! It is able to go on complex journeys to find food and then return directly home (to its starting point) by the shortest route.

am br id

PS

U

ni

op

y

Chapter 9: Vectors

Pr es s

-C

-R

An ant leaves home and its journey is represented by the following list of displacements:  1  −3   6  −1  i + 3j + 6k 10i 4i − k      0  −5  What single displacement takes the ant home? One unit of displacement is 10 cm.

ve rs ity

C

op

y

The ant finds food at this point.

y

ge

U

EXPLORE 9.2

R

C op

ni

ev ie

w

What is the shortest distance home from the point where the ant finds food?

id

ie

w

A system of objects with the properties:

s

-C

-R

am

br

ev

• can be represented by a column of numbers • can be combined using addition • can be multiplied by a number to give another object in the same system

es

is called a vector space.

rs

9.3 The scalar product

y

ve

ev

ie

w

C

ity

op

Pr

y

Do quadratic expressions of the form ax 2 + bx + c , where a, b and c are real constants, form a vector space? Explain your answer.

-R

am

br

ev

id

ie

w

ge

C

U

R

ni

op

We have seen how to add and subtract vectors and also how to multiply a vector by a scalar. It is also possible to multiply two vectors. It is possible to find two different products by multiplying a pair of vectors. The first product gives an answer that is a vector. This is called the vector product or cross product (for example, the cross product of the vectors a and b is a × b). The cross product is not required for Pure Mathematics 3. The second product gives an answer that is a scalar. This is called the scalar product or dot product (the dot product of the vectors a and b, for example, is a ⋅ b).

es

s

-C

Introductory ideas

op

C

w

Directions both towards a point.

id g

e

Directions both away from a point.

θ

y

ity

U

θ

-R s es

am

br

ev

ie

As vectors have magnitude and direction we can think of the scalar product as being multiplication in relation to a particular direction.

-C

R

ev

ie

w

C



either when the direction of each vector is away from a point or when the direction of each vector is towards a point.

ni ve rs



Pr

op y

The angle between two vectors is defined to be the angle made

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225

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

ev ie

w

ge

A useful way to understand this is to think about the work done by a person when they pull an object along a flat surface using a rope:

-R Pr es s

Direction of pull a

y

-C

work done by the person = force needed to pull the object  × distance the object moves

op

Displacement of object b

a θ

x

ni

C op

y

The direction of the pull is not in the same as the direction of the displacement. This means that only some of the force the person uses to pull the object makes it move. In fact, only the x-component of the pull is in the direction of the displacement.

U

R

ev ie

w

C

ve rs ity

The diagram shows vector a representing the pull and vector b representing the displacement of the object.

br

ev

id

ie

w

ge

a is the magnitude of the pull. Using the angle made between the two directions, we can see that the magnitude of the x-component of the pull is x = a cos θ .

Applying this information to the formula, we can see that the cos θ

)

b.

s

(a

es

-C

work done by the person =

-R

am

b is the magnitude of the displacement, which is a measure of the distance moved.

C

rs

y

ni

b cos θ

C

U

R

a⋅b = a

ve

In general, the geometric definition of the scalar product of two vectors is:

ev

ie

w

KEY POINT 9.4

op

226

ity

op

Pr

y

Finding the scalar (number) that is work done by the person is one application of the scalar product.

br

ev

id

ie

w

ge

This formula allows us to calculate the value that represents the impact of one vector on the other, taking account of direction.

-R

 4   , q =  −3  and the angle, θ , between the two vectors is 45°. Find the value of the scalar product, p ⋅ q.     0 

s

es

Pr

42 + ( −3)2 + 02 = 5

y

82 + ( −6)2 + ( −10)2 = 10 2 and q =

U

p =

ni ve rs

p ⋅ q = p q cos θ

-R s es

am

br

ev

ie

id g

w

e

C

 2 = 50 p ⋅ q = ( 10 2 ) (5) cos 45° = ( 10 2 ) (5)   2 

-C

R

ev

ie

w

C

ity

Answer

op

op y

-C

 8 p =  −6   −10

am

WORKED EXAMPLE 9.7

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Chapter 9: Vectors

a

w

ge

Special case 1: Parallel vectors

Pr es s

a⋅b = a b

op

y

This also means that, for example: a ⋅ a = a a or a

2

π radians. 2

b

C op

ni

ie

id

KEY POINT 9.5

w

ge

U

a⋅b = 0

R

a

y

When two vectors, a and b, are perpendicular, θ = 90° or

ev ie

w

C

ve rs ity

Special case 2: Perpendicular vectors a ⋅ b = a b cos 90°

b

-R

am br id

a ⋅ b = a b cos 0

-C

Therefore

ev ie

When two parallel vectors, a and b, have the same direction (not opposite directions), θ = 0° or 0 radians.

-C

-R

am

br

ev

This is very important. When two vectors are perpendicular, their scalar product is zero. It is also true that when the scalar product is zero, the two vectors are perpendicular.

Pr

y

es

s

EXPLORE 9.3

227

ity

a ⋅b = b ⋅a a ⋅ mb = ma ⋅ b = m( a ⋅ b ), where m is a scalar a ⋅ (b + c ) = a ⋅ b + a ⋅ c (a + b ) ⋅ ( c + d ) = a ⋅ c + a ⋅ d + b ⋅ c + b ⋅ d

y op

ge

C

U

R

ni

ev

ve

ie

w

rs

C

op

Show that the scalar product has the following properties:

id

ie

w

Scalar product for component form

-R

am

br

ev

When we write a = a1i + a2 j + a3 k and b = b1i + b2 j + b3 k the properties of the scalar product give an algebraic method of finding the same value that we found using a b cos θ .

es

= a1i(b1i + b2 j + b3 k ) + a2 j(b1i + b2 j + b3 k ) + a3 k( b1i + b2 j + b3 k ) = a1b1i ⋅ i + a1b2 i ⋅ j + a1b3 i ⋅ k + a2b1j ⋅ i + a2b2 j ⋅ j + a2b3 j ⋅ k + a3b1k ⋅ i + a3b2 k ⋅ j + a3b3 k ⋅ k

ni ve rs

ity

Pr

op y C

op

U

i⋅i = j⋅ j = k ⋅k = 1

y

However, as i is parallel to i, j is parallel to j and k is parallel to k then:

ie

id g

w

e

C

Also, as i, j and k are perpendicular to each other, scalar products such as i ⋅ j, j ⋅ k, etc. are all 0.

br

ev

a ⋅ b = a1b1i ⋅ i + a1b2 i ⋅ j + a1b3 i ⋅ k

-R s

+ a3b1k ⋅ i + a3b2 k ⋅ j + a3b3 k ⋅ k

es

am

+ a2b1j ⋅ i + a2b2 j ⋅ j + a2b3 j ⋅ k

-C

w ie ev

R

s

-C

a ⋅ b = ( a1i + a2 j + a3 k )( b1i + b2 j + b3 k )

Copyright Material - Review Only - Not for Redistribution

ve rs ity

In general, the algebraic definition of the scalar product of two vectors is:

Pr es s

-C

-R

a ⋅ b = a1b1 + a2b2 + a3b3

ev ie

am br id

KEY POINT 9.6

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

C

ve rs ity

op

y

 a1   b1    When written in column form, a =  a2  and b =  b2  a3   b3

   , so it is perfectly valid to 

ni

C op

y

We now have a way to find the dot product when we do not know the angle between the two vectors.

U

R

ev ie

w

use the components of column vectors in the algebraic form of the scalar product a ⋅ b = a1b1 + a2b2 + a3b3.

w

ge

Sometimes the two versions of the scalar product are written together like this:

br

ev

id

ie

a ⋅ b = a1b1 + a2b2 + a3b3 = a b cos θ

-C

-R

am

This can be useful as it allows us to find the angle between the two vectors when we know the components of the vectors.

y

es

s

WORKED EXAMPLE 9.8

op

Pr

p = 8 i − 6 j − 10 k and q = 4 i − 3 j

228

rs

Answer

op

ni

Find the value of the scalar product p ⋅ q.

C

U

R

p ⋅ q = p1q1 + p2q2 + p3q3

y

ve

ev

ie

w

C

ity

Show that the angle, θ , between the two vectors is 45°.

p ⋅ q = 8(4) + ( −6)( −3) + ( −10)(0) = 32 + 18 = 50

id

ie

w

ge

Find the magnitude of p and of q using Pythagoras’ theorem.

-R

42 + ( −3)2 + 02 = 5

es Pr

Solve for θ .

ity

1  = 45° θ = cos −1   2 

y -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

ni ve rs

C

op y

50 = cos θ 50 2

w

Make cos θ the subject.

s

-C

50 = ( 10 2 ) (5) cos θ

Use p ⋅ q = p q cos θ .

op

q =

am

br

ev

p = 82 + ( −6)2 + ( −10)2 = 10 2

Copyright Material - Review Only - Not for Redistribution

ve rs ity ge

C

U

ni

op

y

Chapter 9: Vectors

am br id

ev ie

w

WORKED EXAMPLE 9.9

Relative to the origin O, the position vectors of the points A and B are given by  −1    and OB =  3  .      −1 



U

R

ni

ev ie

A

w -R

s

   −5 

 4

Pr ity

op  

OB ⋅ AB = ( −1)( −6) + (3)(5) + ( −1)( −5) = 26

w

rs

C w ie ev

( −1)2 + 32 + ( −1)2 = 11

   

s

-C

-R

am

Use OB ⋅ AB = OB AB cos OBA

es

Make cosOBA the subject.

Pr

op y

26 = cosOBA 86 11

ity

Solve for angle OBA.

Angle OBA = 32.3°

y op -R s

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

ni ve rs

C

op

y

ve ni

U

86

26 = 86 11 cosOBA

w



ge

 −1  =  3 =    −1 



Find the magnitude of OB and of AB using Pythagoras’ theorem.

id

 OB

( −6)2 + 52 + ( −5)2 =

br

 −6  =  5 =    −5 

229

es

C

 −1   Using the given vector OB =  3  ,    −1    find the value of OB ⋅ AB.

es

 

ie ev



ev

am

   −1 

y

-C

 −6   −1   5        AB = OB − OA = 3 − −2 =  5 

 AB

A sketch will help you to make sure you choose a correct pair of directions for your scalar product.

First of all, find the column vector AB .

br

 

Find OB ⋅ AB.

ie

id

B

R



OB and AB are the two directions towards the point B, so these are the vectors we need to use.

ge

O



TIP

y

ve rs ity

Answer

w

C

op

Find angle OBA.

C op

y

Pr es s

-C



-R

 5 OA =  −2   4

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

am br id

EXERCISE 9C

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

1 Find which of the following pairs of position vectors are perpendicular to one another.

-R

For any position vectors that are not perpendicular, find the acute angle between them.

ve rs ity

ni

Find angle BOA.

 

 3

 1  and OB =  7  .    2

y

ev ie

c

C op

2

b

 4  −4    e = −9 , f =  −2       1  −2 

 −5   Relative to the origin O, the position vectors of the points A and B are given by OA =  0 

w

C

op

y

a

 5   −3  c =  12  , d =  −1       13   3

Pr es s

-C

 7  2   a= 8 , b =  −1       −2   3

ge

U

R

3 Given that the vectors 6 i − 2 j + 5 k and ai + 4 j − 2 k are perpendicular, find the value of the constant a.

am

ie

  where k is a constant.  

ev



br



-R

 k   , OQ =  k + 2    −1 

id

 5k OP =  −3   7k + 9

w

4 Relative to the origin O, the position vectors of the points P and Q are given by





s

-C

a Find the values of k for which OP and OQ are perpendicular to one another.

b Given that k = 2, find the angle, θ , between OP and OQ.

Pr

y

5 The diagram shows a cuboid OABCDEFG in which OC = 6 cm, OA = 4 cm and OD = 3 cm. Unit vectors i, j and k are parallel to OC , OA and OD respectively. The point M lies on FB such that FM = 1cm, N is the midpoint of OC and P is the centre of the rectangular face DEFG.

P

rs

w

A

ni

op

B

C

O

i

w

ge

j

ie

-R

D

s

es

Pr

 



C



a Find each of the vectors OM and NG.

ity

C

op y

 

 

N



ni ve rs

y

O

E

-R

C

D

36 cm

w

N

ie O

C

j i

77 cm

s es

-C

Use a scalar product to find angle MAN .

A

i

k

ev

e

am

br

id g



op

M

U

9 The diagram shows a triangular prism, OABCDE . The uniform cross-section of the prism, OAE, is a rightangled triangle with base 77 cm and height 36 cm. The length, AB, of the prism is 60 cm. The unit vectors i, j and k are parallel to OA, OC and OE, respectively. The point N divides the length of the line DB in the ratio 4:1 and M is the midpoint of DE .

 

B

j

k

b Find the angle between the directions OM and NG.

w ie ev

F

E

8 The diagram shows a cube OABCDEFG with side length 4 units. The unit vectors i, j and k are parallel to OA, OC and OD, respectively. The point M is the midpoint of GF and AN : NB is 1:3.

R

M

G

ev

id

br

am

-C

7 Explain why a ⋅ b ⋅ c has no meaning.

C

N

6 The position vector a is such that a = 4 i − 8 j + k. Find a ⋅ j and hence find the angle that a makes with the positive y-axis.

PS

M G

k

U

R

D

y

ve

ie

Using a scalar product, find angle NPM.

ev

F

E

ity

op C

230



es



Copyright Material - Review Only - Not for Redistribution

B A

60 cm

ve rs ity PS

C

U

ni

op

y

Chapter 9: Vectors

V

w

ge

10 The diagram shows a pyramid VOABC . The base of the pyramid, OABC , is a rectangle. The unit vectors i, j and k are parallel to OA, OC and OV , respectively. The position vectors of the points A, B, C and V are given by







ev ie

am br id

 



N

-R

OA = 6 i, OC = 3 j and OV = 9 k.

-C

The points M and N are the midpoints of AB and BV , respectively.

Pr es s



k



b c

The point P lies on OV and is such that PNM is a right angle. Find the position vector of the point P.

B M

O

ve rs ity

ev ie

C

j

  Find the vector MN.

w

C

op

y

a Find the angle between the directions of AN and OC.

ni

C op

y

9.4 The vector equation of a line

A

i

am

br

ev

id

ie

w

ge

U

R

We already know that, in 2-dimensional space, a line can be represented by an equation of the form y = mx + c or ax + by = c . In 3-dimensional space, we need a new approach because introducing the third dimension gives an equation of the form ax + by + cz = d , which is actually the Cartesian equation of a plane. One way around this problem is to write the equation of the line using vectors.

s

es

-C

So what information do we need to be able to do this?

-R

The equation of a line in vector form is valid in both 2- and 3-dimensional space.

op

Pr

y

When a line passes through a fixed given point and continues in a known direction then it is uniquely positioned in space.

231

y

op

w -R

ev

br

es

s

-C O (0, 0, 0)

Pr

op y



ni ve rs

ity

R is any point on the line and so OR represents the position vector of any point on the line.

