NSM10_51_53

Sydney, Melbourne, Brisbane, Perth andassociated companies around the worldAlan McSevenyRob ConwaySteve Wilkes5.1_5.3_Ch00.indd 1 12/7/05 9:50:47 AMUnderstanding is a fountain of life to those who have it.Proverbs 16:22Pearson Education AustraliaA division of Pearson Australia Group Pty LtdLevel 9, 5 Queens RoadMelbourne 3004 Australiawww.pearsoned.com.au/schoolsOffices in Sydney, Brisbane and Perth, and associated companies throughout the world.Copyright Pearson Education Australia(a division of Pearson Australia Group Pty Ltd) 2005First published 2005Reprinted 2006All 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 dataMcSeveny, 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.510.76AcknowledgementsWe 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 publisherstender their apologies and invite copyright owners to contact them. 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 xxiiiReview 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 61:02 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? 43Maths 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 86Maths terms Diagnostic test Revision assignment Working mathematically 87 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 Maths terms Diagnostic test Revision assignment Working mathematically 128Chapter 1Chapter 2Chapter 3Chapter 4 5.1_5.3_PrelimsPage iiiTuesday, July 12, 20059:29 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 Maths terms Diagnostic test Revision assignment Working mathematically 168 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 Maths terms Diagnostic test Revision assignment Working mathematically 198 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 Maths terms Diagnostic test Revision assignment Working mathematically 231 Similarity 237 8:01 Review of similarity 2388:02 Similar triangles 243A 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 Maths terms Diagnostic test Revision assignment Working mathematically 303 Further Algebra 307 10:01 Simultaneous equations involving a quadratic equation 30810:02 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 329Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11 5.1_5.3_PrelimsPage ivTuesday, July 12, 20059:29 AM 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 36812:02 Curves of the form y = ax n and y = a ( x r ) n 37212:03 Curves of the form 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 Maths terms Diagnostic test Revision assignment Working mathematically 412 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 443Maths terms Diagnostic test Revision assignment Working mathematically 444 Answers 449Answers to ID Cards 532Index 533Chapter 12Chapter 13Chapter 14 5.1_5.3_PrelimsPage vTuesday, July 12, 20059:29 AM vi NEW SIGNPOST MATHEMATICS 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 912:01 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 31210:03 Understanding variables 31812:01 Curves of the form y = axn and y = axn + d37012:02 Curves of the form y = axn and y = a(x r)n37412:03 Equations of the form y = a(x r)(x s)(x t) 377The 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 decayDrag and DropsChapter 2: Quadratic equations 1, Quadratic equations 2, Completing the square Chapter 3: Theoretical probability, Maths terms 3, Probability and cardsChapter 4: Compound interest, Depreciation, Maths terms 4, Reducible interestChapter 5: Parabolas, Maths terms 5, Identifying graphsChapter 6: Maths terms 6, Volumes of pyramids, Volumes of cylinders, cones and spheresChapter 7: Maths Terms 7, Box-and-whisker plots, Interquartile rangeChapter 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 CoursebookAppendixesFoundation WorksheetsTechnology ApplicationsYou 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_PrelimsPage viTuesday, July 12, 20059:29 AMviiChapter 13: Maths terms 13, PolynomialsChapter 14: Logarithms, Function notation, Maths terms 14AnimationsChapter 6: The box, Greatest volumeChapter 8: Scale itChapter 9: Unit circleChapter 11: Spin graphsChapter 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_PrelimsPage viiTuesday, July 12, 20059:29 AMviii NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3New 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 CDCoursebookChapter-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 . . . foundationblue . . . Stage 5.3 levelred . . . extensionWorked examples are used extensively and are easy for students to identify. Features of New Signpost Mathematicsprepquiz1 4 9Find the simple interest charged for a loan of:a $620 at 18% pa for 4 years b $4500 at 26% pa for 5 yearsAfter factorising the left-hand side of each equation, solve the following.a x2 + 3x = 0 b m2 5m = 0 c y2 + 2y = 0If (x + 1) and (x + 2) are both factors of x3 + ax2 + bx 10, find the values of a and b.129worked examplesFind the monthly repayments on a loan of $280 000 taken over 20 years at 75% pa.Monthly repayment= 280 $8.06= $2256.805.1_5.3_PrelimsPage viiiTuesday, July 12, 20059:29 AMixImportant 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 Factorisea x2 3x b x2 + 3x + 22 Solvea x(x 4) = 0 b (x 1)(x + 2) = 0Foundation Worksheet 2:01challengefun spotnoitag i t s e vnidiagnostictestassignmentees5.1_5.3_PrelimsPage ixTuesday, July 12, 20059:29 AMx NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3The 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 CDThis 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 WebsiteThe 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.d ism r etshtamsh t amgnidaer5.1_5.3_PrelimsPage xTuesday, July 12, 20059:29 AMxi 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 BookHomework 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 upTeachers 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 InvestigationsMost of this material is also available in the Teachers Resource Centre of the Companion Website.Sample Drag and DropSample Animation5.1_5.3_PrelimsPage xiTuesday, July 12, 20059:29 AMxii NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3Treatment 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 ReferencesWMS5.3.1 Asks questions that could be explored using mathematics in relation to Stage 5.3 content.Assignments B, and throughout the textWMS5.3.2 Solves problems using a range of strategies including deductive reasoning.Assignments B, and throughout the textWMS5.3.3 Uses and interprets formal definitions and generalisations when explaining solutions and or conjectures.Assignments B, and throughout the textWMS5.3.4 Uses deductive reasoning in presenting