MATHSCMATHSC1111YEAR

FOR QUEENSLAND

MATHS QuestMATHS QuestQuest

MQ Maths C Yr 11 - Prelims Page i Wednesday, October 10, 2001 7:16 AM

MQ Maths C Yr 11 - Prelims Page ii Wednesday, October 10, 2001 7:16 AM

MATHSCMATHSC1111YEAR

SUE CAMPBELL NICK SIMPSON

FOR QUEENSLAND

MATHS QuestMATHS QuestQuest

MQ Maths C Yr 11 - Prelims Page iii Wednesday, October 10, 2001 7:16 AM

First published 2001 byJohn Wiley & Sons Australia, Ltd33 Park Road, Milton, Qld 4064

Offices also in Sydney and Melbourne

Typeset in 10.5/12.5pt Times

© John Wiley & Sons Australia, Ltd 2001

National Library of AustraliaCataloguing-in-Publication data

Campbell, Sue, 1951–Maths Quest: Maths C, Year 11 for Queensland.

Includes index.ISBN 0 7016 3623 8.

1. Mathematics. 2. Mathematics — Problems, exercises, etc.I. Simpson, N. P. (Nicholas Patrick), 1957–. II. Title.(Series: Maths Quest series).

510

All rights reserved. No part of this publication may bereproduced, stored in a retrieval system, or transmittedin any form or by any means, electronic, mechanical,photocopying, recording, or otherwise, without the priorpermission of the publisher.

Illustrated by Paul Lennon, Andy Craig, Nives Porcellato and the Wiley Art Department

Cover photograph, internal design images and CD-ROM label: © 2001 Digital Vision

Printed in Singapore byCraft Print International Ltd

10 9 8 7 6 5 4 3 2

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Contents

Introduction ixAcknowledgements xi

CHAPTER 1

�

Number systems: the Real Number System 1

Introduction 2The Real Number System 2Classification of numbers: rational and

irrational 3

Exercise 1A 7

Summary of set notation 8Recurring decimals 8

Exercise 1B 12

Real and complex numbers 13

Investigation — Real number investigations 14Investigation — Other number systems 15

Exercise 1C 17

Surds: a subset of irrational numbers 18

Exercise 1D 21

Simplifying surds 22

Exercise 1E 24

Addition and subtraction of surds 25

Exercise 1F 28

Multiplication of surds 30

Exercise 1G 32

Division of surds 34

Exercise 1H 36

The Distributive Law 38

Exercise 1I 41

Rationalising denominators 43

Exercise 1J 45

Rationalising denominators using conjugate surds 46

Exercise 1K 51

Further properties of real numbers — modulus 53

Exercise 1L 54

Solving equations using absolute values 55

Exercise 1M 57

Solving inequations 58

Exercise 1N 65

Investigation — Approximations for

p

66Investigation — Real numbers — application and modelling 67

Summary 69Chapter review 72

CHAPTER 2

�

Number systems: complex numbers 77

Introduction to complex numbers 78

Exercise 2A 81

Investigation — Complex numbers in quadratic equations 82

Basic operations using complex numbers 82

Investigation — Plotting complex numbers 86

Exercise 2B 88

Conjugates and division of complex numbers 89

Exercise 2C 93

Radians and coterminal angles 95

Exercise 2D 95

Complex numbers in polar form 96

History of mathematics — Abraham de Moivre 103

Exercise 2E 104

Basic operations on complex numbers in polar form 106

Investigation — Multiplication in polar form 106

Exercise 2F 113

Graphics calculator notes for complex numbers 114

Investigation — Complex numbers: applications 117History of mathematics — William Rowan Hamilton 118