  

op



y

To reach R, starting at O, take the path OR = OA + AR.

C

U

OA is a vector we know. To move from A to R, we need to travel in the direction of b.

 

ie

id g

w

e

Multiplying b by different scalars changes the position of R on the line, so simply gives us different points on the line. Using the parameter t to represent the variable scalar

-R s es

am

br

ev

OR = OA + tb

-C

C w ie ev

R

ie

 b1  b =  b2   b3 

R (x, y, z)

am

A (a1, a2, a3)

id

ge

C

U

R

ni

ev



 b1    a point on the line, A( a1, a2 , a3 ), and a direction vector, b =  b2  , that is parallel to the line.  b3 

ve

ie

w

rs

C

ity

Therefore, to form the vector equation of a straight line, we need:

Copyright Material - Review Only - Not for Redistribution

TIP Sometimes Greek letters such as λ and µ are used as parameters.

ve rs ity

r = a + tb

R (x, y, z)

op

C (c1, c2, c3)

y C op

U

R

ni

ev ie

w

C

ve rs ity

A (a1, a2, a3)

Pr es s

-C

Or two points on the line, A( a1, a2 , a3 ) and C ( c1, c2 , c3 ) . One can be used as the fixed given point and the direction can be found from the displacement between the two points.

y



-R

Using lower case notation for vectors, the vector equation of the line is:

ev ie

am br id

KEY POINT 9.7

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

w

ge

O (0, 0, 0)

br

ev

id

ie

The situation is almost the same. In this case, however, we do not know the direction of the line. We can find the direction using the position vectors of the points A and C .

  

s

-C

OR = OA + tAC .

es

  

op

Pr

y

KEY POINT 9.8 232

Any known point can be used for the position vector, so you could also use OR = OC + tAC

-R

am

Using the parameter t as before,

ity rs

y

ve

In each case, the structure is:

ev

general point on line = position of known point + some way along a known direction

w

ge

Parametric form

C

U

R

ni

op

ie

w

C

Using lower case notation for vectors, the vector equation of a line is: r = a + t( c − a )

id

ie

R( x, y, z ) represents any point on the line.

s

-C

-R

am

br

ev

 x The position vector of this point can be written as r =  y  or r = xi + yj + zk.    z  In the equation r = a + tb , using the position vectors in component form:

op C

w

-R

s es

-C

am

br

ev

id g

Comparing the components: x = a1 + tb1 y = a2 + tb2 z = a3 + tb3 These three equations are called the parametric equations of the line.

ie

e

U

 xi + yj + zk = a1i + tb1i + a2 j + tb2 j + a3 k + tb3 k   or xi + yj + zk = ( a + tb ) i + ( a + tb ) j + ( a + tb ) k 1 1 2 2 3 3 

y

ni ve rs

w ie ev

R

 x   a1 + tb1  y  =  a + tb 2    2  z   a3 + tb3 

ity

C

Pr

op y

es

 b1   x   a1   y  =  a  + t  b  or xi + yj + zk = a i + a j + a k + t ( b i + b j + b k ) 1 2 3 1 2 3  2     2   z   a3    b    3 

Simplifying the right-hand side of this:

TIP

Copyright Material - Review Only - Not for Redistribution

ve rs ity ge

C

U

ni

op

y

Chapter 9: Vectors

am br id

ev ie

w

WORKED EXAMPLE 9.10

a the vector equation of the line

Answer

General point on line =

ni U

R

a General point = r

C op

ev ie

w

Using t as the parameter:

y

ve rs ity

C

op

y

b the parametric equations of the line.

Pr es s

-C

Find:

-R

A line passes through the point with position vector 3i – 2 j + 4 k and is parallel to the vector i – j + k.

position of known point

Known direction = i – j + k

+ some way along a known direction.

br

ev

id

ie

w

ge

Known point = 3i – 2j + 4k

w

s es

rs

C

ity

b

Write r as a column vector and collect together the x-, y- and z-components of the right-hand side.

 3  1   r = −2 + t  −1  or r = 3i – 2 j + 4 k + t ( i − j + k )      4  1

Pr

op

y

-C

-R

am

 3  1   r = −2 + t  −1       4  1

y op C

w ie

z = 4+t

ev

id

y = −2 − t

Compare components.

-R

am

br

x = 3+t

ge

U

R

ni

ev

ve

ie

 x   3+t   y  =  −2 − t  or      z   4 + t  xi + yj + zk = (3 + t ) i + (–2 − t ) j + (4 + t ) k

es

s

-C

WORKED EXAMPLE 9.11

ni ve rs



ity

respectively. a Write down the vector AB.

op ie

w

Substitute for a and b.

ev

Expand and simplify.

-R s es

am

br

id g

 AB = b − a  AB = i − 4 j + 8 k − (3i + 2 j − 7 k )  AB = −2 i − 6 j + 15 k

C

U e

Answer a

y

b Find the vector equation of the line AB.

-C

R

ev

ie

w

C

a = 3i + 2 j − 7 k and b = i − 4 j + 8 k,

Pr

op y

The position vectors of the points A and B are given by

Copyright Material - Review Only - Not for Redistribution

233

C

U

ni

op

y

ve rs ity

lines is the angle between the two direction vectors of the Cambridge International AS & Alines. Level Mathematics: Pure Mathematics 2 & 3

General point on line =

Known point = 3i + 4 j − 7 k

position of known point

Pr es s

-C

-R

General point = r

+ some way along a known direction.

y

id

ie

w

ge

The angle between two lines is the angle between the two direction vectors of the lines.

C op

U

ni

y

r = 3i + 2 j − 7 k + t ( −2 i − 6 j + 15 k )

KEY POINT 9.9

REWIND Recall, you could also use b = i − 4 j + 8 k as the known point.

ve rs ity

op C

 r = 3i + 2 j − 7 k + tAB

w ev ie

ev ie

am br id



Known direction = −2 i − 6 j + 15 k

R

w

ge

b Using a = 3i + 4 j − 7 k as the known point, t as the parameter and AB from part a as the known direction:

-R

am

br

ev

WORKED EXAMPLE 9.12

-C

The vector equation of the line L1 is r = 3i + j − k + λ ( i − 8 j + 5 k ).

es

s

a The line L2 is parallel to the line L1 and passes through the point A(0, 2, 5) .

op

Pr

y

Write down the vector equation of the line L2 .

234

op

y

General point on line =

U

R

Known point is 2 j + 5 k

ve

ev

General point is r

rs

Answer a Using µ as the parameter,

ni

ie

w

C

ity

b Find the angle between the line L1 and the line r = 4 i + 3 j − 2 k + µ (5 i − j + 2 k ) .

ge

C

position of known point

The direction of the line L1 is i − 8 j + 5 k

id

ie

w

+ some way along a known direction point.

-R

am

br

ev

The line L2 is therefore r = 2 j + 5k + µ ( i − 8 j + 5k )

Find the scalar product d1 ⋅ d 2

es Pr

ity

y

Make cos θ the subject.

C

U

Solve for θ .

w

e

23 3 10 30

Use d1 ⋅ d 2 = d1 d 2 cos θ .

30

ie

id g

cos θ =

-R s es

am

br

ev

23   cos −1  = 63.727 … ° = 63.7°  3 10 30 

-C

ev

R

23 = 3 10 30 cos θ

Now find the magnitude of d1 and d 2 using Pythagoras’ theorem

op

d 2 = 52 + ( −1)2 + 22 =

ni ve rs

d1 = 12 + ( −8)2 + (5)2 = 3 10

ie

w

C

op y

d1 ⋅ d 2 = (1)(5) + ( −8)( −1) + 5(2) = 23

s

-C

b Find the angle between d1 = i − 8 j + 5 k and d 2 = 5 i − j + 2 k.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

am br id

EXERCISE 9D

w

ge

C

U

ni

op

y

Chapter 9: Vectors

1 Write down the vector equation for the line through the point A that is parallel to vector b when: A(0, −1, 5) and b = 2i + 6j – k

-R

a

A(7, 2, −3) and b = 3i – 4k.

op

y

c

Pr es s

-C

b A( 0, 0, 0 ) and b = 7i – j – k

ve rs ity

3 Show that the line through the points with position vectors 9i + 2j – 5k and i + 7j + k is parallel to the line r = λ (16 i − 10 j − 12 k ).

y

ev ie

w

C

2 Write each of your answers to question 1 in parametric form.

r = 2 i + 13 j + k + t ( i + j − k )

b

r = 10 j + t (2 i + 5 j )

c

r = i − 3 j + t (2 i + 3 j + 4 k )

ev

id

ie

w

ge

U

a

br

R

ni

C op

4 Find the parametric equations of each of the lines.

-R

am

5 a Find the parametric equations of the line L through the points A(0, 4, −2) and B (1, 1, 6).

s

es

6 The line L1 has vector equation r = 2 i – 3 j + k + λ (6 i + j + 3k ) and the line L2 has parametric equations x = µ + 4, y = µ − 7, z = 3µ .

Pr

rs

b Find the angle between the line L1 and the line L2 .

op

y

7 The vector equation of the line L1 is given by

ni

ev

PS

ve

w ie

235

a Show that L1 and L2 are not parallel.

ity

C

op

y

PS

-C

b Hence find the coordinates of the point where the line L crosses the Oxy plane.

id

ie

w

ge

C

U

R

 x   −3   4  y  =  0  + t  −1  .       z   8   −3 

br

ev

a Find the vector equation of the line L2 that is parallel to L1 and which passes through the point A(5, −3, 2) .

-R

es

   5

   3

ity

Pr

   2

a Find a vector equation for the line L passing through A and B.

ni ve rs

op -R s

-C

am

br

ev

ie

id g

w

e

C

U

R

y

b The line through C , perpendicular to L, meets L at the point N. Find the exact coordinates of N .

es

C w

 1  3  0        OA = 1 , OB = −1 and OC =  2  .

ev

ie

s

-C

8 The points A, B and C are position vectors relative to the origin O, given by

op y

PS

am

b Show that A(5, −3, 2) is the foot of the perpendicular from the point B (4, −7, 2) to the line L2 .

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

w

ge

9.5 Intersection of two lines

am br id

ev ie

In 2-dimensional space, two lines are either parallel or they must intersect at a point.

y



Pr es s



they are parallel or they are not parallel and intersect or they are not parallel and do not intersect.

-C



-R

In 3-dimensional space it might be that:

ve rs ity

C

op

Lines that are not parallel and do not intersect are called skew.

y C op

ni

Non-parallel intersecting lines

In 2-dimensional space, to find the point of intersection of a pair of non-parallel lines using their Cartesian equations ( ax + by = c ), we solve the equations of the lines simultaneously. In other words, we put the two equations equal to each other.

id

ie

w

ge

U

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ev ie

w

Parallel lines have the same direction vectors and so if a pair of lines is parallel, this can easily be seen by examining the vector equations of the lines.

am

br

ev

The process is the same in 3-dimensional space and using the vector equations of the lines.

-C

-R

WORKED EXAMPLE 9.13

es

s

Given that the lines with vector equations

op

Pr

y

r = i − 2 j − 4k + λ (−i − j + k )

rs

intersect at the point A, find the coordinates of A.

y C

s (2)

−4 + λ = 1 − 7 µ  (3) Solve these equations to find the value of λ and of µ:

ni ve rs

C

op -R s

-C

am

br

ev

ie

id g

w

e

C

U

R

y

λ =   −1 − µ −4 − 1 − µ = 1 − 7 µ µ =1 6µ = 6

From (1): Substitute into (3):

es

w ie

Pr

−2 − λ = −2 + 2 µ

ity

(1)

op y

1− λ = 2 + µ

es

Now equate the components.

ev

w

ie

= 2i − 2 j + k + µ( i + 2 j − 7k) = 2+µ = −2 + 2µ = 1 − 7µ

ev

r x y z

-R

am

br

id

= i − 2 j − 4k + λ (−i − j + k ) = 1− λ = −2 − λ = −4 + λ

-C

r x y z

ge

U

R

The lines are equal at the point of intersection, so write the vector equation in parametric form so that the components can be equated.

op

ni

Answer

ev

ity

and r = 2 i − 2 j + k + µ ( i + 2 j − 7 k )

ve

ie

w

C

236

Copyright Material - Review Only - Not for Redistribution

ve rs ity The lines intersect at the point A(3, 0, −6).

ev ie

It is good to check that the value of λ and of µ that we have found give us the same point.

Skew lines

y

ve rs ity

U

R

ni

Skew lines have no point of intersection.

id

ie

w

ge

When two lines are skew, they are not parallel and they do not intersect.

br

ev

WORKED EXAMPLE 9.14

-R

am

Show that the lines with vector equations

es

s

-C

r = 4 i + 7 j + 5 k + λ ( −2 i + j + 3k )

op

Pr

y

r = i + j + 10 k + µ ( −6 i + j + 4 k )

ity ve

ie

op

ni

ev

As −2 i + j + 3k   is not a scalar multiple of −6 i + j + 4 k the lines are not parallel.

-C

ie

w

= i + j + 10 k + µ ( −6 i + j + 4 k ) = 1 − 6µ = 1+ µ = 10 + 4µ

ev

am

br

id

ge

r x y z

-R

= 4 i + 7 j + 5 k + λ ( −2 i + j + 3k ) = 4 − 2λ = 7+λ = 5 + 3λ

C

U

R

Writing the equations in parametric form: r x y z

s es Pr

ni ve rs

(1) (2) (3)

ity

4 − 2 λ = 1 − 6µ 7 + λ = 1+ µ 5 + 3λ = 10 + 4µ

Attempt to solve these equations to find the value of λ and of  µ that should give a consistent point:

y C w ie ev

id g

e

-R s es

am

br

When µ = −23:

U

Substitute into (3):

λ =  µ − 6 5 + 3(  µ − 6) = 10 + 4µ µ = −23 λ =  −23 − 6 = −29

op

From (2):

-C

ev

ie

w

C

op y

Now equate the components.

R

237

rs

Answer

w

C

do not intersect.

y

ev ie

w

C

op

y

y = −2 − ( −2) = 0 z = −4 − 2 = −6

x = 2 +1 = 3 y = −2 + 2 = 0 z = 1 − 7 = −6

TIP

C op

-C

x = 1 − ( −2) = 3

µ =1

-R

λ = = −2

Pr es s

Check:

w

ge

µ = 1: λ =   −1 − 1 = −2

am br id

When µ = 1: λ =

C

U

ni

op

y

Chapter 9: Vectors

Copyright Material - Review Only - Not for Redistribution

ve rs ity

µ = = −23

x = 4 − 2( −29) = 62 y = 7 − 29 = −22

x = 1 − 6( −23) = 139

z = 5 + 3( −29) = −82

z = 10 + 4( −23) = −82

-R

λ = = −29

ev ie

am br id

y

-C

y = 1 + ( −23) = −22

Pr es s

Check:

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

TIP

U

R

ni

C op

1 Find whether the following pairs of lines are parallel, intersecting or skew.

y

EXERCISE 9E

ev ie

w

C

ve rs ity

op

The values of λ and µ do not give a consistent point, so there is no point of intersection and the lines are skew.

w

x = µ +1 y= µ+4

br

id

y = 5λ + 1 

ie

x = 3λ + 4

z = 2µ + 5

r = i − 5 j + 4 k + λ (3i − 4 j + 9 k )

r = i − 5 j + 4 k + λ ( −6 i + 8 j − 18 k )

c

r = 7 i + 2 j + 6 k + λ (i + 3 j + 9 k )

r = 4 j + k + µ (8 i + j + 14 k )

d

r = 2 i − 3j + 4k + λ (6i + 4 j   −2 k)

r = 4i − 9 j + 2 k + µ (−i − 8 j − k )

s

es

r = 2 i + 9 j + k + λ ( i – 4 j + 5k )

Pr

2 Given that the lines with vector equations

rs

ity

PS

w

y

ev

ve

ie

and r = 11i + 9 j + pk + µ ( − i − 2 j + 16 k ) intersect at the point P, find the value of p and the position vector of the point P.

