VCE MATHEMATICS UNITS 1 & 2 MATHS Quest 1 111mathsbooks.net/Maths Quest 11 Methods/By...

VCE MAVCE MATHEMATHEMATICS UNITS 1 & 2TICS UNITS 1 & 2

111111MAMATTHS HS QuestQuestMathematical Methods


111111MAMATTHS HS QuestQuestMathematical Methods

JENNIFER NOLAN GEOFF PHILLIPS ROSS ALLEN DAVID PHILLIPSJENNIFER NOLAN GEOFF PHILLIPS ROSS JENNIFER NOLAN GEOFF PHILLIPS ROSS ALLEN DALLEN DAAVID PHILLIPSVID PHILLIPS

Support materialJohn Dowsey • Dennis Fitzgerald • Emily Hui

Carolyn Mews • Vinod Narayan • David Phillips • Peter Swain David Tynan • Ian Younger • Wayne Youngs

Contributing authorsCaroline Denney • George Dimitriadis

TEACHER EDITION

Prelims Teacher Edition Page iii Monday, July 9, 2001 10:07 AM

http://www.jaconline.com.au

First published 2000 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 2000

National Library of AustraliaCataloguing-in-Publication data

Maths Quest. 11: mathematical methods VCE mathematicsunits 1 and 2.

ISBN 0 7016 3397 2 (Student textbook).ISBN 0 7016 3457 X (Teacher edition).

1. Mathematics. I. Nolan, Jennifer.

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.