Summary 119Chapter review 120

CHAPTER 3

�

Matrices 123

Introduction to matrices 124Operations with matrices 126

Exercise 3A 131

Multiplying matrices 133

Exercise 3B 136

History of mathematics — Olga Taussky-Todd 138

Powers of a matrix 139

Investigation — Matrix powers 139

Exercise 3C 140

Investigation — Applications of matrices 141

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vi

Multiplicative inverse and solving matrix equations 142

Exercise 3D 147

The transpose of a matrix 149

Exercise 3E 150

Applications of matrices 150

Exercise 3F 154

Investigation — Matrix multiplication using a graphics calculator 156

Dominance matrices 157

Investigation — Dominance matrices — another application of matrices 157

Exercise 3G 160

Summary 161Chapter review 163

CHAPTER 4

�

An introduction to groups 165

Groups 166

Investigation — Algebraic structures 166

Exercise 4A 168

The terminology of groups 168

History of mathematics — Niels Henrik Abel 171

Exercise 4B 171

Properties of groups 172

Exercise 4C 176

Investigation — Applications of groups — permutations 177

Further examples of groups — transformations 177

History of mathematics — Arthur Cayley 180

Exercise 4D 181

Investigation — Some applications of group theory 182

Summary 185Chapter review 186

CHAPTER 5

�

Matrices and their applications 187

Inverse matrices and systems of linear equations 188

Exercise 5A 190

Solving simultaneous equations — the Gaussian elimination method 191

History of mathematics — Carl Friedrich Gauss 197

Exercise 5B 198

Introducing determinants 198

Exercise 5C 201

Properties of determinants 201

Exercise 5D 204

Inverse of a 3

×

3 matrix 205

Exercise 5E 209

Cramer’s Rule for solving linear equations 210

Exercise 5F 213

Investigation — Solving simultaneous equations 215Investigation — Applications of determinants 216

Summary 217Chapter review 218

CHAPTER 6

�

Transformations using matrices 219

Geometric transformations and matrix algebra 220

Exercise 6A 226

Linear transformations 227

Exercise 6B 230

Linear transformations and group theory 231

Exercise 6C 237

Rotations 237

Exercise 6D 243

Reflections 244

Exercise 6E 251

Dilations 252

History of mathematics — M. C. Escher 258

Exercise 6F 259

Shears 259

Exercise 6G 263

Investigation — Transformations 263

Summary 264Chapter review 265

CHAPTER 7

�

Introduction to vectors 267

Vectors and scalars 268

Exercise 7A 273

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vii

Position vectors in two and three dimensions 278

Exercise 7B 285

Multiplying two vectors — the dot product 290

Exercise 7C 293

History of mathematics — Charles Lutwidge Dodgson 296

Resolving vectors — scalar and vector resolutes 297

Exercise 7D 300

Time-varying vectors 301

Exercise 7E 305

Summary 307Chapter review 309

CHAPTER 8

�

Vector applications 313

Vectors and their applications 314Force diagrams and the triangle

of forces 314

Exercise 8A 321

History of mathematics — Sir Isaac Newton 323

Newton’s First Law of Motion 324

Exercise 8B 331

Momentum 334

Exercise 8C 338

Investigation — Collision momentum 340

Relative velocity 341

Exercise 8D 343

Using vectors in geometry 344

Investigation — Three-dimensional non-zero vectors 346Investigation — Vector geometry 347

Exercise 8E 347

Summary 349Chapter review 350

CHAPTER 9

�

Sequences and series 353

Introduction 354Recognising arithmetic sequences 354

Exercise 9A 357

Finding the terms of an arithmetic sequence 359

Exercise 9B 362

The sum of a given number of terms of an arithmetic sequence 364

Exercise 9C 368

Recognising geometric sequences 370Exercise 9D 374

Finding the terms of a geometric sequence 377

Exercise 9E 381The sum of a given number of terms of a

geometric sequence 384Exercise 9F 387

Applications of geometric sequences 389Compound interest 391

Exercise 9G 394Finding the sum of an infinite geometric

sequence 397Exercise 9H 401

Contrasting arithmetic and geometric sequences through graphs 402

Exercise 9I 407Investigation — Reward time 409Investigation — Changing shape 410The Mandelbrot Set 411Investigation — Draw the Mandelbrot Set 413Fibonacci Sequence 413Investigation — Fibonacci numbers 415Summary 416Chapter review 419

CHAPTER 10� Permutations and combinations 425Introduction 426The addition and multiplication

principles 426Exercise 10A 431

Factorials and permutations 433Exercise 10B 438

Arrangements involving restrictions and like objects 440

Exercise 10C 444Combinations 446

Exercise 10D 452Applications of permutations and

combinations 454Exercise 10E 459

Pascal’s triangle, the binomial theorem and the pigeonhole principle 461

Investigation — Counting paths 462Exercise 10F 461

History of mathematics — Blaise Pascal 470Summary 471Chapter review 473

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CHAPTER 11� Dynamics 477Introduction 478Differentiation and displacement, velocity and

acceleration 479Investigation — Rectilinear motion 479

Exercise 11A 486Interpreting graphs 488

Exercise 11B 493Investigation — Curve fitting using a graphics calculator 495

Algebraic links between displacement, velocity and acceleration 497

Exercise 11C 499Motion under constant acceleration 499

Exercise 11D 503Summary 506Chapter review 507

Answers 511

Index 543

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IntroductionMaths Quest Maths C Year 11 for Queensland is one of the exciting newMaths Quest resources specifically designed for the Queensland seniormathematics syllabuses beginning in 2002. Written and compiled bypractising Maths C teachers, Maths Quest breaks new ground in mathematicstextbook publishing.

