Further Mathematics Third Edition

P1: FXS/ABE P2: FXS 9780521740517agg.xml CUAT013-EVANS September 7, 2008 11:27

i ESSENTIAL Further Mathematics Third edition

PETER JONES MICHAEL EVANS KAY LIPSON

TI-Nspire and Casio ClassPad material prepared in collaboration with Russell Brown Kevin McMenamin SAMPLE

Cambridge University Press • Uncorrected Sample pages • 978-0-521-61328-6 • 2008 © Jones, Evans, Lipson TI-Nspire & Casio ClassPad material in collaboration with Brown and McMenamin P1: FXS/ABE P2: FXS 9780521740517agg.xml CUAT013-EVANS September 7, 2008 11:27

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo Cambridge University Press 477 Williamstown Road, Port Melbourne, VIC 3207, Australia www.cambridge.edu.au Information on this title: www.cambridge.edu.au/0521613280

C Peter Jones, Michael Evans & Kay Lipson 2005

First published 1998 Reprinted 1998 Second edition 1999 Reprinted 2000, 2001, 2002, 2003, 2005 Third edition 2005 Reprinted 2006

Cover designed by Modern Art Production Group Text designed by Sylvia Witte Typeset in India by Techbooks Printed in China through Everbest Printing Company Pty Ltd

National Library of Australia Cataloguing in Publication data Jones, Peter, 1943-. Essential further mathematics. 3rd ed. ISBN-13 978-0-521-74051-7 paperback ISBN-10 0-521-61328-0 paperback 1. Mathematics – Problems, exercises, etc. I. Evans, Michael (Michael Wyndham). II. Lipson, Kay. III. Title 510.76

ISBN-13 978-0-521-74051-7 paperback ISBN-10 0-521-61328-0 paperback

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Contents

Acknowledgements xiv

CORE

CHAPTER 1 — Organising and displaying data 1 1.1 Classifying data 1 1.2 Organising and displaying categorical data 3 1.3 Organising and displaying numerical data 8 1.4 What to look for in a histogram 20 1.5 Stem-and-leaf plots and dot plots 26 Key ideas and chapter summary 34 Skills check 35 Multiple-choice questions 35 Extended-response questions 37

CHAPTER 2 — Summarising numerical data: the median, range, IQR and box plots 40 2.1 Will less than the whole picture do? 40 2.2 The median, range and interquartile range (IQR) 41 2.3 The five-number summary and the box plot 45 2.4 Relating a box plot to distribution shape 52 2.5 Interpreting box plots: describing and comparing distributions 54 Key ideas and chapter summary 57 Skills check 58 SAMPLEMultiple-choice questions 59 Extended-response questions 60

iv Contents

CHAPTER 3 — Summarising numerical data: the mean and the standard deviation 63

3.1 The mean 63 3.2 Measuring the spread around the mean: the standard deviation 67 3.3 The normal distribution and the 68–95–99.7% rule: giving meaning to the standard deviation 73 3.4 Standard scores 79 3.5 Populations and samples 83 Key ideas and chapter summary 88 Skills check 90 Multiple-choice questions 91 Extended-response questions 92

CHAPTER 4 — Displaying and describing relationships between two variables 95 4.1 Investigating the relationship between two categorical variables 95 4.2 Using a segmented bar chart to identify relationships in tabulated data 99 4.3 Investigating the relationship between a numerical and a categorical variable 102 4.4 Investigating the relationship between two numerical variables 104 4.5 How to interpret a scatterplot 107 4.6 Calculating Pearson’s correlation coefficient r 112 4.7 The coefficient of determination 118 4.8 Correlation and causality 121 4.9 Which graph? 122 Key ideas and chapter summary 123 Skills check 124 Multiple-choice questions 125 Extended-response questions 128

