Haese and Harris Publications specialists in mathematics publishing Endorsed by University of Cambridge International Examinations IGCSE Cambridge International Mathematics (0607) Extended Keith Black Alison Ryan Michael Haese Robert Haese Sandra Haese cyan magenta yellow Y:\HAESE\IGCSE01\IG01_00\001IGCSE01_01.CDR Thursday, 30 October 2008 4:36:42 PM PETER 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 Mark Humphries black IGCSE01 IGCSE CAMBRIDGE INTERNATIONAL MATHEMATICS (0607) Keith Black Alison Ryan Michael Haese Robert Haese Sandra Haese Mark Humphries B.Sc.(Hons.), Dip.Ed B.Sc., M.Ed B.Sc.(Hons.), Ph.D B.Sc B.Sc B.Sc.(Hons.) Haese & Harris Publications Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA Telephone: +61 8355 9444, Fax: + 61 8355 9471 Email: info@haeseandharris.com.au Web: www.haeseandharris.com.au National Library of Australia Card Number & ISBN 978-1-921500-04-6 © Haese & Harris Publications 2009 Published by Raksar Nominees Pty Ltd Frank Collopy Court, Adelaide Airport, SA 5950, AUSTRALIA First Edition 2009 Cartoon artwork by John Martin Artwork and cover design by Piotr Poturaj Fractal artwork on the cover copyright by Jarosław Wierny, www.fractal.art.pl Computer software by David Purton, Troy Cruickshank and Thomas Jansson Typeset in Australia by Susan Haese and Charlotte Sabel (Raksar Nominees) Typeset in Times Roman 10 /11 This textbook and its accompanying CD have been endorsed by University of Cambridge International Examinations (CIE) They have been developed independently of the International Baccalaureate Organization (IBO) and are not connected with or endorsed by, the IBO This book is copyright Except as permitted by the Copyright Act (any fair dealing for the purposes of private study, research, criticism or review), no part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher Enquiries to be made to Haese & Harris Publications Copying for educational purposes: Where copies of part or the whole of the book are made under Part VB of the Copyright Act, the law requires that the educational institution or the body that administers it has given a remuneration notice to Copyright Agency Limited (CAL) For information, contact the Copyright Agency Limited Acknowledgements: The publishers acknowledge the cooperation of Oxford University Press, Australia, for the reproduction of material originally published in textbooks produced in association with Haese & Harris Publications While every attempt has been made to trace and acknowledge copyright, the authors and publishers apologise for any accidental infringement where copyright has proved untraceable They would be pleased to come to a suitable agreement with the rightful owner cyan magenta yellow Y:\HAESE\IGCSE01\IG01_00\002IGCSE01_00.CDR Friday, 21 November 2008 12:34:38 PM PETER 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 Disclaimer: All the internet addresses (URL’s) given in this book were valid at the time of printing While the authors and publisher regret any inconvenience that changes of address may cause readers, no responsibility for any such changes can be accepted by either the authors or the publisher black IGCSE01 FOREWORD This book has been written to cover the ‘IGCSE Cambridge International Mathematics (0607) Extended’ course over a two-year period The new course was developed by University of Cambridge International Examinations (CIE) in consultation with teachers in international schools around the world It has been designed for schools that want their mathematics teaching to focus more on investigations and modelling, and to utilise the powerful technology of graphics calculators The course springs from the principles that students should develop a good foundation of mathematical skills and that they should learn to develop strategies for solving open-ended problems It aims to promote a positive attitude towards Mathematics and a confidence that leads to further enquiry Some of the schools consulted by CIE were IB schools and as a result, Cambridge International Mathematics integrates exceptionally well with the approach to the teaching of Mathematics in IB schools This book is an attempt to cover, in one volume, the content outlined in the Cambridge International Mathematics (0607) syllabus References to the syllabus are made throughout but the book can be used as a full course in its own right, as a preparation for GCE Advanced Level Mathematics or IB Diploma Mathematics, for example The book has been endorsed by CIE but it has