ebook img

Pure Mathematics: 1 PDF

322 Pages·2012·10.132 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Pure Mathematics: 1

C ambridge I nternational A and AS Level Mathematics P ure Mathematics 1 Sophie Goldie Series Editor: Roger Porkess Contents Key to symbols in this book vi Questions from the Cambridge International Examinations A & AS level Mathematics papers Introduction vii are reproduced by permission of University of Cambridge International Examinations. The Cambridge A & AS Level Mathematics 9709 syllabus viii Questions from the MEI A & AS level Mathematics papers are reproduced by permission of OCR. We are grateful to the following companies, institutions and individuals you have given permission Chapter 1 Algebra 1 to reproduce photographs in this book. page 106, © Jack Sullivan / Alamy; page 167, © RTimages / Fotolia; page 254, © Hunta / Fotolia; Background algebra 1 page 258, © Olga Iermolaieva / Fotolia Linear equations 6 Every effort has been made to trace and acknowledge ownership of copyright. The publishers will be Changing the subject of a formula 10 glad to make suitable arrangements with any copyright holders whom it has not been possible to contact. Quadratic equations 12 Hachette UK’s policy is to use papers that are natural, renewable and recyclable products and Solving quadratic equations 17 made from wood grown in sustainable forests. The logging and manufacturing processes are expected to conform to the environmental regulations of the country of origin. Equations that cannot be factorised 20 Orders: please contact Bookpoint Ltd, 130 Milton Park, Abingdon, Oxon OX14 4SB. The graphs of quadratic functions 22 Telephone: (44) 01235 827720. Fax: (44) 01235 400454. Lines are open 9.00–5.00, Monday The quadratic formula 25 to Saturday, with a 24-hour message answering service. Visit our website at www.hoddereducation.co.uk Simultaneous equations 29 Much of the material in this book was published originally as part of the MEI Structured Mathematics series. It has been carefully adapted for the Cambridge International A & AS level Inequalities 34 Mathematics syllabus. The original MEI author team for Pure Mathematics comprised Catherine Berry, Bob Francis, Chapter 2 Co-ordinate geometry 38 Val Hanrahan, Terry Heard, David Martin, Jean Matthews, Bernard Murphy, Roger Porkess and Peter Secker. Co-ordinates 38 © MEI, 2012 Plotting, sketching and drawing 39 First published in 2012 by The gradient of a line 39 Hodder Education, a Hachette UK company, The distance between two points 41 338 Euston Road London NW1 3BH The mid-point of a line joining two points 42 Impression number 5 4 3 2 1 The equation of a straight line 46 Year 2016 2015 2014 2013 2012 Finding the equation of a line 49 All rights reserved. Apart from any use permitted under UK copyright law, no part of this The intersection of two lines 56 publication may be reproduced or transmitted in any form or by any means, electronic or Drawing curves 63 mechanical, including photocopying and recording, or held within any information storage and retrieval system, without permission in writing from the publisher or under licence from The intersection of a line and a curve 70 the Copyright Licensing Agency Limited. Further details of such licences (for reprographic reproduction) may be obtained from the Copyright Licensing Agency Limited, Saffron House, 6–10 Kirby Street, London EC1N 8TS. Chapter 3 Sequences and series 75 Cover photo by © Joy Fera / Fotolia Definitions and notation 76 Illustrations by Pantek Media, Maidstone, Kent Arithmetic progressions 77 Typeset in 10.5pt Minion by Pantek Media, Maidstone, Kent Printed in Dubai Geometric progressions 84 A catalogue record for this title is available from the British Library Binomial expansions 95 ISBN 978 1444 14644 8 This eBook does not include the ancillary media that was packaged with the printed version of the book. Contents Key to symbols in this book vi Introduction vii The Cambridge A & AS Level Mathematics 9709 syllabus viii Chapter 1 Algebra 1 Background algebra 1 Linear equations 6 Changing the subject of a formula 10 Quadratic equations 12 Solving quadratic equations 17 Equations that cannot be factorised 20 The graphs of quadratic functions 22 The quadratic formula 25 Simultaneous equations 29 Inequalities 34 Chapter 2 Co-ordinate geometry 38 Co-ordinates 38 Plotting, sketching and drawing 39 The gradient of a line 39 The distance between two points 41 The mid-point of a line joining two points 42 The equation of a straight line 46 Finding the equation of a line 49 The intersection of two lines 56 Drawing curves 63 The intersection of a line and a curve 70 Chapter 3 Sequences and series 75 Definitions and notation 76 Arithmetic progressions 77 Geometric progressions 84 Binomial expansions 95 iii Chapter 4 Functions 106 Chapter 8 Vectors 254 The language of functions 106 Vectors in two dimensions 254 Composite functions 112 Vectors in three dimensions 258 Inverse functions 115 Vector calculations 262 The angle between two vectors 271 Chapter 5 Differentiation 123 The gradient of a curve 123 Answers 280 Finding the gradient of a curve 124 Index 310 Finding the gradient from first principles 126 Differentiating by using standard results 131 Using differentiation 134 Tangents and normals 140 Maximum and minimum points 146 Increasing and decreasing functions 150 Points of inflection 153 The second derivative 154 Applications 160 The chain rule 167 Chapter 6 Integration 173 Reversing differentiation 173 Finding the area under a curve 179 Area as the limit of a sum 182 Areas below the x axis 193 The area between