Further SAMPLEKevin McMenamin Mathematics · 2008. 9. 9. · TI-Nspire & Casio ClassPad material in collaboration with Brown and McMenamin. P1: FXS/ABE P2: FXS 9780521740517agg.xml

P1: FXS/ABE P2: FXS9780521740517agg.xml CUAT013-EVANS September 7, 2008 11:27

i

ESSENTIALFurther

MathematicsThird edition

PETER JONESMICHAEL EVANS

KAY LIPSON

TI-Nspire and Casio ClassPad materialprepared in collaboration with

Russell BrownKevin McMenamin

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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


C A M B R I D G E U N I V E R S I T Y P R E S S

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press477 Williamstown Road, Port Melbourne, VIC 3207, Australia

www.cambridge.edu.auInformation on this title: www.cambridge.edu.au/0521613280

C© Peter Jones, Michael Evans & Kay Lipson 2005

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

Cover designed by Modern Art Production GroupText designed by Sylvia WitteTypeset in India by TechbooksPrinted in China through Everbest Printing Company Pty Ltd

National Library of Australia Cataloguing in Publication dataJones, Peter, 1943-.Essential further mathematics.3rd ed.ISBN-13 978-0-521-74051-7 paperbackISBN-10 0-521-61328-0 paperback1. Mathematics – Problems, exercises, etc. I. Evans,Michael (Michael Wyndham). II. Lipson, Kay. III. Title

510.76

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

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All Victorian Curriculum and Assessment Authority material copyright VCAA. Reproduced by permission ofthe Victorian Curriculum and Assessment Authority Victoria, Australia.Disclaimer: This publication is independently produced for use by teachers and students. Although referenceshave been reproduced with permission of the VCAA the publication is in no way connected with or endorsedby the VCAA.

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Contents

Acknowledgements xiv

CORE

CHAPTER 1 — Organising and displaying data 1

1.1 Classifying data 11.2 Organising and displaying categorical data 31.3 Organising and displaying numerical data 81.4 What to look for in a histogram 201.5 Stem-and-leaf plots and dot plots 26

Key ideas and chapter summary 34Skills check 35Multiple-choice questions 35Extended-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? 402.2 The median, range and interquartile range

(IQR) 412.3 The five-number summary and the box plot 452.4 Relating a box plot to distribution shape 522.5 Interpreting box plots: describing and

comparing distributions 54Key ideas and chapter summary 57Skills check 58Multiple-choice questions 59Extended-response questions 60

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iv Contents

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

3.1 The mean 633.2 Measuring the spread around the mean:

the standard deviation 673.3 The normal distribution and the 68–95–99.7% rule:

giving meaning to the standard deviation 733.4 Standard scores 793.5 Populations and samples 83


CHAPTER 4 — Displaying and describing relationshipsbetween two variables 95

4.1 Investigating the relationship between twocategorical variables 95

4.2 Using a segmented bar chart to identifyrelationships in tabulated data 99

4.3 Investigating the relationship between a numericaland a categorical variable 102

4.4 Investigating the relationship between twonumerical variables 104

4.5 How to interpret a scatterplot 1074.6 Calculating Pearson’s correlation

coefficient r 1124.7 The coefficient of determination 1184.8 Correlation and causality 1214.9 Which graph? 122


CHAPTER 5 — Regression: fitting lines to data 131

5.1 Least squares regression line: the theory 1315.2 Calculating the equation of the least squares

regression line 1335.3 Performing a regression analysis 1405.4 A graphical approach to regression: the three

median line 1535.5 Extrapolation and interpolation 157

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Contents v


CHAPTER 6 — Data transformation 166

6.1 Data transformation 1666.2 Transforming the x axis 1696.3 Transforming the y axis 1836.4 Choosing and applying the appropriate

transformation 189Key ideas and chapter summary 197Skills check 197Multiple-choice questions 197Extended-response questions 200