been developed independently of the Independent Baccalaureate Organization and is not connected with, or endorsed by, the IBO To reflect the principles on which the new course is based, we have attempted to produce a book and CD package that embraces technology, problem solving, investigating and modelling, in order to give students different learning experiences There are non-calculator sections as well as traditional areas of mathematics, especially algebra An introductory section ‘Graphics calculator instructions’ appears on p 11 It is intended as a basic reference to help students who may be unfamiliar with graphics calculators Two chapters of ‘assumed knowledge’ are accessible from the CD: ‘Number’ and ‘Geometry and graphs’ (see pp 29 and 30) They can be printed for those who want to ensure that they have the prerequisite levels of understanding for the course To reflect one of the main aims of the new course, the last two chapters in the book are devoted to multi-topic questions, and investigations and modelling Review exercises appear at the end of each chapter with some ‘Challenge’ questions for the more able student Answers are given at the end of the book, followed by an index The interactive CD contains Self Tutor software (see p 5), geometry and graphics software, demonstrations and simulations, and the two printable chapters on assumed knowledge The CD also contains the text of the book so that students can load it on a home computer and keep the textbook at school The Cambridge International Mathematics examinations are in the form of three papers: one a non-calculator paper, another requiring the use of a graphics calculator, and a third paper containing an investigation and a modelling question All of these aspects of examining are addressed in the book The book can be used as a scheme of work but it is expected that the teacher will choose the order of topics There are a few occasions where a question in an exercise may require something done later in the book but this has been kept to a minimum Exercises in the book range from routine practice and consolidation of basic skills, to problem solving exercises that are quite demanding In this changing world of mathematics education, we believe that the contextual approach shown in this book, with the associated use of technology, will enhance the students’ understanding, knowledge and appreciation of mathematics, and its universal application We welcome your feedback cyan magenta yellow Y:\HAESE\IGCSE01\IG01_00\003IGCSE01_00.CDR Friday, 21 November 2008 12:53:41 PM PETER 95 100 50 75 25 95 KB, AR, PMH, RCH, SHH, MH 100 50 75 25 95 100 50 75 25 info@haeseandharris.com.au www.haeseandharris.com.au 95 100 50 75 25 Email: Web: black IGCSE01 ACKNOWLEDGEMENTS The authors and publishers would like to thank University of Cambridge International Examinations (CIE) for their assistance and support in the preparation of this book Exam questions from past CIE exam papers are reproduced by permission of the University of Cambridge Local Examinations Syndicate The University of Cambridge Local Examinations Syndicate bears no responsibility for the example answers to questions taken from its past question papers which are contained in this publication In addition we would like to thank the teachers who offered to read proofs and who gave advice and support: Simon Bullock, Philip Kurbis, Richard Henry, Johnny Ramesar, Alan Daykin, Nigel Wheeler, Yener Balkaya, and special thanks is due to Fran O'Connor who got us started cyan magenta yellow Y:\HAESE\IGCSE01\IG01_00\004IGCSE01_00.CDR Friday, 21 November 2008 12:18:06 PM PETER 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 The publishers wish to make it clear that acknowledging these teachers, does not imply any endorsement of this book by any of them, and all responsibility for the content rests with the authors and publishers black IGCSE01 USING THE INTERACTIVE CD The interactive Student CD that comes with this book is designed for those who want to utilise technology in teaching and learning Mathematics The CD icon that appears throughout the book denotes an active link on the CD Simply click on the icon when running the CD to access a large range of interactive features that includes: • • • • • • • • spreadsheets printable worksheets graphing packages geometry software demonstrations simulations printable chapters SELF TUTOR INTERACTIVE LINK For those who want to ensure they have the prerequisite levels of understanding for this new