Signpost 10

Sydney, Melbourne, Brisbane, Perth andassociated companies around the world

Alan McSevenyRob ConwaySteve Wilkes

5.1_5.3_Ch00.indd 15.1_5.3_Ch00.indd 1 12/7/05 9:50:47 AM12/7/05 9:50:47 AM

Understanding is a fountain of life to those who have it.

Proverbs 16:22

Pearson Education AustraliaA division of Pearson Australia Group Pty LtdLevel 9, 5 Queens RoadMelbourne 3004 Australiawww.pearsoned.com.au/schools

Offices in Sydney, Brisbane and Perth, and associated companies throughout the world.

Copyright Pearson Education Australia 2005First published 2005

All rights reserved. Except under the conditions described in the Copyright Act 1968 of Australia and subsequent amendments, 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 copyright owner.

Text designed by Pierluigi VidoCover designed by Bob MitchellCover image by Australian Picture LibraryCartoons by Michael BarterTechnical illustrations by Wendy Gorton and Margaret HastieEdited by Janet MauTypeset by Sun Photoset Pty Ltd, BrisbaneSet in Berkeley and Scala SansProduced by Pearson Education AustraliaPrepress work by The Type FactoryPrinted in China (GCC/01).

National Library of AustraliaCataloguing-in-Publication data

McSeveny, A. (Alan).New signpost mathematics 10: stage 5.15.3.

Includes index.For secondary school students.ISBN 0 7339 3678 4.

1. Mathematics - Textbooks. I. Conway, R. (Robert).II. Wilkes, S. (Stephen). III. Title.

510.76

Acknowledgements

We thank the following for their contributions to our text book:

Australian Picture Library: pp. 128, 207, 236, 270.Getty Images: pp. 229, 267.Pearson Education Australia: PEA/ Karly Abery, p. 271; /Kim Nolan, p. 443. Photolibrary.com: pp. 82, 235.Steven Wilkes2005: pp. 77, 80, 134, 167, 194, 197, 200, 201, 314, 319, 323, 411, 414.

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

5.1_5.3_Prelims Page ii Tuesday, July 12, 2005 9:29 AM

iii

Contents

Features of New Signpost Mathematics viiiTreatment of Outcomes xiiMetric Equivalents xviThe Language of Mathematics xvii

ID Card 1 (Metric Units) xviiID Card 2 (Symbols) xviiID Card 3 (Language) xviiiID Card 4 (Language) xixID Card 5 (Language) xxID Card 6 (Language) xxiID Card 7 (Language) xxii

Algebra Card xxiii

Review of Year 9 1

1:01 Basic number skills 2A Order of operations 2B Fractions 2C Decimals 3D Percentages 3E Ratio 4F Rates 5G Significant figures 5H Approximations 6I Estimation 6

1:02 Algebraic expressions 7

How do mountains hear? 10

1:03 Probability 111:04 Geometry 111:05 Indices 141:06 Surds 151:07 Measurement 161:08 Equations, inequations and formulae 171:09 Consumer arithmetic 191:10 Coordinate geometry 201:11 Statistics 221:12 Simultaneous equations 231:13 Trigonometry 241.14 Graphs of physical phenomena 25

Working mathematically 27

Quadratic equations 28

2:01 Solution using factors 292:02 Solution by completing the square 312:03 The quadratic formula 33

How many solutions? 36

2:04 Choosing the best method 37

What is an Italian referee? 39

2:05 Problems involving quadratic equations 39

Temperature and altitude 43Did you know that 2 = 1? 43

Maths terms Diagnostic test Revision assignment Working mathematically 44

Probability 48

3:01 Probability review 49

Chance experiments 55What is the difference between a songwriter and a corpse? 56

3:02 Organising outcomes of compound events 57

Dice football 60

3:03 Dependent and independent events 60

Will it be a boy or a girl? 64

3:04 Probability using tree and dot diagrams 65

Probabilities given as odds 70

3:05 Probability using tables and Venn diagrams 70

Games of chance 76

3:06 Simulation experiments 77

Random numbers and calculator cricket 84Two-stage probability experiments 85Computer dice 86


Consumer Arithmetic 93

4:01 Saving money 94

Financial spreadsheets 96

4:02 Simple interest 97

Why not buy a tent? 99

4:03 Solving simple interest problems 1004:04 Compound interest 102

What is the difference between a book and a bore? 106

4:05 Depreciation 1074:06 Compound interest and depreciation

formulae 109

Compound interest tables 114

4:07 Reducible interest 115

Reducible home loan spreadsheet 119

4:08 Borrowing money 120

A frightening formula 124

4:09 Home loans 125


Chapter 1

Chapter 2

Chapter 3

Chapter 4

5.1_5.3_Prelims Page iii Tuesday, July 12, 2005 9:29 AM

iv

NEW SIGNPOST MATHEMATICS

10

STAGE

5.15.3

Number Plane Graphs 133

5:01 The parabola 134

The graphs of parabolas 139

5:02 Parabolas of the form

y

=

ax

2

+

bx

+

c

140

Why didnt the bald man need his keys? 147

5:03 The hyperbola:

y

=

k/x

1485:04 Exponential graphs:

y

=

a

x

151

The tower of Hanoi 153

5:05 The circle 1545:06 Curves of the form

y

=

ax

3

+

d

156

What is HIJKLMNO? 159

5:07 Miscellaneous number plane graphs 1605:08 Using coordinate geometry to solve

problems 163


Surface Area and Volume 173

6:01 Review of surface area 1746:02 Surface area of a pyramid 1766:03 Surface area of a cone 179

The surface area of a cone 180

6:04 Surface area of a sphere 183

The surface area of a sphere 183How did the raisins win the war with the nuts? 186

6:05 Volume of a pyramid 187

The volume of a pyramid 187

6:06 Volume of a cone 1916:07 Volume of a sphere 193

Estimating your surface area and volume 193

6:08 Practical problems of surface area and volume 195


Statistics 202

7:01 Review of statistics 2037:02 Measures of spread: interquartile range 209

Why did the robber flee from the music store? 214

7:03 Box-and-whisker plots 2157:04 Measures of spread: Standard deviation 2197:05 Comparing sets of data 225


Similarity 237

8:01 Review of similarity 2388:02 Similar triangles 243

A Matching angles 244B Ratios of matching sides 247

Drawing enlargements 252

8:03 Using the scale factor to find unknown sides 253

What happened to the mushroom that was double parked? 258

8:04 Similar triangle proofs 2598:05 Sides and areas of similar figures 2638:06 Similar solids 266

King KongCould he have lived? 271Maths terms Diagnostic test Revision assignment Working mathematically 272

Further Trigonometry 277

9:01 Trigonometric ratios of obtuse angles 2789:02 Trigonometric relationships between

acute and obtuse angles 281

Why are camels terrible dancers? 284

9:03 The sine rule 2859:04 The sine rule: The ambiguous case 2899:05 The cosine rule 291

Why did Toms mother feed him Peters ice-cream? 295

9:06 Area of a triangle 2969:07 Miscellaneous problems 2989:08 Problems involving more than one

triangle 300


Further Algebra 307

10:01 Simultaneous equations involving a quadratic equation 308

10:02 Literal equations: Pronumeral restrictions 311

What small rivers flow into the Nile? 315Fibonacci formula 315

10:03 Understanding variables 316

Number patterns and algebra 320Maths terms Diagnostic test Revision assignment Working mathematically 321

Circle Geometry 325

11:01 Circles 326

Circles in space 329

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

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v

11:02 Chord properties of circles (1) 330

Locating the epicentre of earthquakes 334

11:03 Chord properties of circles (2) 33511:04 Angle properties of circles (1) 33911:05 Angle properties of circles (2) 342

The diameter of a circumcircle 346

11:06 Tangent properties of circles 34611:07 Further circle properties 352

How do you make a bus stop? 356

11:08 Deductive exercises involving the circle 357

How many sections? 360Maths terms Diagnostic test Revision assignment Working mathematically 321

Curve Sketching 235

12:01 Curves of the form

y

=

ax

n

and

y = ax

n

+

d


y

=

ax

n

and

y

=

a

(

x

r

)

n


y

=

a

(

x

r

)(

x

s

)(

x

t

) 37512:04 Circles and their equations 37912:05 The intersection of graphs 380

A parabola and a circle 385Maths terms Diagnostic test Revision assignment Working mathematically 385

Polynomials 389

13:01 Polynomials 39013:02 Sum and difference of polynomials 39213:03 Multiplying and dividing polynomials

by linear expressions 39413:04 Remainder and factor theorems 39613:05 Solving polynomial equations 39813:06 Sketching polynomials 400

How do you find a missing hairdresser? 405

13:07 Sketching curves related to

y

=

P

(

x

) 405


Functions and Logarithms 417

14:01 Functions 41814:02 Inverse functions 422

Quadratic functions and inverses 426

14:03 The graphs of

y

=

f

(

x

),

y

=

f

(

x

) +

k

and

y

=

f

(

x

a

) 427

Where would you get a job playing a rubber trumpet? 430

14:04 Logarithms 43114:05 Logarithmic and exponential graphs 43314:06 Laws of logarithms 435

Logarithmic scales 438

14:07 Simple exponential equations 439

Solving harder exponential equations by guess and check 441

14:08 Further exponential equations 442

Logarithmic scales and the history of calculating 443


Answers 449Answers to ID Cards 532Index 533

Chapter 12

Chapter 13

Chapter 14

5.1_5.3_Prelims Page v Tuesday, July 12, 2005 9:29 AM

vi


10

STAGE

5.15.3

Interactive Student CD

Appendix A 2

A:01 Basic number skills 2A:02 Algebraic expressions 15A:03 Probability 20A:04 Geometry 23A:05 Indices 30A:06 Surds 34A:07 Measurement 38A:08 Equations, inequations and formulae 43A:09 Consumer arithmetic 53A:10 Coordinate geometry 61A:11 Statistics 71A:12 Simultaneous equations 77A:13 Trigonometry 81A:14 Graphs of physical phenomena 87

Appendix B: Working Mathematically 91

B:01 Solving routine problems 91B:02 Solving non-routine problems 91

2:01 Quadratic equations 302:03 The quadratic formula 353:01 Probability review 503:02 Organising outcomes of compound events 584:02 Simple interest 984:04 Compound interest 1044:06 Compound interest formula 1135:02 The parabola

y

= ax2 + bx + c 1445:05 The circle 1545:08 Coordinate geometry 1656:01 Surface area review 1756:02 Surface area of a pyramid 1786:03 Surface area of a cone 1816:05 Volume of a pyramid 1897:02 Inter-quartile range 2117:04 Standard deviation 2218:03 Finding unknown sides in similar triangles 2558:04 Similar triangles proofs 2609:02 Trig. ratios of obtuse angles 2839:03 The sine rule 2879:04 Sine rulethe ambiguous case 2909:05 The cosine rule 2929:07 Sine rule or cosine rule? 2989:08 Problems with more than one triangle 30110:02 Literal equations 312

10:03 Understanding variables 31812:01 Curves of the form y = axn and y = axn + d 37012:02 Curves of the form y = axn and

y = a(x r)n 37412:03 Equations of the form

y = a(x r)(x s)(x t) 377

The material below is found in the Companion Website which is included on the Interactive Student CD as both an archived version and a fully featured live version.