arguments and formal proofs.Assignments B, and throughout the textWMS5.3.5 Links mathematical ideas and makes connections with, and generalisations about, existing knowledge and understanding in relation to Stage 5.3 content.Assignments B, and throughout the textNS4.2 Compares, orders and calculates with integers. Year 9, 1:01NS4.3 Operates with fractions, decimals, percentages, ratios and rates.Year 9, 1:01NS5.1.1 Applies index laws to simplify and evaluate arithmetic expressions and uses scientific notation to write large and small numbers.Year 9, 1:05NS5.1.2 Solves consumer arithmetic problems involving earning and spending money.Year 9, 1:09, 4:014:03NS5.1.3 Determines relative frequencies and theoretical probabilities.Year 9, 1:03, 3:01, 3:06NS5.2.1 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, 1:015.1_5.3_PrelimsPage xiiTuesday, July 12, 20059:29 AMxiiiNS5.2.2 Solves consumer arithmetic problems involving compound interest, depreciation and successive discounts.Year 9, 1:09, 4:044:08NS5.3.1 Performs operations with surds and indices. Year 9, 1:06NS5.3.2 Solves probability problems involving compound events.3:023:05PAS4.3 Uses the algebraic symbol system to simplify, expand and factorise simple algebraic expressions.Year 9, 1:02PAS4.4 Uses algebraic techniques to solve linear equations and simple inequalities.Year 9, 1:08PAS4.5 Graphs and interprets linear relationships on the number plane.Year 9, 1:10PAS5.1.1 Applies the index laws to simplify algebraic expressions.Year 9, 1:05PAS5.1.2 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, 5:01PAS5.2.1 Simplifies, expands and factorises algebraic expressions involving fractions and negative and fractional indices.Year 9, 1:02, 1:05PAS5.2.2 Solves linear and simple quadratic equations, solves linear inequalities and solves simultaneous equations using graphical and analytical methods.Year 9, 1:08, 1:12PAS5.2.3 Uses formulae to find midpoint, distance and gradient and applies the gradientintercept form to interpret and graph straight lines.Year 9, 1:10PAS5.2.4 Draws and interprets graphs including simple parabolas and hyperbolas.5:01, 5:03PAS5.2.5 Draws and interprets graphs of physical phenomena. Year 9, 1:14PAS5.3.1 Uses algebraic techniques to simplify expressions, expand binomial products and factorise quadratic expressions.Year 9, 1:02PAS5.3.2 Solves linear, quadratic and simultaneous equations, solves and graphs inequalities, and rearranges literal equations.Year 9, 1:08, 2:012:06, 10:0110:03PAS5.3.3 Uses various standard forms of the equation of a straight line and graphs regions on the number plane.Year 9, 1:105.1_5.3_PrelimsPage xiiiTuesday, July 12, 20059:29 AMxiv NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3PAS5.3.4 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, 5:045:08PAS5.3.5 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 14DS4.1 Constructs, reads and interprets graphs, tables, charts and statistical information.Year 9, 1:11DS4.2 Collects statistical data using either a census or a sample and analyses data using measures of location and range.Year 9, 1:11, 7:01DS5.1.1 Groups data to aid analysis and constructs frequency and cumulative frequency tables and graphs.Year 9, 1:11, 7:01DS5.2.1 Uses the interquartile range and standard deviation to analyse data.7:027:05MS4.2 Calculates surface area of rectangular and triangular prisms and volume of right prisms and cylinders.6:01MS5.1.1 Uses formulae to calculate the area of quadrilaterals and finds areas and perimeters of simple composite figuresYear 9, 1:07MS5.1.2 Applies trigonometry to solve problems (diagrams given) including those involving angles of elevation and depression.Year 9, 1:13MS5.2.1 Finds areas and perimeters of composite figures. Year 9, 1:07MS5.2.2 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, 6:056:08MS5.2.3 Applies trigonometry to solve problems including those involving bearings.Year 9, 1:135.1_5.3_PrelimsPage xivTuesday, July 12, 20059:29 AMxvThe 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, 8:06MS5.3.2 Applies trigonometric relationships, sine rule, cosine rule and area rule in problem-solving.Year 9, Chapter 9SGS4.4 Identifies congruent and similar two-dimensional figures stating the relevant conditions.8:01SGS5.2.1 Develops and applies results related to the angle sum of interior and exterior angles for any convex polygon.Year 9, 1:04SGS5.2.2 Develops and applies results for proving that triangles are congruent or similar.Year 9, 1:04, 8:02, 8:03SGS5.3.1 Constructs arguments to prove geometrical results. Year 9, 1:04SGS5.3.2 Determines properties of triangles and quadrilaterals using deductive reasoning.Year 9, 1:04SGS5.3.3 Constructs geometrical arguments using similarity tests for triangles8:04#SGS5.3.4 Applies deductive reasoning to prove circle theorems and to solve problems.Chapter 115.1_5.3_PrelimsPage xvTuesday, July 12, 20059:29 AMxvi NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3Metric EquivalentsMonths 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, NovemberLength1 m = 1000 mm= 100 cm= 10 dm1 cm = 10 mm1 km = 1000 mArea1 m2 = 10 000 cm21 ha = 10 000 m21 km2 = 100 haMass1 kg = 1000 g1 t = 1000 kg1 g = 1000 mgVolume1 m3 = 1 000 000 cm3= 1000 dm31 L = 1000 mL1 kL = 1000 L1 m3 = 1 kL1 cm3 = 1 mL1000 cm3 = 1 LTime1 min = 60 s1 h = 60 min1 day = 24 h1 year = 365 days1 leap year = 366 daysIt is importantthat you learnthese factsoff by heart.5.1_5.3_PrelimsPage xviTuesday, July 12, 20059:29 AMxviiThe 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)1m2dm3cm4mm1=2 or 3489ha10m311cm312s942104311 1213min14h15m/s16km/h13 14||15 16|||17g18mg19kg20t17%1819eg20ie21L22mL23kL24C212223 24P(E)di2 23xSee MathsTerms atthe end ofeach chapter.5.1_5.3_PrelimsPage xviiTuesday, July 12, 20059:29 AMxviii NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3See page 532 for answers..ID Card 3 (Language)16 minus 22the sum of6 and 23divide6 by 24subtract2 from 65the quotient of6 and 2632)6the divisoris . . . .732)6the dividendis . . . .86 lots of 29decrease6 by 210the productof 6 and 2116 more than 2122 less than 6136 squared14the squareroot of 36156 take away 216multiply6 by 217average of6 and 218add 6 and 2196 to thepower of 2206 less 221the differencebetween 6 and 222increase6 by 223share6 between 224the total of6 and 2d iWe say six squaredbut we write62.5.1_5.3_PrelimsPage xviiiTuesday, July 12, 20059:29 AMxixSee page 532 for answers.ID Card 4 (Language)1 2 3 45 6 7 89 10 11 1213 14 15 1617 18 19 2021 22 23 24diAll sidesdifferent5.1_5.3_PrelimsPage xixTuesday, July 12, 20059:29 AMxx NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3See page 532 for answers.ID Card 5 (Language)1A............2............3............4............5............ points6C is the ............7........................8............9all angles lessthan 9010one angle 9011one angle greaterthan 9012A, B and C are......... of the triangle.13Use the verticesto name the .14BC is the ......... ofthe right-angled .15a + b + c = .........16BCD = .........17a + b + c + d = .....18Which (a) a < bis true? (b) a = b(c) a > b19a = .............20Angle sum = ............21AB is a ...............OC is a ...............22Name of distancearound the circle..............................23.............................24AB is a ...............CD is an ...............EF is a...............d iABABABPQRSA C B 4 2 0 2 4ABC ABCABCA BCbcaA DBCbabdcab aaA BCOOOBCDFEA5.1_5.3_PrelimsPage xxTuesday, July 12, 20059:29 AMxxiSee page 532 for answers.ID Card 6 (Language)1..................... lines2..................... lines3v .....................h .....................4..................... lines5angle .....................6..................... angle7..................... angle8..................... angle9..................... angle10..................... angle11.....................12..................... angles13..................... angles14..................... angles15..................... angles16a + b + c + d = .....17.....................18..................... angles19..................... angles20..................... angles21b............ an interval22b............ an angle23CAB = ............24CD is p.......... to AB.diABC(lessthan90)(90)(between90 and 180)(180)(between180 and360)(360)a + b = 90aba + b = 180a ba = ba babcda = baba = baba + b = 180abA BCDEABCDA BCA BDC5.1_5.3_PrelimsPage xxiTuesday, July 12, 20059:29 AMxxii NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3See page 532 for answers.ID Card 7 (Language)1a............ D............2b............ C............3a............ M............4p............ m............5area is 1 ............6r............ shapes7............ of a cube8c............-s............9f............10v............11e............12axes of ............13r............14t............15r............16t............17The c............of the dot are E2.18t............19p............ graph20c............ graph21l............ graph22s............ graph23b............ graph24s............ d............d iAD BC am pm100 m100 m43210A B C D E FCars soldMonTuesWedThursFriMoney collectedMonTuesWedThursFriStands for $1070503010M T W T FDollarsMoney collected10080604020Johns height1 2 3 4 5Age (years)Use of timeHobbiesSleepHomeSchoolPeople presentAdultsGirlsBoysSmokingCigarettes smokedLength of life5.1_5.3_PrelimsPage xxiiTuesday, July 12, 20059:29 AMxxiiiAlgebra CardHow 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 + 12mA B C D E F G H I J K L M N O1 3 21 m 3m 5m25x 3x x + 2 x 3 2x + 1 3x 82 1 04 4m 2m 2m33x 5x2x + 7 x 6 4x + 2 x 13 5 08 10m 5m 8m510x 8x x + 5 x + 5 6x + 2 x 54 2 15 8m 7m 6m215x 4x4x + 1 x 9 3x + 3 2x + 45 8 25 2m 10m m27x 2x3x + 8 x + 2 3x + 8 3x + 16 10 07 5m 6m9m39x x2x + 4 x 7 3x + 1 x + 77 6 12 8m 9m 2m66x 5x2x + 6 x 1 x + 8 2x 58 12 05 20m 4m3m312x 4x3x + 10 x 8 5x + 2 x 109 7 01 5m 10m m75x 3x5x + 2 x + 5 2x + 4 2x 410 5 06 9m 7m8m43x 7x5x + 1 x 7 5x + 4 x + 71111 18 7m 8m 4m 4x x3x + 9 x + 6 2x + 7 x 612 4 14 3m 12m 7m27x x10x + 3 x 10 2x + 3 2x + 314---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_PrelimsPage xxiiiTuesday, July 12, 20059:29 AMReview of Year 9 1 1Talk aboutdj`a 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 hereAppendix A 5.1_5.3_Chapter 01Page 1Tuesday, July 12, 20058:48 AM 2 NEW SIGNPOST MATHEMATICS 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 fg (6 + 3) 2 h (10 + 4)2i (19 9)2B | FractionsChange to mixed numerals.a b c dChange to improper fractions.a 5 b 3 c 8 d 66Simplify the fractions.a b c dComplete the following equivalent fractions.a=b=c=d= a+bc+d a 6+ 2 b 4 2 c 4+ 6 d 5 1abcdof a 6 b 2 1 c 1 15 d 10 1abc 6 d 2 1Exercise 1:01AA:01AA:01ACD Appendix1230 10 +30 10 ------------------15 45 +45 5 +------------------1414 7 ---------------Exercise 1:01BA:01B1CD AppendixA:01B2A:01B3A:01B4A:01B5A:01B6A:01B7A:01B8A:01B9174---496------154------118------212---17---34---23---34880------70150---------200300---------250450---------434---24------25---50------27---28------13---120---------5715------115------1320------25---58---310------67---35---612---35---34---310------34---110------38---910------735---47---1825------1516------49---310------710------23---834---12---45---13---12---37---9910------23---38---35---45---34---12---5.1_5.3_Chapter 01Page 2Tuesday, July 12, 20058:48 AMCHAPTER 1 REVIEW OF YEAR 9 3C | DecimalsPut 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)2a 314 10 b 05 1000 c 00003 100 d 38 104a 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 104a 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 dExpress 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 dD | PercentagesExpress as a fraction.a 54% b 203% c 12 % d 91%Express as a percentage.a b c 1 dExpress 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 $32d 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 1:01CA:01C1CD AppendixA:01C2A:01C3A:01C4A:01C5A:01C6A:01C7A:01C8A:01C9A:01C10A:01C11A:01C111234567891045---7200---------58---811------11072064212043508942Exercise 1:01DA:01D1CD AppendixA:01D2A:01D3A:01D4A:01D5114---21120------49---14---23---314---4512---5.1_5.3_Chapter 01Page 3Tuesday, July 12, 20058:48 AM4 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3a 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 isthe percentage (by weight) of flour in the mixture?c Of 32 birds in Rachels aviary, 6 are canaries. What percentage of her birdsare canaries?d When Steve Waugh retired from test cricket in 2003, he had scored32 centuries from 260 innings. In what percentage of his innings did hescore centuries?E | Ratioa Simplify each ratio.i $15 : $25 ii 9 kg : 90 kg iii 75 m: 35 m iv 120 m2: 40 m2b 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 ratioof those who walk to those who ride bicycles.e At the end of their test cricket careers, Steve Waugh had scored 50 fiftiesand 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 ofAustralia 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:01D6A:01D767Exercise 1:01EA:01E1CD AppendixA:01E2115---25.1_5.3_Chapter 01Page 4Tuesday, July 12, 20058:48 AMCHAPTER 1 REVIEW OF YEAR 9 5a 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 | RatesComplete 