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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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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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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
Key ideas and chapter summary 88Skills check 90Multiple-choice questions 91Extended-response questions 92
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
Key ideas and chapter summary 123Skills check 124Multiple-choice questions 125Extended-response questions 128
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
Key ideas and chapter summary 159Skills check 160Multiple-choice questions 160Extended-response questions 162
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
Key ideas and chapter summary 233Skills check 234Multiple-choice questions 235Extended-response questions 237
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
Key ideas and chapter summary 345Skills check 346Multiple-choice questions 346Extended-response questions 348
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
Key ideas and chapter summary 406Skills check 407Multiple-choice questions 408
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
Key ideas and chapter summary 424Skills check 424Multiple-choice questions 424Extended-response questions 426
CHAPTER 15 — Revision: Geometry andtrigonometry 431
15.1 Multiple-choice questions 43115.2 Extended-response questions 438
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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
Key ideas and chapter summary 460Skills check 461Multiple-choice questions 461
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
Key ideas and chapter summary 482Skills check 483Multiple-choice questions 483Extended-response questions 486
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
Key ideas and chapter summary 503Skills check 504Multiple-choice questions 505Extended-response questions 507
CHAPTER 19 — Revision: Graphs and relations 510
19.1 Multiple-choice questions 51019.2 Extended-response questions 514
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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
Key ideas and chapter summary 548Skills check 549Multiple-choice questions 549Extended-response questions 551
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
Key ideas and chapter summary 597Skills check 599Multiple-choice questions 600Extended-response questions 602
CHAPTER 22 — Revision: Business-relatedmathematics 606
22.1 Multiple-choice questions 60622.2 Extended-response questions 610
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
Key ideas and chapter summary 630Skills check 632Multiple-choice questions 632Extended-response questions 636
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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
Key ideas and chapter summary 662Skills check 663Multiple-choice questions 663Extended-response questions 667
CHAPTER 25 — Revision: Networks and decisionmathematics 671
25.1 Multiple-choice questions 67125.2 Extended-response questions 677
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
Key ideas and chapter summary 749Skills check 751Multiple-choice questions 751Extended-response questions 754
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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 …
.
New
Ess
enti
al
Ma
them
ati
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erie
sThe new Essential series for the 2006
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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