Activities and InvestigationsChapter 1 Surd magic square,

Algebraic fractionsChapter 2 Completing the squareChapter 3 Probability investigationChapter 4 Compound interest, Who wants to be a

millionaire?Chapter 5 Investigating parabolas, Curve stitchingChapter 6 The box, Greatest volumeChapter 7 Mean and standard deviationChapter 8 Maths race, Similar figuresChapter 9 Sine rule, Investigating sine curvesChapter 10 Literal equationsChapter 11 CirclesChapter 12 Parabolas, Parabolas in real lifeChapter 14 Radioactive decay

Drag and DropsChapter 2: Quadratic equations 1,

Quadratic equations 2, Completing the square

Chapter 3: Theoretical probability, Maths terms 3, Probability and cards

Chapter 4: Compound interest, Depreciation, Maths terms 4, Reducible interest

Chapter 5: Parabolas, Maths terms 5, Identifying graphs

Chapter 6: Maths terms 6, Volumes of pyramids, Volumes of cylinders, cones and spheres

Chapter 7: Maths Terms 7, Box-and-whisker plots, Interquartile range

Chapter 8: Using the scale factor, Maths terms 8Chapter 9: Maths terms 9, Sine rule, Cosine ruleChapter 10: Maths terms 10, Literal equations, Further

simultaneous equationsChapter 11: Maths terms 11, Parts of a circle, Circle

geometry Chapter 12: Maths terms 12, Transforming curves

Student Coursebook

Appendixes

Foundation Worksheets

Technology Applications

You can access this material by clicking on the links provided on the Interactive Student CD. Go to the Home Page for information about these links.

5.1_5.3_Prelims Page vi Tuesday, July 12, 2005 9:29 AM

vii

Chapter 13: Maths terms 13, PolynomialsChapter 14: Logarithms, Function notation, Maths

terms 14

AnimationsChapter 6: The box, Greatest volumeChapter 8: Scale itChapter 9: Unit circleChapter 11: Spin graphs

Chapter Review QuestionsThese can be used as a diagnostic tool or for revision. They include multiple choice,pattern-matching and fill-in-the-gaps style questions.

DestinationsLinks to useful websites that relate directly to the chapter content.

5.1_5.3_Prelims Page vii Tuesday, July 12, 2005 9:29 AM

viii NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3

New Signpost Mathematics is a completely revised and updated edition of Signpost Mathematics written to meet all of the requirements of the new NSW 710 Mathematics syllabus to be implemented from 2004. It combines the strengths of the previous editions with a number of innovations described below. New Signpost Mathematics also offers considerable additional resources to provide a complete and fully integrated learning package.

New Signpost Mathematics 9 and 10 texts are designed to complete Stages 5.1 to 5.3 of the syllabus. Working with this series, teachers will be able to select an appropriate program of work for all students.

How is New Signpost Mathematics organised?As well as the student coursebook, additional support for both students and teachers is provided: Interactive Student CD free with each coursebook Companion Website Homework Book Teachers Resource printout and CD

CoursebookChapter-opening pages summarise the key content and present the syllabus outcomes addressed in each chapter.

Clear syllabus references are included throughout the text to make programming easier: in the chapter-opening pages, at the start of each main section within each chapter and in the Foundation Worksheet references. For example, Outcome MS532.

Prep Quizzes review skills needed to complete a topic. These anticipate problems and save time in the long run. These quizzes offer an excellent way to start a lesson.

Well-graded exercises Within each exercise, levels of difficulty are indicated by the colour of the question number.

green . . . foundation blue . . . Stage 5.3 level red . . . extension

Worked examples are used extensively and are easy for students to identify.

Features of New Signpost Mathematics

pr

ep quiz

1 4 9

Find the simple interest charged for a loan of:a $620 at 18% pa for 4 years b $4500 at 26% pa for 5 years

After factorising the left-hand side of each equation, solve the following.a x2 + 3x = 0 b m2 5m = 0 c y2 + 2y = 0

If (x + 1) and (x + 2) are both factors of x3 + ax2 + bx 10, find the values of a and b.

1

2

9

worked examples

Find the monthly repayments on a loan of $280 000 taken over 20 years at 75% pa.

Monthly repayment= 280 $8.06= $2256.80

5.1_5.3_Prelims Page viii Tuesday, July 12, 2005 9:29 AM

ix

Important rules and concepts are clearly highlighted at regular intervals throughout the text.

Cartoons are used to give students friendly advice or tips.

Foundation Worksheets provide alternative exercises for students who need to consolidate work at an earlier stage or who need additional work at an easier level. Students can access these on the CD by clicking on the Foundation Worksheet icons. These can also be copied from the Teachers Resource CD or from the Teachers Resource Centre on the Companion Website.

Challenge activities and worksheets provide more difficult investigations and exercises. They can be used to extend more able students.

Fun Spots provide amusement and interest, while often reinforcing course work. They encourage creativity and divergent thinking, and show that Mathematics is enjoyable.

Investigations encourage students to seek knowledge and develop research skills. They are an essential part of any Mathematics course.

Diagnostic Tests at the end of each chapter test students achievement of outcomes. More importantly, they indicate the weaknesses that need to be addressed by going back to the section in the text or on the CD listed beside the test question.

Assignments are provided at the end of each chapter. Where there are two assignments, the first revises the content of the chapter, while the second concentrates on developing the students ability to work mathematically.

The See cross-references direct students to other sections of the coursebook relevant to a particular section.

Extension topics: A selection of extra Mathematics topics is available in Signpost Mathematics 9 & 10 Further Options. These would be ideal to extend students who find Stage 5 easy, and who are looking for further challenges. This principle is included as part of the syllabus. The topics include Fractals, Networks, Matrices, Mathematics of Small Business, Surveying, Navigation, Navigation on Land, Modelling, and Mathematical Investigations.

The table ofvalues lookslike this!

Quadratic equations PAS5321 Factorise

a x2 3x b x2 + 3x + 22 Solve

a x(x 4) = 0 b (x 1)(x + 2) = 0

Foundation Worksheet 2:01

challenge

fun spotnoi

tagitsevni

diagnostic test

assignment

ees

5.1_5.3_Prelims Page ix Tuesday, July 12, 2005 9:29 AM

x NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3

The Algebra Card (see p xxiii) is used to practise basic algebra skills. Corresponding terms in columns can be added, subtracted, multiplied or divided by each other or by other numbers. This is a great way to start a lesson.

The Language of MathematicsWithin the coursebook, Mathematics literacy is addressed in three specific ways:

ID Cards (see pp xviixxii) review the language of Mathematics by asking students to identify common terms, shapes and symbols. They should be used as often as possible, either at the beginning of a lesson or as part of a test or examination.

Maths Terms met during the chapter are defined at the end of each chapter. These terms are also tested in a Drag and Drop interactive that follows this section.

Reading Maths help students to develop maths literacy skills and provide opportunities for students to communicate mathematical ideas. They present Mathematics in the context of everyday experiences.

An Answers section provides answers to all the exercises in the coursebook, including the ID Cards.

Interactive Student CD

This is provided at the back of the coursebook and is an important part of the total learning package.

Bookmarks and links allow easy navigation within and between the different electronic components of the CD that contains: A copy of the student coursebook. Appendix A for review of Year 9 content and skills. Appendix B for a reminder of Working Mathematically strategies. Printable copies of the Foundation Worksheets and Challenge Worksheets, linked from the

coursebook. An archived, offline version of the Companion Website, including:

Chapter Review Questions and Quick Quizzes All the Technology Applications: activities and investigations, drag-and-drops and animations Destinations (links to useful websites)

All these items are clearly linked from the coursebook via the Companion Website. A link to the live Companion Website.

Companion Website

The Companion Website contains a wealth of support material for students and teachers: Chapter Review Questions which can be used as a diagnostic tool or for revision. These are

self-correcting and include multiple-choice, pattern-matching and fill-in-the-gaps style questions. Results can be emailed directly to the teacher or parents.

Quick Quizzes for each chapter. Destinations links to useful websites which relate directly to the chapter content.

di

smretshtam

shtamgnida er

5.1_5.3_Prelims Page x Tuesday, July 12, 2005 9:29 AM

xi

Technology Applications activities that apply concepts covered in each chapter and are designed for students to work independently:

Activities and investigations using technology, such as Excel spreadsheets and The Geometers Sketchpad.

Drag and Drop interactives to improve mastery of basic skills.

Animations to develop key skills by manipulating visually stimulating and interactivedemonstrations of key mathematical concepts.

Teachers Resource Centre provides a wealth of teacher support material and is password protected: Coursebook corrections Topic Review Tests and answers Foundation and Challenge Worksheets and answers Answers to the exercises in the Homework Book

Homework BookThe Homework Book provides a complete homework program linked directly to the coursebook. It features: Enough homework for a whole year Double-sided fill-in worksheets Short examples and brief explanations where needed Puzzles and investigations to liven things up

Teachers resourceThis material is provided as both a printout and as an electronic copy on CD: Electronic copy of the complete Student Coursebook in PDF format Teaching Program, including treatment of syllabus outcomes, in both PDF and editable

Microsoft Word formats Practice Tests and Answers Foundation and Challenge Worksheets and answers Answers to the exercises in the Homework Book Answers to some of the Technology Application Activities and Investigations

Most of this material is also available in the Teachers Resource Centre of the Companion Website.