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/kga 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 figuresState 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 000State 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 3.0300A:01E33Exercise 1:01FA:01FCD AppendixA:01F12Exercise 1:01GA:01GCD AppendixA:01G125.1_5.3_Chapter 01Page 5Tuesday, July 12, 20058:48 AM6 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3H | ApproximationsApproximate 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 275Approximate 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 0005Approximate 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 00359Approximate 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 | EstimationGive 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 089g (093 + 172)(85 17) h ij k lm n o 3.13 184p q rsExercise 1:01HA:01HCD AppendixA:01HA:01HA:01HA:01H12345Exercise 1:01IA:01ICD Appendix1437 182 +78 29 +-----------------------------1016 517 213 148 --------------------------------068 51 025 78 ------------------------116 392 127 658 +----------------------------- 3522179 417 56 426 1056 41 4812 2623---------------------------------1572113 31 --------------------------167215-----------41647----------- +065001-----------075 36 0478-------------------------- 5.1_5.3_Chapter 01Page 6Tuesday, July 12, 20058:48 AMCHAPTER 1 REVIEW OF YEAR 9 71:02 | Algebraic Outcomes PAS43, PAS521, PAS531ExpressionsBeing 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 mDouble-checkyouralgebraskills!Double-checkyouralgebraskills!GeneralisationWhat is the average ofa and b?Answer: Average = SubstitutionFind the value of2x + y2 if x = 3, y = 2.Answer: 2(3) + (2)2= 10a b +2------------Fractions1= = 2= 2x3------x5--- +5 2x 3 x + 15----------------------------------13x15---------5a16b1--------183b2b10a2---------------- 3b2------Simplifying expressions1 3x2 + 5x + x2 3x= 4x2 + 2x2 12xy 8xz= = 123xy82xz---------------3y2z------Products1 5(x + 3) 2(x 5)= 5x + 15 2x + 10= 3x + 252 (3x 1)(x + 7)= 3x2 + 20x 7Factorisation1 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 1:02A:02ACD Appendix1 Wipe that expressionoff your face!5.1_5.3_Chapter 01Page 7Tuesday, July 12, 20058:48 AM8 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3If 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 b2f a2 + c2g h ij k lSimplify 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 10xSimplify these fractions.a b cd e fg h ij k lExpand 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)2q (2m 1)2r (4a + 5)2s (x + y)(x 2y) t (a + 2b)(a 2b) u (m 3n)2Factorise:a 15a 10 b 3m2 6m c 4n + 6mnd 6mn 4m e 10y2 + 5y f 6a2 2a + 4abg x2 49 h 100 a2i 16a2 9b2j x2 + 8x + 12 k x2 x 12 l x2 6x + 8ma2 + 6a + 9 n y2 10y + 25 o 1 4m + 4m2p 2x2 + 7x + 3 q 3m2 + 7m 6 r 6a2 11a + 4s 4n2 + 12n + 9 t 25x2 10x + 1 u 9 24m + 16m2v ab 4a + xb 4x wx2 + ax 2x 2a x 2m2 + 6mn m 3nA:02BA:02CA:02DA:02EA:02GA:02F24ac------3ab3 c -----------3b c 2a--------------ab c +a b c + +2--------------------3c b a -----------342a5------4a5------ +6x7------4x7------ 3y---4y--- +a3---a4--- +2m3-------m5---- 32n------43n------ a3---b4--- m5----2m3------- 9a22x--------xy3a------ 5m----2m---- ab5------a10------ xyz------yz2----- 565.1_5.3_Chapter 01Page 8Tuesday, July 12, 20058:48 AMCHAPTER 1 REVIEW OF YEAR 9 9Factorise these expressions completely.a 2x2 18 b 4x2 + 4x 24 c 3a2 6a 3ab + 6bd 8n2 8n + 2 e 9 9q2f m4 m2g k4 16 h y3 + y2 + y + 1 i x3 x2 x + 1Factorise and simplify:a b cd e fSimplify:a bc de fSimplify each of the following.a bc dFactorise each denominator where possible and then simplify.a bc de fA:02GA:02HA:02HA:02IA:02I783a 12 +3------------------5x 15 x 3 ------------------a 5 +a27a 10 + +------------------------------m2m m21 -----------------n2n 6 n25n 6 + +---------------------------2x2x 3 4x29 --------------------------93x 15 +2------------------4xx 5 +------------ a29 a 3 --------------a 1 +a 3 +------------ 3x 6 +10x---------------x 2 +5x------------ m225 m25m --------------------m 5 +5m------------- a27a 12 + +a25a 4 + +------------------------------a26a 5 + +a212a 35 + +--------------------------------- n23n 4 3n248 ---------------------------n3n n24n +------------------ 101a 4 +------------1a 3 +------------ +32x 1 ---------------54x 3 +--------------- 3x 1 + ( ) x 2 + ( )----------------------------------2xx 2 + ( )-------------------- +5x 3 + ( ) x 5 + ( )----------------------------------3x 3 + ( ) x 4 + ( )---------------------------------- 111a21 --------------1a 1 +------------ +23x 6 +---------------1x24 -------------- 2x2x 6 +-----------------------3x24x 3 + +--------------------------- +6x2x 2 -----------------------3x22x 3 -------------------------- x 1 +x29 --------------x 1 x25x 6 +--------------------------- +n 5 +2n2n 1 +---------------------------n 3 2n25n 3 +------------------------------ 5.1_5.3_Chapter 01Page 9Tuesday, July 12, 20058:48 AM10 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3Fun 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 1Solve:N 5x + 1 = 13 xEH 5x 2(x + 3) = 12Find the value of b2 4ac if:Ha = 4, b = 10, c = 2 I a = 1, b = 5, c = 7R a = 6, b = 9, c = 3 E a = 2, b = 3, c = 5Simplify:R 3x5 2x3A 15x5 5x4W (3x)2T 5x0 (3x)0Factorise:Mx2 81 S x2 8x 9 Ux2 9x Nx2 xy 9x + 9yfunspot1:02x 2 +5------------x 1 3----------- =9x27 5688x 67x2536x8(x 9)(x + 9)x2x(x 9)26x3x2x(x 9)(x y)4959(x 9)(x + 1)1 2 ---5.1_5.3_Chapter 01Page 10Tuesday, July 12, 20058:48 AMCHAPTER 1 REVIEW OF YEAR 9 111:03 | Probability Outcome NS513Using 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 50A 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 pinkA 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, SGS532a bFind x. Give reasons. Find the size of x. Give reasons.Number of matches 48 49 50 51 52Number of boxes 3 6 10 7 4Exercise 1:03A:03ACD AppendixA:03BA:03BA:03C1234 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 1:04A:04ACD Appendix1AE F G HCB D79 xAE F G HCBDx795.1_5.3_Chapter 01Page 11Tuesday, July 12, 20058:48 AM12 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3c dFind the size of x. Find the value of b.Give reasons. Give reasons.e fABDC 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 iic What is the sum of the exterior angles of an octagon?d Find the size of each exterior angle of these regular polygons.i iia bProve BED = ABC + CDE. O is the centre of the circle.Prove that AOC = 2 ABC.