M
ATHS
C
M
ATHS
C
11
11
YEAR
FOR QUEENSLAND
MATHS
Quest
MATHS
Quest
Quest
M
ATHS
C
M
ATHS
C
11
11
YEAR
SUE CAMPBELL NICK SIMPSON
FOR QUEENSLAND
MATHS
Quest
MATHS
Quest
Quest
First published 2001 by John Wiley & Sons Australia, Ltd 33 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 Australia Cataloguing-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 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.
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 by Craft Print International Ltd
10 9 8 7 6 5 4 3 2
Contents
Introduction ix Acknowledgements xi
CHAPTER 1
Number systems: the Real
Number System 1
Introduction 2
The Real Number System 2
Classification of numbers: rational and irrational 3
Exercise 1A 7
Summary of set notation 8 Recurring decimals 8
Exercise 1B 12
Real and complex numbers 13
Investigation — Real number investigations 14
Investigation — 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 66 Investigation — Real numbers — application and modelling 67
Summary 69 Chapter 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 117
History of mathematics — William Rowan Hamilton 118
Summary 119 Chapter review 120
CHAPTER 3
Matrices 123
Introduction to matrices 124 Operations 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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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 161 Chapter 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 185 Chapter 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 215
Investigation — Applications of determinants 216
Summary 217 Chapter 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 264 Chapter review 265
CHAPTER 7
Introduction to vectors 267
Vectors and scalars 268
Exercise 7A 273
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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 307 Chapter review 309
CHAPTER 8
Vector applications 313
Vectors and their applications 314 Force 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 346
Investigation — Vector geometry 347
Exercise 8E 347
Summary 349 Chapter review 350
CHAPTER 9
Sequences and series 353
Introduction 354
Recognising 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 370
Exercise 9D 374
Finding the terms of a geometric sequence 377
Exercise 9E 381
The sum of a given number of terms of a geometric sequence 384
Exercise 9F 387
Applications of geometric sequences 389 Compound interest 391
Exercise 9G 394
Finding the sum of an infinite geometric sequence 397
Exercise 9H 401
Contrasting arithmetic and geometric sequences through graphs 402
Exercise 9I 407
Investigation — Reward time 409 Investigation — Changing shape 410
The Mandelbrot Set 411
Investigation — Draw the Mandelbrot Set 413
Fibonacci Sequence 413
Investigation — Fibonacci numbers 415
Summary 416 Chapter review 419
CHAPTER 10
Permutations and
combinations 425
Introduction 426
The addition and multiplication principles 426
Exercise 10A 431
Factorials and permutations 433
Exercise 10B 438
Arrangements involving restrictions and like objects 440
Exercise 10C 444
Combinations 446
Exercise 10D 452
Applications of permutations and combinations 454
Exercise 10E 459
Pascal’s triangle, the binomial theorem and the pigeonhole principle 461
Investigation — Counting paths 462
Exercise 10F 461
History of mathematics — Blaise Pascal 470
Summary 471 Chapter review 473
CHAPTER 11
Dynamics 477
Introduction 478
Differentiation and displacement, velocity and acceleration 479
Investigation — Rectilinear motion 479
Exercise 11A 486
Interpreting graphs 488
Exercise 11B 493
Investigation — Curve fitting using a graphics calculator 495
Algebraic links between displacement, velocity and acceleration 497
Exercise 11C 499
Motion under constant acceleration 499
Exercise 11D 503
Summary 506 Chapter review 507
Answers 511
Index 543
Introduction
Maths Quest Maths C Year 11 for Queensland is one of the exciting new
Maths Quest resources specifically designed for the Queensland senior mathematics syllabuses beginning in 2002. Written and compiled by
practising Maths C teachers, Maths Quest breaks new ground in mathematics
textbook publishing.
This resource contains: • a student textbook • a student CD-ROM.
Student textbook
Full colour is used throughout to produce clearer graphs and headings, to provide bright, stimulating photographs and to make navigation through the text easier.
Clear, concise theory sections contain worked examples, graphics calculator
tips and highlighted important text and remember boxes.