Edited by Merv Littmann, Margaret Falk, Joy Windowand Jennifer Wright

Illustrated by the Wiley Art Department

Cover photograph: © PhotoDisc 1996

Printed in Singapore byCraft Print International Ltd

10 9 8 7 6 5 4

Prelims MM1&2 Page iv Friday, December 13, 2002 9:53 AM

Contents

CHAPTER 1

�

Linear functions 1

Solving linear equations 2

Exercise 1A 5

Rearrangement and substitution 6

Exercise 1B 9

Career profile: Rick Morris 12

Gradient of a straight line 13

Exercise 1C 16

Equations of the form

y

=

mx

+

c

21

Exercise 1D 21

Sketching linear graphs using intercepts 24

Exercise 1E 26

Simultaneous equations 27

Exercise 1F 30

Using matrices to solve simultaneous equations 31

Perpendicular lines 32

Exercise 1G 32

Formula for finding the equation of a straight line 33

Exercise 1H 35

Distance between two points 37

Exercise 1I 38

Approximating curve length using linear equations 41

Midpoint of a segment 42

Exercise 1J 43

Linear modelling 44

Exercise 1K 46

Summary 48Chapter review 49

CHAPTER 2

�

Quadratic functions 55

Expanding quadratic expressions 56

Exercise 2A 58

Factorising quadratic expressions 59

Exercise 2B 61

Factorising by completing the square 62

Exercise 2C 64

Solving quadratic equations — Null Factor Law 65

Exercise 2D 67

Fixed point iteration 69

Solving quadratic equations — completing the square 70

Solving x

2

+ bx + c = 0 71

Exercise 2E 72

The quadratic formula 73

Exercise 2F 75

The formula that ‘doesn’t work’! 77

The discriminant 78

Exercise 2G 81

Quadratic graphs — turning point form 81

Quadratic graphs — turning point form 82

Exercise 2H 84

Quadratic graphs — intercepts method 86

Exercise 2I 95

Using graphs to solve quadratic equations 97

Exercise 2J 98

Simultaneous quadratic and linear equations 99

Exercise 2K 103


CHAPTER 3

�

Cubic functions 109

Polynomials 110

History of mathematics: Évariste Galois 111

Exercise 3A 112

Expanding 113

Exercise 3B 113

Long division of cubic polynomials 114

Exercise 3C 118

Polynomial values 119

Exercise 3D 120

The remainder and factor theorems 121

Exercise 3E 123

Factorising cubic polynomials 124

Exercise 3F 127

Sum and difference of two cubes 128

Exercise 3G 129

Cubic equations 130

Exercise 3H 132

Solving cubic equations using graphs 133

Cubic graphs — intercepts method 134

Exercise 3I 138

Repeated factors 139

Cubic graphs — using translation 140

Exercise 3J 143

Domain, range, maximums and minimums 144

Exercise 3K 147

Modelling 149

Prelims MM1&2 Page v Friday, June 29, 2001 11:39 AM

vi

Modelling using technology 149

Exercise 3L 152

Career profile: Ashley Hannon 154Fitting a model exactly 155

Finite differences 156

Exercise 3M 159


CHAPTER 4

�

Exponential and logarithmic functions 167

Introduction 168Index laws 168

Exercise 4A 172

Negative and rational powers 174

Exercise 4B 177

Indicial equations 178

Exercise 4C 181

Graphs of exponential functions 182

Exercise 4D 184

A world population model 185

Logarithms 186

Career profile: Alison Hennessy 186

Exercise 4E 189

Solving logarithmic equations 191

Exercise 4F 193

Logarithmic graphs 194

Applications of exponential and logarithmic functions 195

Exercise 4G 197

The Richter scale 199


CHAPTER 5

�

Circular functions 205

Trigonometric ratio revision 206

Exercise 5A 208

The unit circle 212

Exercise 5B 217

Radians 218

Exercise 5C 222

Symmetry 223

Exercise 5D 227

Career profile: Bronwyn Laycock 228

Identities 229

Exercise 5E 232

Further trigonometric identities 233

Sine and cosine graphs 234

Sine and cosine graphs 235

Exercise 5F 239

Tangent graphs 242

Tangent graphs 242

Exercise 5G 245

Solving trigonometric equations 246

Exercise 5H 250

Applications 251

Exercise 5I 253


CHAPTER 6

�

Relations and functions 263

Set notation 264

Exercise 6A 266

Relations and graphs 267

Exercise 6B 270

Domain and range 273

Exercise 6C 277

Interesting relations 279

Types of relations (including functions) 279

Exercise 6D 282

Function notation 284

Exercise 6E 288

Special types of function 289

Exercise 6F 292

A special relation 295

Circles 296

Exercise 6G 298

Functions and modelling 300

Exercise 6H 301


CHAPTER 7

�

Rates of change 311

Identifying rates 312Exercise 7A 314

Career profile: Sean McInnes 316Constant rates 317

Exercise 7B 319Variable rates 322

Exercise 7C 323Average rates of change 325

Exercise 7D 327Instantaneous rates 330

Exercise 7E 332Motion graphs 335

Exercise 7F 338

viiRelating the gradient function to the original

function 343Exercise 7G 343

Relating velocity–time graphs to position–time graphs 344

Exercise 7H 346Rates of change of polynomials 349

Exercise 7I 352Summary 354Chapter review 356

CHAPTER 8� Differentiation 361Introduction to limits 362

Exercise 8A 365Sneaking up on a limit 366Limits of discontinuous, rational and hybrid

functions 367Exercise 8B 369

Differentiation using first principles 371Secants and tangents 371

Exercise 8C 374Finding derivatives by rule 375

Exercise 8D 378Graphs of derivatives 380Antidifferentiation 381Antidifferentiation by rule 382

Exercise 8E 384Deriving the original function from the

gradient function 385Exercise 8F 387


CHAPTER 9� Applications of differentiation 395Rates of change 396

Exercise 9A 399Sketching graphs containing stationary

points 403Exercise 9B 407

Solving maximum and minimum problems 409

Exercise 9C 412When is a maximum not a maximum? 414Applications of antidifferentiation 415

Exercise 9D 417Summary 419Chapter review 420

CHAPTER 10� Introductory probability 423Introduction to probability 424Random outcome experiments 424Estimated probability and expected number

of outcomes 424Exercise 10A 426

Calculating probabilities 428Exercise 10B 432

Tree diagrams and lattice diagrams 434Exercise 10C 437

The Addition Law of probabilities 440Mutually exclusive events 440

Exercise 10D 443Karnaugh Maps and probability tables 446

Exercise 10E 449Conditional probability 452

Exercise 10F 455Independent events 457

Exercise 10G 461Simulation 464

Exercise 10H 467Summary 469Chapter review 471

CHAPTER 11� Combinatorics 475Introduction 476The addition principle 476Multiplication principle 477

Exercise 11A 479Permutations 481

Exercise 11B 483Identification cards 484Factorials 485Stirling’s formula 486

Exercise 11C 487Permutations using nPr 488

Exercise 11D 491Permutations involving restrictions 493

Exercise 11E 496Arrangements in a circle 498

Exercise 11F 500Combinations using nCr 501

Exercise 11G 504Pascal’s triangle 506Applications to probability 507

Exercise 11H 509Summary 511Chapter review 512

Answers 515

IntroductionMaths Quest 11 Mathematical Methods is one of the exciting new MathsQuest resources specifically designed for the new VCE (2000–2003) Math-ematics course. It breaks new ground in Mathematics textbook publishing.

