Maths Quest 11 Mathematical Methods CAS Prelims

VCE mATHEmATiCS UNiTS 1 & 2

m ATHS QUEST QUE ST 11 matheatcal methods metho ds CAS C AS Ti-NSPiRE 2.0 EDiTiON

N O i T i D E D N 2

VCE mATHEmATiCS UNiTS 1 & 2

mATHS QUEST 11 matheatcal methods CAS Ti-NSPiRE 2.0 EDiTiON ROBYN WiLLiAmS

NiCOLAOS KARANiKOLAS

GAYLE ROBERTS

JENNiFER NOLAN

KYLiE BOUCHER

GEOFF PHiLLiPS

CONTRiBUTiNG AUTHORS ROGER WANDER HOWARD LiSTON

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DOUGLAS SCOTT

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mARK BARNES

ANDREW mENTLiKOWSKi

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SUE STOCKDALE

SUPPORT mATERiAL JOHN DOWSEY

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DENNiS FiTZGERALD

PETER SWAiN

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EmiLY HUi

DAViD TYNAN

SimONE RiCHARDSON

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CAROLiNE mEWS

iAN YOUNGER

DiNA ANTONiOU

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ViNOD NARAYAN

WAYNE YOUNGS

NORRENE HiLL

N O i T i D E D N 2

Second revised edition published 2010 by John Wiley & Sons Australia, Ltd 42 McDougall Street, Milton, Qld 4064 First edition published 2009 Typeset in 10/12pt Times LT © John Wiley & Sons Australia, Ltd 2009, 2010 The moral rights of the authors have been asserted. National Library of Australia Cataloguing-in-Publication data

Title:

Edition: ISBN:

Notes: Target Audience: Subjects: Other Authors/ Contributors: Dewey Number:

Maths quest 11 mathematical methods CAS TI-Nspire 2.0 edition / Robyn Williams . . . [et al.]. 2nd ed. 978 1 7424 6447 3 (student pbk.) 978 1 7424 6448 0 (student ebook) 978 1 7424 6450 3 (teacher pbk.) 978 1 7424 6451 0 (teacher ebook) Includes index. For secondary school age. Mathematics — Textbooks. Mathematics — Problems exercises, etc. Williams, Robyn (Robyn Ellen) 1967– 510

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Screenshots from TI-Nspire reproduced with permission of Texas Instruments Illustrated by Aptara Typeset in India by Aptara Printed in Singapore by Craft Print International Ltd 10 9 8 7 6 5 4 3 2

Contents 2H

Introduction ix About eBookPLUS xi Acknowledgements xii

2i

CHAPTER 1

Lnear functons 1a

1b

1C

1d

1e

1F

1G

1H

2J

1

Solving linear equations and inequations 1 Exercise 1A 5 Rearrangement and substitution 6 Exercise 1B 9 Gradient o a straight line 11 Exercise 1C 13 Sketching linear unctions 16 Exercise 1D 19 Simultaneous equations 21 Exercise 1E 24 Finding the equation o a straight line 25 Exercise 1F 27 Distance between two points and midpoint o a segment 29 Exercise 1G 30 Linear modelling 32 Exercise 1H 34

Summary 36 Chapter review 37 eBookPLUS activities

42

Quadratc functons

2b

2C

2d

2e

2F

2G

2l

Summary 102 Chapter review 104 eBookPLUS activities 108 CHAPTER 3

Cubc and quartc functons 3a

3b

3C

3d

CHAPTER 2

2a

2k

3e

43

Polynomials 43 Exercise 2A 45 Expanding quadratic expressions 46 Exercise 2B 48 Factorising quadratic expressions 49 Exercise 2C 52 Factorising by completing the square 53 Exercise 2D 56 Solving quadratic equations — Null Factor Law 56 Exercise 2E 59 Solving quadratic equations — completing the square 61 Exercise 2F 64 The quadratic ormula 65 Exercise 2G 68