CHAPTER 5 — Regression: fitting lines to data 131 5.1 Least squares regression line: the theory 131 5.2 Calculating the equation of the least squares regression line 133 SAMPLE5.3 Performing a regression analysis 140 5.4 A graphical approach to regression: the three median line 153 5.5 Extrapolation and interpolation 157

Contents v

Key ideas and chapter summary 159 Skills check 160 Multiple-choice questions 160 Extended-response questions 162

CHAPTER 6 — Data transformation 166

6.1 Data transformation 166 6.2 Transforming the x axis 169 6.3 Transforming the y axis 183 6.4 Choosing and applying the appropriate transformation 189 Key ideas and chapter summary 197 Skills check 197 Multiple-choice questions 197 Extended-response questions 200

CHAPTER 7 — Time series 204 7.1 Time series data 204 7.2 Smoothing a time series plot (moving means) 210 7.3 Smoothing a time series plot (moving medians) 215 7.4 Seasonal indices 220 7.5 Fitting a trend line and forecasting 207 Key ideas and chapter summary 233 Skills check 234 Multiple-choice questions 235 Extended-response questions 237

CHAPTER 8 — Revision of the core 239 8.1 Displaying, summarising and describing univariate data 239 8.2 Displaying, summarising and describing relationships in bivariate data 243 8.3 Regression and data transformation 245 8.4 Time series 249 8.5 Extended-response questions 253

MODULE 1 — Number patterns and SAMPLEapplications

CHAPTER 9 — Arithmetic and geometric sequences 259

vi Contents

9.1 Sequences 259 9.2 Arithmetic sequences 260 9.3 The nth term of an arithmetic sequence and its applications 264 9.4 The sum of an arithmetic sequence and its applications 274 9.5 Geometric sequences 281 9.6 The nth term of a geometric sequence 285 9.7 Applications modelled by geometric sequences 289 9.8 The sum of the terms in a geometric sequence 294 9.9 The sum of an infinite geometric sequence 297 9.10 Rates of growth of arithmetic and geometric sequences 302 Key ideas and chapter summary 307 Skills check 308 Multiple-choice questions 309 Extended-response questions 310

CHAPTER 10 — Difference equations 312 10.1 Introduction 312 10.2 The relationship between arithmetic and geometric sequences and difference equations 320 10.3 First-order difference equations 322 10.4 Solving first-order difference equations that generate arithmetic sequences 324 10.5 Solving difference equations that generate geometric sequences 325 10.6 Solution of general first-order difference equations (optional) 327 10.7 Summary of first-order difference equations 328 10.8 Applications of first-order difference equations 329 10.9 The Fibonacci sequence 338 Key ideas and chapter summary 345 Skills check 346 Multiple-choice questions 346 SAMPLEExtended-response questions 348 CHAPTER 11 — Revision: Number patterns and applications 350

11.1 Multiple-choice questions 350 11.2 Extended-response questions 355 Cambridge University Press • Uncorrected Sample pages • 978-0-521-61328-6 • 2008 © Jones, Evans, Lipson TI-Nspire & Casio ClassPad material in collaboration with Brown and McMenamin P1: FXS/ABE P2: FXS 9780521740517agg.xml CUAT013-EVANS September 7, 2008 11:27

Contents vii

MODULE 2 — Geometry and trigonometry

CHAPTER 12 — Geometry 360

12.1 Properties of parallel lines–areview 360 12.2 Properties of triangles–areview 362 12.3 Properties of regular polygons–areview 364 12.4 Pythagoras’ theorem 367 12.5 Similar figures 371 12.6 Volumes and surface areas 375 12.7 Areas, volumes and similarity 382 Key ideas and chapter summary 387 Skills check 389 Multiple-choice questions 390

CHAPTER 13 — Trigonometry 392 13.1 Defining sine, cosine and tangent 392 13.2 The sine rule 396 13.3 The cosine rule 401 13.4 Area of a triangle 404 Key ideas and chapter summary 406 Skills check 407 Multiple-choice questions 408