This resource contains:• a student textbook with accompanying CD-ROM and • a teacher edition with accompanying CD-ROM.

Student textbookFull colour is used throughout to produce clearer graphs and headings, toprovide bright, stimulating photos and to make navigation through the texteasier.

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

Worked examples in a Think/Write format provide clear explanation of keysteps and suggest presentation of solutions.

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.

Career profiles and History of mathematics place mathematical concepts incontext.

Investigations, often suggesting the use of technology, provide further dis-covery learning opportunities.

Each chapter concludes with a summary and chapter review exercise contain-ing examination style questions (multiple choice, short answer and analysis)which help consolidate students’ learning of new concepts.

Technology is fully integrated (in line with VCE 2000 recommendations). Aswell as graphics calculators, Maths Quest 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’ calcu-lators using the Graphlink software provided.

111111MATHS QuestQuest

JENNIFER NOLAN GEOFF PHILLIPS ROSS ALLEN DAVID PHILLIPS JENNIFER NOLAN GEOFF PHILLIPS ROSS ALLEN DAVID PHILLIPS

JAC

ARANDA

MA

THSQUES

T


Mathematical MethodsMathematical Methods

INTE

RACTIVE

C

D- ROM

Cabri

Geometry

EXCEL

Spreadsheet

Mathca

d

Graph

icsCalculator

ixWorkSHEET and Test yourself icons link to editable Word 97 documents, andmay be completed on screen or printed and completed.

SkillSHEET icons link to printable pages designed to help students reviserequired concepts, and contain additional examples and problems.

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

Teacher edition with accompanying CD-ROMThe teacher edition textbook contains everything in the student textbook andmore. To support teachers assisting students in class, answers appear in rednext to most questions in the exercises. Each exercise is annotated withrelevant study design points. A readily accessible Work program lists allavailable resources and provides curriculum coverage information.

The accompanying teacher CD-ROM contains everything in the studentCD-ROM and more. Two tests per chapter, fully worked solutions toWorkSHEETs, the work program and other curriculum advice in editableWord 97 format are provided.

Web site: www.jaconline.com.au/maths

The Maths Quest Web site will provide support in the form of additionaltechnology files, worksheets, links to other sites, assessment materialsincluding practice examinations and more.

Maths Quest is a rich collection of teaching and learning resources withinone package.

Maths Quest 11 Mathematical Methods provides ample material, such asexercises, analysis questions, investigations, worksheets and technology files,from which teachers may set school assessed coursework (SAC).