Worked examples in a Think/Write format provide a clear explanation of key steps and suggest how solutions can be presented.
Exercises contain many carefully graded skills and application problems, including multiple-choice questions. Cross-references to relevant worked examples appear beside the first ‘matching’ question throughout the exercises.
Investigations, often suggesting the use of technology, provide further learning opportunities.
Each chapter concludes with asummary and chapter review exercise
contain-ing questions that help consolidate students’ learncontain-ing of new concepts.
Technology is fully integrated within the resource. As well as use of graphics
calculators, the Maths Quest for Queensland seriesfeatures computer algebra
systems, spreadsheets, dynamic geometry software and several graphing packages. Not only does the text promote these technologies as learning tools, but demonstration versions of the programs (with the exception of Microsoft Excel) are also included, as well as hundreds of supporting files on
the free accompanying CD-ROM.
Student CD-ROM
The accompanying CD-ROM contains the entire student textbook plus addi-tional exercises. Students may work from the CD on laptops, school or home computers, and cut and paste material for revision or assignments.
Clearly labelled icons within the electronic version of the text hyperlink to
hundreds of technology files for programs such as Mathcad, Excel and Cabri Geometry 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 RQ U E E N S L A N D
F O RQ U E E N
S L A N D MATHS
Quest
MATHS
Quest YEAR
11
11
MATHS
C
M
ATHS
C
SUE CAMPBELL NICK SIMPSON
IN
TERACTIVE
C
D- ROM
EXC
EL Spreadsheet
Mathc ad
Grap hicsCa
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Cabr
i Geometry
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WorkSHEET and Test yourself icons link to editable Word 97 documents that may be completed on screen, or printed and completed later.
SkillSHEET icons link to printable pages that contain additional examples and problems designed to help students revise required concepts.
Minimum system requirements
Windows 95/98 or NT Macintosh
Processor: Pentium Macintosh OS 7.6
CD-ROM drive speed: 4x Processor: PowerPC
16 MB RAM CD-ROM drive speed: 4x
speakers 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 for the 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 is
provided under ‘Frequently Asked Questions’ in the ‘Contact Us’ section.
• Either email or write to John Wiley & Sons Australia, Ltd explaining the problem, and providing details of the type of computer and the amount of RAM you have, the processor type and the CD-ROM speed. If you return the disk, please package it appropriately to protect it during transit.
Email: [email protected]
Address: Multimedia Assistant
John Wiley & Sons Australia, Ltd PO Box 1226
MILTON QLD 4064
Work
SHEET
3.2
test
test
CHAPTER
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yourselfourself
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SkillS
HEET
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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 to reproduce 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/Greg Pease) • 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 Photo Library) • © 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 product in any way.
Third party software — registered full version ordering information
Full versions of third party software may be obtained by contacting the companies listed below.
Texas Instruments 83 and 89 Graphlink software
Material reproduced with permission of the publisher. © Texas Instruments Incorporated.
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 or calculator suppliers.
xii
Mathcad Explorer*
Reproduced with permission of Mathsoft www.mathsoft.com
Distributed in Australia by 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
Graphmatica*
Reproduced with permission of kSoft, Inc.
345 Montecillo Dr., Walnut Creek, CA 94595–2654
e-mail: [email protected]
Web: www.pair.com/ksoft
Software included is for evaluation purposes only. The user is expected to register shareware if use exceeds 30 days. Order forms are available at www.pair.com/ksoft/register.txt
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
e-mail: [email protected]
Web: www.aamt.edu.au
GrafEq and Poly*
Evaluation copies of GrafEq™ and Poly™ have been included with per-mission from Pedogoguery Software.
e-mail: [email protected]
Web: www.peda.com
Equation Grapher with Regression Analyser*
Reproduced with permission of MFSoft International.
e-mail: [email protected]
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 the United States and/or other countries.
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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 to rectify any error or omission in subsequent reprints will be welcome. In such cases, 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 Analyser and Mathcad Explorer student edition are now available in Australia from:
Geoff Phillips Publications
8 Wattletree Avenue, Wonga Park, Victoria 3115 Phone: (03) 9722 2505
Fax: (03) 9722 2545