The discriminant 70 Exercise 2H 74 Graphs o quadratic unctions as power unctions (turning point orm) 75 Exercise 2I 78 Graphs o quadratic unctions (intercepts method) 80 Exercise 2J 89 Using technology to solve quadratic equations 91 Exercise 2K 94 Simultaneous quadratic and linear equations 95 Exercise 2L 100

3F

3G

3H

3i

3J

3k

3l

109

Expanding 109 Exercise 3A 111 Long division o polynomials 111 Exercise 3B 114 Polynomial values 115 Exercise 3C 117 The remainder and actor theorems 118 Exercise 3D 120 Factorising polynomials 121 Exercise 3E 123 Sum and dierence o two cubes 124 Exercise 3F 125 Solving polynomial equations 126 Exercise 3G 129 Cubic graphs — intercepts method 129 Exercise 3H 134 Quartic graphs — intercepts method 136 Exercise 3I 138 Graphs o cubic unctions in power unction orm 139 Exercise 3J 142 Domain, range, maximums and minimums 143 Exercise 3K 146 Modelling using technology 148 Exercise 3L 150

3m

Finite dierences 152 Exercise 3M 157

Summary 159 Chapter review 162 eBookPLUS activities 167 ExAm PRACTiCE 1

5e

5F

5G

5H

Based on Chapters 1–3 168 CHAPTER 4

Relatons, functons and transforatons 170 4a

4b

4C

4d

4e

4F

4G

4H

4i

4J

Set notation 170 Exercise 4A 172 Relations and graphs 173 Exercise 4B 175 Domain and range 177 Exercise 4C 181 Types o relations (including unctions) 183 Exercise 4D 185 Power unctions (hyperbola, truncus and square root unction) 187 Exercise 4E 195 Function notation 195 Exercise 4F 199 Special types o unction (including hybrid unctions) 200 Exercise 4G 204 Inverse relations and unctions 206 Exercise 4H 207 Circles 208 Exercise 4I 210 Functions and modelling 212 Exercise 4J 213

Summary 216 Chapter review 218 eBookPLUS activities 225

Logarithms 244 Exercise 5E 247 Solving logarithmic equations 248 Exercise 5F 250 Logarithmic graphs 251 Exercise 5G 252 Applications o exponential and logarithmic unctions 253 Exercise 5H 255


Crcular functons 6a

6b

6C

6d

6e

6F

6G

6H

6i

267

Trigonometric ratio revision 267 Exercise 6A 269 The unit circle 271 Exercise 6B 275 Radians 276 Exercise 6C 279 Symmetry 280 Exercise 6D 284 Identities 285 Exercise 6E 288 Sine and cosine graphs 289 Exercise 6F 295 Tangent graphs 297 Exercise 6G 300 Solving trigonometric equations 301 Exercise 6H 308 Applications 309 Exercise 6I 311


CHAPTER 5

Eponental and logarthc functons 226 5a

5b

5C

5d

vi

Introduction 226 Index laws 227 Exercise 5A 231 Negative and rational powers 232 Exercise 5B 235 Indicial equations 235 Exercise 5C 238 Graphs o exponential unctions 238 Exercise 5D 243

Contents


matrces 7a

7b

7C

7d

326

Addition and subtraction o matrices Exercise 7A 334 Multiplying matrices 335 Exercise 7B 337 Solving matrix equations 339 Exercise 7C 344 Matrices and transormations 345 Exercise 7D 348

326



Based on Chapters 1–7 358

Applcatons of dfferentaton 10a

CHAPTER 8

Rates of change 8a

8b

8C

8d

8e

8F

8G

8H

8i

360

Identiying rates 360 Exercise 8A 362 Constant rates 363 Exercise 8B 365 Variable rates 368 Exercise 8C 369 Average rates o change 370 Exercise 8D 373 Instantaneous rates 375 Exercise 8E 377 Motion graphs (kinematics) 379 Exercise 8F 382 Relating the gradient unction to the original unction 387 Exercise 8G 388 Relating velocity–time graphs to position–time graphs 389 Exercise 8H 391 Rates o change o polynomials 394 Exercise 8I 399


