**11**

### Ross

**Brodie**

### Stephen

**Swift**

**NewQMaths**

**B**

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00 NQM11B SB TXT.prelim.indd 1

**New QMaths 11B**

**3rd Edition**
**Ross Brodie**

**Stephen Swift**

Publishing editor: Kirstie Pickett Developmental editor: Roger Walter

Editor: Anna Beth McCormack

Senior designer: Sue Dani and Santiago Villamizar

Cover designer: Sue Dani

Cover: Corbis / O. Alamany & E. Vicens

Text designer: Sue Dani Text illustrator: Paulene Meyer

Photo research: Corrina Tauschke Production controller: Jo Vraca

Typeset in 11/14pt Esperanto by Phillip Idczak and Amanda Griffin

Any URLs contained in this publication were checked for currency during the production process. Note, however,

that the publisher cannot vouch for the ongoing currency of URLs.

© 2008 R Brodie, S Swift

ALL RIGHTS RESERVED. No part of this work covered by the

copyright herein may be reproduced, transmitted, stored or used in any form or by any means graphic, electronic, or mechanical,

including but not limited to photocopying, recording, scanning, digitising, taping, Web distribution, information networks, or

information storage and retrieval systems, except as permitted by the Australian Copyright Act 1968 (the Act), without the prior

written permission of the publisher.

For product information and technology assistance,

**in Australia call 1300 790 853;**
**in New Zealand call 0508 635 766**

For permission to use material from this text

or product, please email **aust.permissions@cengage.com**

**National Library of Australia Cataloguing-in-Publication Data**

Ross Brodie, Stephen Swift New QMaths 11B

For secondary school age.

ISBN 9780170136570

Mathematics - Textbooks. 510

**Cengage Learning Australia**

Level 7, 80 Dorcas Street

South Melbourne, Victoria Australia 3205

**Cengage Learning New Zealand**

Unit 4B Rosedale Office Park

331 Rosedale Road, Albany, North Shore 0632, NZ

For learning solutions, visit **cengage.com.au**

Printed in China by C&C Offset Printing Co Ltd.

1 2 3 4 5 6 7 12 11 10 09 08

00 NQM11B SB TXT.prelim.indd 2

**iii**

### Contents

Contents

### Introduction vi

### Symbols and abbreviations

### viii

**1**

### Triangle applications

### 1

1.1 Using right-angled triangles 2

1.2 Applying directions and bearings 9

1.3 The sine rule 12

1.4 The cosine rule 16

1.5 Trig applications 20

1.6 Working in three dimensions 28

Chapter summary 34

Chapter review 35

**2**

### Working with data

### 37

2.1 Collecting statistical data 38

2.2 Organising and displaying information 51 2.3 Finding data positions and centres 63

2.4 Measuring spread 73

2.5 Analysing data 79

Chapter summary 87

Chapter review 89

**3**

### Straight line functions

### 93

3.1 Review of linear equations 94

3.2 Functions and relations 102

3.3 Sequences and progressions 109

3.4 Gradient and intercepts of a linear function 117 3.5 Equations of linear functions 126

3.6 Distances and midpoints 138

3.7 Finding simultaneous solutions of linear functions 143

Chapter summary 149

Chapter review 151

**4**

### Waves and functions

### 155

4.1 Features of periodic functions 156 4.2 Predictions using periodic functions 162 4.3 Models of periodic functions 166

Chapter summary 170

Chapter review 171

00 NQM11B SB TXT.prelim.indd iii

**iv** New QMaths 11B

**5**

### Gradients of curves

### 173

5.1 Finding average rates of change 174 5.2 Finding instantaneous rates of change 188 5.3 Practical applications of rates of change 194 5.4 Mathematical applications of rates of change 199

Chapter summary 208

Chapter review 210

**6**

### Parabolic functions

### 215

6.1 Solving quadratic equations 216 6.2 Algebraic solution of quadratics 223 6.3 Sketching quadratic functions 232 6.4 Finding simultaneous solutions 241

Chapter summary 246

Chapter review 247

**7**

### Differential calculus

### 251

7.1 Finding derivatives as limits 252 7.2 Rules for fi nding derivatives 263 7.3 Practical applications of derivatives 271 7.4 Mathematical applications of derivatives 276

Chapter summary 285

Chapter review 286

**8**

### Applying trigonometric functions

### 289

8.1 Degrees and radians 290

8.2 Trigonometric ratios and the unit circle 295

8.3 Trigonometric graphs 306

8.4 Trigonometric functions and applications 309

Chapter summary 320

Chapter review 322

**9**

### Growth, decay and logarithms

### 325

9.1 Exponential functions 326

9.2 Solving exponential equations 340

9.3 Logarithms 346

9.4 Modelling using exponential functions 350

Chapter summary 354

Chapter review 355

00 NQM11B SB TXT.prelim.indd iv

### Chapter Review

**v**
Contents

**10**

### Working with probability

### 359

10.1 Review of probability 360

10.2 Finding probabilities 367

10.3 Using binomial probability 374

Chapter summary 381

Chapter review 382

**11**

### Reversing derivatives

### 385

11.1 Finding indefi nite integrals 386 11.2 Graphing indefi nite integrals 390 11.3 Applying indefi nite integrals 393

Chapter summary 396

Chapter review 397

### Appendix

### 398

### Answers

### 402

### Index 438

00 NQM11B SB TXT.prelim.indd v

**vi** New QMaths 11B

### Introduction

The *New QMaths B* textbooks are designed to assist students to build competence in the advanced
computational and algebraic methods and procedures they need for further study and lifelong learning, to
increase mathematical modelling and problem-solving skills, and to provide a basis for sound mathematical
arguments and decisions. Students are encouraged to further develop key competencies in contexts that
arise naturally from the general objectives and learning experiences of the subject.