CHAPTER 14 — Applications of geometry and trigonometry 410 14.1 Angles of elevation and depression, bearings, and triangulation 410 14.2 Problems in three dimensions 417 14.3 Contour maps 421 Key ideas and chapter summary 424 Skills check 424 Multiple-choice questions 424 Extended-response questions 426

CHAPTER 15 — Revision: Geometry and trigonometry 431 15.1 Multiple-choice questions 431 SAMPLE15.2 Extended-response questions 438

viii Contents

MODULE 3 — Graphs and relations

CHAPTER 16 — Constructing and interpreting linear graphs 441

16.1 The gradient of a straight line 441 16.2 The general equation of a straight line 443 16.3 Finding the equation of a straight line 445 16.4 Equation of a straight line in intercept form 449 16.5 Linear models 450 16.6 Simultaneous equations 452 16.7 Problems involving simultaneous linear equations 456 16.8 Break-even analysis 458 Key ideas and chapter summary 460 Skills check 461 Multiple-choice questions 461

CHAPTER 17 — Graphs 465 17.1 Line segment graphs 465 17.2 Step graphs 468 17.3 Non-linear graphs 470 17.4 Relations of the form y = kxn for n = 1, 2, 3, −1, −2 472 17.5 Linear representation of non-linear relations 475 Key ideas and chapter summary 482 Skills check 483 Multiple-choice questions 483 Extended-response questions 486

CHAPTER 18 — Linear programming 488 18.1 Regions defined by an inequality 488 18.2 Regions defined by two inequalities 490 18.3 Feasible regions 492 18.4 Objective functions 493 Key ideas and chapter summary 503 Skills check 504 Multiple-choice questions 505 Extended-response questions 507 SAMPLECHAPTER 19 — Revision: Graphs and relations 510 19.1 Multiple-choice questions 510 19.2 Extended-response questions 514

Contents ix

MODULE 4 — Business related mathematics

CHAPTER 20 — Principles of financial mathematics 521

20.1 Percentage change 521 20.2 Simple interest 526 20.3 Compound interest 534 20.4 Reducing balance loans 546 Key ideas and chapter summary 548 Skills check 549 Multiple-choice questions 549 Extended-response questions 551

CHAPTER 21 — Applications of financial mathematics 553 21.1 Percentage changes and charges 553 21.2 Bank account balances 558 21.3 Hire purchase 561 21.4 Inflation 567 21.5 Depreciation 571 21.6 Applications of Finance Solvers 581 Key ideas and chapter summary 597 Skills check 599 Multiple-choice questions 600 Extended-response questions 602

CHAPTER 22 — Revision: Business-related mathematics 606 22.1 Multiple-choice questions 606 22.2 Extended-response questions 610

MODULE 5 — Networks and decision mathematics

CHAPTER 23 — Undirected graphs 614 23.1 Introduction and definitions 614 23.2 Planar graphs and Euler’s formula 619 23.3 Complete graphs 622 23.4 Euler and Hamilton paths 623 SAMPLE23.5 Weighted graphs 626 Key ideas and chapter summary 630 Skills check 632 Multiple-choice questions 632 Extended-response questions 636

x Contents

CHAPTER 24 — Directed graphs 639

24.1 Introduction, reachability and dominance 639 24.2 Network flows 645 24.3 The critical path problem 649 24.4 Allocation problems 656 Key ideas and chapter summary 662 Skills check 663 Multiple-choice questions 663 Extended-response questions 667

CHAPTER 25 — Revision: Networks and decision mathematics 671

25.1 Multiple-choice questions 671 25.2 Extended-response questions 677

MODULE 6 — Matrices and applications

CHAPTER 26 — Matrices and applications 1 690 26.1 What is a matrix? 690 26.2 Using matrices to represent information 696 26.3 Matrix arithmetic: addition, subtraction and scalar multiplication 699 26.4 Matrix arithmetic: the product of two matrices 706 Key ideas and chapter summary 715 Skills check 717 Multiple-choice questions 717 Extended-response questions 720