C

U

R

ni

op

238

C

op

y

-C

b

-R

am

z = 4λ − 3

ev

a

ge

State the position vector of the common point of any lines that intersect.

w

ge

3 Three lines, L1, L2 and L3 , have vector equations

id

ie

r = 16 i − 4 j − 6 k + λ ( −12 i + 4 j + 3k )

br

ev

r = 16 i + 28 j + 15 k + µ (8 i + 8 j + 5 k )

-R

am

r = i + 9 j + 3k + ν (4 i − 12 j − 8 k )

s

-C

The 3 points of intersection of these lines form an acute-angled triangle.

es

For this triangle, find:

op

y

4 The line L1 passes through the points (3, 7, 9) and ( −1, 3, 4) .

C

U

a Find a vector equation of the line L1.

w

e

b The line L2 has vector equation

ie

id g

r = i + 2 j + k + µ (3 j + 2 k ).

-R s es

am

br

ev

Show that L1 and L2 do not intersect.

-C

ie ev

R

ity

the length of each side.

ni ve rs

c

w

C

b the size of each of the interior angles

Pr

op y

a the position vector of each of the three vertices

Copyright Material - Review Only - Not for Redistribution

• Always carry out the check with both λ and µ as it is easy to make a mistake. • Re-read the question to check whether you were asked to find the position vector or the coordinates of the point of intersection and give your answer in the correct form.

ve rs ity

C

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op

y

Chapter 9: Vectors

ev ie

am br id

w

ge

5 The point A has coordinates (1, 0, 5) and the point B has coordinates ( −1, 2, 9).  a Find the vector AB.

Find the acute angle between the line AB and the line L with vector equation r = i + 3 j + 4 k + µ ( − i − 2 j + 3k ) .

Pr es s

-C

c

-R

b Write down a vector equation of the line AB.

ev ie

w

C

ve rs ity

op

y

d Find the point of intersection of the line AB and the line L.

The magnitude (length or size) is worked out using Pythagoras’ theorem:

Two vectors are parallel if one is a scalar multiple of the other.



Problem solving with angles will involve the use of the scalar product.

-R

am



s

239

Vector equation of a line: r = a + tb



To find the point of intersection of two intersecting lines, solve the parametric equations of the lines simultaneously.

ity



y op y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

es

s

-C

-R

am

br

ev

id

ie

w

ge

C

U

R

ni

ev

ve

rs

C

es

b cos θ

Pr

a ⋅ b = a1b1 + a2 b2 + a3b3 = a

op

y

-C

When a = a1i + a2 j + a3 k and b = b1 i + b2 j + b3 k

w

ev

ie

a12 + a2 2 + a32 .

br

id

When a = a1i + a2 j + a3 k , a =

ie

y



U

For position vectors OA = a and OB = b, AB = b − a.

w







ge

R



C op

ni

Checklist of learning and understanding

Copyright Material - Review Only - Not for Redistribution

ve rs ity

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

1

w ev ie

am br id

END-OF-CHAPTER REVIEW EXERCISE 9

-R

Relative to the origin O, the position vectors of points A, B and C are given by

Pr es s

   1 

   7

 

It is given that AB = OC . Find the angle OAB.

[4]

b

Find the vector equation of the line AC.

[3]

With respect to the origin O, the position vectors of the points A and B are given by  −2   OA =  0 

y C op

ni

w

ge

U

   4





Find whether or not the vectors OA and OB are perpendicular.

ie

s

es

r = 7 i − 12 j + 7 k + µ ( −3i + 10 j − 5k ).

Pr

y op

[2]

The vector equation of the line L is given by

ity

Show that the lines AB and L intersect and find the position vector of the point of intersection.

3

rs

H

w

op

y

ve k

U

R

N

G

F

ni

ev

ie

E

C

M

B

w

A

ie

i

id

O

ge

j

D

[5]

C

C

[1]

-R

-C

ii Find a vector equation of the line AB. c

[2]

ev

i

 Write down the vector AB.

am

b

 6

id

a

 1  and OB =  −1  .

br

R

 

240

 .  

a

ev ie

w

2

4 −m   − m( m + 1)

ve rs ity

C

op

y

-C

 3m    2        1 OA = 3 , OB = and OC = 

-R

am

br

ev

The diagram shows a prism, ABCDEFGH, with a parallelogram-shaped uniform cross-section.

 

The point E is such that OE is the height of the parallelogram. The point M is such that OM is parallel to DC and N is the midpoint of DE . The side OD has a length of 5 units. The unit vectors i, j and k are parallel to OA, OM and OE, respectively.

s



es

-C

   



 



 

Pr

 

Express the vectors AH and NH in terms of i, j and k.

[2]

b

Use a vector method to find angle AHN .

[5]

c

Write down a vector equation of the line AH.

[2]

op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

y

ni ve rs

ity

a

ev

ie

w

C

op y

The position vectors of the points A, E , C and M are given by OA = 9 i, OM = 15 j and OE = 12 k.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

w ev ie

 8  2  6 OA =  3  , OB =  n  and OC =  9       −1   0  2





ve rs ity

where n is a constant.

 

Find the value of n for which AB = CB .

[4]

b

In this case, use a scalar product to find angle ABC .

[3]

a

Given the vectors 8 i − 2 j + 5 k and i + 2 j + pk are perpendicular, find the value of the constant p.

b

The line L1 passes through the point ( −3, 1, 5) and is parallel to the vector 7 i − j − k.

ni

C op

y

a

ge

R

5

   

U

ev ie

w

C

op

y

-C



-R

The position vectors of A, B and C relative to an origin O are given by

Pr es s

4

am br id

ge

C

U

ni

op

y

Chapter 9: Vectors

ie

id

[4]

Pr

es

The origin O and the points A, B and C are such that OABC is a rectangle. With respect to O, the position vectors of the points A and B are −4 i + pj − 6 k and −10 i − 2 j − 10 k. a

Find the value of the positive constant p.

b

Find a vector equation of the line AC.

c

Show that the line AC and the line, L, with vector equation

ity

[3]

rs

[3]

y

ve

y op C w ie

Show that L1 and L2 do not intersect.

s

-C

am

r = i − 2 j + 2 k + µ ( i + 8 j − 3k ).

-R

br

ev

ii The line L2 has vector equation

6

[2]

w

i Write down a vector equation of the line L1.

op

U

R

ni

ev

r = 3i + 7 j + k + µ ( −4 i − 4 j − 3k )

ge

C

intersect and find the position vector of the point of intersection. d

[5] [4]

id

ie

w

Find the acute angle between the lines AC and L.





ev

Relative to the origin O, the points A, B, C and D have position vectors

br

7

[3]





-R

Use a scalar product to show that angle ABC is a right angle.

s

[2]

c

The point E is the midpoint of the line AD. Find a vector equation of the line EC .

[4]

Pr

es

b

With respect to the origin O, the points A, B, C , D have position vectors given by





ity

8

[3]

  Show that AD = kBC , where k is a constant, and explain what this means. 



ni ve rs

OA = −3i + j + 8 k, OB = −10 i + 2 j + 15 k, OC = 5 i − 2 j − 2 k, OD = i + 6 j + 10 k. a

Calculate the acute angle between the lines AB and CD.

b

Prove that the lines AB and CD intersect.

c

The point E has position vector 5 i + 3 j + 4 k. Show that the perpendicular distance from E to the line CD is equal to 5 .

op

y

[4] [5]

C

U

w

e

ev

ie

id g

es

s

-R

br am -C

R

ev

ie

w

C

op y

-C

a

am

OA = 4 i + 2 j − k, OB = 2 i − 2 j + 5 k, OC = 2 j + 7 k, OD = −6 i + 22 j + 9 k.

Copyright Material - Review Only - Not for Redistribution

[5]

241

ve rs ity

w ev ie

am br id

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C

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3





Find the vectors PQ and PS.

b

Find the coordinates of R.

c

Show that PQRS is in fact a square and find the length of the side of the square.

[3]

d

The point T is the centre of the square. Find the coordinates of T .

[3]

e

The point V has coordinates (5, 17.5, −13.5).

[3]

ve rs ity

y op C w

[2]

Pr es s

-C

a

-R

PQRS is a rhombus. The coordinates of the vertices P, Q and S are (9, 2, 4), ( −0.5, 6, 6.5) and (4.5, −4, −3.5) , respectively.

9

U

R

ni

ii Verify that T is the foot of the perpendicular from V to PR.

ge

iii Describe the solid VPQRS .

y

[3]

C op

ev ie

i Find a vector equation of the line VT .

[2] [1]

id

ie

w

10 The points P and Q have coordinates (0, 19, −1) and ( −6, 26, −11) , respectively.

am

br

ev

The line L has vector equation

b

 Find the magnitude of PQ.

es

[2]

Pr

y

c

Find the obtuse angle between PQ and L.

[3]

d

Calculate the perpendicular distance from Q to the line L.

[5]

w

rs

ity

op C

ve

Write down a vector equation of the line AB.

b

The point P lies on the line AB. The point Q has coordinates (0, −5, 7).

C

w

ge

Given that PQ is perpendicular to AB, find a vector equation of the line PQ.

[6]

ie

[1]

ev

id

Find the shortest distance from Q to the line AB.

br

c

[3]

op

y

a

ni

R

ev

ie

11 The points A and B have coordinates (7, 1, 6) and (10, 5, 1), respectively.

U

242

[1]

s

Show that the point P lies on the line L.

-C

a

-R

r = 3i + 9 j + 2 k + λ (3i − 10 j + 3k ).

-R

am

12 The line L1 has vector equation

-C

r = 3i + 2 j + 5 k + λ (4 i + 2 j + 3k).

es

s

The points A(3, p, 5) and B ( q, 0, 2), where p and q are constants, lie on the line L1. [2]

b

Show that L1 and L2 intersect and find the position vector of the point of intersection.

[5]

c

Find the acute angle between L1 and L2 .

Find the value of p and the value of q.

Pr ity

y

ni ve rs

r = 3 j + k + µ (7 i + j + 7 k ).

-R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

C

The line L2 has vector equation

op

op y

a

Copyright Material - Review Only - Not for Redistribution

[4]

op

y

ve rs ity ni

C

U

ev ie

w

ge

-R

am br id

Pr es s

-C y

ni

C op

y

ve rs ity

op C w ev ie

Chapter 10 Differential equations

y op w

ge

C

U

R

ni

ev

ve

rs

w

C

ity

op

Pr

y

es

s

-C

-R

am

br

ev

id

ie

w

ge

U

R ie

243

id

ie

P3 This chapter is for Pure Mathematics 3 students only.

br

am

es

s

-C

-R

formulate a simple statement involving a rate of change as a differential equation find, by integration, a general form of solution for a first order differential equation in which the variables are separable use an initial condition to find a particular solution interpret the solution of a differential equation in the context of a problem being modelled by the equation.

Pr

ity

op

y

ni ve rs -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

C

op y

■ ■ ■ ■

ev

In this chapter you will learn how to:

Copyright Material - Review Only - Not for Redistribution

ve rs ity

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C

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Check your skills

IGCSE / O Level Mathematics

Express direct and inverse variation in algebraic terms and use this form of expression to find unknown quantities. Understand and apply ● laws of indices ● laws of logarithms

Chapter 5

Integrate ● standard functions ● using substitution ● using parts

ve rs ity

IGCSE / O Level Mathematics

1 The weight, w newtons, of a sphere is directly proportional to the cube of its radius, r cm. When the radius of the sphere is 60 cm, its weight is 4320 newtons. Find an equation connecting w and r.

ni

U

3 Find:

∫ 3 − 1 dx sin x dx ∫ cos x ∫ xe dx.

ie

w

ge

ev

id

b

br

-R

am

s

es

c



4 Find

2

dx. ∫ (x − 2)( 2 )(3 )( 3x + 1)

Pr

y

ity

op

y op

b

12e x → k as x → ∞. 3 + ex Find the value of the constant k.

ie ev

id br

-R

am

-C

What is a differential equation?

s

dy d2 y or as one of its terms is called a dx dx 2 dy differential equation. When the equation has the first derivative as one of its terms, dx dy it is called a fi rst order differential equation. For example, the equation = 3x 2 is an dx alternative way of representing the relationship y = x 3 + C , where C is a constant.

es

ie

id g

w

e

C

U

op

y

To solve a differential equation, we integrate to find the direct relationship between x and y, such as y = x 3 + C . The direct equation y = x 3 + C is called the general solution dy of the differential equation = 3x 2 . To find a particular solution, we must have further dx information that will enable us to find the value of C .

-R s es

am

br

ev

The solution of an algebraic equation is a number or set of numbers. The solution of a differential equation is a function.

-C

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

An equation that has a derivative such as

2

5 a Sketch the graph of y = e x .

C

U

Sketch graphs of standard functions e.g. involving ln f( x ) and e f( x ) (including understanding end behaviour as x → ± ∞).

ge

R

Chapter 1

w

ni

ev

ve

ie

w

rs

C

244

1

a

Decompose a compound fraction with denominator ● ( + b )( )( cx + d )( )( ex + f ) ● ( + b )( )( cx + d )2 ● ( + b )( )( cx 2 + d ) into its partial fractions and integrate the result.