(Hint: AO = BO = CO (radii).)A:04BA:04CAE FCB D55x130BAC D b36B ACDFEx105xy14023A BD CEABDC O5.1_5.3_Chapter 01Page 12Tuesday, July 12, 20058:48 AMCHAPTER 1 REVIEW OF YEAR 9 13a bO 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 bIn 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 bFind the value of YZ. i Find an expression for AB.ii Find an expression for BC.iii Hence, find the value of x.A:04DA:04FA:04GA:04E4ABCOABCF DE5ABCDWXYZ6XWZY10 cm12 cm15 cmBA CD 4 cm 8 cmx cm5.1_5.3_Chapter 01Page 13Tuesday, July 12, 20058:48 AM14 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.31:05 | Indices Outcomes NS511, PAS511, PAS521Write in index form.a a a a b 2 2 2 2c n n n n n d 10 10 10Simplify, giving your answers in index form.a 24 25b a3 a2c m m4d 106 102e a10 a2f y4 y3g b3 b h 105 102i (m3)4j (a2)3k (x4)2l (105)2ma0 3 n b0 + c0o 6y0p e6 e6q 6a2 5 r 6m3 3 s 6a 5a t (4x4)2Simplify.a 6a4 5ab3b 7a2b2 8a3b c 4a2b3 6a2b4d 10a7 a3b3e (7x3)2f (2m2)4g (x2y3)3h (5xy2)4i 30a5 5a3j 100x4 20x k 36a3b4 12a2b4l 8y7z2 y7z2Rewrite without a negative index.a 41b 101c x1d 2a1e 52f 23g m3h 5x2Rewrite each of the following with a negative index.a b c de f g hFind the value of the following.a b c dRewrite, using fractional indices.a b c dSimplify these expressions.a x4 x2b 5m3 m2c 4n2 3n3d e fg h iWrite these numbers in scientific (or standard) notation.a 148 000 000 b 68 000 c 0000 15 d 0000 001 65Write these as basic numerals.a 62 104b 115 106c 74 103d 691 105Exercise 1:05A:05ACD AppendixA:05BA:05BA:05CA:05CA:05DA:05DA:05EA:05EA:05CA:05D1234513---18---1a---3x---124-----1106--------1y4-----5n3-----6912---3612---813---2713---7a y35 m 16x86y43y12--- 12x32---6x12--- 27x6( )13---5a44a510a8-----------------------6m42m ( )33m28m5-------------------------------9x32x3( )36x63x2 ------------------------------12---9105.1_5.3_Chapter 01Page 14Tuesday, July 12, 20058:48 AMCHAPTER 1 REVIEW OF YEAR 9 151:06 | Surds Outcome NS531Indicate whether each of the following is rational or irrational.a 6 b 131 c d 5162e f g hEvaluate each of the following to one decimal place.a b c dSimplify:a b c de f g hi j k lSimplify:a b cd e fg h iSimplify:a b cd e fExpand and simplify:a bc de fg hi jk lRewrite each fraction with a rational denominator.a b cd e fExercise 1:06A:06ACD AppendixA:06BA:06CA:06DA:06EA:06AA:06B135--- 323074927 5 2 + 11 3 32-------35 2 5 7 3 2 3 6 202----------426---------- 130 5 4981------7 ( )22 3 2 5 3 ( )28 6 2 So5, 37,2 + 3,11 -10are all surds.475 3 8 1804 3 7 3 6 5 5 2 2 +8 18 + 5 32 50 24 2 54 53 2 5 2 4 7 9 5 96 12 7 5 ( )22 4 3 18 12--------------------------- 3 2 3 5 ( )62 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 + ( )25 3 ( )22 3 3 2 + ( )25 2 ( ) 5 2 + ( )7 3 + ( ) 7 3 ( ) 5 3 2 2 ( ) 5 3 2 2 + ( )713-------55-------62-------13 2----------32 6----------2 5 +2 5----------------5.1_5.3_Chapter 01Page 15Tuesday, July 12, 20058:48 AM16 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.31:07 | Measurement Outcomes MS511, MS521, MS522Find the perimeter of the following figures. (Answer to 1 dec. pl.)a b cFind the area of each plane shape. (Answer to 2 dec. pl.)a b cd e fFind the area of the following shaded figures (correct to 3 sig. figs.).a b cFind the surface area of the following solids.Exercise 1:07A:07ACD AppendixA:07BA:07BA:07C1104 m28 m96 m62 m56 cm135O86 m(Use = 3142)256 m27 m106 cm48 m96 m36 cm78 cm51 cm28 cm42 cmA BD CAC = 36 cmBD = 64 cm341 m52 m101 m9 m3 m8 m965 km517 km314 km4113 cm6 cm68 cmRectangular prisma12 m5 m15 m15 m4 mTrapezoidal prismb10 m9 m126 mx mTriangular prism(Note: Use Pythagorastheorem to find x).c5.1_5.3_Chapter 01Page 16Tuesday, July 12, 20058:48 AMCHAPTER 1 REVIEW OF YEAR 9 17For 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.)abc 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 FormulaeSolve the following.a a + 7 = 25 b m 6 = 1 c 5x = 75 d 10 y = 12e 3p = 7 f g 2x + 3 = 7 h 8m + 5 = 21i 5y + 2 = 3 j 9k 1 = 5 k 5 + 3x = 11 l 15 2q = 8Solve 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 5mSolve 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) = 5Solve these equations.abcdefSolve these equations involving fractions.a b cd e fg h ij k lA:07DA:07EA:07E556 m22 m22 cm24 cm684 m18 m67Exercise 1:08A:08ACD AppendixA:08BA:08CA:08CA:08A1n4--- 3 =2345x2------ 10 =2a3------ 6 =3m5------- 4 =n 1 +5------------ 2 =x 4 2----------- 1 =2p 5 +3--------------- 1 =5a3---a3--- + 4 =2x5------x5--- 3 =5p3------p3--- 8 =q2---q3--- 6 =2k3------k4--- 10 =3x4------x2--- 15 =m 6 +3-------------2m 4 +4----------------- =n 3 2------------3n 5 4--------------- =5x 1 3---------------3 x 2----------- =x 3 +2------------x 5 +5------------ + 8 =m 2 +5-------------m 3 +6------------- 1 =3a 4 +2---------------a 1 3------------ 2a 3 +4--------------- =5.1_5.3_Chapter 01Page 17Tuesday, July 12, 20058:48 AM18 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3a 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, theresult is 6. What is the number?iii A rectangle is four times as long as it is wide. If it has a perimeter of340 m, what are its dimensions?Write the set of x that has been graphed below.a bc dSolve these inequations and show the solution to each on a number line.a x + 7 > 11 b a 5 < 3 c 10 y 8d 3m 21 e 15 < 4x fg 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 op q rs t ua 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 bc dA:08DA:08EA:08EA:08FA:08FA:08G674 3 2 1 0 1 2 3 2 1 0 1 2 3 4 53 4 5 6 7 8 9 10 6 5 4 3 2 1 0 18m4---- < 1x2--- 1 6 < +3x4------ 5 > 1 52y3------ < 6 p 1 4------------ 2 4 x 3----------- > 1x2---x3--- 5 > +a4---a2--- < 6 +x2---2x3------ < 3 912---10Sa1 r ----------- =F 329C5------- + =The subject goeson the left.11xa--yb-- + 1 = ay2x =TBy--- = ay by 1 =5.1_5.3_Chapter 01Page 18Tuesday, July 12, 20058:48 AMCHAPTER 1 REVIEW OF YEAR 9 191:09 | Consumer Arithmetic Outcomes NS512, NS522a 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 $18.75b 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 1:09A:09ACD AppendixA:09DA:09BA:09C112---2345.1_5.3_Chapter 01Page 19Tuesday, July 12, 20058:48 AM20 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3a 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, PAS533GeometryFind 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 moneycan I pay in fish?Yes, but theres a catchIll need it in whiting!A:09EA:09F56Exercise 1:10A:10ACD AppendixA:10CA:10BHorizontallines havea gradient ofzero (m = 0).1235.1_5.3_Chapter 01Page 20Tuesday, July 12, 20058:48 AMCHAPTER 1 REVIEW OF YEAR 9 21Sketch each of these lines on a number plane.a y = 3x + 4 b 2x + 3y = 12 c y = 3x d x = 2Find the gradient and y-intercept of the lines:a y = 3x + 5 b y = x 2 c y = 2x + 5Write 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 linepassing 