CHAPTER 27 — Matrices and applications II 722 27.1 The inverse matrix 722 27.2 Applications of the inverse matrix: solving simultaneous linear equations 729 27.3 Matrix powers 737 27.4 Transition matrices and their applications 739 Key ideas and chapter summary 749 Skills check 751 Multiple-choice questions 751 SAMPLEExtended-response questions 754

Contents xi

CHAPTER 28 — Revision: Matrices and applications 756

Multiple-choice questions 756 Extended-response questions 760

Appendix TI-nspire 763 Appendix ClassPad 768 Answers 771

SAMPLE

Cambridge University Press • Uncorrected Sample pages • 978-0-521-61328-6 • 2008 © Jones, Evans, Lipson TI-Nspire & Casio ClassPad material in collaboration with Brown and McMenamin Ess Maths IN-BOOK BROCHUR.qxd 11/3/05 7:37 PM Page xii Quark08 Quark08:Books:CUAT006_From Pooja:CUAT013:

CHAPTER The new Essential 1 CORE Organising and series for the 2006 displaying data

What is the difference between categorical and numerical data? What is a frequency table, how is it constructed and when is it used? study design What is the mode and how do we determine its value? What are bar charts, histograms, stem plots and dot plots? How are they constructed and when are they used? How do you describe the features of bar charts, histograms and stem plots when writing a statistical report?

In each chapter you will find … 1.1 Classifying data Statistics is a science concerned with understanding the world through data. The first step in this process is to put the data into a form that makes it easier to see patterns or trends. Some data The data contained in Table 1.1 is part of a larger set of data collected from a group of university students. Chapter9–Arithmetic and geometric sequencesTable 1.1 Student237 data Height Weight Age Sex Plays sport Pulse rate How to use a graphics calculator to generate the terms of an arithmetic sequence(cm) on the (kg) (years) M male 1 regularly (beats/min) Home screen Ffemale 2 sometimes 3rarely , , , , ,... 173 57 18 M 2 86 Generate the first five terms of the arithmetic sequence: 2 7 12 17 22 179 58 19 M 2 82 167 62 18 M 1 96 Steps 195 84 18 F 1 71 1 Start on the Home screen. Clear. Enter the value of the 173 64 18 M 3 90 first term 2. Press Í. 184 74 22 F 3 78 175 60 19 F 3 88 140 50 34 M 3 70

Source: www.statsci.org/data/oz/ms212.html. Used with permission.

2 The common difference for this sequence is 5. So, type 1 in + 5. Press Í. The second term in the sequence, 7, is generated.

a vibrant full colour text with a 3 Pressing Í again generates the next term, 12. clear layout that makes maths

4 Pressing Í again generates the next term, 17. more accessible for students Keep pressing Í until the required number of terms is generated.

Being able to recognise an arithmetic sequence is another skill that you need to develop. The key idea here is that the successive terms in an arithmetic sequence differ by a constant amount ‘Using a graphics calculator’ (the common difference). boxes within chapters explain Example 1 Testing for an arithmetic sequence a Is the sequence 20, 17, 14, 11, 8,...arithmetic? Chapter4—Displaying and describing relationships between two variables 107 how to do problems using the Solution Strategy: Subtract successive terms in the sequence to see whether they differ by a constantClearly, traffic volume is a very good predictor of carbon monoxide levels in the air. Knowing amount. If they do, the sequence is arithmetic. the traffic volume will enable us to predict carbon monoxide levels with a high degree of TI-83/Plus and TI-84 graphics 1 Write down the terms of the sequence. 20, 17, 14, 11, 8,... accuracy. This contrasts with the next example, which concerns the ability to predict 2 Subtract successive terms. 17 − 20 =−3 mathematical ability from verbal ability. 14 − 17 =−3 calculators, and include screen 11 − 14 =−3 and so on Example 3 Calculating and interpreting the coefficient of determination 3 Write down your conclusion Sequence is arithmetic as terms differ by a shots to further assist students constant amount. Scores on tests of verbal and mathematical ability are linearly related with: rmathematical, verbal =+0.275

Determine the value of the coefﬁcient of determination, write it in percentage terms, and interpret. In this relationship, mathematical ability is the DV.