*New QMaths 11B* provides ample opportunity for students to develop skills and demonstrate performance
in the dimensions of communication and justifi cation, knowledge and procedures, use of technology, use of
mathematical instruments and modelling and problem solving. It is a gender-inclusive and culturally inclusive
text that accommodates a variety of student learning styles through a modern approach that will engage
students and provide impetus for the affective objectives of the Mathematics B syllabus. The Mathematics B
texts are sequenced to allow for the systematic development of mathematical concepts through the course
and are designed to complement the sequencing in the *New QMaths C* textbooks for those students studying
both subjects.

The* New QMaths B* textbooks provide considerable support for the use of appropriate technologies and
application of concepts in a variety of situations from life-related to purely abstract tasks. Students are
provided with opportunities to demonstrate initiative and there is variation from simple, single-step tasks to
ones that are complex in nature.

### What you will fi nd in

*New QMaths 11B*

### and the CD-ROM

The textbook includes:

■ modern full-colour layout to help you to distinguish between examples, exercises, defi nitions and investigations

■ clear defi nitions and explanations of important terms and methods

■ artwork, photographs and maps in colour for maximum clarity and appeal

■ new content and exercises to improve development of concepts and skills

■ extensive explanations to help you to get the most from graphics calculators

■ detailed instructions in the text for the Casio and Texas Instruments graphics calculators and on the CD-ROM for the Sharp graphics calculator; the authors have assumed that calculators are in default settings—different results may be obtained if this is not the case

■ explanations of the use of spreadsheets and other technology

■ modelling and problem-solving practice within exercises to develop concepts beyond basic knowledge and procedures

■ a chapter summary and chapter review at the end of each chapter to assist in revision for end-of-term and end-of-semester assessment; the review questions are labelled to indicate the exercises where concepts were introduced

■ a clear linkage from the text to the syllabus at the beginning of each chapter.

00 NQM11B SB TXT.prelim.indd vi

**vii**
Introduction

### The CD-ROM includes:

The entire student text with hot-link icons from the text to the additional features shown below.

■ Additional exercises (with answers) for every exercise to give opportunities for extra practice of concepts

■ Graphics calculator programs in downloadable form

■ Graphics calculator instructions for the Sharp graphics calculator

■ Extra material as background or extension for interested students and their teachers

■ Alternative solutions to some examples

■ Detailed proofs of important theorems

■ Spreadsheets for investigations and exercises, including interactive spreadsheets

■ A complete syllabus guide for Years 11 and 12 with appropriate sections linked to chapter openings

■ Teacher notes and copy masters

■ Hot-links from individual review questions back to the exercise where the concept was introduced

■ An extensive glossary explaining important terms with examples

Additional Exercise

Calculator Program

Extra Material

Spreadsheet

Syllabus Guide

**Chapter 3.0**

**Ex 3.11**
Teacher Notes

Calculator Instructions

Alternative Method

Detailed Proof

00 NQM11B SB TXT.prelim.indd vii

**viii** New QMaths 11B

### Symbols and abbreviations

While every effort has been made to ensure that calculator instructions are accurate, results may vary when graphics calculators are not set in default mode or as a result of manufacturers’ changes.

= is equal to

≠ is not equal to

≈ is approximately equal to

√ square root % percentage

⬍ is less than

⬎ is greater than

⭐ is less than or equal to

⭓ is greater than or equal to

⫾ plus or minus

兺 summation

*Q*3 3rd quartile

*P*87 87th percentile

*D*4 4th decile

1 : 650 000 at a scale of 1 to 650 000

π pi (approximately 3.141 592 7) 52°15′42′′ 52 degrees 15 minutes 42 seconds

∠ angle

䉭 triangle

*x* mean of *x* values

σ standard deviation 5 : 8 ratio of 5 to 8 / per

157°M magnetic bearing of 157° 123°T true bearing of 123° LHS left-hand side RHS right-hand side

僆 is a member of

### Acknowledgements

We would like to thank the following for permission to reproduce copyright material:

AAP Image, pp. 79, 165, 174, 373, 395; Auscape/ Nicholas Birks, p. 75/ Doug Lee-Peter Arnold, p. 27; Corbis, p. 22, 67, 109, 188, 273; Dale Mann/Retrospect, p. 353, 363; Fotolia, pp. 20, 22, 26, 71, 101, 199, 231, 241, 319, 325, 338; Getty Images, pp. 273, 274, 318, 333, 393; iStockphoto, pp. 1, 21, 30, 33, 37, 39, 51, 54, 61, 83, 93, 137, 155, 173, 215, 251, 289, 352, 359, 385; Jupiter Images Corporation, pp. 276; Newspix / News Ltd / 3rd Party Managed Reproduction & Supply Rights, pp. 47, 97, 195, 365; Photolibrary, pp. 14, 16, 62, 91, 148, 157, 169, 179, 329, 337, 339, 350, 366, 372, 379; Photolibrary/Alamy, p. 64; © Sport the Library/Tom Putt, p. 164; Steve Lovegrove, p. 362.

Every effort has been made to trace and acknowledge copyright. However, if any infringement has occurred the publishers tender their apologies and invite the copyright holders to contact them.

00 NQM11B SB TXT.prelim.indd viii