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 toy = 3x 8Sketch 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 6a 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:10DA:10HA:10IA:10EA:10EA:10FA:10GA:10HA:10I4567891014---12---11125.1_5.3_Chapter 01Page 21Tuesday, July 12, 20058:48 AM22 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.31:11 | Statistics Outcomes DS41, DS42, DS511In 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 1a 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 polygona For the scores 5, 1, 8, 4, 3, 5, 5, 2, 4, find:i the range ii the mode iii the mean iv the medianb 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 medianc 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 totalof the frequency column?Use your table in question 3 to draw on the same diagram:a a cumulative frequency histogram b a cumulative frequency polygonThe 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 1:11A:11ACD AppendixA:11BA:11CA:11DA:11E123455.1_5.3_Chapter 01Page 22Tuesday, July 12, 20058:48 AMCHAPTER 1 REVIEW OF YEAR 9 231:12 | Simultaneous Outcome PAS522EquationsFind the value of y if:a x + y = 12 and x = 3b 2x 4y = 1 and x = 4Find the value of x if:a y = 5x 4 and y = 21b 3x + y = 12 and y = 6a 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 bc de fSolve these simultaneous equations by the substitution method.a b c dSolve these simultaneous equations by the elimination method.a b c dA 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 1:12A:12ACD AppendixA:12AA:12BA:12CA:12DA:12AA:12AA:12D123y = 2y = 2x 6x + y + 6 = 0y = 12x62268 4 4 8yx4y 2 =y 2x 6 = y 2 =x y 6 + + 0 =y12--- x =x y 6 + + 0 = y12--- x =y 2x 6 = y 2x 6 =x y 6 + + 0 =y 2 =y12--- x =52x y + 12 =3x 2y + 22 = 4x 3y 13 =2x y 9 + = y x 2 =2x y + 7 = 4a b 3 =2a 3b + 11 =65x 3y 20 =2x 3y + 15 = 4a 3b 11 =4a 2b + 10 = 3c 4d + 16 =7c 2d 60 = 2x 7y + 29 =3x 5y + 16 =785.1_5.3_Chapter 01Page 23Tuesday, July 12, 20058:48 AM24 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.31:13 | Trigonometry Outcomes MS512, MS523, MS532Write 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 AUse the triangle on the right to give, correct to three decimal places, the value of:a sin A b cos A c tan AUse 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 3527Find the value of x (correct to two decimal places) for each triangle. (All measurements are in metres.)a b cd e fg hi jExercise 1:13A:13ACD AppendixA:13AA:13AA:13BA:13C1CB A131252345426531x48160x71149x15625520'x8377315'x6375608'xFor these fourtriangles, you need tofind the hypotenuse.10561x9727x17635935'x2764312'x5.1_5.3_Chapter 01Page 24Tuesday, July 12, 20058:48 AMCHAPTER 1 REVIEW OF YEAR 9 25For each figure, find the size of angle .(Answer to the nearest minute. Measurements are in centimetres.)ab c da 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 PAS525PhenomenaThe 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:13DA:13EA:13EMake sureyour calculatoris set todegrees mode.6526328722335107941921 m78Exercise 1:14A:14ACD Appendix110 11 3 4 noonTimeDistance (km)1 2 5AB10203040050JamesCallum5.1_5.3_Chapter 01Page 25Tuesday, July 12, 20058:48 AM26 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3f How far does James travel?g How far apart are James and Callum when Callum is at town A?h How far does Callum travel?a What did the baby weigh at birth?b What was the babys weight at 4 weeks of age?c By how much did the babys weightincrease in the first two weeks of age?d By how much did the babys weightincrease from 2 weeks of age to 4 weeks of age?e Considering your answer to parts c and d, in which period, (02) or (24) was the babys rate of growth the greatest?Water is poured into each container at a constant rate. The graphs indicate the height of the water in each container against time. Match each graph with the correct container.a bc dA:14BA:14B0 2 4 6 8 10Age (weeks)Weight (g)30004000500060007000Babys weightincrease23hthtABhthtCD5.1_5.3_Chapter 01Page 26Tuesday, July 12, 20058:48 AMCHAPTER 1 REVIEW OF YEAR 9 27Chapter 1 | Working Mathematically1 Complete a table of values for each matchstick pattern below, and hence find the rule for each, linking the number of coloured triangles (t) to the number of matches (m).abc2 Divide this shape into three pieces that have the same shape.3 Ryan answered all 50 questions in a maths competition in which he received 4 marks for each correct answer but lost one mark for each incorrect answer.a What is Ryans score if he answered 47 questions correctly?b How many answers did he get right if his score was 135?4 It takes 3 min 15 s to join two pieces of pipe. How long would it take to join 6 pieces of pipe into one length?5 A number of cards can be shared between 4 people with no remainder. When shared between 5 or 6 people, there are two cards left over. If there are fewer than 53 cards, how many are there?6 From August 2003 to August 2004, the unemployment rate fell from 60% to 56%.a If the number of unemployed in August 2004 was 576 400, how many were unemployed in August 2003? Answer correct to four significant figures.b If 576 400 represents 56% of the total workforce, what is the size of the total workforce? Answer correct to four significant figures.c If the rate of 56% is only correct to one decimal place, the rate could really be from 555% to 565%. How many people does this approximation range of 01% represent?assignment1Appendix Bt 1 3 5mt 2 4 6mt 1 3 6m%60555065Aug2003Nov Feb2004May AugUnemployment rateTrendSource: Australian Bureau of Statistics, Labour Force, October 2004.1 Surd magicsquare2 Algebraicfractions5.1_5.3_Chapter 01Page 27Tuesday, July 12, 20058:48 AMQuadraticEquations 2 28x2 8x + 7 = 0(x 7)(x 1) = 0 x 7 = 0 or x 1 = 0x = 7 or 1... I just hopeits easy! Chapter Contents 2:01 Solution using factors PAS532 2:02 Solution by completing the square PAS532 2:03 The quadratic formula PAS532 Investigation: How many solutions?2:04 Choosing the best method PAS532 Fun Spot: What is an Italian referee?2:05 Problems involving quadratic equations PAS532 Investigation: Temperature and altitudeFun Spot: Did you know that 2 = 1?Maths Terms, Diagnostic Test, Revision Assignment, Working Mathematically Learning Outcomes PAS532 Solves linear, quadratic and simultaneous equations, solves and graphs inequalities, and rearranges literal equations.Working Mathematically Stage 5.31 Questioning, 2 Applying Strategies, 3 Communicating, 4 Reasoning, 5 Reflecting 5.1_5.3_Chapter 02Page 28Tuesday, July 12, 20058:49 AM CHAPTER 2 QUADRATIC EQUATIONS 29 2:01 | Solution Using Factors Outcome PAS532 A quadratic equation is one in which the highest power of the unknown pronumeral is 2. So equations such as: x 2 + 4 x + 3