Solution a wealth of worked examples The coefficient of determination is: r 2 = (0.275)2 = 0.0756 ... or 0.076 × 100 = 7.6% that support theory explanations Therefore, only 7.6% of the variation observed in scores on the test of mathematical ability can be explained by the variation in scores obtained on the test of verbal ability. within chapters Clearly, scores on the verbal ability test are not good predictors of the scores on the mathematical ability test; 92.4% of the variation in mathematical ability is explained by other factors. carefully graduated exercises Exercise 4G 1 For each of the following values of r, calculate the value of the coefﬁcient of determination and convert to a percentage (correct to one decimal place). that include a number of easier a r = 0.675 b r = 0.345 c r =−0.567 d r =−0.673 e r = 0.124

2a For the relationship described by the scatterplot Chapter6—Data transformation 173 lead-in questions to provide shown opposite, the coefficient of determination = 0.8215. DetermineReview the value of the correlation coefficient r Key ideas and chapter summary students with a greater (correct to three decimal places). Data transformation This means changing the scale on either the x or y axis. It is performed when a residual plot shows that the underlyingb For the relationship described by the scatterplot shown opportunity for immediate relationship in a set of bivariate data is clearly non-linearopposite,. the coefficient of determination = 0.1243. x2 or y2 transformation The square transformation stretches out the upper endDetermine of the value of the correlation coefficient r the scale on an axis. (correct to three decimal places). success log x or log y transformation The log transformation compresses the upper end of the scale on an axis. 1 1 or transformation The reciprocal transformation compresses the upper end x y of the scale on an axis to a greater extent than the log transformation. chapter summaries at the end of Residual plots Residual plots are used to assess the effectiveness of each data transformation. Coefficient of determination (r2) The transformation which results in a linear relationship each chapter provide students and which has the highest value of the coefficient of determination is considered to be the best transformation. The circle of transformations The circle of transformations provides guidance in with a coherent overview choosing the transformations that can be used to linearise various types of scatterplots. See page 166.

Skills check

Having completed this chapter you should be able to: 1 1 recognise which of the x2,logx, , y2,logy or transformations might be used to chapter reviews that include key x y linearise a bivariate relationship apply each of these transformations to a data set ideas and chapter summary and use residual plots and the coefﬁcient of determination r 2 to decide which transformation gives the best model for the relationship use the transformed variable as part of a regression analysis to give a model for the skills check lists, and multiple- relationship choice and extended-response Multiple-choice questions 1 The missing data values, a and b,inthe table are: value 1 2 3 4 questions (value)2 a 4 9 16 log(value) 0 b 0.477 0.602

New Essential Mathematics Series A a = 0, b = 0.5 B a = 1, b = 0.5 C a = 1, b = 0.301 D a = 1, b = 0.602 E a = 1, b = 0.693 Glossary