This resource contains:• a student textbook • a student CD-ROM.

Student textbookFull colour is used throughout to produce clearer graphs and headings, toprovide bright, stimulating photographs and to make navigation through thetext easier.

Clear, concise theory sections contain worked examples, graphics calculatortips and highlighted important text and remember boxes.

Worked examples in a Think/Write format provide a clear explanation of keysteps and suggest how solutions can be presented.

Exercises contain many carefully graded skills and application problems,including multiple-choice questions. Cross-references to relevant workedexamples appear beside the first ‘matching’ question throughout theexercises.

Investigations, often suggesting the use of technology, provide furtherlearning opportunities.

Each chapter concludes with a summary and chapter review exercise contain-ing questions that help consolidate students’ learning of new concepts.

Technology is fully integrated within the resource. As well as use of graphicscalculators, the Maths Quest for Queensland series features computer algebrasystems, spreadsheets, dynamic geometry software and several graphingpackages. Not only does the text promote these technologies as learningtools, but demonstration versions of the programs (with the exception ofMicrosoft Excel) are also included, as well as hundreds of supporting files onthe free accompanying CD-ROM.

Student CD-ROMThe accompanying CD-ROM contains the entire student textbook plus addi-tional exercises. Students may work from the CD on laptops, school or homecomputers, and cut and paste material for revision or assignments.

Clearly labelled icons within the electronic version of the text hyperlink tohundreds of technology files for programs such as Mathcad, Excel and CabriGeometry to allow further exploration of ‘what if’ scenarios.

TI-83 Graphics Calculator programs can be downloaded to students’calculators using the Graphlink software provided.

F O R Q U E E N S L A N DF O R Q U E E N S L A N D

MATHS QuestMATHS Quest 1111YEAR

MATHSCMATHSC

SUE CAMPBELL

NICK SIMPSON

INTE

RACTIVE

C

D- ROM

EXCELSpreadsheet

Mathca

d

Graphic

s Calculator

Cabri

Geometry

MQ Maths C Yr 11 - Prelims Page ix Wednesday, October 10, 2001 7:16 AM

xWorkSHEET and Test yourself icons link to editable Word 97 documents thatmay be completed on screen, or printed and completed later.

SkillSHEET icons link to printable pages that contain additional examplesand problems designed to help students revise required concepts.

Minimum system requirementsWindows 95/98 or NT MacintoshProcessor: Pentium Macintosh OS 7.6CD-ROM drive speed: 4x Processor: PowerPC16 MB RAM CD-ROM drive speed: 4xspeakers 16 MB RAM

Programs included

Mathcad Explorer: a computer algebra system and graphing program

Graphmatica: an excellent graphing utility

Equation grapher and regression analyser: like a graphics calculator forthe PC

GrafEq: graphs any relation, including complicated inequalities

Poly: for visualising 3D polyhedra and their nets

TI Graphlink 83 and 89: calculator screen capture and program transfer

Cabri Geometry II: dynamic geometry program

Adobe® Acrobat® Reader 4.0

Trouble-shooting

If you have problems with the operation of this CD-ROM:• Check that you have the right equipment (see Minimum system

requirements).

• Visit www.jaconline.com.au to check if the answer to your problem isprovided under ‘Frequently Asked Questions’ in the ‘Contact Us’ section.

• Either email or write to John Wiley & Sons Australia, Ltd explaining theproblem, and providing details of the type of computer and the amount ofRAM you have, the processor type and the CD-ROM speed. If you returnthe disk, please package it appropriately to protect it during transit.

Email: multimedia@johnwiley.com.au

Address: Multimedia AssistantJohn Wiley & Sons Australia, LtdPO Box 1226MILTON QLD 4064

WorkS

HEET 3.2testtestCH

APTERyyourselfourself

testyyourselfourself

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llSHEET 8.5

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Acknowledgements

The authors and publisher would like to thank the following copyright holders,organisations and individuals for their assistance and for permission toreproduce copyright material in this book.