Sample Drag and Drop

Sample Animation

5.1_5.3_Prelims Page xi Tuesday, July 12, 2005 9:29 AM

xii NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3

Treatment of OutcomesEach outcome relevant to the Year 10 text is listed on the left-hand side. The places where these are treated are shown on the right. Where part of an outcome has been treated in Year 9, this is also indicated.

The outcomes for Chapters 11 to 14 are optional topics as further preparation for the Mathematics Extension courses in Stage 6. These are indicated by the # symbol. The syllabus strand Working Mathematically involves questioning, applying strategies, communicating, reasoning and reflecting. These are given special attention in the assignment at the end of each chapter, but are also an integral part of each chapter.

Outcome Text References

WMS5.3.1 Asks questions that could be explored using mathematics in relation to Stage 5.3 content.

Assignments B, and throughout the text

WMS5.3.2 Solves problems using a range of strategies including deductive reasoning.


WMS5.3.3 Uses and interprets formal definitions and generalisations when explaining solutions and or conjectures.


WMS5.3.4 Uses deductive reasoning in presenting arguments and formal proofs.


WMS5.3.5 Links mathematical ideas and makes connections with, and generalisations about, existing knowledge and understanding in relation to Stage 5.3 content.


NS4.2 Compares, orders and calculates with integers. Year 9, 1:01

NS4.3 Operates with fractions, decimals, percentages, ratios and rates.

Year 9, 1:01

NS5.1.1 Applies index laws to simplify and evaluate arithmetic expressions and uses scientific notation to write large and small numbers.

Year 9, 1:05

NS5.1.2 Solves consumer arithmetic problems involving earning and spending money.

Year 9, 1:09, 4:014:03

NS5.1.3 Determines relative frequencies and theoretical probabilities.

Year 9, 1:03, 3:01, 3:06

NS5.2.1 Rounds decimals to a specified number of significant figures, expresses recurring decimals in fraction form and converts rates from one set of units to another.

Year 9, 1:01

5.1_5.3_Prelims Page xii Tuesday, July 12, 2005 9:29 AM

xiii

NS5.2.2 Solves consumer arithmetic problems involving compound interest, depreciation and successive discounts.

Year 9, 1:09, 4:044:08

NS5.3.1 Performs operations with surds and indices. Year 9, 1:06

NS5.3.2 Solves probability problems involving compound events.

3:023:05

PAS4.3 Uses the algebraic symbol system to simplify, expand and factorise simple algebraic expressions.

Year 9, 1:02

PAS4.4 Uses algebraic techniques to solve linear equations and simple inequalities.

Year 9, 1:08

PAS4.5 Graphs and interprets linear relationships on the number plane.

Year 9, 1:10

PAS5.1.1 Applies the index laws to simplify algebraic expressions.

Year 9, 1:05

PAS5.1.2 Determines the midpoint, length and gradient of an interval joining two points on the number plane and graphs linear and simple non-linear relationships from equations.

Year 9, 1:10, 5:01

PAS5.2.1 Simplifies, expands and factorises algebraic expressions involving fractions and negative and fractional indices.

Year 9, 1:02, 1:05

PAS5.2.2 Solves linear and simple quadratic equations, solves linear inequalities and solves simultaneous equations using graphical and analytical methods.

Year 9, 1:08, 1:12

PAS5.2.3 Uses formulae to find midpoint, distance and gradient and applies the gradientintercept form to interpret and graph straight lines.

Year 9, 1:10

PAS5.2.4 Draws and interprets graphs including simple parabolas and hyperbolas.

5:01, 5:03

PAS5.2.5 Draws and interprets graphs of physical phenomena. Year 9, 1:14

PAS5.3.1 Uses algebraic techniques to simplify expressions, expand binomial products and factorise quadratic expressions.

Year 9, 1:02

PAS5.3.2 Solves linear, quadratic and simultaneous equations, solves and graphs inequalities, and rearranges literal equations.

Year 9, 1:08, 2:012:06, 10:0110:03

PAS5.3.3 Uses various standard forms of the equation of a straight line and graphs regions on the number plane.

Year 9, 1:10

5.1_5.3_Prelims Page xiii Tuesday, July 12, 2005 9:29 AM

xiv NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3

PAS5.3.4 Draws and interprets a variety of graphs including parabolas, cubics, exponentials and circles and applies coordinate geometry techniques to solve problems.

5:01, 5:02, 5:045:08

PAS5.3.5 Analyses and describes graphs of physical phenomena.

Year 9

#PAS5.3.6 Uses a variety of techniques to sketch a range of curves and describes the features of curves from the equation.

Chapter 12

#PAS5.3.7 Recognises, describes and sketches polynomials, and applies the factor and remainder theorems to solve problems.

Chapter 13

#PAS5.3.8 Describes, interprets and sketches functions and uses the definition of a logarithm to establish and apply the laws of logarithms.

Chapter 14

DS4.1 Constructs, reads and interprets graphs, tables, charts and statistical information.

Year 9, 1:11

DS4.2 Collects statistical data using either a census or a sample and analyses data using measures of location and range.

Year 9, 1:11, 7:01

DS5.1.1 Groups data to aid analysis and constructs frequency and cumulative frequency tables and graphs.

Year 9, 1:11, 7:01

DS5.2.1 Uses the interquartile range and standard deviation to analyse data.

7:027:05

MS4.2 Calculates surface area of rectangular and triangular prisms and volume of right prisms and cylinders.

6:01

MS5.1.1 Uses formulae to calculate the area of quadrilaterals and finds areas and perimeters of simple composite figures

Year 9, 1:07

MS5.1.2 Applies trigonometry to solve problems (diagrams given) including those involving angles of elevation and depression.

Year 9, 1:13

MS5.2.1 Finds areas and perimeters of composite figures. Year 9, 1:07

MS5.2.2 Applies formulae to find the surface area of right cylinders and volume of right pyramids, cones and spheres and calculates the surface area and volume of composite solids.

Year 9, 1:07, 6:01, 6:056:08

MS5.2.3 Applies trigonometry to solve problems including those involving bearings.

Year 9, 1:13

5.1_5.3_Prelims Page xiv Tuesday, July 12, 2005 9:29 AM

xv

The above material is independently produced by Pearson Education Australia for use by teachers. Although curriculum references have been reproduced with the permission of the Board of Studies NSW, the material is in no way connected with or endorsed by them. For comprehensive course details please refer to the Board of Studies NSW Website www.boardofstudies.nsw.edu.au.

MS5.3.1 Applies formulae to find the surface area of pyramids, right cones and spheres.

6:026:04, 6:08, 8:05, 8:06

MS5.3.2 Applies trigonometric relationships, sine rule, cosine rule and area rule in problem-solving.

Year 9, Chapter 9

SGS4.4 Identifies congruent and similar two-dimensional figures stating the relevant conditions.

8:01

SGS5.2.1 Develops and applies results related to the angle sum of interior and exterior angles for any convex polygon.

Year 9, 1:04

SGS5.2.2 Develops and applies results for proving that triangles are congruent or similar.

Year 9, 1:04, 8:02, 8:03

SGS5.3.1 Constructs arguments to prove geometrical results. Year 9, 1:04

SGS5.3.2 Determines properties of triangles and quadrilaterals using deductive reasoning.

Year 9, 1:04

SGS5.3.3 Constructs geometrical arguments using similarity tests for triangles

8:04

#SGS5.3.4 Applies deductive reasoning to prove circle theorems and to solve problems.

Chapter 11

5.1_5.3_Prelims Page xv Tuesday, July 12, 2005 9:29 AM

xvi NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3

Metric Equivalents

Months of the year30 days each has September, April, June and November.All the rest have 31, except February alone,Which has 28 days clear and 29 each leap year.

SeasonsSummer: December, January, FebruaryAutumn: March, April, MayWinter: June, July, AugustSpring: September, October, November

Length

1 m = 1000 mm= 100 cm= 10 dm

1 cm = 10 mm1 km = 1000 m

Area

1 m2 = 10 000 cm2

1 ha = 10 000 m2

1 km2 = 100 ha

Mass

1 kg = 1000 g1 t = 1000 kg1 g = 1000 mg

Volume

1 m3 = 1 000 000 cm3

= 1000 dm3

1 L = 1000 mL1 kL = 1000 L1 m3 = 1 kL

1 cm3 = 1 mL1000 cm3 = 1 L

Time

1 min = 60 s1 h = 60 min

1 day = 24 h1 year = 365 days

1 leap year = 366 days

It is importantthat you learnthese factsoff by heart.

5.1_5.3_Prelims Page xvi Tuesday, July 12, 2005 9:29 AM

xvii

The Language of MathematicsYou should regularly test your knowledge by identifying the items on each card.

See page 532 for answers. See page 532 for answers.

ID Card 1 (Metric Units) ID Card 2 (Symbols)

1

m

2

dm

3

cm

4

mm

1

=

2

or

3

4

8

9

ha

10

m311

cm312

s

9

4210

4311 12

13

min

14

h

15

m/s

16

km/h

13 14

||15 16

|||

17

g

18

mg

19

kg

20

t

17

%

18

19

eg

20

ie

21

L

22

mL

23

kL

24

C

21

22

23 24

P(E)

di

2 23

x

See MathsTerms atthe end of

each chapter.

5.1_5.3_Prelims Page xvii Tuesday, July 12, 2005 9:29 AM

xviii NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3

See page 532 for answers.

.

ID Card 3 (Language)

1

6 minus 2

2

the sum of6 and 2

3

divide6 by 2

4

subtract2 from 6

5

the quotient of6 and 2

63

2)6the divisor

is . . . .

73

2)6the dividend

is . . . .