CHAPTER 7 — Time series 204

7.1 Time series data 2047.2 Smoothing a time series plot (moving

means) 2107.3 Smoothing a time series plot (moving

medians) 2157.4 Seasonal indices 2207.5 Fitting a trend line and forecasting 207


CHAPTER 8 — Revision of the core 239

8.1 Displaying, summarising and describingunivariate data 239

8.2 Displaying, summarising and describingrelationships in bivariate data 243

8.3 Regression and data transformation 2458.4 Time series 2498.5 Extended-response questions 253

MODULE 1 — Number patterns andapplications

CHAPTER 9 — Arithmetic and geometricsequences 259

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vi Contents

9.1 Sequences 2599.2 Arithmetic sequences 2609.3 The nth term of an arithmetic sequence

and its applications 2649.4 The sum of an arithmetic sequence

and its applications 2749.5 Geometric sequences 2819.6 The nth term of a geometric

sequence 2859.7 Applications modelled by geometric

sequences 2899.8 The sum of the terms in a geometric

sequence 2949.9 The sum of an infinite geometric sequence 2979.10 Rates of growth of arithmetic and geometric

sequences 302Key ideas and chapter summary 307Skills check 308Multiple-choice questions 309Extended-response questions 310

CHAPTER 10 — Difference equations 312

10.1 Introduction 31210.2 The relationship between arithmetic and geometric

sequences and difference equations 32010.3 First-order difference equations 32210.4 Solving first-order difference equations that

generate arithmetic sequences 32410.5 Solving difference equations that generate

geometric sequences 32510.6 Solution of general first-order difference equations

(optional) 32710.7 Summary of first-order difference equations 32810.8 Applications of first-order difference

equations 32910.9 The Fibonacci sequence 338


CHAPTER 11 — Revision: Number patterns andapplications 350

11.1 Multiple-choice questions 35011.2 Extended-response questions 355

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Contents vii

MODULE 2 — Geometry and trigonometry

CHAPTER 12 — Geometry 360

12.1 Properties of parallel lines – a review 36012.2 Properties of triangles – a review 36212.3 Properties of regular polygons – a review 36412.4 Pythagoras’ theorem 36712.5 Similar figures 37112.6 Volumes and surface areas 37512.7 Areas, volumes and similarity 382

Key ideas and chapter summary 387Skills check 389Multiple-choice questions 390

CHAPTER 13 — Trigonometry 392

13.1 Defining sine, cosine and tangent 39213.2 The sine rule 39613.3 The cosine rule 40113.4 Area of a triangle 404


CHAPTER 14 — Applications of geometry andtrigonometry 410

14.1 Angles of elevation and depression, bearings, andtriangulation 410

14.2 Problems in three dimensions 41714.3 Contour maps 421


CHAPTER 15 — Revision: Geometry andtrigonometry 431


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viii Contents

MODULE 3 — Graphs and relations

CHAPTER 16 — Constructing and interpreting lineargraphs 441

16.1 The gradient of a straight line 44116.2 The general equation of a straight line 44316.3 Finding the equation of a straight line 44516.4 Equation of a straight line in intercept form 44916.5 Linear models 45016.6 Simultaneous equations 45216.7 Problems involving simultaneous linear

equations 45616.8 Break-even analysis 458


CHAPTER 17 — Graphs 465

17.1 Line segment graphs 46517.2 Step graphs 46817.3 Non-linear graphs 47017.4 Relations of the form y = kxn for

n = 1, 2, 3, −1, −2 47217.5 Linear representation of non-linear relations 475


CHAPTER 18 — Linear programming 488

18.1 Regions defined by an inequality 48818.2 Regions defined by two inequalities 49018.3 Feasible regions 49218.4 Objective functions 493