-C

Chapter 8

x + ln 3 4 Find and simplify an expression for y in terms of x. ln y − ln(20 − y ) =

y

2

C op

y

Pr es s

-R

What you should be able to do

-C

Where it comes from

op C w ev ie

R

ev ie

am br id

w

PREREQUISITE KNOWLEDGE

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Chapter 10: Differential equations

am br id

ev ie

w

ge

In this chapter, we will learn the initial skills needed to master the solution of one type of first order differential equations. These simple skills will be the foundation for further study of this subject.

ev ie

w

C

ve rs ity

op

y

Pr es s

-C

-R

You might wonder about the usefulness of such seemingly abstract equations in the real world. In fact, differential equations are excellent mathematical models for many real-life situations and are essential to solve many problems. First order differential equations can model rates of change, in other words, the rate at which one quantity is increasing or decreasing with respect to another quantity. There are many examples in the physical world that can be modelled using such relationships. A few examples are the rate of increase of the number of bacteria in a culture, simple kinematics problems, the rate of decrease of the value of a car and the rate at which a hot liquid cools.

ni

C op

y

10.1 The technique of separating the variables

w

ge

U

R

dy Solving a differential equation such as = 6( x + 1)2 can be done using an dx anti-differentiation approach.

br



-C

-R

am



ev

id

ie

We know, from the Pure Mathematics 1 Coursebook, Chapter 9, that if we integrate both sides with respect to x, we have: d dx = 6( x + 1)2 dx

s

6( x + 1)3 +C 3 y = 2( x + 1)3 + C

op

Pr

y

es

y=

rs

C w

245

dx =

∫ 6(x + 1) dx and from 2

dy anti-differentiation we can see that, if we integrate with respect to x we obtain an dx expression for y .

y

w

∫ dy = ..., which is y = …

op

= ... . is equivalent to

C

ni

dy

∫ dx d

ge

This means that

U

R

ev

ve

ie



ity

Notice that the first line on the left-hand side is

d

br

ev

id

ie

This result is generally true.

-R

am

KEY POINT 10.1

s

es

dy dy = 6( x + 1)2 is easy to solve because is equal to a dx dx function that we know how to integrate. This is not always the case.

op

y

ni ve rs

The differential equation

dy = xy when y . 0 ? dx As it is written, we cannot integrate both sides with respect to x and find a solution.

-R s es

am

br

ev

ie

id g

w

e

C

U

How do we solve the differential equation

-C

w ie ev

R

Pr



ity



C

op y

-C

dy Integrating an expression of the form f( y ) with respect to x is equivalent to integrating f( y ) dx with respect to y. dy Therefore f( y ) dx is equivalent to f( y ) d y. y dx

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

ev ie

w

ge

Separating the variables in the expression xy means that we are then able to integrate. dy = xy dx 1 dy =x y dx

-R

Separate the variables x and y.

-C

Form integrals of both sides with respect to x.

ie ev

id

br x

This is the general solution of the differential equation as we do not know the value of the arbitrary constant C .

-R

am

+C 2

where D = eC

y

es

s

-C

y = De 2

y

The constants of integration can be combined as one term and we only need to write a constant on one side.

w

ge

R

x2

y=e2

C op

x2 +C 2

ve rs ity

ln y =

The modulus of y is not required as we know that y . 0.

ni

x2 +B 2

Integrate each side.

U

ln y + A =

ev ie

w

C

op

y

Pr es s

dy dx ∫ x dx ∫ 1y ddy x ∫ y dy ∫ x dx

op

y

ve ni

Using this information:

w ie

x2 + ln 2 e2

ev

This is the particular solution of the differential equation.

-R

am

x2

C

U

ge

y=

x2 02 + C and so ln 2 = + C , C = ln 2 2 2

id

ln y =

br

R

ev

ie

w

rs

C

ity

op

Pr

When we have further information, we can find the particular solution of the differential solution given by that information. dy = xy given that y = 2 when x = 0 . For example, solve dx

246

y = e 2 × e ln 2

-C

Note that you often need to write your answer as an expression for y in terms of x. This commonly means using laws of indices or logarithms.

s

x2

Pr

op y

es

y = 2e 2

y op

C

U

w ie ev -R s es

∫ f( y) dy

∫ gg((x ) dx ∫ g(x ) dx

e

dx

(Note: Other variables may be used such as v and t.)

id g

d

am

∫ f(

dy = g( x ) dx

-C

R

f( y )

br

ev

ie

w

ni ve rs

C

ity

By separating the variables we are rearranging the differential equation to the form dyy f( y ) = g( x ). dx

Copyright Material - Review Only - Not for Redistribution

ve rs ity am br id

C

dy : dx

w

ge

So, in the case where the derivative is ●

form an expression in x only on one side of the equation



form an expression in y ×

ev ie

U

ni

op

y

Chapter 10: Differential equations

-R

dyy on the other side of the equation. dx

op

y

Pr es s

-C

This gives us a form of the equation that can be integrated on both sides, one side with respect to x and the other side with respect to y.

w

C

ve rs ity

WORKED EXAMPLE 10.1

dy 1 + x = . dx 2y

y

ev ie

The variables x and y satisfy the differential equation

U

R

ni

C op

It is given that y = 3 when x = 0.

ie ev

id

Answer

w

ge

Solve the differential equation and obtain an expression for y in terms of x.

-R

am

br

Even though the question does not ask for it directly, you are expected to use the initial condition that you have been given to find a particular solution.

es ity rs ve

y

op

U

d

Form integrals of both sides with respect to x.

ni

dy = 1+ x dx

w

ge

C

∫ ( 1 + x ) dx ∫ 2 y dy = ∫ ( 1 + x ) dx dx =

ie

Integrate each side.

br

x2 +C 2 02 9= 0+ + C, 2 x2 y2 = x + +9 2

-R

y = 3 when x = 0 .

Pr

es

s

C =9

ity

1

 2 x2 y= x+ +9   2

You can check this by differentiating.

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

ni ve rs

C

op y

-C

am

y2 = x +

ev

id



w

2 . It was not Here we have moved the 2y essential to move the 2, we could have moved the y only. This would be just as valid as it would also give expressions that can be integrated.

Pr

y op C w ie

R

ev

2y

Separate the variables x andy.

s

-C

dy 1 + x = dx 2y

Copyright Material - Review Only - Not for Redistribution

247

ve rs ity

ev ie

w

ge

am br id

WORKED EXAMPLE 10.2

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

dy =0 dx

op

dy = −( x 2 11) y3 dx 1 dy ( 2 11) =− ** 3 y dx x2

1 1 = − x +C 2 y2 x

− 1) dx

y

ni

Integrate each side.

w ie

−2 −

ev

∫ (−−x

− 1) dx

U

dy = =

∫ (−−x

Here we are assuming that y ≠ 0 .

ge

−3 −

dx = =

−2

-R

am



d

br

R

∫ ∫y

−3

Is this the complete solution?

ity

op

Pr

y

es

s

-C

Earlier in the solution (at line **) we assumed that y ≠ 0. However, when y = 0, the original differential equation dyy 0 becomes = 2 = 0 . Since y = 0 is the x-axis, it is true that its gradient is 0 at all points and so y = 0 is also dx x a solution of the differential equation and must be included for the general solution to be complete. 1 1 The complete general solution is, therefore, − 2 = − x + C or y = 0. x 2y

rs

y

ve

ni

op

dy In Worked example 10.2, we saw that, when rearranging to the form f( y ) = g( x ) , there dx 1 are cases where f( y ) is a function such as, for example, 3 . In these cases, a value (or y values) of y must be excluded for the separation of the variables to take place.

br

ev

id

ie

w

ge

C

U

R

ev

ie

w

C

248

Separate the variables x and y.

Form integrals of both sides with respect to x.

id

ev ie

w

C

ve rs ity

x2

dy = 0. dx

C op

y

( x 2 + 11)) y3 + x 2

Pr es s

-C

Answer

-R

Find the general solution of the differential equation ( x 2 + 11) y3 + x 2

es

s

-C

-R

am

As a final step, when finding the general solution in these cases, we must check if the value (or values) we have excluded when separating the variables gives (or give) a valid solution of the original differential equation. When they do, we should include them.

Pr

op y

WORKED EXAMPLE 10.3

ity

dx x((300 300 − x ) = and x = 50 when t = 0. Find the particular solution of the dt 600 differential equation, giving your answer as an expression for x in terms of t.

op

Separate the variables. The derivative is with respect to t and so the expression in x needs to be moved to the left-hand side.

-R s es

am

br

ev

ie

id g

w

e

C

U

dx x((300 300 − x ) = dt 600

y

ni ve rs

Answer

-C

R

ev

ie

w

C

It is given that, for 0 , x , 300 ,

Copyright Material - Review Only - Not for Redistribution

ve rs ity 1 dx 1 = x((300 300 − x ) dt 600 1 d 1 t x( x) 1 1 x x( x)

∫ ∫ dt ∫ dt

Pr es s

-C

1 1 1 1  x 300  x x 1 1 t +C ( ln x − ln( − x ) = 600 300 1 ln x − ln(300 ln( − x) = t + C 2 x 1 ln = t +C 300 − x 2 50 C = ln , C = ln l 0.2 300 − 50 x 1 ln = t + ln 0.2 300 − x 2



y C op

ni

x = 50 when t = 0 .

s es ity

ni

1

y

ie ev

t

op

1

t

60e 2

x=

U

t

1

t

br

-R

am

EXERCISE 10A

ev

id

ie

w

ge

1 + 0.2e 0 2

C

w

1

x(1 + 0.2e 2 ) = 60e 2

t

rs

1

t

249

ve

op C

1

x + 0. 0.2e 2e 2 x = 60 60e 2

Pr

y

t

x = 0. 0.2e 2e 2 ((300 300 − x )

R

ev -R

am

br

id

ie

w

ge

U

R

-C

1

t x = e 2 × e ln 0.2 300 − x 1

es

s

-C

1 Find the general solution of each of the following differential equations. dy ds dA = 2 πr a = x3 − 6 b c s2 = cos(t + 5) dx dt dr dyy dV dy y +1 =1 e f x + sec x sin y =2 d V −1 = d x dt dx x

ni ve rs

ity

Pr

op y

op

xy

C

b

dy = y given y = 5 when x = 0 dx

U

cos2 x

e

a

y

2 Find the particular solution for each of the following differential equations, using the values given.

br

ev

ie

id g

w

dy x3 − x = given y = 4 when x = 3 d 1− y

-R

s es

am

dy = e x − y and y = ln 2 when x = 0 , obtain an expression for y in 3 Given that dx terms of x.

-C

C w ie ev

R

Write the expression on the left in terms of its partial fractions.

The modulus signs are not required as we know that 0 , x , 300 .

1

t + ln 0.2 x = e2 300 − x

Note that it is simpler to leave the 600 on the right-hand side here.

Integrate each side.

ve rs ity

y

op C w

-R

dt



ev ie

ev ie

Form integrals of both sides with respect to t.

am br id



w

ge

C

U

ni

op

y

Chapter 10: Differential equations

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

w

a Write down the differential equation satisfied by y.

ev ie

ge

4 The gradient of a curve at the point ( x, y ) is 5 y3x. x

1 A B in the form + . (2 − x )( x + 1) 2 − x x +1

TIP

Pr es s

-C

5 a Express

-R

b Given that the curve passes through the point (1, 1), find y2 as a function of x.

y

b The variables x and t satisfy the differential equation

Solve this differential equation and obtain an expression for x in terms of t.

ii

State what happens to the value of x when t becomes large.

br

ev

id

ie

w

dw = 0.001(100 − w )2 . dt

C op

U

ge

6 The variables w and t satisfy the differential equation

y

i

ni

w

It is given that t = 0 when x = 0.

ev ie

R

PS

1).

ve rs ity

C

op

dx = (2 − = − x)(x dt

xɺ is an alternative dx . notation for dt

op

Pr

y

es

s

-C

-R

am

Given that when t = 0 w = 0 , show that the solution to the differential equation 1000 can be written in the form w = 100 − . t + 10 dx 7 Solve the differential equation 4 = (3x 2 − x ) cos t given that x = 0.4 when dt t = 0. Give your answer as an expression for x in terms of t.

8 Find the general solution of the differential equation xy

ity

C

dy x 2 − 1 = y +1 . dx e

w

rs

dy 1 + y2 = dx 4 + x 2 and passes through the point (0, 1). Give your answer in the form y = f( x ).

op

w

ge

10 Find the general solution of the equation x 3

C

dy + 2 y = 1. 1 dx

U

R

ni

ev

ve

ie

9 Find the equation of the curve that satisfies the equation

y

250

dx = 4x cos2 t given that x = 1 when t = 0. dt Give your answer as an expression for x in terms of t. dt . 2

s

dx t ( e t + 55) = given that x = 1 when t = 0. dt x2

ni ve rs

ity

13 At a point with coordinates ( x, y ) the gradient of a particular curve is directly proportional to xy2. At a particular point P with coordinates (1, 3) it is known that the gradient of this curve is 6.

y

Find and simplify an expression for y in terms of x.

ie

id g

w

e

C

U

op

14 The variables x and t satisfy the differential equation dx (3x + 2 x 3 ) = k (3x 2 + x 4 ) dt for x . 0 where k is a constant. When t = 0 , x = 1 and when t = 0.5, x = 2 .

es

s

-R

br

3x 2 + x 4 . 4

am

that 72t =

ev

Solve the differential equation, finding the exact value of k , and show

-C

R

ev

ie

w

C

PS

t2

Pr

op y

b Solve the differential equation

∫ te

es

-C

12 a Use the substitution u = t 2 to find

-R

am

br

ev

id

ie

11 Solve the differential equation

Copyright Material - Review Only - Not for Redistribution

ve rs ity

C

1

∫ x lnx dx.

w

ln x to find

am br id

15 a Use the substitution u

ev ie

ge

U

ni

op

y

Chapter 10: Differential equations

M

-R

b Given that x . 0, y . 0 , find the general solution of the differential equation dy y ln x = . dx x

Pr es s

It is important that, after treatment with this type of medicine, the person stays out of bright light until the amount of medicine remaining in the skin is at a safe level.

ve rs ity

w

C

op

y

-C

16 A type of medicine that reacts to strong light can be used to treat some skin problems.

ev ie

Jonty is treated with a medicine of this type.

ni

C op

y

After the treatment, 65 mg remains in the treated area of skin.

br

ev

id

ie

w

ge

U

R

The rate at which the medicine disappears from the treated area of Jonty’s skin dm = −0.05 m, where m is the amount of is modelled by the differential equation dt medicine, in milligrams, in the skin t hours after the treatment.

-R

am

Jonty is safe to go out in bright light when no more than 6 mg of the medicine remains in his skin.

TIP Notice that the rate at which the medicine disappears is with respect to time but the words ‘with respect to time’ are usually not included.

y

es

s

-C

How long must Jonty wait before he goes out in bright light?

ity

Newton’s law of cooling or heating

251

w

rs

C

op

Pr

EXPLORE 10.1

y

op

U

R

ni

ev

ve

ie

Many differential equations arise from modelling natural laws and relationships. One example is Newton’s law of cooling or heating:

es

s

T0 as the temperature of the object when t = 0

as the constant temperature of the surroundings

k

as the constant of proportionality.

ity

Pr

s

ni ve rs

y

op

C

-R es

s

eC .

ev

ie

id g

br

T − s = Ae kt where A

am



w

e



Why it is necessary to define the variables and constants. dT Why the differential equation = k (T − ) , k . 0 is satisfied by T . dt Other possible correct ways to represent Newton’s law of cooling or heating as a differential equation with this set of defined variables and constants. Why the general solution of the given form of the differential equation may be written as:

U



ev -R

as time

Discuss the following.