8

6 lots of 2

9

decrease6 by 2

10

the productof 6 and 2

11

6 more than 2

12

2 less than 6

13

6 squared

14

the squareroot of 36

15

6 take away 2

16

multiply6 by 2

17

average of6 and 2

18

add 6 and 2

19

6 to thepower of 2

20

6 less 2

21

the differencebetween 6 and 2

22

increase6 by 2

23

share6 between 2

24

the total of6 and 2

di

We say six squaredbut we write

62.

5.1_5.3_Prelims Page xviii Tuesday, July 12, 2005 9:29 AM

xix



1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

17 18 19 20

21 22 23 24

di

All sidesdifferent

5.1_5.3_Prelims Page xix Tuesday, July 12, 2005 9:29 AM

xx NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3



1

A

............

2

............

3

............

4

............

5

............ points

6

C is the ............

7

............

............

8

............

9

all angles lessthan 90

10

one angle 90

11

one angle greaterthan 90

12

A, B and C are......... of the triangle.

13

Use the verticesto name the .

14

BC is the ......... ofthe right-angled .

15

a + b + c = .........

16

BCD = .........17

a + b + c + d = .....

18

Which (a) a < bis true? (b) a = b

(c) a > b

19

a = .............

20

Angle sum = ............

21

AB is a ...............OC is a ...............

22

Name of distancearound the circle..............................

23

.............................

24

AB is a ...............CD is an ...............

EF is a...............

di

A

B

A

B

A

B

P

QR

S

A C B 4 2 0 2 4

A

B

C A

B

C

A

B

C

A B

C

b

ca

A D

B

C

b

a

b

d

c

aba

a

A B

C

OO

O

BC

D

FEA

5.1_5.3_Prelims Page xx Tuesday, July 12, 2005 9:29 AM

xxi



1

..................... lines

2

..................... lines

3

v .....................h .....................

4

..................... lines

5

angle .....................

6

..................... angle

7

..................... angle

8

..................... angle

9

..................... angle

10

..................... angle

11

.....................

12

..................... angles

13

..................... angles

14

..................... angles

15

..................... angles

16

a + b + c + d = .....

17

.....................

18

..................... angles

19

..................... angles

20

..................... angles

21

b............ an interval

22

b............ an angle

23

CAB = ............

24

CD is p.......... to AB.

di

A

B C

(lessthan90) (90)

(between90 and 180)

(180) (between180 and360)

(360)

a + b = 90

ab

a + b = 180

a b

a = b

a ba bc

d

a = b

a

ba = b

a

b

a + b = 180a

b

A B

C

D

E

A

B C

D

A B

C

A B

D

C

5.1_5.3_Prelims Page xxi Tuesday, July 12, 2005 9:29 AM

xxii NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3



1

a............ D............

2

b............ C............

3

a............ M............

4

p............ m............

5

area is 1 ............

6

r............ shapes

7

............ of a cube

8

c............-s............

9

f............

10

v............

11

e............

12

axes of ............

13

r............

14

t............

15

r............

16

t............

17

The c............of the dot are E2.

18

t............

19

p............ graph

20

c............ graph

21

l............ graph

22

s............ graph

23

b............ graph

24

s............ d............

di

AD BC am pm

100 m

100

m

43210A B C D E F

Cars soldMonTuesWedThursFri

Money collectedMonTuesWedThursFri

Stands for $10

70503010

M T W T F

Dol

lars

Money collected

10080604020

Johns height

1 2 3 4 5Age (years)

Use of time

HobbiesSleep

HomeSchool

People present

Adul

ts

Girl

s

Boys

Smoking

Cigarettes smokedLen

gth

of lif

e

5.1_5.3_Prelims Page xxii Tuesday, July 12, 2005 9:29 AM

xxiii

Algebra Card

How to use this cardIf the instruction is column D + column F, then you add corresponding terms in columns D and F.eg 1 m + (3m) 2 (4m) + 2m 3 10m + (5m)

4 (8m) + 7m 5 2m + 10m 6 (5m) + (6m)7 8m + 9m 8 20m + (4m) 9 5m + (10m)

10 (9m) + (7m) 11 (7m) + (8m) 12 3m + 12m

A B C D E F G H I J K L M N O

1 3 21 m 3m 5m2 5x 3x x + 2 x 3 2x + 1 3x 8

2 1 04 4m 2m 2m3 3x 5x2 x + 7 x 6 4x + 2 x 1

3 5 08 10m 5m 8m5 10x 8x x + 5 x + 5 6x + 2 x 5

4 2 15 8m 7m 6m2 15x 4x4 x + 1 x 9 3x + 3 2x + 4

5 8 25 2m 10m m2 7x 2x3 x + 8 x + 2 3x + 8 3x + 1

6 10 07 5m 6m 9m3 9x x2 x + 4 x 7 3x + 1 x + 7

7 6 12 8m 9m 2m6 6x 5x2 x + 6 x 1 x + 8 2x 5

8 12 05 20m 4m 3m3 12x 4x3 x + 10 x 8 5x + 2 x 10

9 7 01 5m 10m m7 5x 3x5 x + 2 x + 5 2x + 4 2x 4

10 5 06 9m 7m 8m4 3x 7x5 x + 1 x 7 5x + 4 x + 7

11 11 18 7m 8m 4m 4x x3 x + 9 x + 6 2x + 7 x 6

12 4 14 3m 12m 7m2 7x x10 x + 3 x 10 2x + 3 2x + 3

14---

2m3

-------

x6---

x2---

18---

m4----

x3---

x4---

13---

m4----

2x7

------

2x5

------

120------

3m2

-------

x10------

x5---

35---

m5----

2x3

------

x3---

27---

3m7

-------

2x5

------

3x5

------

38---

m6----

5x6

------

2x3

------

920------

2m5

-------

3x4

------

x7---

34---

3m5

-------

3x7

------

3x7

------

710------

4m5

-------

x6---

2x9

------

110------

m5----

x5---

x3---

25---

m3----

3x4

------

x6---

5.1_5.3_Prelims Page xxiii Tuesday, July 12, 2005 9:29 AM

Review of Year 9

1

1

Talk aboutdja` vu!

Chapter Contents

1:01

Basic number skills

NS42, NS43, NS5.21

A

Order of operations

B

Fractions

C

Decimals

D

Percentages

E

Ratio

F

Rates

G

Significant figures

H

Approximations

I

Estimation

1:02

Algebraic expressions

PAS43, PAS521, PAS531

Fun Spot: How do mountains hear?1:03

Probability

NS5.13

1:04

Geometry

SGS5.21, SGS5.22, SGS531, SGS532

1:05

Indices

NS511, PAS511, PAS521

1:06

Surds

NS531

1:07

Measurement

MS511,

MS521,

MS522

1:08

Equations, inequations and formulae

PAS44, PAS522, PAS532

1:09

Consumer arithmetic

NS512, NS522

1:10

Coordinate geometry

PAS45, PAS512, PAS523, PAS533

1:11

Statistics

DS41, DS42, DS511

1:12

Simultaneous equations

PAS522

1:13

Trigonometry

MS512,

MS523,

MS532

1:14

Graphs of physical phenomena

PAS525

Working Mathematically

Learning Outcomes

As this is a review chapter, many outcomes are addressed. These include the Stage 5 outcomes of NS511, NS512, NS513, NS521, NS522, NS531, PAS511, PAS512, PAS521, PAS522, PAS523, PAS525, PAS531, PAS532, PAS533, MS511, MS512, MS521, MS522, MS523, MS532, DS511, SGS521, SGS522, SGS531, SGS532.Working Mathematically Stage 5.31 Questioning, 2 Applying Strategies, 3 Communicating, 4 Reasoning, 5 Reflecting

Note: A complete review of Year 9 content is found in Appendix A located on the Interactive Student CD.

Click here

Appendix A

5.1_5.3_Chapter 01 Page 1 Tuesday, July 12, 2005 8:48 AM

2


10

STAGE

5.15.3

This chapter is a summary of the work covered in

New Signpost Mathematics 9, Stage 5.15.3

. For an explanation of the work, refer to the cross-reference on the right-hand side of the page which will direct you to the Appendix on the Interactive Student CD.

1:01

|

Basic Number Skills

Outcomes NS42, NS43, NS521

Rational numbers:

Integers, fractions, decimals and percentages (both positive and negative) are rational numbers. They can all be written as a terminating or recurring decimal. The following exercises will remind you of the skills you should have mastered.

A | Order of operations

Answer these questions without using a calculator.

a

4

(5

3)

b

6

(9

4)

c

4

+

(3

+

1)

d

6

+

4

2

e

9

3

4

f

16

+

4

4

g

10

4

4

7

h

30

3

+

40

2

i

5

8

+

6

5

j

5

2

2

k

3

10

2

l

3

2

+

4

2

m

6

+

3

4

+

1

n

8

+

4

2

+

1

o

6

(

6

6)

a

6

(5

4)

+

3

b

27

(3

+

6)

3

c

16

[10

(6

2)]

d e f

g

(6

+

3)

2

h

(10

+ 4)2 i (19 9)2

B | Fractions

Change to mixed numerals.

a b c d

Change to improper fractions.

a 5 b 3 c 8 d 66

Simplify the fractions.

a b c d

Complete the following equivalent fractions.

a = b = c = d =

a + b c + d

a 6 + 2 b 4 2 c 4 + 6 d 5 1

a b c d of

a 6 b 2 1 c 1 15 d 10 1

a b c 6 d 2 1

Exercise 1:01A

A:01A

A:01A

CD Appendix

1

230 10+30 10------------------

15 45+45 5+

------------------

1414 7---------------

Exercise 1:01B

A:01B1

CD Appendix

A:01B2

A:01B3

A:01B4

A:01B5

A:01B6

A:01B7

A:01B8

A:01B9

174---

496

------

154

------

118

------

212---

17---

34---

23---

34880------

70150---------

200300---------

250450---------

434---

24------

25---

50------

27---

28------

13---

120---------

5 715------

115------

1320------

25---

58---

310------

67---

35---

6 12---

35---

34---

310------

34---

110------

38---

910------

7 35---

47---

1825------

1516------

49---

310------

710------

23---

8 34---

12---

45---

13---

12---

37---

9 910------

23---

38---

35---

45---

34---

12---


CHAPTER 1 REVIEW OF YEAR 9 3

C | Decimals

Put in order, smallest to largest.a {0606, 06, 066, 0066} b {153, 0153, 1053}c {07, 0017, 7, 077} d {35, 345, 305, 34}Do not use your calculator to do these.a 7301 + 2 b 305 + 04c 0004 + 31 d 6 + 03 + 002e 867 67 f 912 1015g 8 3112 h 1623 3a 0012 3 b 003 02 c 045 13 d (005)2

a 314 10 b 05 1000 c 00003 100 d 38 104

a 015 5 b 106 4 c 1535 5 d 001 4a 13 3 b 91 11 c 14 9 d 6 7a 4804 10 b 16 100 c 09 1000 d 65 104

a 84 04 b 0836 008 c 75 0005 d 14 05Express as a simplified fraction or mixed numeral.a 3017 b 004 c 086 d 16005Express as a decimal.