CHAPTER 19 — Revision: Graphs and relations 510


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Contents ix

MODULE 4 — Business related mathematics

CHAPTER 20 — Principles of financial mathematics 521

20.1 Percentage change 52120.2 Simple interest 52620.3 Compound interest 53420.4 Reducing balance loans 546


CHAPTER 21 — Applications of financialmathematics 553

21.1 Percentage changes and charges 55321.2 Bank account balances 55821.3 Hire purchase 56121.4 Inflation 56721.5 Depreciation 57121.6 Applications of Finance Solvers 581


CHAPTER 22 — Revision: Business-relatedmathematics 606


MODULE 5 — Networks and decisionmathematics

CHAPTER 23 — Undirected graphs 614

23.1 Introduction and definitions 61423.2 Planar graphs and Euler’s formula 61923.3 Complete graphs 62223.4 Euler and Hamilton paths 62323.5 Weighted graphs 626


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x Contents

CHAPTER 24 — Directed graphs 639

24.1 Introduction, reachability and dominance 63924.2 Network flows 64524.3 The critical path problem 64924.4 Allocation problems 656


CHAPTER 25 — Revision: Networks and decisionmathematics 671


MODULE 6 — Matrices and applications

CHAPTER 26 — Matrices and applications 1 690

26.1 What is a matrix? 69026.2 Using matrices to represent information 69626.3 Matrix arithmetic: addition, subtraction and scalar

multiplication 69926.4 Matrix arithmetic: the product of two

matrices 706Key ideas and chapter summary 715Skills check 717Multiple-choice questions 717Extended-response questions 720

CHAPTER 27 — Matrices and applications II 722

27.1 The inverse matrix 72227.2 Applications of the inverse matrix:

solving simultaneous linear equations 72927.3 Matrix powers 73727.4 Transition matrices and their applications 739


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Contents xi

CHAPTER 28 — Revision: Matrices andapplications 756

Multiple-choice questions 756Extended-response questions 760

Appendix TI-nspire 763

Appendix ClassPad 768

Answers 771

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In each chapter you will find …

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study design

C H A P T E R

1CORE

Organising anddisplaying data

What is the difference between categorical and numerical data?

What is a frequency table, how is it constructed and when is it used?

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?

1.1 Classifying dataStatistics 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 dataThe data contained in Table 1.1 is part of a larger set of data collected from a group of

university students.

Table 1.1 Student data

Height Weight Age Sex Plays sport Pulse rate(cm) (kg) (years) M male 1 regularly (beats/min)

F female 2 sometimes3 rarely

173 57 18 M 2 86179 58 19 M 2 82167 62 18 M 1 96195 84 18 F 1 71173 64 18 M 3 90184 74 22 F 3 78175 60 19 F 3 88140 50 34 M 3 70

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

1

Chapter 9 – Arithmetic and geometric sequences 237

How to use a graphics calculator to generate the terms of an arithmetic sequence on theHome screen

Generate the first five terms of the arithmetic sequence: 2, 7, 12, 17, 22, . . .

Steps1 Start on the Home screen. Clear. Enter the value of the

first term 2. Press Í.

2 The common difference for this sequence is 5. So, type

in + 5. Press Í. The second term in the sequence,

7, is generated.

3 Pressing Í again generates the next term, 12.

4 Pressing Í again generates the next term, 17.

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

(the common difference).

Example 1 Testing for an arithmetic sequence

a Is the sequence 20, 17, 14, 11, 8, . . . arithmetic?

Solution

Strategy: Subtract successive terms in the sequence to see whether they differ by a constant

amount. If they do, the sequence is arithmetic.

1 Write down the terms of the sequence.

2 Subtract successive terms.

20, 17, 14, 11, 8, . . .

17− 20= −314− 17 = −311− 14= −3 and so on

3 Write down your conclusion Sequence is arithmetic as terms differ by aconstant amount.

Chapter 4 — Displaying and describing relationships between two variables 107

Clearly, traffic volume is a very good predictor of carbon monoxide levels in the air. Knowing

the traffic volume will enable us to predict carbon monoxide levels with a high degree of

accuracy. This contrasts with the next example, which concerns the ability to predict

mathematical ability from verbal ability.