-C

R

ev

ie

w

C

op y

-C

am

t



ie

id

T as the temperature of the object

br

We define

w

ge

C

The rate of change of temperature of an object is proportional to the difference in temperature of the object and its surroundings.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ge

w

ev ie

Pr es s

-C



-R



How many arbitrary constants there are in the general solution of any first order differential equation. Why the modulus in the general solution is required. How the general solution can be used to represent the temperature of an object cooling down or warming up.

am br id



C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

op

y

10.2 Forming a differential equation from a problem

ni

C op

y

For example, when you are given a problem such as:

The rate of increase of the number of bacteria in a dish is k times the number of bacteria in the dish. The number of bacteria present in the dish at time t is x…

ie

w

ge

U

R

ev ie

w

C

ve rs ity

The key to forming a differential equation from a problem is the successful interpretation of the language used in the question.

-R

The rate of increase of x is k times x.

s

-C

am

br

ev

id

Replacing the words the number of bacteria in a dish by x the problem becomes:

C

op

ni

ev

Velocity is defined to be the rate of change of displacement with respect to time.

id

a Form a differential equation to represent this data.

br

ev

b Initially, the particle has a displacement of 2 metres from O.

ie

w

ge

U

R

The velocity of a particle at time t seconds is inversely proportional to its displacement, s metres, from an origin, O.

y

ve

ie

w

rs

WORKED EXAMPLE 10.4

C

252

ity

op

Pr

y

es

This rate of increase of x is with respect to time, t, and is much easier to translate into a dx differential equation: = kx . dt

-R

am

After 1 second, the particle has a displacement of 5 metres from O.

s

-C

Solve the differential equation giving your answer in the form t = f( s ) .

Pr

op y

es

c Find t when the displacement of the particle from O is 10 metres.

ity

y

ds ds 1 ds k and so ∝ , which means that = dt dt s dt s

ev

ie

id g

w

e

C

U

op

At the moment, you do not have enough information to find the constant of proportionality, k

es

s

-R

br am -C

ev

R

v=

ni ve rs

a The velocity of a particle at time t seconds is inversely proportional to its displacement, s metres, from the origin, O.

ie

w

C

Answer

Copyright Material - Review Only - Not for Redistribution

ve rs ity

w

ge

Separate the variables s and t.

ev ie

ds k = dt s ds s =k dt

-R

am br id ds

Pr es s ve rs ity

s = 2 when t = 0 .

C op

y

s = 5 when t = 1.

w

ge

U

R

ev ie

w

s2 = kt + C 2 s2 C = 2, = kt + 2 2 52 = k + 2, k = 10.5 2 s2 21 = t+2 2 2 s2 − 4 t= 21

Integrate each side.

ni

C

op

y

-C

∫ s t dt ∫ k dt ∫ s ds ∫ k dt

Form integrals of both sides with respect to t.

-R

am

br

ev

id

ie

Write in the form t = f ( s ) .

s es

Pr

100 − 4 = 4.5714... = 4.57 seconds correct to 3 significant figures. 21

C

rs ve

ie

w

WORKED EXAMPLE 10.5

ev

253

ity

t=

op

y

-C

c Find t when s = 10

U

R

ni

op

The population density, P, of a town can be modelled as a continuous variable.

y

b

C

U

ni

op

y

Chapter 10: Differential equations

w

ge

C

In the year when records began:

the population density, P, of the town was increasing at the rate of 0.5 people per hectare per year



the population density of the town was 2 people per hectare.

br

ev

id

ie



-C

-R

am

It is suggested that, t years after records began, the rate of increase of the population density is inversely proportional to the square of the population density.

es

s

a Form a differential equation to model this information.

Pr

3.5

15

20

4

5.5

7

y

P

10

C

U

5

-R s es

am

br

ev

ie

id g

w

e

Comment on how good a fit the model is to the actual data in the longer term.

-C

ie ev

R

t

op

ni ve rs

ity

c The table shows the actual population density of the town, every 5 years for the first 20 years after records began.

w

C

op y

b Solve the differential equation, giving your answer in the form P = f(t ) .

Copyright Material - Review Only - Not for Redistribution

ve rs ity

a The rate of increase of P is inversely proportional to P 2 :

k ,k =2 4

C op

ni U

Form integrals of both sides with respect to t.

w

ie

dP =2 dt dP

Separate the variables P and t.

ge

-R

am

br

ev

∫ P t dt ∫ 2 dt ∫ P dP 2 ∫ dt

es Pr

8 3 = 6t + 8

op

y

ve

6t + 8

C

am

br

P

10

3.5

4

3.4

4.1

15

20

5.5

7

4.6

5.0

s

-C

Model prediction (to 2 significant figures)

5

w

id

t

ie

c c..................................................

-R

ge

3

Write in the form P = f(t ) .

rs

= 2t +

P=

R

ity

=C

ni

ev

ie

w

C

254

= 2t + C

U

op

y

P3 3 8 3 P3 3 P3

s

-C

Integrate each side. t = 0, P = 2

ev

R

P2

y

ve rs ity

0.5 =

dP 2 = 2 dt P

b

Pr es s

dP k = 2 dt P dP 2 = 2 dt P

dP = 0.5 and P = 2 , therefore: dt

id

ev ie

w

C

op

y

When t = 0 ,

-R

dP k = 2 dt P

-C

dP 1 ∝ 2 dt P

ev ie

am br id

Answer

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

y op -R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

ev

ie

w

ni ve rs

C

ity

Pr

op y

es

In the longer term, the model is not a particularly good fit as the values it generates when time increases are under-estimates of the known data.

Copyright Material - Review Only - Not for Redistribution

ve rs ity ge

C

U

ni

op

y

Chapter 10: Differential equations

am br id

ev ie

w

WORKED EXAMPLE 10.6

-R

The value, $V , of a motorbike as it gets older can be modelled as a continuous variable.

Pr es s

-C

A model is suggested to estimate the rate of decrease of the value of the motorbike. This model is that the value V when the motorbike is t months old decreases at a rate proportional to V .

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a Write down a differential equation that is satisfied by V .

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b The motorbike had a value of $10 000 when it was new.

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Solve the differential equation, showing that V = 10 000e− kt , where k is a positive constant.

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c After 3 years the motorbike has lost half of its value when new.

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dV dV ∝ V and so = − kV , where k is a positive constant. dt dt

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Integrate each side. t = 0 , V = 10 000

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For this particular solution, V is positive and so the modulus is not required.

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ln 10 000 = C

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ln V = − kt kt + C

Form integrals of both sides with respect to t.

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∫ 1 ∫ dV = ∫ − k dt

Separate the variables V and t.

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dV = − kV dt 1 dV = −k V dt 1 d dt = − k dt

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b

ln V = −k kt + ln 10 000

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Write in the form V = f(t ) .

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V = e − kt + ln 10 000

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V = 10 000e − kt

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− ln 0.5 3

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ln 0.55 = −3k

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t = 3, V = 5000

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5000 = 10 000e 5000 = e −3 k 10 000

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− ln 0.5   3

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V = 10 000e

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correct to 3 significant figures

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V = $6299.605 … . = $6300

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c V = 10 000e− kt

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V = = e − kt × × e ln 10 000

k=

This describes exponential decay. The rate is decreasing and so the constant of proportionality is negative.

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a The rate of decrease of V is proportional to V .

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Answer

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Find the value of the motorbike when it was 2 years old.

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EXERCISE 10B M

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1 Write the following as differential equations. Do not solve the equations.

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a For height h and time t:

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The rate of decrease of the height of water in a tank is proportional to the square of the height.

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b For number of infected cells n and time t:

For velocity v and time t:

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The rate of increase of the number of infected cells in a body is proportional to the number of infected cells in the body at that time.

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The rate of decrease of the velocity of a particle moving in a fluid is proportional to the product of v and v + 1.

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d For volume V and time t:

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The rate at which oil is leaking from an engine is proportional to the volume of oil in the engine at that time.

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e For cost C and time t:

The rate of change of the speed of a skydiver just after he jumps out of a

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plane is inversely proportional to e 2 .

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2 A large tank is full of water. The tank starts to leak.

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For speed v and time t:

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f

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The rate at which the cost of an item increases is proportional to the cube of the cost at that time.

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t seconds after the tank starts to leak the depth of water in the tank is x metres.

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The rate of decrease of the depth of water is proportional to the square root of the depth of water.

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a Write down a differential equation relating x, t and a constant of proportionality.

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3 The cost of designing an aircraft, $A per kilogram, at time t years after 1950 can be modelled as a continuous variable. The rate of increase of A is directly proportional to A.

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a Write down a differential equation that is satisfied by A.

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After 25 years, the cost of designing an aircraft was $48 per kilogram. Find the cost of designing an aircraft after 60 years.

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b In 1950, the cost of designing an aircraft was $12 per kilogram. Show that A = 12e kt , where k is a positive constant.

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Find the time it takes for the depth of the water to halve.

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b At the instant when water starts to leak from the tank, the depth of water is 6.25 metres. Given that 120 seconds after the tank starts to leak, the depth of water is 4 metres, solve the differential equation, expressing t in terms of x.

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4 A liquid is heated so that its temperature is x °C after t seconds. It is given that the rate of increase of x is proportional to (100 − x ). The initial temperature of the liquid is 25 °C.

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Chapter 10: Differential equations

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a Form a differential equation relating x, t and a constant of proportionality, k, to model this information.

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After 180 seconds the temperature of the liquid is 85 °C. Find the temperature of the liquid after 195 seconds.

d The model predicts that x cannot exceed a certain temperature. Write down this maximum temperature. 10 cm

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5 The diagram shows an inverted cone filled with liquid paint.

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b Solve the differential equation and obtain an expression for x in terms of t and k.

15 cm

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An artist cuts a small hole in the bottom of the cone and the liquid paint drips out at a rate of 16 cm 3 per second. At time t seconds after the hole is cut, the paint in the cone is an inverted cone of depth h cm. dV 4 = πh2 . a Show that dh 9 dh . b Hence find an expression for dt c Solve the differential equation in part b, giving t in terms of h.

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6 The size of a population, P, at time t minutes is to be modelled as a continuous variable such that the rate of increase of P is directly proportional to P.

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a Write down a differential equation that is satisfied by P.

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d Find the length of time it takes for the depth of the paint to fall to 7.5 cm.

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The size of the population is 1.5P0 after 2 minutes. Find when the population will be 3P0 .

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7 A ball in the shape of a sphere is being filled with air. After t seconds, the radius of the ball is r cm. The rate of increase of the radius is inversely proportional to the square root of its radius. It is known that when t = 4 the radius is increasing at the rate of 1.4 cm s−1 and r = 7.84 .

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b The initial size of the population is P0 . Show that P = P0 e kt , where k is a positive constant.

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How much air was in the ball at the start?

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8 Maria makes a large dish of curry on Monday, ready for her family to eat on Tuesday. She needs to put the dish of curry in the refrigerator but must let it cool to room temperature, 18 °C, first. The temperature of the curry is 94 °C when it has finished cooking.

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At 6 pm, Maria places the hot dish in a sink full of cold water and keeps the water at a constant temperature of 7 °C by running the cold tap and stirring the dish from time to time. After 10 minutes, the temperature of the curry is 54 °C.

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b Solve the differential equation and obtain an expression for r in terms of t.

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a Form a differential equation relating r and t to model this information.

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TIP You must define your own variables in this question as they have not been defined for you.

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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It is given that the rate at which the curry cools down is proportional to the difference between the temperature of the curry and the temperature of the water in the sink.

9 The half-life of a radioactive isotope is the amount of time it takes for half of the isotope in a sample to decay to its stable form.

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At what time can Maria put her dish of curry in the refrigerator?

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It is given that the rate of decrease of the mass, m, of the carbon-14 in a sample is proportional to its mass.

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A sample of carbon-14 has initial mass m0. What fraction of the original amount of carbon-14 would be present in this sample after 2500 years? 10 Anya carried out an experiment and discovered that the rate of growth of her hair was constant. At the start of her experiment, her hair was 20 cm long. After 20 weeks, her hair was 26 cm long. Form and solve a differential equation to find a direct relationship between time, t, and the length of Anya’s hair, L.

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11 A bottle of water is taken out of a refrigerator. The temperature of the water C. The bottle of water is taken outside to drink. The air in the bottle is 4 °C temperature outside is constant at 24 °C.

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Carbon-14 is a radioactive isotope that has a half-life of 5700 years.

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After 2 minutes the temperature of the water in the bottle is 11°C.

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It is given that the rate at which the water in the bottle warms up is proportional to the difference in the air temperature outside and the temperature of the water in the bottle.

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a How long does it take for the water to warm 20 °C ?

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12 The number of customers, n, of a food shop t months after it opens for the first time can be modelled as a continuous variable. It is suggested that the number of customers is increasing at a rate that is proportional to the square root of n.

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b According to the model, what temperature will the water in the bottle eventually reach if the air temperature remains constant and the water is not drunk?

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a Form and solve a differential equation relating n and t to model this information.

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13 Doubling time is the length of time it takes for a quantity to double in size or value.

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a What is the doubling time of the bacteria in the culture?

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The number of bacteria in a liquid culture can be modelled as a continuous variable and grows at a rate proportional to the number of bacteria present. Initially, there are 5000 bacteria in the culture. After 3.5 hours there are a million bacteria.

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b What assumption does the model make about the growth of the bacteria and how realistic is this assumption?

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The food shop has a capacity of serving 14 000 customers per month. Show that the model predicts the shop will have reached its capacity sometime in the 12th month.

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b Initially, n = 0, and after 6 months the food shop has 3600 customers. Find how many complete months it takes for the number of customers to reach 6800.

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EXPLORE 10.2

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Chapter 10: Differential equations

Modelling growth

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dP = kP. dt

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It is thought that the maximum capacity of the Earth is about 10 billion people.

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Is an exponential growth model appropriate for modelling the population of the planet Earth in the long term?

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This means that

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Many of the models we have been using have assumed exponential growth. This type of growth is unlimited as the rate of growth of a population is proportional to the size of the population at that time.

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In 1987, the population of the Earth was about 5 billion people and the growth rate of the population was about 1.84%.

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Some people think that the problem of the Earth’s increasing population can be solved by sending some people to live on other planets, such as Mars. In 2031, the first people to live on Mars will launch into space. The environment on Mars is different to Earth and many challenges must be overcome. This will all take time.

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When a population is modelled as growing exponentially at a constant rate of r % per 70 year, it takes about years for the population to double. r ● Use the data given so far to investigate whether going to live on Mars is a good solution to the population problem. ● Comment on any assumptions you have made in your investigation.

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There is statistical evidence that the population of the Earth is not growing at a constant rate each year. In 2017, the growth rate had decreased to 1.11% per year. The population needs resources to survive and so the growth rate has decreased as resources have decreased.

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A different model, logistic growth, can be used when resources are limited. This model takes into account the changing growth rates of the population.

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Here are sketch graphs of exponential and logistic growth models:

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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Exponential growth

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Population size

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Maximum possible population

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Logistic growth

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Population size

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Maximum possible population

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Time

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Time

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For example, for a population of size P,

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dP rP(( k − P ) = dt k



Comment on the limitations of this model.

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Show that as time increases this model suggests that P tends to k.