a b c d

Express these recurring decimals as fractions.a 05555 b 0257 257 2 c dExpress these recurring decimals as fractions.a 08333 b 0915 151 5 c d

D | Percentages

Express as a fraction.a 54% b 203% c 12 % d 91%Express as a percentage.

a b c 1 d

Express as a decimal.a 16% b 86% c 3% d 18 %Express as a percentage.a 047 b 006 c 0375 d 13a 36% of 400 m b 9% of 84 g c 8 % of $32

d At the local Anglican church, the offertories for 2005 amounted to $127 000. If 68% of this money was used to pay the salary of the two full-time ministers,how much was paid to the ministers?

Exercise 1:01C

A:01C1

CD Appendix

A:01C2

A:01C3A:01C4A:01C5A:01C6A:01C7A:01C8A:01C9

A:01C10

A:01C11

A:01C11

1

2

3

4

5

6

7

8

9

1045---

7200---------

58---

811------

1107 2 06 42

120435 089 42

Exercise 1:01D

A:01D1

CD Appendix

A:01D2

A:01D3

A:01D4

A:01D5

114---

21120------

49---

14---

23---

314---

4

5 12---


4 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3

a 9% of Lukes money was spent on fares. If $5.40 was spent on fares, how much money did Luke have?

b 70% of Alanas weight is 175 kg. How much does Alana weigh?c Lyn bought a book for a reduced price of 70 cents. This was 14% of the

books recommended retail price. What was the recommended retail price?d 54 minutes of mathematics lesson time was lost in one week because of

other activities. If this represents 30% of the allocated weekly time for mathematics, what is this allocated time?

a Express 85 cents as a percentage of $2.b 4 kg of sugar, 9 kg of flour and 7 kg of mixed fruit were mixed. What is

the percentage (by weight) of flour in the mixture?c Of 32 birds in Rachels aviary, 6 are canaries. What percentage of her birds

are canaries?d When Steve Waugh retired from test cricket in 2003, he had scored

32 centuries from 260 innings. In what percentage of his innings did hescore centuries?

E | Ratio

a Simplify each ratio.i $15 : $25 ii 9 kg : 90 kg iii 75 m : 35 m iv 120 m2 : 40 m2

b Find the ratio in simplest terms of 56 m to 40 cm.c Naomi spends $8 of $20 she was given by her grandparents and saves the rest.

What is the ratio of money spent to money saved?d Three-quarters of the class walk to school while ride bicycles. Find the ratio

of those who walk to those who ride bicycles.e At the end of their test cricket careers, Steve Waugh had scored 50 fifties

and 32 hundreds from 260 innings, while Mark Waugh had scored 47 fiftiesand 20 hundreds from 209 innings.i Find the ratio of the number of hundreds scored by Steve to the number

scored by Mark.ii Find the ratio of the number of times Steve scored 50 or more to the

number of innings.f Express each ratio in the form X : 1.

i 3 : 5 ii 2 : 7 iii 10 : 3 iv 25 : 4g Express each ratio in f in the form 1 : Y.

a If x : 15 = 10 : 3, find the value of x.b If the ratio of the populations of Africa and Europe is 5 : 4, find the

population of Africa if Europes population is 728 million.c The ratio of the average population density per km2 of Asia to that of

Australia is 60 : 1. If the average in Asia is 152 people per km2, what isthe average in Australia?

d The ratio of the population of Sydney to the population of Melbourne is 7 : 6. If 4.2 million people live in Sydney, how many people live in Melbourne?

A:01D6

A:01D7

6

7

Exercise 1:01E

A:01E1

CD Appendix

A:01E2

1

15---

2



a If 84 jellybeans are divided between Naomi and Luke in the ratio 4 : 3,how many jellybeans does each receive?

b The sizes of the anglesof a triangle are in theratio 2 : 3 : 4. Find thesize of each angle.

c A total of 22 millionpeople live in the citiesof Tokyo and Moscow.If the ratio of thepopulations of Tokyo and Moscow is 6 : 5, what is the population of each city?

d At Christ Church, Cobargo, in 1914, there were 60 baptisms. The ratio ofmales to females who were baptised was 3 : 2. How many of each were baptised?

F | Rates

Complete these equivalent rates.a 5 km/min = . . . km/h b 8 km/L = . . . m/mLc 600 kg/h = . . . t/day d 2075 cm3/g = . . . cm3/kg

a At Cobargo in 1915, the Rector, H. E. Hyde, travelled 3396 miles by horseand trap. Find his average speed (to the nearest mile per hour) if it tooka total of 564 hours to cover the distance.

b Over a period of 30 working days, Adam earned $1386. Find his averagedaily rate of pay.

c Sharon marked 90 books in 7 hours. What rate is this in minutes per book?d On a hot day, our family used an average of 36 L of water per hour.

Change this rate to cm3 per second (cm3/s).

G | Significant figures

State the number of significant figures in each of the following.a 21 b 46 c 252 d 0616e 1632 f 106 g 3004 h 203i 106 j 5004 k 05 l 0003m 0000 32 n 006 o 0006 p 30q 250 r 260 s 13000 t 640u 41 235 v 600 (to nearest w 482 000 (to nearest x 700 (to nearesty 1600 hundred) thousand) ten)z 16 000

State the number of significant figures in each of the following.a 30 b 300 c 03 d 003e 0.030 f 0.0030 g 0.0300 h 3.0300

A:01E33

Exercise 1:01F

A:01F

CD Appendix

A:01F

1

2

Exercise 1:01G

A:01G

CD Appendix

A:01G

1

2



H | Approximations

Approximate each of the following correct to one decimal place.a 463 b 081 c 317 d 0062e 15176 f 8099 g 099 h 12162i 0119 j 47417 k 035 l 275

Approximate each of the following correct to two decimal places.a 0537 b 2613 c 7134 d 1169e 120163 f 8399 g 412678 h 00756i 04367 j 100333 k 0015 l 0005

Approximate each number correct to: i 1 sig. fig. ii 2 sig. figs.a 731 b 849 c 063 d 258e 416 f 00073 g 00828 h 305i 0009 34 j 00098 k 752 l 00359

Approximate each of the following numbers correct to the number ofsignificant figures indicated.a 23 (1 sig. fig.) b 1463 (3 sig. figs.) c 215 (2 sig. figs.)d 093 (1 sig. fig.) e 407 (2 sig. figs.) f 7368 94 (3 sig. figs.)g 0724 138 (3 sig. figs.) h 5716 (1 sig. fig.) i 31685 (4 sig. figs.)j 0007 16 (1 sig. fig.) k 078 (1 sig. fig.) l 0007 16 (2 sig. figs.)

Approximate each of the following numbers correct to the number of decimalplaces indicated.a 561 (1 dec. pl.) b 016 (1 dec. pl.) c 0437 (2 dec. pl.)d 1537 (1 dec. pl.) e 8333 (2 dec. pl.) f 413789 (1 dec. pl.)g 7198 (1 dec. pl.) h 30672 (3 dec. pl.) i 999 (1 dec. pl.)j 47998 (3 dec. pl.) k 0075 (2 dec. pl.) l 00035 (3 dec. pl.)

I | Estimation

Give estimates for each of the following.a 127 58 b 055 210 c 178 51 0336d 156 2165 e (462 + 217) 421 f 78 52 + 217 089

g (093 + 172)(85 17) h i

j k l

m n o 3.13 184

p q r

s

Exercise 1:01H

A:01H

CD Appendix

A:01H

A:01H

A:01H

A:01H

1

2

3

4

5

Exercise 1:01I

A:01I

CD Appendix

1

437 182+78 29+

-----------------------------

1016 517213 148

--------------------------------

068 51025 78------------------------

116 392127 658+----------------------------- 3522 179

417 56 426 1056

41 48122623

---------------------------------

1572

113 31--------------------------

167215-----------

41647

-----------+

065001-----------

075 360478

--------------------------



1:02 | Algebraic Outcomes PAS43, PAS521, PAS531Expressions

Being able to use algebra is often important in problem-solving. Below is a reminder of the skills you have met up to Year 9.

Write an expression for:a the sum of 3a and 4bb the product of 3a and 4bc the difference between k and m, if k > md the difference between k and m, if k < me the average of x, y and zf twice the sum of m and 5g the square of the difference between a and bh the square root of the sum of 5m and 4ni the next even number after n, if n is evenj the sum of three consecutive integers, if the first one is m

Double-checkyour

algebraskills!

Double-checkyour

algebraskills!

GeneralisationWhat is the average ofa and b?

Answer: Average =

SubstitutionFind the value of2x + y2 if x = 3, y = 2.