course, printable chapters of assumed knowledge are provided for Number (see p 29) and Geometry and Graphs (see p 30) SELF TUTOR is an exciting feature of this book The Self Tutor icon on each worked example denotes an active link on the CD Simply ‘click’ on the Self Tutor (or anywhere in the example box) to access the worked example, with a teacher’s voice explaining each step necessary to reach the answer Play any line as often as you like See how the basic processes come alive using movement and colour on the screen Ideal for students who have missed lessons or need extra help Example Self Tutor A die has the numbers 0, 0, 1, 1, and It is rolled twice Illustrate the sample space using a 2-D grid Hence find the probability of getting: a a total of 2-D grid roll b two numbers which are the same There are £ = 36 possible outcomes 1 0 a P(total of 5) = 36 b P(same numbers) = 0 1 fthose with a 10 36 g fthose circled g roll See Chapter 25, Probability, p.516 GRAPHICS CALCULATORS cyan magenta yellow Y:\HAESE\IGCSE01\IG01_00\005IGCSE01_00.CDR Friday, 21 November 2008 12:54:09 PM PETER 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 The course assumes that each student will have a graphics calculator An introductory section ‘Graphics calculator instructions’ appears on p 11 To help get students started, the section includes some basic instructions for the Texas Instruments TI-84 Plus and the Casio fx-9860G calculators black IGCSE01 SYMBOLS AND NOTATION USED IN THIS BOOK N the set of positive integers and zero, f0, 1, 2, 3, g > is greater than ¸ or > is greater than or equal to Z the set of integers, f0, §1, §2, §3, g < is less than Z+ the set of positive integers, f1, 2, 3, g · or is less than or equal to Q the set of rational numbers un the nth term of a sequence or series the set of positive rational numbers, fx j x > 0, x Q g f : x 7! y f is a function under which x is mapped to y f (x) the image of x under the function f R the set of real numbers R+ f ¡1 the inverse function of the function f the set of positive real numbers, fx j x > 0, x R g loga x logarithm to the base a of x + Q sin, cos, tan the circular functions fx1 , x2 , g the set with elements x1 , x2 , A(x, y) the point A in the plane with Cartesian coordinates x and y n(A) the number of elements in the finite set A fx j the set of all x such that is an element of = is not an element of ? or f g the empty (null) set U the universal set bB CA the angle between CA and AB [ union ¢ABC the triangle whose vertices are A, B and C \ intersection the vector v µ is a subset of v ¡ ! AB ½ is a proper subset of A0 the complement of the set A jaj ¡ ! j AB j the magnitude of vector a ¡ ! the magnitude of AB P(A) probability of event A P(A0 ) probability of the event “not A” p n a an , a2 , ( AB b A a to the power of n , nth root of a p n (if a > then a > 0) p a a to the power 12 , square root of a p (if a > then a > 0) jxj the modulus or absolute value of x, that is n x for x > 0, x R ¡x for x < 0, x R the line segment with end points A and B the distance from A to B the line containing points A and B the angle at A the vector represented in magnitude and direction by the directed line segment from A to B x1 , x2 , observations of a variable f1 , f2 , frequencies with which the observations x1 , x2 , x3 , occur x mean of the values x1 , x2 , §f sum of the frequencies f1 , f2 , ´ identity or is equivalent to r Pearson’s correlation coefficient ¼ is approximately equal to » = r2 coefficient of determination is congruent to k is parallel to cyan magenta yellow Y:\HAESE\IGCSE01\IG01_00\006IGCSE01_00.CDR Friday, 21 November 2008 12:06:59 PM PETER 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 is perpendicular to black IGCSE01 Table of contents SYMBOLS AND NOTATION USED IN THIS BOOK Basic calculations Basic functions Secondary function and alpha keys Memory Lists Statistical graphs Working with functions Two variable analysis 12 13 17 17 19 21 22 26 ASSUMED KNOWLEDGE (NUMBER) 29 Number types Operations and brackets HCF and LCM Fractions Powers and roots Ratio and proportion Number equivalents Rounding numbers Time CD CD CD CD CD CD CD CD CD 50 25 32 33 35 37 39 40 42 45 47 95 100 50 75 25 95 The distributive law The product (a+b)(c+d) Difference of two squares Perfect squares expansion Further expansion Algebraic common factors Factorising with common factors Difference of two squares factorisation Perfect squares factorisation 100 A B C D E F G H I 50 ALGEBRA (EXPANSION AND FACTORISATION) 31 75 25 