Example 3 Calculating and interpreting the coefficient of determination

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

The coefficient of determination is:

r 2 = (0.275)2 = 0.0756 . . . or 0.076× 100= 7.6%

Therefore, only 7.6% of the variation observed in scores on the test of mathematical ability canbe explained by the variation in scores obtained on the test of verbal ability.

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.

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).

a r = 0.675 b r = 0.345 c r = −0.567 d r = −0.673 e r = 0.124

2 a For the relationship described by the scatterplot

shown opposite, the coefficient of determination = 0.8215.

Determine the value of the correlation coefficient r

(correct to three decimal places).

b For the relationship described by the scatterplot shown

opposite, the coefficient of determination = 0.1243.

Determine the value of the correlation coefficient r

(correct to three decimal places).

a vibrant full colour text with aclear layout that makes mathsmore accessible for students

‘Using a graphics calculator’boxes within chapters explainhow to do problems using the TI-83/Plus and TI-84 graphicscalculators, and include screenshots to further assist students

a wealth of worked examplesthat support theory explanationswithin chapters

carefully graduated exercisesthat include a number of easierlead-in questions to providestudents with a greateropportunity for immediatesuccess

chapter reviews that include keyideas and chapter summary andskills check lists, and multiple-choice and extended-responsequestions

chapter summaries at the end ofeach chapter provide studentswith a coherent overview

Appendices that include a TI-83/84 Plus help guide and step-by-step worked examples usingTI-89 Graphics Calculators

a comprehensive glossary ofmathematical terms with pagereferences to assist in the ‘openbook’ exam

revision chapters to helpconsolidate student knowledge

Review

Chapter 6 — Data transformation 173

Key ideas and chapter summary

Data transformation This means changing the scale on either the x or y axis. It is

performed when a residual plot shows that the underlying

relationship in a set of bivariate data is clearly non-linear.

x2 or y2 transformation The square transformation stretches out the upper end of

the scale on an axis.

log x or log y transformation The log transformation compresses the upper end of the

scale on an axis.1

xor

1

ytransformation The reciprocal transformation compresses the upper end

of the scale on an axis to a greater extent than the log

transformation.

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

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

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:

recognise which of the x2, log x,1

x, y2, log y or

1

ytransformations might be used to

linearise a bivariate relationship

apply each of these transformations to a data set

use residual plots and the coefficient of determination r2 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

relationship

Multiple-choice questions

1 The missing data values, a and b, in the table are:

value 1 2 3 4

(value)2 a 4 9 16

log(value) 0 b 0.477 0.602

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

AActivity (CPA): [p. 595] A task to be completed aspart of a project. Activities are represented by theedges in the project diagram.

Acute angle: An angle less than 90◦.

Adjacency matrix: [pp. 561, 586] A square matrixshowing the number of edges joining each pair ofvertices in a graph.

Algorithm: A step-by-step procedure for solving aparticular problem that involves applying the sameprocess repeatedly. Examples include Prim’salgorithm and the Hungarian algorithm.

Allocation problem: [p. 602] A problem thatinvolves finding the best way to match a givennumber of objects (people, machines, etc.) to a givennumber of activities.

Alternate angles: [p. 320]

Angle of depression: [p. 371] The angle between thehorizontal and a direction below the horizontal.

Angle of elevation: [p. 371] The angle between thehorizontal and a direction above the horizontal.

Angle sum of a triangle: [p. 322] In triangleABC, <A + <B + <C = 180◦.

Annuity equation: [pp. 531, 534] The equation:

A = PRn − Q(Rn − 1)

R − 1, where R = 1 + r

100.