Answer: 2(3) + (2)2

= 10

a b+2

------------

Fractions

1

=

=

2

=

2x3

------

x5---+

5 2x 3 x+15

----------------------------------

13x15

---------

5a1

6b1--------

183b2b

10a2----------------

3b2

------

Simplifying expressions

1 3x2 + 5x + x2 3x= 4x2 + 2x

2 12xy 8xz

=

=

123xy82 xz

---------------

3y2z------

Products

1 5(x + 3) 2(x 5)= 5x + 15 2x + 10= 3x + 25

2 (3x 1)(x + 7)= 3x2 + 20x 7

Factorisation

1 5a2b 10a= 5a(ab 2)

2 x2 + 3x 10= (x + 5)(x 2)

3 ab 3a + xb 3x= a(b 3) + x(b 3)= (b 3)(a + x)

Exercise 1:02

A:02A

CD Appendix

1 Wipe that expressionoff your face!



If a = 3, b = 5 and c = 6, find the value of:a 2a + 3b b a + b + c c 2b cd ab + bc e ac b2 f a2 + c2

g h i

j k l

Simplify these expressions.a 5a + 3b a + b b 5ab 2ba c 3x2 + x x2 + xd 5x 3y e 6ab 3a f 2m 5mng 15a 5 h 24m 12m i 10a2b 5abj n 3n k 15m 10n l 12xy2 8x2ym 6a 7 2a n 20y 2 5y o 7x + 2 4x 10x

Simplify these fractions.

a b c

d e f

g h i

j k l

Expand and simplify these products.a 3(2a + 1) 5a b 10m 2(m + 5) c 6a (a 5) + 10d 3(2n 1) + 2(n + 5) e 4(2a 1) 3(a + 5) f 6(1 2x) (3 10x)g (x + 3)(x + 7) h (y 4)(y 1) i (k 7)(k + 9)j (2p + 3)(p 5) k (6x + 1)(3x 2) l (3m 1)(2m 5)m (m 7)(m + 7) n (3a 4)(3a + 4) o (10 3q)(10 + 3q)p (a + 8)2 q (2m 1)2 r (4a + 5)2

s (x + y)(x 2y) t (a + 2b)(a 2b) u (m 3n)2

Factorise:a 15a 10 b 3m2 6m c 4n + 6mnd 6mn 4m e 10y2 + 5y f 6a2 2a + 4abg x2 49 h 100 a2 i 16a2 9b2

j x2 + 8x + 12 k x2 x 12 l x2 6x + 8m a2 + 6a + 9 n y2 10y + 25 o 1 4m + 4m2

p 2x2 + 7x + 3 q 3m2 + 7m 6 r 6a2 11a + 4s 4n2 + 12n + 9 t 25x2 10x + 1 u 9 24m + 16m2

v ab 4a + xb 4x w x2 + ax 2x 2a x 2m2 + 6mn m 3n

A:02B

A:02C

A:02D

A:02E

A:02G

A:02F

2

4ac

------

3ab3 c-----------

3b c2a

--------------

ab c+ a b c+ +2

--------------------

3cb a-----------

3

42a5

------

4a5

------+ 6x7

------

4x7

------

3y---

4y---+

a3---

a4---+ 2m

3-------

m5----

32n------

43n------

a3---

b4--- m

5----

2m3

------- 9a2

2x--------

xy3a------

5m----

2m----

ab5------

a10------

xyz

------

yz2-----

5

6



Factorise these expressions completely.a 2x2 18 b 4x2 + 4x 24 c 3a2 6a 3ab + 6bd 8n2 8n + 2 e 9 9q2 f m4 m2

g k4 16 h y3 + y2 + y + 1 i x3 x2 x + 1

Factorise and simplify:

a b c

d e f

Simplify:

a b

c d

e f

Simplify each of the following.

a b

c d

Factorise each denominator where possible and then simplify.

a b

c d

e f

A:02G

A:02H

A:02H

A:02I

A:02I

7

83a 12+

3------------------

5x 15x 3

------------------

a 5+a2 7a 10+ +------------------------------

m2 mm2 1-----------------

n2 n 6n2 5n 6+ +---------------------------

2x2 x 34x2 9

--------------------------

93x 15+

2------------------

4xx 5+------------ a

2 9a 3--------------

a 1+a 3+------------

3x 6+10x

---------------

x 2+5x

------------m2 25m2 5m--------------------

m 5+5m

-------------

a2 7a 12+ +a2 5a 4+ +

------------------------------

a2 6a 5+ +a2 12a 35+ +--------------------------------- n

2 3n 43n2 48

---------------------------

n3 nn2 4n+------------------

101

a 4+------------

1a 3+------------+ 3

2x 1---------------

54x 3+---------------

3x 1+( ) x 2+( )----------------------------------

2x x 2+( )--------------------+

5x 3+( ) x 5+( )----------------------------------

3x 3+( ) x 4+( )----------------------------------

111

a2 1--------------

1a 1+------------+ 2

3x 6+---------------

1x2 4--------------

2x2 x 6+-----------------------

3x2 4x 3+ +---------------------------+ 6

x2 x 2-----------------------

3x2 2x 3--------------------------

x 1+x2 9--------------

x 1x2 5x 6+---------------------------+ n 5+

2n2 n 1+---------------------------

n 32n2 5n 3+------------------------------



Fun Spot 1:02 | How do mountains hear?Work out the answer to eachquestion and put the letter for that part in the box that is above the correct answer.

Simplify:T 7x + x T 7x xE 7x x I 7x xI 3(x + 1) (x + 3)O (x 1)2 + 2x 1

Solve:N 5x + 1 = 13 x

E

H 5x 2(x + 3) = 12

Find the value of b2 4ac if:H a = 4, b = 10, c = 2 I a = 1, b = 5, c = 7R a = 6, b = 9, c = 3 E a = 2, b = 3, c = 5

Simplify:R 3x5 2x3 A 15x5 5x4 W (3x)2 T 5x0 (3x)0

Factorise:M x2 81 S x2 8x 9 U x2 9x N x2 xy 9x + 9y

funspot

1:02

x 2+5

------------

x 13

-----------=

9x2 7 5 68 8x 6

7x2

53 6x8

(x

9)(

x +

9) x2

x(x

9

) 2 6x 3x 2x

(x

9)(

x

y)

49 5

9

(x

9)(

x +

1)1 2---



1:03 | Probability Outcome NS513

Using the figures shown in the table, find the probability of selecting at randoma matchbox containing:

a 50 matches b 48 matches c more than 50 d at least 50

A single dice is rolled. What is the probability of getting:a a five? b less than 3? c an even number? d less than 7?

A bag contains 3 red, 4 white and 5 blue marbles. If one is selected from the bagat random, find the probability that it is:a white b red or white c not red d pink

A pack of cards has four suits, hearts and diamonds (both red), and spades andclubs (both black). In each suit there are 13 cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10,Jack, Queen and King. The Jack, Queen and King are called court cards.

A card is drawn from a standard pack. What is the probability that the card is:a red? b not red? c a six? d not a six?e a court card? f a red Ace? g a spade? h a red thirteen?i either a red five or a ten? j either a heart or a black Ace?k either a blue five or a seven? l either a heart or a black card?

In each of these cases, the events may not be mutually exclusive.m either a court card or a diamond?n either a number larger than two or a club?o either a heart or a five?p either a Queen or a black court card?q either a number between two and eight or

an even-numbered heart?

1:04 | Geometry Outcomes SGS521, SGS522, SGS531, SGS532

a b

Find x. Give reasons. Find the size of x. Give reasons.

Number of matches 48 49 50 51 52

Number of boxes 3 6 10 7 4

Exercise 1:03

A:03A

CD Appendix

A:03B

A:03B

A:03C

1

2

3

4

Since there are 4 suits with 13 cards in each suit, the number of cards in a standard pack is 52. (In some games a Joker is also used.)

Exercise 1:04

A:04A

CD Appendix

1 A

E F G H

C

B D

79 x

A

E F

G H C

B

D

x

79



c d

Find the size of x. Find the value of b.Give reasons. Give reasons.

e f

ABDC is a parallelogram. Find Find the value of x and y.the size of x. Give reasons. Give reasons.

a What is the sum of the interior angles of:i a hexagon? ii a decagon?

b What is the size of each interior angle in these regular polygons?i ii

c What is the sum of the exterior angles of an octagon?d Find the size of each exterior angle of these regular polygons.

i ii

a b

Prove BED = ABC + CDE. O is the centre of the circle.Prove that AOC = 2 ABC.(Hint: AO = BO = CO (radii).)

A:04B

A:04C

A

E

F

C

B

D55

x

130

B

A

C D b

36

BA

CD

FE

x

105

x

y

140

2

3 A B

D C

E

A

B

D C

O



a b

O is the centre and OC AB. ABC is any triangle. D is theProve that OCA OBC and midpoint of BC, and BE and CFhence that AC = BC. are perpendiculars drawn to AD,

produced if necessary.Prove that BED CFD andhence that BE = CF.

a b

In ABC, a perpendicular drawn WXYZ is a parallelogram,from B to AC bisects ABC. ie WX | | ZY and WZ | | XY.Prove that ABC is isosceles. Prove WXY = YZW

(Hint: Use congruent triangles.)

a b

Find the value of YZ. i Find an expression for AB.ii Find an expression for BC.iii Hence, find the value of x.