WorkS

HEET 3.2testtestCH

APTERyyourselfourself

testyyourselfourself

1Ski

llSHEET 16.5


111111MATHS QuestQuest

JENNIFER NOLAN GEOFF PHILLIPS ROSS ALLEN DAVID PHILLIPS JENNIFER NOLAN GEOFF PHILLIPS ROSS ALLEN DAVID PHILLIPS

JAC

ARANDA

MA

THSQUES

T

TEACHERTEACHER EDITION

EDITION

Mathematical MethodsMathematical Methods

INTE

RACTIVE

C

D- ROM

http://www.jaconline.com.au

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

• © PhotoDisc (cover) (pages 39, 55, 68, 76[upper], 98[upper], 103, 109, 110,112, 148[upper], 153, 166, 185, 198[upper], 199, 210[right], 253, 254, 263,264[five], 299, 301, 302[two], 311, 333, 334, 338, 342, 348, 353[two], 397,398, 399, 412, 414, 417, 421, 422[two], 423, 427, 433[two], 451, 463, 473,474, 480, 484, 492[top], 498, 509, 514) • © ANT Photo Library (pages 196/Silvestris, 197/C. and S. Pollitt, 198[lower]/Dave Watts, 204/Silvestris) •© Digital Vision (pages 211[upper], 396[right]) • © Australian Picture Library(pages 11/Pacific Stock, 20[lower]/Larry Muluehill, 47/J. Carnemolla[upper],47/R. Landau[lower], 54/Zefa, 149/Gary Bell, 262) • © Coo-ee PictureLibrary (pages 210[left], 251) • © 1999 Corbis page 76[lower] • © DigitalStock (pages 1, 13[two], 23, 98[lower], 148[lower], 269, 312, 395, 396[left],400, 418, 456, 475, 497) • © Stockbyte page 439 • © The Photo Library —Sydney (pages 20[upper]/David Messent, 40/Phillip Hayson, 46/EdwardCross, 111/Science Photo Library, 167/Eye of Science/SPL, 168/Eye ofScience/SPL, 205, 211[lower]/David Madison, 267/Charlie Dass, 468/Nicholas Pinturas) • Reproduced with permission of Victoria Police (pages361, 362) • Photos courtesy Malcolm Cross (pages 186, 401, 438, 492[middle& lower], 496), Ashley Hannan (page 154), Bronwyn Laycock and CSIRO(page 228), Sean McInnes (page 316) and Rick Morris (page 12)

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/us/product/software/ticonnect/features/features.html

Note

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

Prelims MM1&2 Page x Friday, December 20, 2002 6:37 AM

http://education.ti.com/us/product/software/ticonnect/features/features.html

xi

Mathcad Explorer

Reproduced with permission of Mathsoft www.mathsoft.com

Distributed in Australia by

1. Hearne Scientific Software Pty Ltd

Level 6, 552 Lonsdale Street, Melbourne 3000

e-mail: [email protected]

Web: www.hearne.com.au/

Phone: (03) 9602 5088

2. Geoff Phillips Publications

8 Wattletree Avenue, Wonga Park 3115


Phone: (03) 9722 2505

Fax: (03) 9722 2545

Graphmatica

Reproduced with permission of kSoft, Inc.


Web: www.pair.com/ksoft

345 Montecillo Dr., Walnut Creek, CA 94595-2654.

Software 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

Distributed in Australia by Geoff Phillips Publications (see previous entry).

Cabri Geometry II

Reproduced with permission of Cabri.

Leibniz

Cabri-géomètre

46, avenue Félix Viallet

38031 Grenoble Cedex FRANCE

Web: www.cabri.net

Distributed by AAMT (Australian Association of Mathematics Teachers)

Phone: (08) 8363 0288

Fax: (08) 8362 9288


Web: www.aamt.edu.au

Prelims MM1&2 Page xi Friday, December 13, 2002 9:54 AM

http://www.mathsoft.com

http://www.hearne.com.au/

http://www.pair.com/ksoft

http://www.cabri.net

http://www.pair.com/ksoft/register.txt

http://www.aamt.edu.au

xii

DERIVE™

Screen shots reprinted by permission of Soft Warehouse, Inc.

DERIVE™

isa trademark of Soft Warehouse, Inc worldwide and is a registered trademarkof Soft Warehouse, Inc in the United States of America and various othercountries.

Soft Warehouse, Inc.

3660 Waialae Avenue, Suite 304

Honolulu, Hawaii 96816 USA


Web: www.derive.com

Selected list of

DERIVE™

dealers in Australia:

EdSoft Pty Ltd www.edsoft.com.au

Hearne Scientific Software Pty Ltd www.hearne.com.au

MathStat Software www.mathstat.com.au

GrafEq and Poly

Evaluation copies of GrafEq™ and Poly™ have been included with permis-sion from Pedogoguery Software.


Web: www.peda.com


Equation Grapher and Regression Analyser

Reproduced with permission of MFSoft International.


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 registered trademark of the Microsoft Corporation in theUnited States and/or other countries.

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.

Prelims MM1&2 Page xii Friday, June 29, 2001 11:40 AM

http://www.edsoft.com.au

http://www.hearne.com.au

http://www.mathstat.com.au

http://www.peda.com

http://www.mfsoft.com

http://www.derive.com

VCE MATHEMATICS UNITS 1 & 2 MATHS Quest 1 111mathsbooks.net/Maths Quest 11 Methods/By...

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