Assignment problem: [p. 602] See allocation Appendices that include a TI- A problem. Activity (CPA): [p. 595] A task to be completed as part of a project. Activities are represented by the B edges in the project diagram. 83/84 Plus help guide and step- Bar chart: [p. 4] A statistical graph used to display ◦ Acute angle: An angle less than 90 . the frequency distribution of categorical data. Adjacency matrix: [pp. 561, 586] A square matrix Bearing: [p. 371] See true bearing. showing the number of edges joining each pair of Bipartite graph (bigraph): [p. 562] Agraph whose by-step worked examples using vertices in a graph. set of vertices can be split into two subsets, X and Y, Algorithm: A step-by-step procedure for solving a in such a way that each edge of the graph joins a particular problem that involves applying the same vertex in X and a vertex in Y. process repeatedly. Examples include Prim’s Bivariate data: [p. 85] Data associated with two TI-89 Graphics Calculators algorithm and the Hungarian algorithm. related variables. Allocation problem: [p. 602] A problem that Book value: [p. 526] The value of an item after involves finding the best way to match a given depreciation. number of objects (people, machines, etc.) to a given number of activities. Box plot (standard): [p. 41] Agraphical representation of a five number summary. SAMPLEAlternate angles: [p. 320] Box plot (with outliers): [p. 42] A modified form of Angle of depression: [p. 371] The angle between the the standard box plot in which possible outliers are horizontal and a direction below the horizontal. shown. Possible outliers are defined as data values revision chapters to help Angle of elevation: [p. 371] The angle between the greater than Q3 + 1.5 × IQR and less than horizontal and a direction above the horizontal. Q1 − 1.5 × IQR. Angle sum of a triangle: [p. 322] In triangle Break-even analysis: [p. 418] Finding the point ◦ ABC,

Explaining icons in the book ... Calculator icons Links to Teacher CD-ROM

Gives an extra hint in extended-response Indicates that a skillsheet is available to questions that a numerical approach is provide further practice and examples in this required area. If students are having difficulty they can approach their teacher who can access this material on the Teacher CD-ROM. Indicates that there is an explanation in Appendix B as to how this example may be Links to Student CD-ROM done using TI-89 Graphics Calculators

Live links to interactive files on the Student CD-ROM.

What teachers and students will find on the CD-ROMs ... Teacher CD-ROM Student CD-ROM The Essential Further Mathematics Teacher CD-ROM The textbook includes a Student CD-ROM that contains contains a wealth of time-saving assessment and a PDF of the book, interactive multiple-choice questions classroom resources including: and unique drag-and-drop activities. Technology applets such as PowerPoint and Excel activites are also modifiable chapter tests and answers containing included. multiple-choice and short-answer questions chapter review assignments with extended problems that can be given to students in class or can be completed at home printable versions of the multiple-choice questions from the Student CD-ROM print-ready skillsheets to revise the prerequisite knowledge and skills required for the chapter editable Exam Question Sets from which teachers can create their own exams.

Additional resources ... Solutions Supplements Websites www.essentialmaths.com.au The Essential Further Mathematics Third Edition Teacher Website Solutions Supplement book provides solutions to the This dynamic website enables teachers to interact with extended-response questions, highlighting the process each other through teacher forums, to send questions as well as the answer. to the authors and to obtain updates. SAMPLEStudent Website This free student website contains a student forum allowing keen mathematics students to interact with each other, as well as interactive tests and links to other useful sites.

Acknowledgements

Cambridge University Press and the authors would like to acknowledge all the reviewers who provided invaluable feedback throughout the development of this text. In particular we would like to thank Cathy Ashworth (Sandringham Secondary College), Anthony Gale (Catholic Regional College, Sydenham), Tim Grant (St Bernard’s College, Essendon), David Greenwood (Trinity Grammar School, Kew), Fran Petrie (Melbourne High School), Paul Rice (St Bernard’s College, Essendon), Inna Smith (St Michael’s Grammar School, St Kilda), Kyle Staggard (Bendigo Senior Secondary College), Leah Whifﬁn (Bendigo Senior Secondary College), and Joe Wilson (Mill Park Secondary College). We also acknowledge the work of James Wan and James Hillis who checked all of the answers in this book.

SAMPLE

Cambridge University Press • Uncorrected Sample pages • 978-0-521-61328-6 • 2008 © Jones, Evans, Lipson TI-Nspire & Casio ClassPad material in collaboration with Brown and McMenamin