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where the constant r is the maximum natural growth rate (birth rate – death rate) and the constant k is the maximum capacity of the limited environment.

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Checklist of learning and understanding

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Chapter 10: Differential equations

Integrate both sides

Solve to find a general solution.



Substitute any given conditions to find a particular solution.



Check your answer by differentiating.

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Other points to remember

Take care with modulus signs in expressions such as ln f( x ) .



When finding a general solution, check to see if any values excluded when the variables are separated are possible solutions.

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261

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y op C w ev ie

dy

∫ f( y x dx ∫ g(x ) dx ∫ f( y) dy ∫ g(x ) dx.

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dyy = g( x ). dx

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Separate the variables f( y )

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General procedure to solve a differential equation using the technique of separating the variables

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

1

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dyy = x2e y+2x dx and it is given that y = 0 when x = 0.

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The variables x and y satisfy the differential equation

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END-OF-CHAPTER REVIEW EXERCISE 10

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dx x( 2500 x) x = . dt 5000 It is given that x = 500 when t = 0.

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The number of birds, x, in a particular area of land is recorded every year for t years. x is to be modelled as a continuous variable. The rate of change of the number of birds over time is modelled by

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Solve the differential equation and obtain an expression for y in terms of x.

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a Find an expression for x in terms of t.

b How many birds does the model suggest there will be in the long term?

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a Form and solve a differential equation to find the equation of this curve. 3 4) . b Find the gradient of the curve at the point ( −3,

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[1]

50

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dx. ∫ (5 − x )(10 )( − x )

[3]

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Given that x , 5 , find

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[1]

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iii According to the model, approximately how many grams of chemical C are produced by the reaction?

[1]

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Solve this equation, giving x in terms of t.

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Show that x and t satisfy the differential equation 50

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dx = (5 − x )( )(10 10 − x ) . dt

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b A chemical reaction takes place between two substances A and B. When this happens, a third substance, C , is produced. After t hours there are 5 − x grams of A, there are 10 − x grams of B and there are x grams of C present. The rate of increase of x is proportional to the product of 5 − x and 10 − x. Initially, x = 0 and the rate of increase of x is 1 gram per hour. i

The variables x and y are related by the differential equation

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3 when x = 1, find a relationship between x and y.

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dyy (2 x + 1) y 4 − 1 = , y= dx xy3

y3 dy. y4 − 1

[1]

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Given that



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y 4 − 1 to find

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Use the substitution u =

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dyy xe x = . dx 5 y 4 It is given that y = 4 when x = 0 . Find a particular solution of the differential equation and hence find the value of y when x = 3.5 . [8]

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[6]

The gradient of a curve is such that, at the point ( x, y ) , the gradient of the curve is proportional to x y . At the point (3, 4) the gradient of this curve is −5.

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dy = 1 + y4 dx obtaining an expression for y 4 in terms of x. y3

[1]

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Given that y = 2 when x = 0, solve the differential equation

3

[9]

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The height, h metres, of a cherry tree is recorded every year for t years after it is planted. It is thought that the height of the tree is increasing at a rate proportional to 8 − h. When the tree is planted it is 0.5 metres tall and after 5 years it is 2 metres tall.

9

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b According to the model, what will the height of the cherry tree be when it is fully grown?

[1]

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a Form and solve a differential equation to model this information. Give your answer in the form h = f(t ) .

The variables x and y satisfy the differential equation

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Chapter 10: Differential equations

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dyy = e3( x + y ) and y = 0 when x = 0 . dx

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Solve the differential equation and obtain an expression for y in terms of x.

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10 At the start of a reaction, there are x grams of chemical X present. At time t seconds after the start, the rate of decrease of x is proportional to xt.

[5]

c Initially, there are 150 grams of chemical X present and after 10 seconds this has decreased to 120 grams. Find the time after the start of the reaction when the amount of chemical X has decreased to 1 gram.

[3]

dP .

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∫ P(5

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P P)

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dP Given that P = 3 when t = 0, solve the differential equation = P (5 − P ), obtaining an expression dt for P in terms of t.

[4]

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Describe what happens to P when t → ∞.

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Using partial fractions find

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b Solve the differential equation obtaining a relation between x, t and k.

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a Using k as constant of proportionality, where k . 0, form a differential equation to model this reaction rate.

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12 The population of a country was 50 million in 2000 and 60 million in 2010. The rate of increase of the population is modelled by

[5]

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dyy x sin x = , y = −1 when x = 0, find a relationship between x and y. dx y

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f( x). x

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given that x = 0 when y = 0. Give your answer in the form y

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dyy tan x = 3y dx e

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[3]

14 Solve the differential equation

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15 There are 2000 mice in a field. At time t hours, x mice are infected with a disease. The rate of increase of the number of mice infected is proportional to the product of the number of mice infected and the number of mice not infected. Initially 500 mice are infected and the disease is spreading at a rate of 50 mice per hour.

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a Form and solve a differential equation to model this data. Give your answer in the form t = f( x ).

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b Find how long it takes for 1900 mice to be infected.

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d dx.

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Given that

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∫ x sisinn x

[7]

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Use the substitution u = x 2 to find

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dP kP (100 − P ) = dt 100 Use the model to predict the population of the country in 2025.

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

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16 The variables x and y are related by the differential equation dyy x = 1 − y2 . dx

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When x = 2 , y = 0. Solve the differential equation, obtaining an expression for y in terms of x.

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Cambridge International A Level Mathematics 9709 Paper 31 Q6 November 2012

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dx = ( + 2 ) sin2 2θ , dθ

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17 The variables x and θ satisfy the differential equation

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1 π giving your answer correct to 3 significant figures. π, [9] 4 Cambridge International A Level Mathematics 9709 Paper 31 Q8 November 2015

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x when θ =

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and it is given that x = 0 when θ = 0. Solve the differential equation and calculate the value of

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Chapter 11 Complex numbers

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P3 This chapter is for Pure Mathematics 3 students only.

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understand the idea of a complex number, recall the meaning of the terms real part, imaginary part, modulus, argument and conjugate and use the fact that two complex numbers are equal if and only if both real and imaginary parts are equal carry out operations of addition, subtraction, multiplication and division of two complex numbers expressed in Cartesian form x + iy use the result that, for a polynomial equation with real coefficients, any non-real roots occur in conjugate pairs represent complex numbers geometrically by means of an Argand diagram carry out operations of multiplication and division of two complex numbers expressed in polar form r(cos θ + isin θ ) = re iθ find the two square roots of a complex number understand in simple terms the geometrical effects of conjugating a complex number and of adding, subtracting, multiplying and dividing two complex numbers illustrate simple equations and inequalities involving complex numbers by means of loci in an Argand diagram.

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■ ■ ■ ■ ■ ■ ■ ■

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In this chapter you will learn how to:

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

IGCSE / O Level Mathematics

Add, subtract and multiply pairs of binomial expressions.

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ev Copy the diagram and shade the region that is less than 4 cm from A and nearer to B than to C .

C

op

y

4 a Solve tan x = −1 for −π , x , π radians. 3 b Find π − sin −1   .  4

ev

ie

w

5 The point A has position vector a = 3i + 4 j and the point B has position vector b = 5 i − 7 j. Find:

-R

a the magnitude of a

es

s

b the angle made between a and the positive x-axis



c BA

op

y

ity

ni ve rs

To understand the structure of a complex number we must first introduce a new set of numbers, the set of imaginary numbers.

-R

s es

am

br

ev

ie

id g

w

e

C

U

The number system we use today has developed over time as new sets of numbers have been needed to model new situations. Early mathematicians used natural numbers for counting. Later, zero and negative numbers were accepted as valid numbers and the set of integers was established. In Pythagoras’ time, mathematicians understood that rational numbers existed as they were the values in between integers. Pythagorean mathematicians thought that all numbers could be written as rational numbers. These same mathematicians were shocked by the discovery of irrational numbers such as 2 and

-C

)

C

Pr

op y C w

5

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s

es

am

) (2 +

B

Pr ity

rs

ve

ni

U ge id br

Find the magnitude and direction of a vector. Know and use the relationship AB = b − a when the position vector of A is a and the position vector of B is b.



-C

5 4− 3 5+2 3

C op

ni

y op C w ie ev



(2 −

A

it is always the same distance away from a fixed point is a circle it is always equidistant from two fixed points is a perpendicular bisector.

Understand the convention for positive and negative angles and use trigonometry to find the size of an angle that is not acute.



32 − 3 8

a

3 The diagram shows a triangle ABC .

U

-C



R ie

2 Simplify.

c

ge id

br am



( a + bx ) + (2 a − 3bx )

y

-C

y

ev ie

R

a

b

Understand, for example, that the locus of a point which moves so that

What are complex numbers?

ev

1 Simplify each of the following. b ( a + bx )(2 a − 3bx )

Draw simple loci.

Pure Mathematics 1 Coursebook, Chapter 5

Chapter 9

R

Check your skills

Manipulate expressions involving surds and calculate with surds.

op

Assumed prior knowledge (not in the IGCSE / O Level course)

w

C

w

What you should be able to do

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Where it comes from

IGCSE / O Level Mathematics

266

ev ie

am br id

PREREQUISITE KNOWLEDGE

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ve rs ity

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U

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op

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Chapter 11: Complex numbers

am br id

ev ie

w

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they did not trust them as valid quantities at the time. Even as late as the 17th and 18th centuries, some famous mathematicians still doubted that negative numbers were of any use and thought that subtracting something from nothing was an impossible operation!

In the set of natural or counting numbers (ℕ) there are no negatives and no zero. In the set of integers (ℤ) there are no part or fractional values. In the set of rational numbers (ℚ) there are no irrational quantities such as π. In the set of real numbers (ℝ) there are no numbers that when squared result in a negative value.



C op

y

Until now, the numbers with which we have been working have all been real numbers. That is, they belong to the set of real numbers, which includes the rational and irrational numbers. Real numbers can be represented on a 1-dimensional diagram called a number line.

U

R

ni

ev ie

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C



ve rs ity

op

y



Pr es s

-C



-R

We will start with some simple facts.

es

s

-C

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am

br

ev

id

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w

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In the 17th century, the set of imaginary numbers (I) was accepted as a valid number set. René Descartes described the collection of number sets that had been accepted as valid before that point as the set of real numbers. The new set, which he did not like very much, he described as imaginary. He intended this to be an insult as he thought they were not of great importance. Many other famous mathematicians agreed with him. They were, however, quite wrong about that. This name has often confused students who are new to this area of study.

op

Pr

y

So what is an imaginary number and how is it related to a complex number?

rs

op

ni U

R

ev

KEY POINT 11.1

y

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w

C

ity

In the new set of imaginary numbers there are numbers that when squared result in a negative value.

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am

br

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C

The set of complex numbers combines the real numbers and the imaginary numbers into a 2-dimensional rather than a 1-dimensional number system. Descartes described the numbers in the set of complex numbers as the sum of a real part and an imaginary part. Complex numbers are represented as points on a plane rather than points on a number line because of their 2 -dimensional nature.

op

y

ni ve rs

11.1 Imaginary numbers

w

e

ie ev

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s es



Engineers call this quantity j as i is used in engineering for a different quantity. Just as with some other important numbers, i is described as a letter rather than a digit.

am



br

Note that:

−1.

id g

This means that i =

C

U

In the set of imaginary numbers we introduce a new number i and we define i2 = −1.

-C

R

ev

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w

C

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Pr

op y

es

s

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Clearly, accepting new sets of numbers as being valid is something mathematicians have struggled with over time. You might not feel comfortable with this new idea. The set of real, imaginary and complex numbers does have application in the real world. Without imaginary and, therefore, complex numbers there would be no mobile telephones, for example, as they are used to find audio signals. Complex numbers are used to generate fractals, which are used in the study of weather systems and earthquakes. Fractals are also used to make realistic computer-generated images for movies.

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267

ve rs ity

ge

Consider the equation x 2 + 1 = 0 .

am br id

ev ie

How many solutions does this equation have in the set of real numbers? x 2 = −1

x = ± −1

No solutions in the set of real numbers.

-R

x2 + 1 = 0

w

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

x 2 = −1

x = ± −1 = ± i

Two solutions in the set of imaginary numbers.

y

x2 + 1 = 0

Pr es s

-C

How many solutions does this equation have in the set of imaginary numbers?

C

ve rs ity

op

We can use laws of surds to extend the use of i to find other imaginary numbers.

w ev

br

-R

am

5 × −1

s

When the multiplier is a simple surd, it is better to write the i before the surd to avoid any confusion with 5i or we can write ( 5 ) i.

Pr

2 × 9 × −1

=

2 ×3×i = 3 2 i

)

ni

(

y

=

op

w ie ev

The imaginary number i is called the unit imaginary number as it has a coefficient of 1.

C

U

R

2 × 9 × −1

ity

−18 =

C

c

rs

op

y

es

5 × −1 = i 5

ve

-C

−5 =

=

268

−18

When solving x 2 + 1 = 0 we needed ± in the solution. Why do we not need it here?

−9 = 9 × −1

= 9 × −1 = 3i b

c

ie

id

Answer a

−5

C op

b

U

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ni

−9

a

y

Write the following numbers in their simplest form.

ge

ev ie

w

WORKED EXAMPLE 11.1

id

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All other imaginary numbers are multiples of i.

-R

am

br

ev

Imaginary numbers are therefore of the form bi or iy , where b is a real number or y is a real number.

es

s

-C

We calculate with imaginary numbers in the same way that we calculate with real numbers.

Pr

op y

TIP

ni ve rs

-R s es

am

br

ev

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id g

w

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C

U

op

y

You need to demonstrate your ability to work with these numbers without a calculator. It is therefore important to practise writing down your method steps. Working without a calculator will also help to improve your understanding and recall of this subject (and do not forget that not all calculators are able to calculate with complex numbers).

-C

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ity

Many calculators are able to calculate in the set of complex numbers. You must practise without a calculator but can use one to check your answer.

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TIP Simplest form for an imaginary number means using simple surd form and i.

ve rs ity

C

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ni

op

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Chapter 11: Complex numbers

w

Without using a calculator, find: −8i + ( − 4i)3

Answer

9i+16i = 4i

25 5 = 4 2

y

1 1 1 = = −1 3 6 = i ( −1)3 ( i2 )

C op

i −6 =

br

EXERCISE 11A

ev

id

ie

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ge

U

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d

25i = 4i

-R

am

Do not use a calculator in this exercise.