A:04D

A:04F

A:04G

A:04E4

A

B C

O A

B

C

F DE

5

A

B

C D

W

X Y

Z

6 X

W Z Y

10 cm

12 cm

15 cm

B

A C D4 cm 8 cm

x cm



1:05 | Indices Outcomes NS511, PAS511, PAS521

Write in index form.a a a a b 2 2 2 2c n n n n n d 10 10 10

Simplify, giving your answers in index form.a 24 25 b a3 a2 c m m4 d 106 102

e a10 a2 f y4 y3 g b3 b h 105 102

i (m3)4 j (a2)3 k (x4)2 l (105)2

m a0 3 n b0 + c0 o 6y0 p e6 e6

q 6a2 5 r 6m3 3 s 6a 5a t (4x4)2

Simplify.a 6a4 5ab3 b 7a2b2 8a3b c 4a2b3 6a2b4

d 10a7 a3b3 e (7x3)2 f (2m2)4

g (x2y3)3 h (5xy2)4 i 30a5 5a3

j 100x4 20x k 36a3b4 12a2b4 l 8y7z2 y7z2

Rewrite without a negative index.a 41 b 101 c x1 d 2a1

e 52 f 23 g m3 h 5x2

Rewrite each of the following with a negative index.

a b c d

e f g h

Find the value of the following.

a b c d

Rewrite, using fractional indices.

a b c d

Simplify these expressions.a x4 x2 b 5m3 m2 c 4n2 3n3

d e f

g h i

Write these numbers in scientific (or standard) notation.a 148 000 000 b 68 000 c 0000 15 d 0000 001 65

Write these as basic numerals.a 62 104 b 115 106 c 74 103 d 691 105

Exercise 1:05

A:05A

CD Appendix

A:05B

A:05B

A:05C

A:05C

A:05D

A:05D

A:05E

A:05E

A:05CA:05D

1

2

3

4

513---

18---

1a---

3x---

124-----

1106--------

1y4-----

5n3-----

6

912---

3612---

813---

2713---

7

a y3 5 m 16x

8

6y4 3y12---

12x32---

6x12---

27x6( )13---

5a4 4a510a8

-----------------------

6m4 2m( )33m2 8m5

-------------------------------

9x3 2x3( )36x6 3x 2

------------------------------

12---

9

10



1:06 | Surds Outcome NS531

Indicate whether each of the following is rational or irrational.

a 6 b 131 c d 5162

e f g h

Evaluate each of the following to one decimal place.a b c d

Simplify:

a b c d

e f g h

i j k l

Simplify:

a b c

d e f

g h i

Simplify:

a b c

d e f

Expand and simplify:

a b

c d

e f

g h

i j

k l

Rewrite each fraction with a rational denominator.

a b c

d e f

Exercise 1:06

A:06A

CD Appendix

A:06B

A:06C

A:06D

A:06E

A:06A

A:06B

135--- 3

23 07 49

27 5 2+ 11 3 3

2-------

3

5 2 5 7 3 2 3 6

20

2----------

42

6---------- 130 5 49

81------

7( )2 2 3 2 5 3( )2 8 6 2

So 5, 37,2 + 3,

11 - 10are all surds.

4

75 3 8 180

4 3 7 3 6 5 5 2 2+

8 18+ 5 32 50 24 2 54

5

3 2 5 2 4 7 9 5 96 12

7 5( )2 2 4 3 1812

--------------------------- 3 2 3 5( )6

2 1+( ) 2 5+( ) 5 3( ) 5 2( )2 3+( ) 5 3( ) 5 3+( ) 5 2+( )2 3 1( ) 3 7+( ) 5 2 2 3( ) 3 2 5 3+( )

3 2+( )2 5 3( )22 3 3 2+( )2 5 2( ) 5 2+( )7 3+( ) 7 3( ) 5 3 2 2( ) 5 3 2 2+( )

71

3-------

5

5-------

6

2-------

1

3 2----------

3

2 6----------

2 5+

2 5----------------



1:07 | Measurement Outcomes MS511, MS521, MS522

Find the perimeter of the following figures. (Answer to 1 dec. pl.)a b c

Find the area of each plane shape. (Answer to 2 dec. pl.)a b c

d e f

Find the area of the following shaded figures (correct to 3 sig. figs.).a b c

Find the surface area of the following solids.

Exercise 1:07

A:07A

CD Appendix

A:07B

A:07B

A:07C

1

104 m28 m

96 m

62 m56 cm

135O

86 m

(Use = 3142)

2

56 m

27 m 106 cm 48

m 96 m

36 cm

78 cm51 cm

28 cm

42 cm

A B

D CAC = 36 cmBD = 64 cm

3

41 m

52 m

101 m

9 m3 m

8 m

965 km

517 km

314 km

4

113 cm

6 cm

68 cm

Rectangular prism

a 12 m5 m

15 m

15 m

4 m

Trapezoidal prism

b10 m

9 m 126 m

x m

Triangular prism(Note: Use Pythagoras

theorem to find x).

c



For each of the following cylinders, find i the curved surface area, ii the area ofthe circular ends, and iii the total surface area. (Give answers correct to two decimal places.)a b c

Find the volume of each prism in question 4.

Find the volume of each cylinder in question 5.

1:08 | Equations, Outcomes PAS44, PAS522, PAS532Inequations and Formulae

Solve the following.a a + 7 = 25 b m 6 = 1 c 5x = 75 d 10 y = 12

e 3p = 7 f g 2x + 3 = 7 h 8m + 5 = 21

i 5y + 2 = 3 j 9k 1 = 5 k 5 + 3x = 11 l 15 2q = 8

Solve the following.a 5m + 2 = 4m + 7 b 3x 7 = 2x 3 c 5x + 2 = 6x 5d 2a + 3 = 3a 5 e 3m 2 = 5m 10 f q + 7 = 8q + 14g 10 2x = x + 4 h 3z + 7 = z + 10 i 13 2m = 9 5m

Solve these equations involving grouping symbols.a 5(a + 1) = 15 b 4(x 3) = 16 c 3(2x + 5) = 33d 3(5 2a) = 27 e 4(3 2x) = 36 f 3(2m 5) = 11g 3(a + 2) + 2(a + 5) = 36 h 2(p + 3) + p + 1 = 31i 4(2b + 7) = 2(3b 4) j 4(2y + 3) + 3(y 1) = 2yk 3(m 4) (m + 2) = 0 l 2m 3(1 m) = 22m 5(y 3) 3(1 2y) = 4 n 4(2x 1) 2(x + 3) = 5

Solve these equations.

a b c d e f

Solve these equations involving fractions.

a b c

d e f

g h i

j k l

A:07D

A:07E

A:07E

5

56 m

22 m

22 cm

24 cm

684 m

18 m

6

7

Exercise 1:08

A:08A

CD Appendix

A:08B

A:08C

A:08C

A:08A

1

n4--- 3=

2

3

45x2

------ 10= 2a3

------ 6= 3m5

------- 4= n 1+5

------------ 2= x 42

----------- 1=2p 5+

3--------------- 1=

5a3---

a3---+ 4= 2x

5------

x5--- 3=

5p

3------

p

3--- 8=

q2---

q3--- 6= 2k

3------

k4--- 10= 3x

4------

x2--- 15=

m 6+3

-------------

2m 4+4

-----------------=n 3

2------------

3n 54

---------------=5x 1

3---------------

3 x2

-----------=

x 3+2

------------

x 5+5

------------+ 8= m 2+5

-------------

m 3+6

------------- 1= 3a 4+2

---------------

a 13

------------

2a 3+4

---------------=



a Translate these into an equation, using n as the unknown number.i A certain number is multiplied by 8, then 11 is added and the result is 39.ii I think of a number, double it, add 7 and the result is 5.iii I think of a number, add 4 and then divide the result by 10. The answer is 7.

b Solve each of the following problems by first forming an equation.i If 5 is added to 3 times a certain number, the result is 38.

What is the number?ii If one quarter of a certain number is added to half the same number, the

result is 6. What is the number?iii A rectangle is four times as long as it is wide. If it has a perimeter of

340 m, what are its dimensions?

Write the set of x that has been graphed below.a b

c d

Solve these inequations and show the solution to each on a number line.a x + 7 > 11 b a 5 < 3 c 10 y 8

d 3m 21 e 15 < 4x f

g 2x + 1 > 5 h 7 3n > 4 i 5x + 6 > x + 18j 3x 5 < x + 6 k 3 a < 5 2a l 3(m + 4) < 2(m + 6)

m n o

p q r

s t u

a If s = ut + at2, find s if u = 9, t = 4 and a = 7.b Given F = p + qr, find F if p = 23, q = 39 and r = 09.c For the formula T = a + (n 1)d, find T if a = 92, n = 6 and d = 13.

a Given that V = LBH, evaluate B when V = 432, L = 12 and H = 09.

b It is known that . Find a when S = 25 and r = 06.

c . Find C if F = 77.

d If v2 = u2 + 2as, find a if v = 21, u = 16 and s = 03.e Given that T = a + (n 1)d, find d if T = 246, a = 88 and n = 4.

Change the subject of each formula to y.

a b

c d

A:08D

A:08E

A:08E

A:08F

A:08F

A:08G

6

7

4 3 2 1 0 1 2 3 2 1 0 1 2 3 4 5

3 4 5 6 7 8 9 10 6 5 4 3 2 1 0 1

8

m4----

< 1

x2--- 1 6 1 5

2y3

------ < 6

p 14

------------ 2 4 x

3-----------

> 1

x2---

x3--- 5>+ a

4---

a2---

< 6+ x2---

2x3

------ < 3

9 12---

10

S a1 r-----------=

F 32 9C5

-------+=

The subject goeson the left.

11xa--

yb--+ 1= ay2 x=

T By---= ay by 1=



1:09 | Consumer Arithmetic Outcomes NS512, NS522

a Michelle is paid $8.40 per hour and time-and-a-half for overtime. If a normal days work is 7 hours, how much would she be paid for 10 hours work in one day?

b Jake receives a holiday loading of 17 % on four weeks normal pay.If he works 37 hours in a normal week and is paid $9.20 per hour,how much money does he receive as his holiday loading?

c In a week, a saleswoman sells $6000 worth of equipment. If she is paid$150 plus 10% commission on sales in excess of $4000, how muchdoes she earn?

d A waiter works from 5:00 pm till 1:30 am fourdays in one week. His hourly rate of pay is $14.65and he gets an average of $9.20 as tips per working night. Find his income for the week.

a Find the net pay for the week if Saransh earns $420.80, is taxed $128.80, pays $42.19 for superannuation and his miscellaneous deductionstotal $76.34. What percentage of his gross pay did he pay in tax? (Answer correct to decimal place of 1 per cent.)

b Find the tax payable on a taxable income of $40 180 if the tax is $2380 plus 30 cents for each $1 in excess of $20 000.

c James received a salary of $18 300 and from investments an income of $496. His total tax deductions were $3050. What is his taxable income?

d Toms taxable income for the year was $13 860. Find the tax which must be paid if it is 17 cents for each $1 in excess of $6000.

a An item has a marked price of $87.60 in two shops. One offers a 15% discount, and the other a discount of $10.65. Which is the better buy,and by how much?

b Emma bought a new tyre for $100, Jade bought one for $85 and Diane bought a retread for $58. If Emmas tyre lasted 32 000 km, Jades 27 500 km, and Dianes 16 000 km, which was the best buy? (Assume that safety and performance for the tyres are the same.)

c Alice wants to get the best value when buying tea. Which will she buy if Pa tea costs $1.23 for 250 g, Jet tea costs $5.50 for 1 kg and Yet tea costs$3.82 for 800 g?

a Find the GST (10%) that needs to be added to a base price of:i $75 ii $6.80 iii $18.75

b For each of the prices in part a, what would the retail price be? (Retail price includes the GST.)

c How much GST is contained in a retail price of:i $220? ii $8.25? iii $19.80?