9a

9b

9C

9d

9e

9F

10C

10d

410

Introduction to limits 410 Exercise 9A 413 Limits o discontinuous, rational and hybrid unctions 415 Exercise 9B 418 Dierentiation using frst principles 420 Exercise 9C 423 Finding derivatives by rule 423 Exercise 9D 430 Antidierentiation 432 Exercise 9E 435 Deriving the original unction rom the gradient unction 437 Exercise 9F 439

Rates o change 449 Exercise 10A 453 Sketching graphs containing stationary points 455 Exercise 10B 460 Solving maximum and minimum problems 462 Exercise 10C 465 Applications o antidierentiation 467 Exercise 10D 471



introductory probablty 11a

11b

11C

11d

CHAPTER 9

Dfferentaton

10b

449

11e

11F

11G

11H

11i

482

Introduction to experimental probability 482 Exercise 11A 484 Calculating probabilities 486 Exercise 11B 490 Tree diagrams and lattice diagrams 491 Exercise 11C 494 The Addition Law o Probabilities 496 Exercise 11D 500 Karnaugh maps and probability tables 502 Exercise 11E 505 Conditional probability 508 Exercise 11F 511 Transition matrices and Markov chains 513 Exercise 11G 518 Independent events 520 Exercise 11H 524 Simulation 526 Exercise 11I 528


Contents

vii

CHAPTER 12

Cobnatorcs 12a

12b

12C

12d

12e

12F

viii

12G

540

The addition and multiplication principles 540 Exercise 12A 543 Permutations 544 Exercise 12B 546 Factorials 548 Exercise 12C 549 Permutations using P 550 Exercise 12D 553 Permutations involving restrictions Exercise 12E 558 Arrangements in a circle 559 Exercise 12F 561

Contents

12H

Combinations using C 561 Exercise 12G 564 Applications to probability 566 Exercise 12H 569 n

r


Based on Chapters 1–12 578

n

r

555

Answers 580 Index 638

introducton Maths Quest 11 Mathematical Methods CAS Second edition is specifcally designed or the VCE Mathematical Methods CAS course and based on the award-winning Maths Quest series. This resource contains: • a student textbook with accompanying eBookPLUS and • a teacher edition with accompanying eGuidePLUS.

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A selection o questions are tagged as technology-ree to indicate to students that they should avoid using their calculators or other technologies to assist them in fnding a solution. Exam practice sections contain exam style questions, including time and mark allocations or each question. Fully worked solutions are available on the eBookPLUS or students. Each chapter concludes with a summary and chapter review exercise containing examinationstyle questions (multiple choice, short answer and extended response), which help consolidate students’ learning o new concepts. Also included are questions rom past VCE exams along with relevant Exam tips. Technology is ully integrated (in line with VCE recommendations).

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introducton

ix

WorkSHEET icons link to editable Word documents, and may be completed on screen or printed and completed by hand. SkillSHEET icons link to printable pages designed to help students revise required concepts, and contain additional examples and problems. Interactivity icons link to dynamic animations which help students to understand difcult concepts. eLesson icons link to videos or animations designed to elucidate concepts in ways other than what the teacher can achieve in the classroom. Tutorial icons link to one-way engagement activities which explain the worked examples in detail or students to view at home or in the classroom. Test yoursel tests are also available and answers are provided or students to receive instant eedback.

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introducton

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Acknowledgeents The authors and publisher would like to thank the ollowing copyright holders, organisations and individuals or their assistance and or permission to reproduce copyright material in this book.

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xii

Acknowledgeents

Maths Quest 11 Mathematical Methods CAS Prelims

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