−16 + −81

c

100i 4 − 16i2 4

ve b

4x 2 + 7 = 0

c

12 x 2 + 3 = 0

−5 6i2

w ie

br

ev

id

EXPLORE 11.1

d

C

U

ni

64 =0 25

269

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ie ev

R

x2 +

d

)3

9i − ( i 2

rs

b

ity

−5i2 + 3i8

3 Solve. a

−90

y

op w

C

2 Simplify. a

c

Pr

y

es

s

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1 Write the following numbers in their simplest form. 36 − a −144 b 81

op

ev ie

w

c

ve rs ity

−8i + ( − 4i)3 = −8i + 64i = 56i

C

b

(3i2 ) + (3i)2 = 3i2 + 9i2 = 12i2 = −12

d i −6 .

ni

op

y

a

9i + 16i 4i

c

-R

b

Pr es s

(3i2 ) + (3i)2

-C

a

ev ie

am br id

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WORKED EXAMPLE 11.2

-C

-R

am

Without using a calculator, find i200.

es

s

11.2 Complex numbers

C

ity

Pr

op y

A complex number is defined as a number of the form x + iy , where x and y are real values and i is the unit imaginary number.

ni ve rs

op C

e



the real number x is called the real part of z, Re z the real number y is called the imaginary part of z, Im z.

U



y

A complex number is often denoted by the letter z so z = x + iy :

ie

id g

w

When x = 0, then z = iy and z is completely imaginary.

-R s es

am

br

ev

When y = 0, then z = x and z is completely real.

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w

Note: Other general cases are common too, such as a + bi, where a and b are real.

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TIP Sometimes iy is considered to be Im z.

ve rs ity

am br id

ev ie

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The relationship between the set of complex numbers (ℂ) and other sets of numbers is often represented using nested rings.

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Equal complex numbers

0

Pr es s

-C

-R

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

op

y

So when z1 = x + iy and z2 = a + bi then z1 = z2 means that x = a and y = b. TIP

C

ve rs ity

Complex conjugate

y

ni U

R

WORKED EXAMPLE 11.3

Sometimes z * is written as z .

C op

ev ie

w

The complex conjugate of z = x + iy is defined as z* = x − iy.

br

Answer

ev

id

ie

w

ge

Solve the equation 5 z 2 + 14 z + 13 = 0.

es

s

-C

−14 ± 142 − 4(5)(13) 10 −14 ± −64 10

z =

−14 ± 8i 10

y op

ni

C

U

7 4i 7 4i + or − − 5 5 5 5

Note that the two roots are complex conjugates of each other.

w

ie -R

am

br

ev

id

z=−

rs

−14 ± 64 × −1 10

ve

z =

ge

R

ev

ie

w

C

270

ity

op

z =

Pr

y

z =

-R

am

Using the quadratic formula a = 5, b = 14, c = 13.

s

-C

WORKED EXAMPLE 11.4

es

(2 x + y ) + i( y − 5) = 0

Pr

op y

Find the value of x and the value of y.

and y−5= 0

(2)

C

(1)

ev -R s

5 2

es

am

Substituting into (1): 2 x + 5 = 0, x = −

ie

id g

br

From (2): y = 5

w

e

2x + y = 0

U

Therefore

op

(2 x + y ) + i( y − 5) = 0 + 0i

y

ni ve rs

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

-C

R

ev

ie

w

C

ity

Answer

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ve rs ity

C

U

ni

op

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Chapter 11: Complex numbers

w

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Calculating with complex numbers

ni

U

id

z1z1* = (2 + 3i)(2 − 3i)

Expand the brackets and simplify.

2

i2 = −1

z1z1* = 4 + 9

-R

-C

am

br

z1z1* = 4 − 6i + 6i − 9i z1z1* = 13

ie

w

ity

C

z2 (8 − i)(2 − 3i) = z1 (2 + 3i)(2 − 3i)

Pr

es

Multiply the numerator and denominator by the complex conjugate of z1.

Expand the brackets and simplify.

i2 = −1

op

ni

ev

ve

z2 16 − 24i − 2i + 3i2 = z1 13

C

U

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z2 16 − 24i − 2i − 3 = z1 13

id

ie

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We have already found 13 so we did not show full working. You should usually show all your working to demonstrate you understand the method you are using.

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br

ev

z2 13 − 26i = = 1 − 2i z1 13

-C

am

271

rs

y op

s

z2 8−i = z1 2 + 3i

Division

y

w

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z1z2 = 19 + 22i

i2 = −1 It is important to show all your working out clearly.

y

R

z1z2 = 16 + 22i + 3

Expand the brackets and simplify.

ie

ev ie

z1z2 = 16 − 2i + 24i − 3i

2

C op

z1z2 = (2 + 3i)(8 − i)

Multiplication

ev

C

z1 − z2 = −6 + 4i

Collect like terms.

ve rs ity

op

y

z1 + z2 = 10 + 2i z1 − z2 = 2 + 3i − (8 − i)

Subtraction

ev ie

Collect like terms.

z1 + z2 = 2 + 3i + 8 − i

Addition

w

Pr es s

-C

For example, when z1 = 2 + 3i and z2 = 8 − i we have:

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am br id

The processes we need to use are the same as those we use when manipulating expressions involving surds.

es

s

KEY POINT 11.2

Pr

-R s es

-C

am

br

ev

ie

id g

w

e

C

U

op

ie ev

R

y

ni ve rs

ity

x 2 + y 2 is also real.

w

C

op y

When we multiply a complex number by its conjugate the result is always a real number as, if we let z = x + iy, and so z* = x − iy, then zz* = ( x + iy )( x − iy ) = x 2 + y 2 . Since x and y are real,

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

am br id

EXERCISE 11B

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Do not use a calculator in this exercise.

-R

z1 = 5 − 3i and z2 = 1 + 2i

1

a

z 2 + 2 z + 13 = 0

b

z 2 + 4z + 5 = 0

c

2z2 − 2z + 5 = 0

d

z 2 − 6 z + 15 = 0

e

3z 2 + 8 z + 10 = 0 f

2 z 2 + 5z + 4 = 0

ve rs ity

ev ie

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C

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y

Pr es s

-C

Write each of the following in the form x + iy , where x and y are real values. z1 b z1* − z2 c z1z2 d a z1 + z2* z2 2 Solve these equations.

c

( x − y ) + (2 x − y )i = −1

y C op

ni

( x + 2 y ) + i(3x − y ) = 1 + 10i

( x + y − 4) + 2 xi = (5 − y )i

b

U

a

ie

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R

3 Find the value of x and the value of y.

b

(4 − 5i)(4 + 5i)

c

(7 − 3i)2

d

( 3 − i )4

e

17 − i 3+i

f

−1 + 11i 6 − 5i

g

13(1 + i) 2 + 3i

h

(3 − 2i)2 5+i

s es Pr

-C

1 2 6 + i is a solution of the equation 5 z 2 − 2 z + 5 = 0. 5 5 b Write down the other solution of the equation.

β = 2 + 3i

α =−

5 31 − i 2 2

y op

d

β = 1 − 5i

C

α = 1 + 5i

β =−

w

α = 2 − 3i

b

ie

c

U

β = 7i

ge

α = −7i

id

a

ni

6 Find the quadratic equations that have the following roots.

5 31 + i 2 2

ev

7 Find the complex number z satisfying the equation z + 4 = 3iz*.

am

PS

ve

rs

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5 a Show that z =

br

w ie

PS

R

ev

-R

(1 + 3i)(2 − i)

am

a

y op C

272

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br

id

4 Write each of the following in the form x + iy , where x and y are real values.

-C

s

8 Find the complex number u, given that u(3 − 5i) = 13 + i.

es

PS

-R

Give your answer in the form x + iy , where x and y are real.

9 z = 5 + i 3 is a root of a quadratic equation. Find this quadratic equation.

M

10 In an electrical circuit, the voltage (volts), current (amperes) and impedance (ohms) are related by the equation

Pr

ity

ni ve rs

y

voltage = current × impedance

-R s es

am

br

ev

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id g

w

e

Find the current.

C

U

op

The voltage in a particular circuit is 240 V and the impedance is 48 + 36i ohms.

-C

R

ev

ie

w

C

op y

PS

Copyright Material - Review Only - Not for Redistribution

TIP ( z − α )( z − β ) = 0 z 2 − (α + β ) z + αβ = 0

ve rs ity

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Chapter 11: Complex numbers

DID YOU KNOW?

w

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11.3 The complex plane

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am br id

ev ie

As complex numbers have two dimensions, one real and one imaginary, a diagrammatical representation must have two dimensions. This means it has to be a plane and not a line. Jean-Robert Argand (1768 −1822) is credited for the diagram used to do this.

-C

Argand diagrams

Pr es s

The complex number x + iy can be represented by the Cartesian coordinates ( x, y ).

For example, we can represent the numbers z1 = 2 + 3i and z2 = 5 − 3i by the points A(2, 3) and B (5, − 3) on an Argand diagram.

ve rs ity

4

We can also represent the numbers z1 = 2 + 3i and  2 z2 = 5 − 3i by the position vectors OA =   and  3  5 . OB =   −3 

id

ev

-R s 4

5

Re(z)

273

y

ni

B

op

–3

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ie

B

es 3

–2

ev

Re(z)

Pr

2

5

ity

1

4

rs

w

C

op

y

1

–1 O –1

3

w

ge

C

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–4

id

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Modulus-argument form

-R

am

br

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When a complex number is written as x + iy where x and y are real values, we say it is in Cartesian form. There are other ways of writing complex numbers.

es

s

-C

The vector representation of a complex number on an Argand diagram helps us to understand a different representation called the modulus-argument form.

ity

The modulus of a complex number x + iy is the magnitude  x of the position vector   .  y

y

y

2

θ

id g

O

Re(z)

ev es

s

-R

br am -C

x

ie

Also note that the plural of modulus is moduli.

w

e

Notation: The modulus of z = x + iy is z .

op

x +y .

U

Therefore, the modulus of x + iy is defined to be

2

C

ni ve rs

C w ie ev

Im(z)

Pr

op y

Before we do this, we need to define the modulus and the argument of a complex number.

R

When drawing an Argand diagram, make sure you choose equal scales for the real and imaginary axes. This will give you a good picture and not stretch any shape you draw.

–4

-C

2

2

–3

A

3

1

–2

br

4

y

–1 O –1

C op

ni

1



am

Im(z)

TIP

2

ge

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R



A

3

w

C w ev ie

Im(z)

ie

op

y

The horizontal axis is called the real axis and the vertical axis is called the imaginary axis.

Real numbers can be ordered along a number line. Complex numbers cannot be ordered as they are represented by points in a plane rather than on a line.

Copyright Material - Review Only - Not for Redistribution

ve rs ity

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Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

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 x The argument of a complex number x + iy is the direction of the position vector   .  y

op

y

Notation: The argument of z = x + iy is arg z.

Pr es s

-C

-R

More precisely, it is the angle made between the positive real axis and the position vector. The principal argument, θ , of a complex number is an angle such that −π , θ ø π. Sometimes it might be more convenient to give θ as an angle such that 0 ø θ , 2 π. Usually, the argument is given in radians.

C

ve rs ity

We can use trigonometry to find the argument. For example, when θ is acute, tan θ =

y . x

y

ni

WORKED EXAMPLE 11.5

C op

ev ie

w

A diagram must be drawn first to check the position of the complex number.

U

R

Find the modulus and argument of each of the following complex numbers.

d

− 4 − 3i

ev

br

Answer

12 − 5i

ie

c

w

−3 + 4i

b

id

ge

5 + 12i

a

-R

ity

op

From the definition of the modulus. The diagram shows that the angle is in the first quadrant and so the angle is acute and positive.

Pr

Im(z)

rs

w

C

274

s

52 + 122 = 169 = 13

es

-C

5 + 12i =

y

a

am

A simple sketch of the Argand diagram is always necessary when finding the argument.

5

y op w From the definition of the argument.

-R

s

( −3)2 + 42 =

From the definition of the modulus.

es

25 = 5

Pr

−3 + 4i =

op y

The diagram shows that the angle is in the second quadrant and is obtuse and positive. This is calculated by subtracting the value of the acute angle α from π. Remember that the principal argument is −π , θ ø π.

U

R

y

θ

α

O

Re(z)

-R s

-C

am

br

ev

ie

id g

θ = π−α

w

e

3

op

ev

4

C

ie

w

ni ve rs

C

ity

Im(z)

es

b

-C

am

br

12  = 1.1760... arg(5 + 12i) = tan −1   5  = 1.18 radians correct to 3 significant figures

ie

id

ge

Re(z)

ev

θ O

C

U

R

ni

ev

ve

ie

12

Copyright Material - Review Only - Not for Redistribution

ve rs ity -R

am br id

12 − 5i = 122 + ( −5)2 = 169 = 13

Pr es s

-C

Re(z)

ni

ev ie

α

y

w

12 O

The diagram shows that the angle is in the fourth quadrant and is acute and negative. Again, recall that the principal argument is −π , θ ø π.

ve rs ity

C

op

y

Im(z)

From the definition of the modulus.

C op

c

From the definition of the argument.

ev ie

4 arg( −3 + 4i) = π − tan −1   = 2.214...  3 = 2.21 radians correct to 3 significant figures

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ge

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Chapter 11: Complex numbers

-R

am

br

ev

id

ie

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ge

U

R

5

( −4)2 + ( −3)2 =

s es

ve

op

y

The diagram shows the angle is in the third quadrant and is obtuse and negative. Again recall that the principal argument is − π , θ ø π.

ni

ev

Im(z)

C

O θ

Re(z)

ie

α

w

4

id

ge

U

R

s es

3 arg( − 4 − 3i) = −π + tan −1   = −2.498...  4 = −2.50 radians correct to 3 significant figures

-R s es

-C

am

br

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id g

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C

U

op

ev

R

From the definition of the argument.

y

ity

Pr

θ = −π + α

ni ve rs

w

C

op y

-C

-R

am

br

ev

3

ie

275

From the definition of the modulus.

25 = 5

rs

C w

− 4 − 3i =

ity

= − 0.395 radians correct to 3 significant figures

ie

d

From the definition of the argument.

Pr

5  = − 0.3947... arg(12 − 5i) = − tan −1   12 

op

y

-C

θ = −α

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ve rs ity

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

Im(z)

r

-R

am br id

ev ie

w

ge

Using trigonometry, we can now write a complex number in modulus-argument form. x x cos θ = and sin θ = . r r Therefore, x = r cos θ and y = r sin θ .

y

z = r cos θ + i( r sin θ )

op

z = r(cos θ + isin θ )

C

y C op

ni

w

w

ge

U

WORKED EXAMPLE 11.6

id

ie

u = 6 − 3i and w = −7 + 5i

-C

Answer

-R

am

br

ev

Write each of u and w in modulus-argument form.

es

The diagram shows the angle to be acute and negative.

y op

Re(z)

w

C

U 3

es

s

-C

-R

am

br

ev

id

ie

α

ge

R

ni

6 O

ve

ev

ie

w

rs

Im(z)

θ = −α

Pr From the definition of the argument.

ity

C

op y

3 arg(6 − 3i) = − tan −1    6 = − 0.4636...

ni ve rs

= − 0.464 radians correct to 3 significant figures

w

op C

e

w

From the definition of the modulus.

ev

ie

id g

74

-R s es

am

=

( −7)2 + 52

br

r= w =

U

w = −7 + 5i

y

u = 3 5(cos( − 0.464) + i sin( − 0.464))

-C

ie

From the definition of the modulus.