Exercise 1:09

A:09A

CD Appendix

A:09D

A:09B

A:09C

1

12---

2

3

4



a What is meant by the expressionbuying on terms?

b Find the amount Jason will pay for a fishing line worth $87 if he pays $7 deposit and $5.69 per month for 24 months. How muchextra does he pay in interest charges?

c Nicholas was given a discount of 10% on the marked price of a kitchen table. If the discount was$22, how much was the marked price?

d A factorys machinery depreciates at a rate of 15% per annum. If it is worth $642 000, what will be its value after one year?

e The price of a book was discounted by 20%. A regular customer was given a further discount of 15%. If the original price was $45, what was the final price of the book?

a The cost price of a DVD player was $180 and it was sold for $240.What was:i the profit as a percentage of the cost price?ii the profit as a percentage of the selling price?

b A new car worth $32 000 was sold after two years for $24 000. What was:i the loss as a percentage of the original cost price?ii the loss as a percentage of the final selling price?

1:10 | Coordinate Outcomes PAS45, PAS512, PAS523, PAS533Geometry

Find the gradient of the line that passes through the points:a (1, 2) and (1, 3)b (1, 7) and (0, 0)c (3, 2) and (5, 2)

Find the midpoint of the intervaljoining:a (2, 6) and (8, 10)b (3, 5) and (4, 2)c (0, 0) and (7, 0)

a Find the distance between(1, 4) and (5, 2).

b A is the point (5, 2). B is the point (2, 6). Find the distance AB.c Find the distance between the origin and (6, 8).d Find the distance AB between A(2, 1) and B(5, 3).

Im flounderingfor money

can I pay in fish?Yes, but theres a catch

Ill need it in whiting!

A:09E

A:09F

5

6

Exercise 1:10

A:10A

CD Appendix

A:10C

A:10B

Horizontallines have

a gradient ofzero (m = 0).

1

2

3



Sketch each of these lines on a number plane.a y = 3x + 4 b 2x + 3y = 12 c y = 3x d x = 2

Find the gradient and y-intercept of the lines:a y = 3x + 5 b y = x 2 c y = 2x + 5

Write each equation in question 4 in the general form.

Write the equation of the line that has:a gradient 5 and y-intercept 2b gradient 0 and y-intercept 4c gradient 2 and passes through (0, 5)d gradient 1 and passes through (2, 3).

Write the equation of the line that passesthrough:a (2, 1) and (4, 2)b (1, 5) and (3, 1)

a Which of the lines y = 3x, 2x + y = 3 and y = 3x 1 are parallel?b Are the lines y = 2x 1 and y = 2x + 5 parallel?c Show that the line passing through (1, 4) and (4, 2) is parallel to the line

passing through (4, 0) and (1, 2).d Which of these lines are parallel to the y-axis?

{y = 4, y = x + 1, x = 7, y = 5x 5, x = 2}

a Are y = 4x and y = x 2 perpendicular?b Are y = 3 and x = 4 perpendicular?c Which of the lines y = x + 1, y = x 1 and y = x 7 are perpendicular?d Show that the line passing through (0, 5) and (3, 4) is perpendicular to

y = 3x 8

Sketch the regions corresponding to the inequations.a x > 2 b y 1 c x + y < 2d y 2x e y < x 1 f 2x + 3y 6

a Graph the region described by the intersection of y > x + 1 and y < 3.b Graph the region described by the union of y > x + 1 and y < 3.

A:10D

A:10H

A:10I

A:10E

A:10E

A:10F

A:10G

A:10H

A:10I

4

5

6

7

8

9

10 14---

12---

11

12



1:11 | Statistics Outcomes DS41, DS42, DS511

In a game, a dice was rolled 50 times, yielding the results below. Organise theseresults into a frequency distribution table and answer the questions.

5 4 1 3 2 6 2 1 4 55 1 3 2 6 3 2 4 4 16 2 5 1 6 6 6 5 3 26 3 4 2 4 1 4 2 4 42 3 1 5 4 2 2 3 2 1

a Which number on the dice was rolled most often?b Which number had the lowest frequency?c How often did a 3 appear?d For how many throws was the result an odd number?e On how many occasions was the result greater than 3?

Use the information in question 1 to draw, on separate diagrams:a a frequency histogram b a frequency polygon

a For the scores 5, 1, 8, 4, 3, 5, 5, 2, 4, find:i the range ii the mode iii the mean iv the median

b Use your table from question 1 to find, for the scores in question 1:i the range ii the mode iii the mean iv the median

c Copy your table from question 1 and add a cumulative frequency column.i What is the cumulative frequency of the score 4?ii How many students scored 3 or less?iii Does the last figure in your cumulative frequency column equal the total

of the frequency column?

Use your table in question 3 to draw on the same diagram:a a cumulative frequency histogram b a cumulative frequency polygon

The number of cans of drink sold by a shop each day was as follows:30 28 42 21 54 47 36 37 22 1825 26 43 50 23 29 30 19 28 2040 33 35 31 27 42 26 44 53 5029 20 32 41 36 51 46 37 42 2728 31 29 32 41 36 32 41 35 4129 39 46 36 36 33 29 37 38 2527 19 28 47 51 28 47 36 35 40The highest and lowest scores are circled.a Tabulate these results using classes of 1622, 2329, 3036, 3743, 4450,

5157. Make up a table using these column headings: Class, Class centre, Tally, Frequency, Cumulative frequency.

b What was the mean number of cans sold?c Construct a cumulative frequency histogram and cumulative frequency

polygon (or ogive) and find the median class.d What is the modal class?e Over how many days was the survey held?

Exercise 1:11

A:11A

CD Appendix

A:11B

A:11C

A:11D

A:11E

1

2

3

4

5



1:12 | Simultaneous Outcome PAS522Equations

Find the value of y if:a x + y = 12 and x = 3b 2x 4y = 1 and x = 4

Find the value of x if:a y = 5x 4 and y = 21b 3x + y = 12 and y = 6

a Does the line 2x 4y = 12pass through the point(14, 4)?

b Does the point (4, 8) lie on the line 6x 2y = 7?

Use the graph to solve these pairs of simultaneous equations.

a b

c d

e f

Solve these simultaneous equations by the substitution method.

a b c d

Solve these simultaneous equations by the elimination method.

a b c d

A theatre has 2100 seats. All of the rows of seats in the theatre have either 45 seatsor 40 seats. If there are three times as many rows with 45 seats as there are with 40 seats,how many rows are there?

Fiona has three times as much money asJessica. If I give Jessica $100, she will havetwice as much money as Fiona. How much did Jessica have originally?

Exercise 1:12

A:12A

CD Appendix

A:12A

A:12B

A:12C

A:12D

A:12A

A:12A

A:12D

1

2

3

y = 2

y = 2x 6x + y + 6 = 0

y = 12x

6

2

2

6

8 4 4 8

y

x

4

y 2=

y 2x 6= y 2=

x y 6+ + 0=

y 12---x=

x y 6+ + 0= y 12---x=

y 2x 6=

y 2x 6=

x y 6+ + 0= y 2=

y 12---x=

5

2x y+ 12=

3x 2y+ 22= 4x 3y 13=

2x y 9+= y x 2=

2x y+ 7= 4a b 3=

2a 3b+ 11=

6

5x 3y 20=

2x 3y+ 15= 4a 3b 11=

4a 2b+ 10= 3c 4d+ 16=

7c 2d 60= 2x 7y+ 29=

3x 5y+ 16=

7

8



1:13 | Trigonometry Outcomes MS512, MS523, MS532

Write down the formula for:a sin b cos c tan Use the triangle on the right to find the value of eachratio. Give each answer as a fraction.a sin A b cos A c tan A

Use the triangle on the right to give, correct to three decimal places, the value of:a sin A b cos A c tan A

Use your calculator to find (correct to three decimal places) the value of:a sin 14 b sin 8 c sin 8530 d sin 3027e cos 12 f cos 6 g cos 8815 h cos 6050i tan 45 j tan 7 k tan 8707 l tan 3527

Find the value of x (correct to two decimal places) for each triangle. (All measurements are in metres.)a b c

d e f

g h

i j

Exercise 1:13

A:13A

CD Appendix

A:13A

A:13A

A:13B

A:13C

1

C

B A

13

12

5

2

3

4

5

4265

31

x481

60x 711

49

x

1562

5520'

x

8377315'

x637

5608'x

For these fourtriangles, you need tofind the hypotenuse.

105

61x

97

27x

1763

5935'x

2764312'

x



For each figure, find the size of angle .(Answer to the nearest minute. Measurements are in centimetres.)

a

b c d

a The angle of depression of an object ona level plain is observed to be 19 fromthe top of a 21 m tower. How far fromthe foot of the tower is the object?

b The angle of elevation of the top of a vertical cliff is observed to be 23 froma boat 180 m from the base of the cliff.What is the height of the cliff?

a A ship sails south for 50 km, then 043until it is due east of its starting point.How far is the ship from its startingpoint (to the nearest metre)?

b If the town of Buskirk is 15 km north and 13 km east of Isbister, find thebearing of Buskirk from Isbister.

1:14 | Graphs of Physical Outcome PAS525Phenomena

The travel graph shows the journeys of James and Callumbetween town A and town B.(They travel on the same road.)a How far from A is Callum when

he commences his journey?b How far is James from B at

2:30 pm?c When do James and Callum

first meet?d Who reaches town B first?e At what time does Callum

stop to rest?

A:13D

A:13E

A:13E

Make sureyour calculator

is set todegrees mode.

6

52 63

2872

2335

107 94

19

21 m

7

8

Exercise 1:14

A:14A

CD Appendix

1

10 11 3 4noonTime

Signpost 10

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Transcript of Signpost 10