CD CD CD CD CD CD CD Angles Lines and line segments Polygons Symmetry Constructing triangles Congruence Interpreting graphs and tables A B C D E F G 95 30 magenta SETS 57 A B C D E F Set notation Special number sets Interval notation Venn diagrams Union and intersection Problem solving Review set 2A Review set 2B 57 60 61 63 65 69 72 73 ALGEBRA (EQUATIONS AND INEQUALITIES) 75 A B C D E F G Solving linear equations Solving equations with fractions Forming equations Problem solving using equations Power equations Interpreting linear inequalities Solving linear inequalities Review set 3A Review set 3B 75 80 83 85 87 88 89 91 92 LINES, ANGLES AND POLYGONS 93 A B C D E Angle properties Triangles Isosceles triangles The interior angles of a polygon The exterior angles of a polygon Review set 4A Review set 4B 93 98 100 103 106 107 109 GRAPHS, CHARTS AND TABLES 111 A Statistical graphs B Graphs which compare data C Using technology to graph data Review set 5A Review set 5B 112 116 119 120 122 EXPONENTS AND SURDS 123 A B C D E Exponent or index notation Exponent or index laws Zero and negative indices Standard form Surds 123 126 129 131 134 yellow Y:\HAESE\IGCSE01\IG01_00\007IGCSE01_00.CDR Friday, 21 November 2008 12:29:38 PM PETER 95 11 100 50 GRAPHICS CALCULATOR INSTRUCTIONS cyan 48 49 51 54 55 56 ASSUMED KNOWLEDGE (GEOMETRY AND GRAPHS) 75 25 A B C D E F G H I Expressions with four terms Factorising xX¡+¡bx¡+¡c Splitting the middle term Miscellaneous factorisation Review set 1A Review set 1B 75 A B C D E F G H J K L M 100 TABLE OF CONTENTS black IGCSE01 Table of contents F Properties of surds G Multiplication of surds H Division by surds Review set 6A Review set 6B 12 COORDINATE GEOMETRY 255 A B C D E Plotting points Distance between two points Midpoint of a line segment Gradient of a line segment Gradient of parallel and perpendicular lines Using coordinate geometry Review set 12A Review set 12B 256 258 261 263 A B C D E F Formula substitution Formula rearrangement Formula derivation More difficult rearrangements Simultaneous equations Problem solving Review set 7A Review set 7B 148 150 153 155 158 164 166 167 THE THEOREM OF PYTHAGORAS 169 13 ANALYSIS OF DISCRETE DATA 275 A B C D E Pythagoras’ theorem The converse of Pythagoras’ theorem Problem solving Circle problems Three-dimensional problems Review set 8A Review set 8B 170 176 177 181 185 187 188 A B C D E F G MENSURATION (LENGTH AND AREA) 277 278 282 285 288 290 292 293 295 191 Length Perimeter Area Circles and sectors Review set 9A Review set 9B 192 194 196 201 206 207 cyan magenta F Variables used in statistics Organising and describing discrete data The centre of a discrete data set Measuring the spread of discrete data Data in frequency tables Grouped discrete data Statistics from technology Review set 13A Review set 13B 267 270 272 273 14 STRAIGHT LINES 297 A Vertical and horizontal lines B Graphing from a table of values C Equations of lines (gradient-intercept form) D Equations of lines (general form) E Graphing lines from equations F Lines of symmetry Review set 14A Review set 14B 297 299 301 304 307 308 310 311 15 TRIGONOMETRY 313 A B C D E F 314 316 322 327 330 331 336 337 Labelling sides of a right angled triangle The trigonometric ratios Problem solving The first quadrant of the unit circle True bearings 3-dimensional problem solving Review set 15A Review set 15B yellow Y:\HAESE\IGCSE01\IG01_00\008IGCSE01_00.CDR Friday, November 2008 9:47:47 AM PETER 95 339 100 A Simplifying algebraic fractions 50 339 75 16 ALGEBRAIC FRACTIONS 25 95 100 50 75 25 95 100 50 231 239 75 A Surface area B Volume 25 231 11 MENSURATION (SOLIDS AND CONTAINERS) 209 211 214 217 218 222 224 226 228 229 95 Percentage Profit and loss Simple interest Reverse percentage problems Multipliers and chain percentage Compound growth Speed, distance and time Travel graphs Review set 10A Review set 10B 100 A B C D E F G H 50 209 75 10 TOPICS IN ARITHMETIC 25 245 248 249 253 254 147 A B C D C Capacity D Mass E Compound solids Review set 11A Review set 11B FORMULAE AND SIMULTANEOUS EQUATIONS 137 139 142 143 145 black IGCSE01 Table of contents 354 355 359 364 365 18 SIMILARITY 367 A B C D Similarity Similar triangles Problem solving Area and volume of similar shapes Review set 18A Review set 18B 367 370 373 376 380 381 19 INTRODUCTION TO FUNCTIONS 383 A B C D E F 383 385 389 391 393 395 398 399 20 TRANSFORMATION GEOMETRY 401 A B C D E F G H 402 404 406 408 410 413 416 417 419 420 Translations Rotations Reflections Enlargements