0, x 2 + 5 x

0, x 2 25

0 and 2 x 2 3 x + 7

0are all quadratic equations.The solving of a quadratic equation depends on the following observation (called the Null Factor Law ).Factorise: 1 x 2 + 4 x + 3 2 x 2 5 x + 4 3 x 2 + 5 x 4 6 x 2 3 x 5 x 2 9 6 4 x 2 25Solve: 7 x + 2

0 8 3 x 1

0 9 5 x

0 10 2 x + 3

0prepquiz2:01 A quadratic equation is an equation of the second degree.If ab0, then at least one of a and b must be zero.worked examplesSolve the quadratic equations:1 a (x 1)(x + 7)0 b 2x(x + 3)0 c (2x 1)(3x + 5)02 a x2 + 4x + 30 b x2 490 c 2x2 + 9x 503 a x2 + x12 b 5x22x c 6x25x + 6Solutions1 a If (x 1)(x + 7)0 b If 2x(x + 3)0 c If (2x 1)(3x + 5)0then either then either then eitherx 10 or x + 70 2x0 or x + 30 2x 10 or 3x + 50 x1 or x7 x0 or x3 2x1 or 3x5 xor x2 To solve these equations, they are factorised first so they look like the equations in example 1.a x2 + 4x + 30 b x2 490 or x2 490(x + 3)(x + 1)0 (x 7)(x + 7)0 So x249So x + 30 So x 70 x7 or 7or x + 10 or x + 70 ie x7x3 or 1 x7 or 7c 2x2 + 9x 50(2x 1)(x + 5)0So 2x 10or x + 50xor 512---5 3------A quadratic equation can have two solutions.||||||To factorise anexpression like2x2 9x 5,you can use theCROSS METHOD.|||12---continued 5.1_5.3_Chapter 02Page 29Tuesday, July 12, 20058:49 AM30 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3Find the two solutions for each equation. Check by substitution to ensure your answers are correct.a x(x 5)0 b x(x + 7)0 c 2x(x + 1)0d 5a(a 2)0 e 4q(q + 5)0 f 6p(p 7)0g (x 2)(x 1)0 h (x 7)(x 3)0 i (a 5)(a 2)0j (y + 3)(y + 4)0 k (t + 3)(t + 2)0 l (x + 9)(x + 5)0m (a 6)(a + 6)0 n (y + 8)(y 7)0 o (n + 1)(n 1)0p (a + 1)(2a 1)0 q (3x + 2)(x 5)0 r 2x(3x 1)0s (4x 1)(2x + 1)0 t (3a 4)(2a 1)0 u (6y 5)(4y + 3)0v 6x(5x 3)0 w (9y + 1)(7y + 2)0 x (5x 1)(5x + 1)0After factorising the left-hand side of each equation, solve the following.a x2 + 3x0 b m2 5m0 c y2 + 2y0d 6x2 + 12x0 e 9n2 3n0 f 4x2 + 8x0g x2 40 h a2 490 i y2 360j a2 10 k n2 1000 l m2 640mx2 + 3x + 20 n a2 5a + 60 o y2 + 12y + 350p a2 10a + 210 q x2 10x + 160 r m2 11m + 240s h2 + h 200 t x2 + 2x 350 u a2 4a 450v x2 + x 560 wy2 8y + 70 x a2 + 9a 1003 Before these equations are solved, all the terms are gathered to one side of the equation, letting the other side be zero.a x2 + x12 b 5x22x c 6x25x + 6x2 + x 120 5x2 2x0 6x2 5x 60(x + 4)(x 3)0 x(5x 2)0 (3x + 2)(2x 3)0So x + 40 So x0 So 3x + 20or x 30 or 5x 20 or 2x 30 x4 or 3 x0 or xor Of course, you can always check your solutions by substitution. For example 3a above:Substituting x4 Substituting x3x2 + x12 x2 + x12L.H.S.(4)2 + (4) L.H.S.(3)2 + (3) 16 49 + 3 1212 R.H.S.R.H.S. Both x4 and x3 are solutions.|||||||||25---23---32---To solve a quadratic equation: gather all the terms to one side of the equation factorise solve the two resulting simple equations. L.H.S.left-hand sideR.H.S.right-hand sideExercise 2:01Quadratic equations PAS5321 Factorisea x2 3x b x2 + 3x + 22 Solvea x(x 4)0 b (x 1)(x + 2)0Foundation Worksheet 2:01125.1_5.3_Chapter 02Page 30Tuesday, July 12, 20058:49 AMCHAPTER 2 QUADRATIC EQUATIONS 31Factorise and solve the following.a 2x2 + x 10 b 3x2 + 7x + 20c 3x2 + 17x + 100 d 2x2 11x + 120e 2x2 x 100 f 2x2 11x 210g 4x2 + 21x + 50 h 4x2 19x 50i 4x2 21x + 50 j 5x2 + 16x + 30k 2x2 + 13x 240 l 7x2 + 48x 70m 4x2 4x 30 n 6x2 x 10o 9x2 + 9x + 20 p 10x2 + 9x + 20q 12x2 7x + 10 r 10x2 13x + 40Gather all the terms to one side of each equation and then solve.a x23x b m28m c x25xd x25x 4 e a22a + 15 f y23y 2g m29m 18 h n27n + 18 i h24h + 32j x2 + x2 k y2 + 2y3 l x2 7x10my2 + 3y18 n t2 + 3t28 o y2 + 2y15p 2x2 + x1 q 2x2 x15 r 4m2 3m6s 3x213x 14 t 5p217p 6 u 2x211x 52:02 | Solution by Completing Outcome PAS532the SquareThis method depends upon completing an algebraic expression to form a perfect square, that is, an expression of the form (x + a)2 or (x a)2.These are harder tofactorise! Check answers by substitution.34worked examplesWhat must be added to the following to make perfect squares?1 x2 + 8x 2 x2 5xSolutionsBecause (x + a)2x2 + 2ax + a2, the coefficient of the x term must be halved to give the value of a.1 x2 + 8x + . . . 2 x2 5x + . . .Half of 8 is 4, so the perfect square is: Half of 5 is , so the perfect square is:x2 + 8x + 42(x + 4)2x2 5x + ( )2(x )2Now, to solve a quadratic equation using this technique, we follow the steps in the example below.x2 + 4x 21 0x2 + 4x21x2 + 4x + 22 21 + 22 (x + 2)225x + 2x2 5 x3 or 752---52---52---25 Move the constant to the R.H.S.Add (half of x coefficient)2 to both sides.5.1_5.3_Chapter 02Page 31Tuesday, July 12, 20058:49 AM32 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3Note that the previous example could have been factorised to give (x 3)(x + 7)0, which, of course, is an easier and quicker way to find the solution. The method of completing the square, however, can determine the solution of quadratic equations that cannot be factorised. This can be seen in the examples below.worked examplesSolve:1 x2 + 6x + 10 2 x2 3x 50 3 3x2 4x 10Solutions1 x2 + 6x + 10x2 + 6x1x2 + 6x + 32 1 + 32(x + 3)28x + 3 x3 ie x3 + or 3 (x 017 or 583)3 3x2 4x 1 0x2 x 0x2 xx2 x + ( )2+ ( )2(x )2

x x ie or (x 155 or 022)When the coefficientof x2 is not 1,we first of alldivide each termby that coefficient.2 x2 3x 50x2 3x5x2 3x + ( )2 5 + ( )2(x )27x x ie or (x 419 or 119)32---32---32---14---32--- 714---32---292----------x3 29 +2------------------- =3 29 2------------------- Note:You can use the following fact to check your answers.For the equation:ax2 + bx + c0the two solutions must add up to equal In example 1, (017) + (583)6 [or]In example 3, 155 + (022)133 []ba-- 6 1------43---882 2 2 2 Note that the solution involves a square root, ie the solution is irrational. Using your calculator, approximations may be found.43---13---43---13---43---23---13---23---23---79---23---73-------23---73-------x2 7 +3---------------- =2 7 3----------------5.1_5.3_Chapter 02Page 32Tuesday, July 12, 20058:49 AMCHAPTER 2 QUADRATIC EQUATIONS 33What number must be inserted to complete the square?a x2 + 6x + . . .2(x + . . .)2b x2 + 8x + . . .2(x + . . .)2c x2 2x + . . .2(x . . .)2d x2 4x + . . .2(x . . .)2e x2 + 3x + . . .2(x + . . .)2f x2 7x + . . .2(x . . .)2g x2 + 11x + . . .2 = (x + . . .)2h x2 x + . . .2(x . . .)2i x2 + + . . .2(x + . . .)2j x2 + . . .2(x . . .)2Solve the following equations, leaving your answers in surd form.a (x 2)23 b (x + 1)22 c (x + 5)25d (x 1)210 e (x 3)27 f (x + 2)211g (x + 3)28 h (x + 10)212 i (x 3)218j (x +)25 k (x )23 l (x + 1 )212m (x 1)22 n (x + 3)24 o (x )2