Illustrative material

• Australian Picture Library/Corbis (pp. 118, 171, 180, 296) • © 2001 Corbis(pp. 286, 352, 414 [right], 459 [upper], 509) • © 2001 Digital Stock/Corbis (pp. 78, 123, 137, 188, 342, 395, 475, 478 [middle], 479, 508)• © 2001 Digital Vision (cover, internal design images and CD-ROM label;pp. 177, 184, 421 [lower], 422, 446) • Getty Images/Stone (p. 445/GregPease) • http://alepho.clark.edu~djoyce/explorer.html (pp. 91, 353, 411) • © Image Disk Photography (p. 313) • © John Wiley & Sons Australia(pp. 328, 421 [upper]) • © 2001 PhotoAlto (p. 426) • © 2001 PhotoDisc, Inc.(pp. 1, 2, 15, 77, 126, 141 [two], 142, 155, 157, 160, 164, 166, 187, 220,227, 237, 244, 252, 267, 268, 275, 288, 289, 300, 311, 314, 315 [two], 325,326, 332, 339, 358, 362, 363, 364, 380, 383, 414 [left], 423, 425, 429, 432,453, 459 [lower], 460, 462, 467, 473, 476, 477, 478 [upper and lower], 499,503, 505, 510) • © 2001 Photo-Essentials (p. 219) • Photolibrary.com(pp. 197/Science Source/PRI, 323/Science Photo Library, 470/Science PhotoLibrary) • © 2001 Stockbyte (pp. 165, 376 [eight]).

Software

The authors and publisher would like to thank the following software pro-viders for their assistance and for permission to use their materials. However,the use of such material does not imply that the providers endorse this productin any way.

Third party software — registered full version ordering information

Full versions of third party software may be obtained by contacting thecompanies listed below.

Texas Instruments 83 and 89 Graphlink software

Material reproduced with permission of the publisher. © Texas InstrumentsIncorporated.

TI 83 and 89 Graphlink Software available from Texas Instruments

Web: http://education.ti.com

Note

: A Graphlink cable can be purchased from educational booksellers orcalculator suppliers.

MQ Maths C Yr 11 - Prelims Page xi Wednesday, November 5, 2003 9:57 AM

xiiMathcad Explorer*

Reproduced with permission of Mathsoft www.mathsoft.comDistributed in Australia by Hearne Scientific Software Pty LtdLevel 6, 552 Lonsdale Street, Melbourne 3000e-mail: info@hearne.com.au Web: www.hearne.com.auPhone: (03) 9602 5088

Graphmatica*

Reproduced with permission of kSoft, Inc.345 Montecillo Dr., Walnut Creek, CA 94595–2654e-mail: ksoft@pair.com Web: www.pair.com/ksoftSoftware included is for evaluation purposes only. The user is expected toregister shareware if use exceeds 30 days. Order forms are available atwww.pair.com/ksoft/register.txt

Cabri Geometry II

Reproduced with permission of Cabri.LeibnizCabri-géomètre46, avenue Félix Viallet38031 Grenoble Cedex FRANCE Web: www.cabri.netDistributed by AAMT (Australian Association of Mathematics Teachers)Phone: (08) 8363 0288Fax: (08) 8362 9288e-mail: aamtinc@nexus.edu.auWeb: www.aamt.edu.au

GrafEq and Poly*

Evaluation copies of GrafEq™ and Poly™ have been included with per-mission from Pedogoguery Software.e-mail: peda@peda.com Web: www.peda.com

Equation Grapher with Regression Analyser*

Reproduced with permission of MFSoft International.e-mail: info@mfsoft.com Web: www.mfsoft.com

Microsoft® Excel

Screen Shots reproduced by permission of Microsoft Corporation. Note: Microsoft Software was used only in Screen Dumps.Microsoft Excel is a registered trademark of the Microsoft Corporation in theUnited States and/or other countries.

MQ Maths C Yr 11 - Prelims Page xii Wednesday, October 10, 2001 7:16 AM

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Every effort has been made to trace the ownership of copyright material.Information that will enable the publisher to trace the copyright holders or torectify any error or omission in subsequent reprints will be welcome. In suchcases, please contact the Permission Section of John Wiley & Sons Australia,who will arrange for the payment of the usual fee.

*GrafEq, Poly, Graphmatica, Equation Grapher with Regression Analyserand Mathcad Explorer student edition are now available in Australia from:

Geoff Phillips Publications8 Wattletree Avenue, Wonga Park, Victoria 3115Phone: (03) 9722 2505Fax: (03) 9722 2545e-mail: gphillips@bigpond.com

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