and reductions Stretches Transforming functions The inverse of a transformation Combinations of transformations Review set 20A Review set 20B 422 423 427 429 cyan magenta 95 100 50 75 25 95 100 50 75 25 Quadratic equations The Null Factor law The quadratic formula Quadratic functions 25 A B C D 421 95 21 QUADRATIC EQUATIONS AND FUNCTIONS 22 TWO VARIABLE ANALYSIS 455 A Correlation B Line of best fit by eye C Linear regression Review set 22A Review set 22B 456 459 461 466 467 23 FURTHER FUNCTIONS 469 A B C D 469 473 475 480 481 481 Cubic functions Inverse functions Using technology Tangents to curves Review set 23A Review set 23B 24 VECTORS 483 A B C D E F G H 484 485 486 489 491 496 497 499 501 503 Directed line segment representation Vector equality Vector addition Vector subtraction Vectors in component form Scalar multiplication Parallel vectors Vectors in geometry Review set 24A Review set 24B 25 PROBABILITY 505 A B C D E F G H I J 506 507 510 512 513 515 519 522 524 Introduction to probability Estimating probability Probabilities from two-way tables Expectation Representing combined events Theoretical probability Compound events Using tree diagrams Sampling with and without replacement Mutually exclusive and non-mutually exclusive events K Miscellaneous probability questions Review set 25A Review set 25B 50 Mapping diagrams Functions Function notation Composite functions Reciprocal functions The absolute value function Review set 19A Review set 19B 431 438 441 445 446 447 451 453 yellow Y:\HAESE\IGCSE01\IG01_00\009IGCSE01_00.CDR Friday, November 2008 9:48:16 AM PETER 95 A The mean of continuous data B Histograms C Cumulative frequency Review set 17A Review set 17B Graphs of quadratic functions Axes intercepts Line of symmetry and vertex Finding a quadratic function Using technology Problem solving Review set 21A Review set 21B 100 353 344 75 17 CONTINUOUS DATA 100 50 346 348 351 352 75 25 Multiplying and dividing algebraic fractions C Adding and subtracting algebraic fractions D More complicated fractions Review set 16A Review set 16B E F G H I J B black 527 528 530 531 IGCSE01 10 26 SEQUENCES 533 A B C D 534 535 537 539 544 545 547 A Circle theorems B Cyclic quadrilaterals Review set 27A Review set 27B 547 556 561 562 28 EXPONENTIAL FUNCTIONS AND EQUATIONS 565 Rational exponents Exponential functions Exponential equations Problem solving with exponential functions Exponential modelling Review set 28A Review set 28B 566 568 570 573 576 577 578 29 FURTHER TRIGONOMETRY 579 A B C D E 579 583 585 588 The unit circle Area of a triangle using sine The sine rule The cosine rule Problem solving with the sine and cosine rules F Trigonometry with compound shapes G Trigonometric graphs H Graphs of y¡=¡a¡sin(bx) and y¡=¡a¡cos(bx) Review set 29A Review set 29B 591 593 595 599 601 602 30 VARIATION AND POWER MODELLING 605 magenta 32 INEQUALITIES 639 A Solving one variable inequalities with technology B Linear inequality regions C Integer points in regions D Problem solving (Extension) Review set 32A Review set 32B 639 641 644 645 647 648 33 MULTI-TOPIC QUESTIONS 649 34 INVESTIGATION AND MODELLING QUESTIONS 661 A Investigation questions B Modelling questions 661 669 ANSWERS 673 INDEX 752 yellow Y:\HAESE\IGCSE01\IG01_00\010IGCSE01_00.CDR Friday, 21 November 2008 12:30:32 PM PETER 95 100 50 75 95 100 50 75 25 95 100 50 75 25 95 100 625 627 50 A Logarithms in base a B The logarithmic function 75 625 25 629 630 634 636 637 606 612 615 619 622 623 31 LOGARITHMS cyan C Rules for logarithms D Logarithms in base 10 E Exponential and logarithmic equations Review set 31A Review set 31B 25 Direct variation Inverse variation Variation modelling Power modelling Review set 30A Review set 30B A B C D E Number sequences Algebraic rules for sequences Geometric sequences The difference method for sequences Review set 26A Review set 26B 27 CIRCLE GEOMETRY A B C D Table of contents black IGCSE01 ... entered by pressing MATH 4: p ( p we press MATH cyan magenta yellow y:HAESE IGCSE0 1IG01_001 4IGCSE0 1_00.CDR Thursday, October 2008 3:48:09 PM PETER 95 100 50 75 ) 25 81 MATH 95 p x 100 5:... either the authors or the publisher black IGCSE0 1 FOREWORD This book has been written to cover the IGCSE Cambridge International Mathematics (0607) Extended course over a two-year period The... yellow Y:HAESE IGCSE0 1IG01_000 6IGCSE0 1_00.CDR Friday, 21 November 2008 12:06:59 PM PETER 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 95 100 50 75 25 is perpendicular to black IGCSE0 1 Table