Solve the following equations by completing the square. Also find approximations for your answers, correct to two decimal places.a x2 + 2x 10 b x2 2x 50 c x2 4x 80d x2 + 6x 80 e x2 6x + 20 f x2 + 4x + 10g x2 + 10x5 h x2 + 2x4 i x2 12x1j x2 + 5x + 20 k x2 + 7x 30 l x2 + x 30mx2 + 9x + 30 n x2 + 3x 50 o x2 11x + 50p x2 x3 q x2 + 3x2 r x2 5x1s 2x2 4x 10 t 2x2 + 3x 40 u 2x2 8x + 10v 3x2 + 2x 30 w 5x2 4x 30 x 4x2 x 202:03 | The Quadratic Formula Outcome PAS532As we have seen in the previous section, a quadratic equation is one involving a squared term.In fact, any quadratic equation can be represented by the general form of a quadratic equation:ax2 + bx + c0where a, b, c are all integers, and a is not equal to zero.If any quadratic equation is arranged in this form, a formula using the values of a, b and c can be used to find the solutions.Exercise 2:0215x2------2x3------212---23---12---12---12---13---59---3The quadratic formula for ax2 + bx2 + c0 is:xb b24ac 2a------------------------------------- =5.1_5.3_Chapter 02Page 33Tuesday, July 12, 20058:49 AM34 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3NOTE:This proof uses themethod of completingthe square.This formula is veryuseful if you cantfactorise an expression.PROOF OF THE QUADRATIC FORMULAax2 + bx + c00

xx2ba--xca-- + +x2ba--x +ca-- x2ba--xb2a------( J| 2+ +b2a------( J| 2ca-- xb2a------ +( J| 2b24ac 4a2--------------------xb2a------ + b24ac 2a----------------------------b b24ac 2a-------------------------------------worked examplesSolve the following by using the quadratic formula.1 2x2 + 9x + 40 2 x2 + 5x + 10 3 3x22x + 2 4 2x2 + 2x + 70Solutions1 For the equation 2x2 + 9x + 40, 2 For x2 + 5x + 10,a2, b9, c4. a1, b5, c1.Substituting these values into the Substituting into the formula gives:formula:x or xor 4x Since there is no rational equivalent to the answer may be left as: or Approximations for these answers maybe found using a calculator. In this casethey would be given as:x 021 or 479 (to 2 dec. pl.)b b24ac 2a-------------------------------------5 524 1 1 2 1 ---------------------------------------------------5 25 4 2--------------------------------5 212-----------------------21x5 21 +2------------------------ =5 21 2------------------------b b24ac 2a-------------------------------------9 924 2 4 2 2 ---------------------------------------------------9 81 32 4-----------------------------------9 494-----------------------9 74----------------24---164------12---5.1_5.3_Chapter 02Page 34Tuesday, July 12, 20058:49 AMCHAPTER 2 QUADRATIC EQUATIONS 35Use the quadratic formula to solve the following equations. All have rational answers.a x2 + 5x + 60 b x2 + 6x + 80c x2 + 10x + 90 d x2 3x 100e x2 2x 150 f x2 + 4x 120g x2 9x + 140 h x2 8x + 120 i x2 6x + 50j 3x2 + 7x + 20 k 2x2 + 11x + 50 l 4x2 + 11x + 60m 2x2 5x 30 n 5x2 9x 20 o 3x2 5x + 20p 6x2 + 7x + 20 q 6x2 + 7x 30 r 8x2 14x + 30Solve the following, leaving your answers in surd form. (Remember: A surd is an expression involving a square root.)a x2 + 4x + 20 b x2 + 3x + 10 c x2 + 5x + 30d x2 + x 10 e x2 + 2x 20 f x2 + 4x 10g x2 2x 10 h x2 7x + 20 i x2 6x + 30j x2 10x 90 k x2 8x + 30 l x2 5x + 70m 2x2 + 6x + 10 n 2x2 + 3x 10 o 2x2 7x + 40p 3x2 + 10x + 20 q 3x2 9x + 20 r 5x2 + 4x 20s 4x2 x + 10 t 3x2 3x 10 u 4x2 3x 20v 2x2 + 11x 50 w 2x2 9x + 80 x 5x2 + 2x 103 The equation 3x22x + 2 mustfirst be written in the formax2 + bx + c0,ie 3x2 2x 20So a3, b2, c2.Substituting these values gives:x So xor (ie x 122 or 055 to 2 dec. pl.)You shouldlearn thisformula!4 For 2x2 + 2x + 70,a2, b2, c7.Substituting these values gives:x But is not real!So 2x2 + 2x + 70 has no real solutions.2 224 2 7 2 2 ---------------------------------------------------2 52 4--------------------------52 The solutions of the equation ax2 + bx + c0are given by:xb b24ac 2a------------------------------------- =b b24ac 2a-------------------------------------2 ( ) 2 ( )24 3 2 ( ) 2 3 ---------------------------------------------------------------------------2 4 24 +6------------------------------2 286--------------------2 28 +6-------------------2 28 6-------------------Exercise 2:03The quadratic formula PAS5321 Evaluate if:a a1, b3, c2 b a2, b5, c22 Solve:a x2 + 5x + 20 b x2 3x 10b b24ac 2a--------------------------------------Foundation Worksheet 2:03125.1_5.3_Chapter 02Page 35Tuesday, July 12, 20058:49 AM36 NEW SIGNPOST MATHEMATICS 10 STAGE 5.15.3Use the formula to solve the following and give the answers as decimal approximations correct to two decimal places.a x2 4x + 10 b x2 6x + 30 c x2 + 8x 50d x2 + 9x + 10 e x2 + 2x 50 f x2 + 3x 10g x2 + 20 h x2 7x2 i x26x 11j 2x2 + x 20 k 2x2 5x 20 l 3x2 + 9x + 50m 2x27x 2 n 5x2 3x4 o 6x2x + 3Investigation 2:03 | How many solutions?Consider these three quadratic equations:A x2 + 6x + 50 B x2 + 6x + 90 C x2 + 6x + 120If we use the formula to solve each of these, we get:A x B x x1 or 5 x3C x = = [has no real solution] x has no real solutionsLooking at these equations, it appears that a quadratic equation may have two, one or no solutions. The key is the part of the formula under the square root sign.ExercisesBy evaluating b2 4ac for each equation, determinehow many solutions it will have.1 x2 + 4x + 30 2 x2 + 4x + 40 3 x2 + 4x + 504 x2 x 20 5 x2 x0 6 x2 x + 207 4x2 12x + 90 8 4x2 12x + 70 9 4x2 12x + 11010 5x2 x + 70 11 5x2 x 70 12 9x2 + 6x + 103investigation2:036 624 1 5 2 1 ---------------------------------------------------6 624 1 9 2 1 ---------------------------------------------------6 162-----------------------6 02--------------------6 42----------------62--- 6 624 1 12 2 1 ------------------------------------------------------6 12 2--------------------------12 The number of solutions is determined by b2 4ac.If b2 4ac is positive then the equation will have 2 solutions zero then the equation will have 1 solution negative then the equation will have no solution. b2 4ac is called the discriminant.5.1_5.3_Chapter 02Page 36Tuesday, July 12, 20058:49 AMCHAPTER 2 QUADRATIC EQUATIONS 372:04 | Choosing the Outcome PAS532Best MethodSome quadratic equations may appear in a different form from those we have seen so far, but they can always be simplified to the general form ax2 + bx + c0. They may then be factorised, or the formula applied, to solve them.Factorise: 1 5x2 10x 2 x2 5x 14 3 x2 81 4 x2 + 5x + 6Solve: 5 (x 2)(x + 7)0 6 (2x 3)(3x + 1)07 x2 160 8 3x2 12x09 x2 3x + 2010 Write down the formula for the solution of the equation: ax2 + bx + c0.prepquiz2:04worked examplesSolve the following equations.1 x2 2x + 13x + 6 2 x(x 5)6 3Solutions1 In this e