The annuity equation can be used to determineeither the amount owed on a reducing balance loanwith regular repayments, or the value of aninvestment (annuity) with regular payments orwithdrawals.

Area of a triangle: [p. 364] See also Heron’sformula.

Arithmetic sequence: [p. 236] A sequence whosesuccessive terms differ by a constant amount (d)called the common difference. Given the value of thefirst term in an arithmetic sequence (a), there arerules for finding the nth term and the sum of the firstn terms.

Assignment problem: [p. 602] See allocationproblem.

BBar chart: [p. 4] A statistical graph used to displaythe frequency distribution of categorical data.

Bearing: [p. 371] See true bearing.

Bipartite graph (bigraph): [p. 562] A graph whoseset of vertices can be split into two subsets, X and Y,in such a way that each edge of the graph joins avertex in X and a vertex in Y.

Bivariate data: [p. 85] Data associated with tworelated variables.

Book value: [p. 526] The value of an item afterdepreciation.

Box plot (standard): [p. 41] A graphicalrepresentation of a five number summary.

Box plot (with outliers): [p. 42] A modified form ofthe standard box plot in which possible outliers areshown. Possible outliers are defined as data valuesgreater than Q3 + 1.5 × IQR and less thanQ1 − 1.5 × IQR.

Break-even analysis: [p. 418] Finding the pointwhere the revenue of a business first equals the costsof running the business. Past this point, the businessis running at a profit: profit = revenue − costs.

CCapacity of a cut: [p. 592] The sum of thecapacities (weights) of the edges directed from X to Ythat the cut passes through.

Categorical data: [p. 2] Data obtained whenclassifying or naming some quality or attribute, forexample: place of birth, hair colour.

Causal relationship: [p. 108] When the change inone variable in a relationship can be said to be thedirect result of a change in a second variable, therelationship is said to be causal. A high correlationbetween two variables does not necessarily mean thatthe variables are causally related.

705

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Gives an extra hint in extended-responsequestions that a numerical approach isrequired

Indicates that there is an explanation inAppendix B as to how this example may bedone using TI-89 Graphics Calculators

Explaining icons in the book ...

What teachers and students will find on the CD-ROMs ...

Calculator icons

Indicates that a skillsheet is available toprovide further practice and examples in thisarea. If students are having difficulty they canapproach their teacher who can access thismaterial on the Teacher CD-ROM.

Links to Teacher CD-ROM

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

Links to Student CD-ROM

The Essential Further Mathematics Teacher CD-ROMcontains a wealth of time-saving assessment andclassroom resources including:

modifiable chapter tests and answers containingmultiple-choice and short-answer questionschapter review assignments with extendedproblems that can be given to students in class orcan be completed at homeprintable versions of the multiple-choice questionsfrom the Student CD-ROMprint-ready skillsheets to revise the prerequisiteknowledge and skills required for the chaptereditable Exam Question Sets from which teacherscan create their own exams.

Teacher CD-ROM

The textbook includes a Student CD-ROM that containsa PDF of the book, interactive multiple-choice questionsand unique drag-and-drop activities. Technologyapplets such as PowerPoint and Excel activites are alsoincluded.

Student CD-ROM

Additional resources ...

The Essential Further Mathematics Third EditionSolutions Supplement book provides solutions to theextended-response questions, highlighting the processas well as the answer.

Solutions Supplements

Teacher WebsiteThis dynamic website enables teachers to interact witheach other through teacher forums, to send questionsto the authors and to obtain updates.

Student WebsiteThis free student website contains a student forumallowing keen mathematics students to interact witheach other, as well as interactive tests and links toother useful sites.

Websites www.essentialmaths.com.au

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AcknowledgementsCambridge 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.

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Documents

Further SAMPLEKevin McMenamin Mathematics · 2008. 9. 9. · TI-Nspire & Casio ClassPad material in collaboration with Brown and McMenamin. P1: FXS/ABE P2: FXS 9780521740517agg.xml