Pr

45 = 3 5

ity

y

op

=

62 + ( −3)2

C

276

s

u = 6 − 3i r= u =

ev

x

When a complex number is written as z = r(cos θ + isin θ ) where r is z and θ is arg z, we say it is in modulus-argument form.

R

ev ie

θ O

ve rs ity

where r is z and θ is arg z.

R

y

Pr es s

-C

Substituting these expressions into the Cartesian form of a complex number, we have z = x + iy

Copyright Material - Review Only - Not for Redistribution

Re(z)

ve rs ity

w

ge

ev ie

The diagram shows the angle to be obtuse and positive.

Re(z)

ve rs ity

ev ie

w

C

θ = π−α 5 arg( −7 + 5i) = π − tan −1    7 = 2.5213...

From the definition of the argument.

y

O

op

y

7

-R

θ

α

Pr es s

-C

am br id

Im(z)

5

C

U

ni

op

y

Chapter 11: Complex numbers

U

74(cos(2.52) + i sin(2.52))

ie ev

br

Exponential form

w

ge

w=

id

R

ni

C op

= 2.52 radians correct to 3 significant figures

-C

-R

am

The mathematician Leonhard Euler (1707–1783) discovered the relationship cos θ + isin θ = e iθ .

es

s

This is the basis for another representation, called the exponential form.

Pr

y

z = r(cos θ + isin θ )

op

z = re iθ

rs

The coordinates ( r, θ ), where r is z and θ is arg z, are called the polar coordinates.

ve

ie

w

C

ity

where r is z and θ is arg z.

277

y

op

C

U

R

ni

ev

When a complex number is written as z = re iθ , where r is z and θ is arg z, we say it is in exponential form.

br

ev

id

ie

w

ge

The modulus-argument and exponential forms of a complex number are polar forms. A polar form uses polar coordinates; a Cartesian form uses Cartesian coordinates.

es

c Write z3 in

Pr

b Write z2 in

ev



exponential form



Cartesian form.



Cartesian form.

-R s es

-C

am

br

ev

ie

id g

w

e

C

U

R

modulus-argument form

y

exponential form.



op



ni ve rs

modulus-argument form

ie



ity

op y

a Write z1 in

C

π π z3 = 2  cos + isin   12 12 

s

-C



z2 = 5e 3

z1 = 1 + i

w

-R

am

WORKED EXAMPLE 11.7

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ve rs ity

ev ie

w

ge am br id

Answer

-R

The diagram shows the angle to be acute and positive.

-C

1

y

ve rs ity

1

2 π 4

From the definition of the argument.

w

ge

arg(1 + i) = tan −1( 1 ) =

From the definition of the modulus.

ni

R

z1 = 12 + 12 =

U

ev ie

w

Re(z)

y

op C

θ O

C op

Im(z)

Pr es s

a

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ev

id

ie

Modulus-argument form:

br

π π 2  cos + isin   4 4

-R

am

z1 =

z1 =

s

π 3 Modulus-argument form:

rs

ni

op

y

Comparing the modulus-argument form with x + iy .

es

s

-C

Pr ity

y C s es

-C

-R

br

ev

ie

5 3 5 + i  2  2

am

z2 =

Write in Cartesian form, x + iy.

id g

Cartesian form:

Evaluate.

op

5 3 2

π 3

w

y=

y = 5 sin

ni ve rs

5 2

π 3

U

x = 5 cos

Re(z)

e

w

C

op y

π 3 x

ie ev

R

y

5

O

x=

-R

am

br

ev

id

ie

w

ge

Im(z)

C

U

ev

π π z2 = 5 cos + isin   3 3

R

ity

z2 = 5 θ =

ve

b

Pr

2e 4

y op

ie

w

C

278



es

-C

Exponential form:

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ve rs ity

w

ge

am br id

ev ie

π 12

z3 = 2 θ =

c

C

U

ni

op

y

Chapter 11: Complex numbers



-R

Exponential form: z3 = 2e 12

ve rs ity

2

y

x

Evaluate.

U

br

w

 6+ 2 +i  2

6− 2  2

y

es

s

-C

am

Cartesian form: z3 =

Write in Cartesian form, x + iy.

ie

id

6− 2 2

ev

6+ 2 y= 2

x=

w

rs

u = 5e2i and w = 10e −3i

y op ie

w

ge

id

uw = 5 × e2i × 10 × e −3i

ev

Using laws of indices.

br

es

w 10 × e −3i = u 5 × e2i w = 2 × e −3i − 2i u w = 2e −5i u

Pr

Using laws of indices.

−5 is less than −π and so this argument is not a principal argument. The equivalent principal argument, in this example, is θ = 2 π − 5

Re(z)

ie

id g

w

e

α O

ev es

s

-R

br

am

op

U

θ

w = 2e(2π−5)i u

-C

y

ity

Im(z)

ni ve rs

op y C w ie ev

The exponential form of a complex number is very useful when multiplying or dividing two complex numbers.

s

-C

uw = 50e − i

ii

TIP

C

am

uw = 50 × e

2i − 3i

-R

a i

C

U

R

ni

ev

ve

ie

a Find the modulus and argument of: w ii i uw u w b Show u, w, uw and on a single Argand diagram. u Answer

R

279

ity

op

Pr

WORKED EXAMPLE 11.8

C

y

π π y = 2 sin 12 12

ge

R

x = 2 cos

Re(z)

ni

O

-R

ev ie

w

π 12

C op

C

op

y

Pr es s

-C

Im(z)

Comparing the modulus-argument form with x + iy .

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

am br id

ev ie

A ≈ ( −2, 4.5) B ≈ ( −10, 1.5)

w = 10(cos( −3) + i sin( −3)) = −9.90 − 1.41i

uw = 50(cos( −1) + i sin( −1)) = 27.0 − 42.1i

-R

D ≈ (0.5, 2)

Pr es s

-C

y C op

D

w

ge

O

Re(z)

br

ev

id

ie

A

U

R

ni

ev ie

w

C

ve rs ity

op

y

C

B

It is not essential to find the Cartesian form, but you might find it easier to plot in the correct quadrant if you do.

C ≈ (27, 42)

w = 2(cos(2π − 5) + i sin(2π − 5)) = 0.567 + 1.92i u Im(z)

TIP

w

u = 5(cos 2 + i sin 2) = −2.08 + 4.55i

b

s

-C

-R

am

Using laws of indices and the exponential form of a complex number, we can derive some very important and useful results.

2

1

2

1

2

1

−θ 2 )

2

rs

w

op

y

ve

z1z2 = r1r2 = z1 z2 and arg ( z1z2 ) = θ1 + θ 2 = arg z1 + arg z2, z and the complex number 1 is such that: z2

ie

w

ge

z and arg  1  = θ1 − θ 2 = arg z1 − arg z2.  z2 

id

z1 z1 r = 1 = z2 r2 z2

C

U

ni

ie ev

R

1

ity

1

C

280

Pr

op

y

es

Let the complex number z1 be such that z1 = r1 and arg z1 = θ1 and let the complex number z2 be such that z2 = r2 and arg z2 = θ2 then: z r e iθ r z1 = r1e iθ , z2 = r2 e iθ , z1z2 = r1e iθ × r2 e iθ = r1r2 e i(θ +θ ) and 1 = 1 iθ = 1 e i(θ z2 r2 r2 e Therefore, the complex number z1z2 is such that:

-R

am

br

ev

These results greatly simplify the multiplication and division of complex numbers.

es

s

to multiply two complex numbers we multiply the moduli and add the arguments to divide two complex numbers we divide the moduli and subtract the arguments.

Pr

op y

ni ve rs

y

The complex numbers u and w are given by u = 9 + 12i and w = −5 − 3i. u w

By definition for the moduli of u and w.

225 = 15

( −5)2 + ( −3)2 =

34

s

w =

ev

92 + 122 =

am

u =

br

Answer

ie

id g

w

e

b

op

a uw

C

U

Find the modulus and argument of:

-C

R

ev

ie

w

C

ity

WORKED EXAMPLE 11.9

-R



es



-C

We can see that, when in exponential form,

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

From the diagram, the angle for u is acute and positive.

-R

am br id

Im(z)

w

ge

C

U

ni

op

y

Chapter 11: Complex numbers

y

θ

op

O

9

ve rs ity

Re(z)

C op

From the diagram, the angle for w is obtuse and negative.

Re(z)

ev

O θ

br

α

ie

id

5

w

ge

U

Im(z)

R

y

12  = 0.9272... arg u = tan −1   9  = 0.927 radians correct to 3 significant figures

ni

s

-C

-R

am

3

es

Pr

ity

281

rs

uw = u w = 15 34

Using z1z2 = r1r2 = z1 z2

y op

Using arg( z1z2 ) = θ1 + θ 2 = arg z1 + arg z2

U

R

ni

ev

C

arg( uw ) = arg u + arg w = 0.9272… + ( −2.601…)

ie

id

u u 15 34 15 = = = w w 34 34

ev -R

br

u arg   = arg u − arg w = 0.9272… − ( −2.601…)  w = 3.528 radians

z Using arg  1  = θ1 − θ 2 = arg z1 − arg z2  z2 

Pr

es

s

-C

However, 3.528… is more than π and so this argument is not a principal argument. The equivalent principal argument, in this example, is

ity

Im (z)

ni ve rs

op y

θ = −(2 π − 3.528…) = −2.754… = −2.75 radians correct to 3 significant figures

α

C

U

O θ

w

e

ev

ie

id g

es

s

-R

br am -C

C w ie ev

R

z z1 r = 1 = 1 z2 r2 z2

Using

am

b

w

ge

= −1.67 radians correct to 3 significant figures

y

ie

a

3 arg w = −π + tan −1   = −2.601  5 = − 2.60 radians correct to 3 significant figures

ve

w

C

op

y

θ = −π + α

op

C w ev ie

Pr es s

-C

12

Copyright Material - Review Only - Not for Redistribution

Re (z)

ve rs ity

ev ie

am br id

EXERCISE 11C

w

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

u = 5 − 2i

1

-C

-R

a On an Argand diagram, show the points A, B and C representing the complex numbers u, u* and − u, respectively.

Pr es s

b The points ABCD form a rectangle. Write down the complex number represented by the point D. z1 = 1 + 5i and z2 = −7 − i

a On an Argand diagram, show the points P and Q representing the complex numbers z1 and z2 , respectively.

ve rs ity

w

C

op

y

2

b Write down the complex number represented by the point R, the midpoint of PQ.

4

2

3

Re(z)

y

ity

C

rs

C w

k(1 − i), where k . 0

Pr

1

–4

ve

ie

i

es

B

–2 –3

−1 − i 3

s

1

–3 –2 –1 O –1

f

-R

am op

y

-C

2

282

C op

−24 − 7i

br

Im(z) 4 A 3

h

w

5 + 2i

−9 − 40i

8 + 15i

ie

g

e

c

ev

60 − 11i

ni

d

b 5i

U

−12 + 5i

ge

a

id

R

ev ie

3 Find the modulus and argument of each of the following complex numbers.

y

op

ni

ev

a Write each of the complex numbers shown in the Argand diagram in modulus-argument form.

C

U

R

b Show that ABC is a right-angled, isosceles triangle.

3π 3π  + i sin 5 cos  8 8 

ev

id

ie

b

w

π π 3 cos + i sin   3 3

br

a

ge

5 Write each of these complex numbers in Cartesian form.



es Pr

z . w π 8 The complex number a + bi is denoted by z. Given that z = 5 and arg z = , find the value of 6 a and the value of b.

ni ve rs

U

op

y

Find and simplify an expression for

ev

b Show that z −

1 = i(2 r sin θ ). z

es

s

-R

br

am

a Show that z 2 = r 2 (cos 2θ + i sin 2θ ).

-C

P

ie

id g

w

e

9 Given that z = r(cos θ + isin θ ), find and simplify an expression for z a zz * b z* 10 z = r(cos θ + i sin θ )

C

PS

ity

b Write z in exponential form. c

R

ev

ie

w

C

op y

3 − 7i π π 7 w = 5  cos + isin  and z =  6 6 5 − 2i a Write w in exponential form.

s

-C

-R

am

− e iπ d 3e 4 2 6 Given that z(4 − 9i) = 8 + 3i, find the value of z 2 and the value of arg z 2 .

c

Copyright Material - Review Only - Not for Redistribution

ve rs ity

ev ie

am br id

EXPLORE 11.2

w

ge

C

U

ni

op

y

Chapter 11: Complex numbers

Pr es s

the geometrical effects of combining two complex numbers by addition, subtraction, multiplication and division.

ev ie

w

C

ve rs ity

op

y



the geometrical relationship between a complex number and its conjugate

-C



-R

Using Argand diagrams and position vectors, investigate the transformations of the plane that are:

11.4 Solving equations

ni

C op

y

We have already solved quadratic equations with complex roots.

ie

id

b 2 − 4 ac

Roots

,0

2 real and equal roots (repeated roots)

α and β α and β , α = β

s

-C

=0

2 real and distinct roots

-R

br

ev

Nature of roots

am

0 real roots

op

Pr

y

es

.0

w

ge

U

R

From the Pure Mathematics 1 Coursebook, Chapter 1, we know that when solving quadratic equations for solutions that are real:

283

y

Roots

ve

Nature of roots

α and β

=0

2 real and equal roots (repeated roots)

α and β , α = β

,0

2 complex roots of the form x ± iy with y ≠ 0 (a complex conjugate pair). This means that one root is the conjugate of the other root.

α and α *

C

w

-R

am

br

ev

ie

U

ge

op

2 real and distinct roots

ni

.0

id

R

ev

ie

b 2 − 4 ac

rs

w

C

ity

We can now update this for solving quadratic equations for solutions that are complex:

es

s

-C

Using the set of complex numbers to find solutions, a quadratic equation always has two roots. TIP

Pr

op y

Higher order polynomial equations

ni ve rs

We will now consider solving these equations for complex roots.

-R s es

-C

am

br

ev

ie

id g

w

e

C

U

op

y

The Fundamental Theorem of Algebra states that a polynomial equation of degree n has n complex roots.

R

ev

ie

w

C

ity

We solved cubic and quartic equations for real roots in Chapter 1.

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The degree of a quadratic equation is 2, the degree of a cubic equation is 3… the degree of a polynomial equation is the same as its largest power.

ve rs ity

ge

C

U

ni

op

y

Cambridge International AS & A Level Mathematics: Pure Mathematics 2 & 3

ev ie

am br id

w

WORKED EXAMPLE 11.10

Answer

f(4) = 43 − 3(42 ) − 3(4) − 4 = 64 − 48 − 12 − 4 = 64 − 64 = 0 ✓

ve rs ity

w

C

op

y

a Let f( z ) = z 3 − 3z 2 − 3z − 4 then

Pr es s

-C

b Find all complex solutions of z 3 − 3z 2 − 3z − 4 = 0.

-R

a Show that z = 4 is a root of the equation z 3 − 3z 2 − 3z − 4 = 0.

y C op

ni

Multiply out and compare.

z 3 + ( a − 4) z 2 + (1 − 4a ) z − 4

ity

w

ge

C

U

ni

Given.<