two curves 197 The area between a curve and the y axis 202 The reverse chain rule 203 Improper integrals 206 Finding volumes by integration 208 Chapter 7 Trigonometry 216 Trigonometry background 216 Trigonometrical functions 217 Trigonometrical functions for angles of any size 222 The sine and cosine graphs 226 The tangent graph 228 Solving equations using graphs of trigonometrical functions 229 Circular measure 235 The length of an arc of a circle 239 The area of a sector of a circle 239 Other trigonometrical functions 244 iv Chapter 8 Vectors 254 Vectors in two dimensions 254 Vectors in three dimensions 258 Vector calculations 262 The angle between two vectors 271 Answers 280 Index 310 vv Key to symbols in this book ● ? This symbol means that you want to discuss a point with your teacher. If you are working on your own there are answers in the back of the book. It is important, however, that you have a go at answering the questions before looking up the answers if you are to understand the mathematics fully. ● This symbol invites you to join in a discussion about proof. The answers to these questions are given in the back of the book. ! This is a warning sign. It is used where a common mistake, misunderstanding or tricky point is being described. This is the ICT icon. It indicates where you could use a graphic calculator or a computer. Graphical calculators and computers are not permitted in any of the examinations for the Cambridge International A & AS Level Mathematics 9709 syllabus, however, so these activities are optional. This symbol and a dotted line down the right-hand side of the page indicates material that you are likely to have met before. You need to be familiar with the material before you move on to develop it further. This symbol and a dotted line down the right-hand side of the page indicates material which is beyond the syllabus for the unit but which is included for completeness. vi Introduction This is the first of a series of books for the University of Cambridge International Examinations syllabus for Cambridge International A & AS Level Mathematics 9709. The eight chapters of this book cover the pure mathematics in AS level. The series also contains a more advanced book for pure mathematics and one each for mechanics and statistics. These books are based on the highly successful series for the Mathematics in Education and Industry (MEI) syllabus in the UK but they have been redesigned for Cambridge users; where appropriate new material has been written and the exercises contain many past Cambridge examination questions. An overview of the units making up the Cambridge International A & AS Level Mathematics 9709 syllabus is given in the diagram on the next page. Throughout the series the emphasis is on understanding the mathematics as well as routine calculations. The various exercises provide plenty of scope for practising basic techniques; they also contain many typical examination questions. An important feature of this series is the electronic support. There is an accompanying disc containing two types of Personal Tutor presentation: examination-style questions, in which the solutions are written out, step by step, with an accompanying verbal explanation, and test yourself questions; these are multiple-choice with explanations of the mistakes that lead to the wrong answers as well as full solutions for the correct ones. In addition, extensive online support is available via the MEI website, www.mei.org.uk. The books are written on the assumption that students have covered and understood the work in the Cambridge IGCSE syllabus. However, some of the early material is designed to provide an overlap and this is designated ‘Background’. There are also places where the books show how the ideas can be taken further or where fundamental underpinning work is explored and such work is marked as ‘Extension’. The original MEI author team would like to thank Sophie Goldie who has carried out the extensive task of presenting their work in a suitable form for Cambridge International students and for her many original contributions. They would also like to thank Cambridge International Examinations for their detailed advice in preparing the books and for permission to use many past examination questions. Roger Porkess Series Editor vii The Cambridge A & AS Level Mathematics syllabus P2 Cambridge AS Level IGCSE P1 S1 Mathematics Mathematics M1 M1 S1 S2 A Level P3 Mathematics S1 M1 M2 vviiiiii 1 Algebra P1 1 B Sherlock Holmes: ‘Now the skillful workman is very careful indeed a c k … He will have nothing but the tools which may help him in doing g r o his work, but of these he has a large assortment, and all in the most u n d perfect order.’ a lg A. Conan Doyle e b r a Background algebra Manipulating algebraic expressions You will often wish to tidy up an expression, or to rearrange it so that it is easier to read its meaning. The following examples show you how to do this. You should practise the techniques for yourself on the questions in Exercise 1A. Collecting terms Very often you just need to collect like terms together, in this example those in x, those in y and those in z. ● ? What are ‘like’ and ‘unlike’ terms? EXAMPLE 1.1 Simplify the expression 2x + 4y − 5z − 5x − 9y + 2z + 4x − 7y + 8z. SOLUTION Collect like Expression = 2x + 4x − 5x + 4y – 9y − 7y + 2z + 8z − 5z terms = 6x − 5x + 4y − 16y + 10z − 5z Tidy up = x − 12y + 5z This cannot be simplified further and so it is the answer. Removing brackets Sometimes you need to remove brackets before collecting like terms together. 1

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.