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Mathematics Extension 1Preliminary Course

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Margaret Grove

Mathematics Extension 1 Preliminary Course

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Text 2010 Grove and Associates Pty LtdIllustrations and design 2010 McGraw-Hill Australia Pty LtdAdditional owners of copyright are acknowledged in on-page credits

Every effort has been made to trace and acknowledge copyrighted material. The authors and publishers tender theirapologies should any infringement have occurred.

Reproduction and communication for educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this work,whichever is the greater, to be reproduced and/or communicated by any educational institution for its educationalpurposes provided that the institution (or the body that administers it) has sent a Statutory Educational notice toCopyright Agency Limited (CAL) and been granted a licence. For details of statutory educational and other copyrightlicences contact: Copyright Agency Limited, Level 15, 233 Castlereagh Street, Sydney NSW 2000. Telephone:(02) 9394 7600. Website: www.copyright.com.au

Reproduction and communication for other purposes Apart from any fair dealing for the purposes of study, research, criticism or review, as permitted under the Act, no part ofthis publication may be reproduced, distributed or transmitted in any form or by any means, or stored in a database orretrieval system, without the written permission of McGraw-Hill Australia including, but not limited to, any network orother electronic storage.

Enquiries should be made to the publisher via www.mcgraw-hill.com.au

National Library of Australia Cataloguing-in-Publication DataAuthor: Grove, Margaret.Title: Maths in focus: mathematics extension preliminary course/Margaret Grove.Edition: 2nd ed.ISBN: 9780070278585 (pbk.)

Target Audience: For secondary school age.Subjects: MathematicsProblems, exercises, etc. MathematicsTextbooks.Dewey Number: 510.76

Published in Australia byMcGraw-Hill Australia Pty Ltd Level 2, 82 Waterloo Road, North Ryde NSW 2113Publisher: Eiko BronManaging Editor: Kathryn FairfaxProduction Editor: Natalie CrouchEditorial Assistant: Ivy ChungArt Director: Astred HicksCover and Internal Design: Simon Rattray, Squirt CreativeCover Image: Corbis Proofreaders: Terence Townsend and Ron BuckCD-ROM Preparation: Nicole McKenzieTypeset in ITC Stone serif, 10/14 by diacriTechPrinted in China on 80 gsm matt art by iBook


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PREFACE ix

ACKNOWLEDGEMENTS ix

CREDITS ix

FEATURES OF THIS BOOK ix

SYLLABUS MATRIX x

STUDY SKILLS xi

Chapter 1: Basic Arithmetic 2

INTRODUCTION 3

REAL NUMBERS 3DIRECTED NUMBERS 9FRACTIONS, DECIMALS AND PERCENT AGES 12POWERS AND ROOTS 19ABSOLUTE VALUE 37TEST YOURSELF 1 41CHALLENGE EXERCISE 1 43

Chapter 2: Algebra and Surds 44

INTRODUCTION 45SIMPLIFYING EXPRESSIONS 45BINOMIAL PRODUCTS 51FACTORISATION 55COMPLETING THE SQUARE 69ALGEBRAIC FRACTIONS 71SUBSTITUTION 73SURDS 76TEST YOURSELF 2 90CHALLENGE EXERCISE 2 93

Chapter 3: Equations 94

INTRODUCTION 95SIMPLE EQUA TIONS 95SUBSTITUTION 100INEQUATIONS 103EQUATIONS AND INEQUATIONS INVOLVING ABSOLUTE VALUES 107EXPONENTIAL EQUATIONS 114QUADRATIC EQUATIONS 118FURTHER INEQUATIONS 125

QUADRATIC INEQUATIONS 129SIMULTANEOUS EQUATIONS 132TEST YOURSELF 3 138CHALLENGE EXERCISE 3 139

Contents


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Chapter 4: Geometry 1 140

INTRODUCTION 141NOTATION 141TYPES OF ANGLES 142PARALLEL LINES 149TYPES OF TRIANGLES 153CONGRUENT TRIANGLES 159SIMILAR TRIANGLES 163PYTHAGORAS THEOREM 171TYPES OF QUADRILATERALS 177POLYGONS 184AREAS 188TEST YOURSELF 4 195CHALLENGE EXERCISE 4 197

Practice Assessment Task Set 1 199

Chapter 5: Functions and Graphs 204

INTRODUCTION 205FUNCTIONS 205GRAPHING TECHNIQUES 216LINEAR FUNCTION 224QUADRATIC FUNCTION 228ABSOLUTE VALUE FUNCTION 234THE HYPERBOLA 242CIRCLES AND SEMI-CIRCLES 246OTHER GRAPHS 254LIMITS AND CONTINUITY 260FURTHER GRAPHS 264

REGIONS 277TEST YOURSELF 5 287CHALLENGE EXERCISE 5 288

Chapter 6: Trigonometry 290

INTRODUCTION 291TRIGONOMETRIC RATIOS 291RIGHT-ANGLED TRIANGLE PROBLEMS 299APPLICATIONS 308EXACT RATIOS 318ANGLES OF ANY MAGNITUDE 322TRIGONOMETRIC EQUATIONS 336

TRIGONOMETRIC IDENTITIES 342NON-RIGHT-ANGLED TRIANGLE RESULTS 347APPLICATIONS 358AREA 362TRIGONOMETRY IN THREE DIMENSIONS 365SUMS AND DIFFERENCES OF ANGLES 367FURTHER TRIGONOMETRIC EQUATIONS 374

TEST YOURSELF 6 385CHALLENGE EXERCISE 6 387


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Chapter 7: Linear Functions 390

INTRODUCTION 391DISTANCE 391MIDPOINT 396GRADIENT 398EQUATION OF A STRAIGHT LINE 408PARALLEL AND PERPENDICULAR LINES 412INTERSECTION OF LINES 417PERPENDICULAR DISTANCE 422ANGLE BETWEEN TWO LINES 426RATIOS 430


Chapter 8: Introduction to Calculus 438

INTRODUCTION 439GRADIENT 440DIFFERENTIATION FROM FIRST PRINCIPLES 449SHORT METHODS OF DIFFERENTIATION 465TANGENTS AND NORMALS 471FURTHER DIFFERENTIATION AND INDICES 476COMPOSITE FUNCTION RULE 478PRODUCT RULE 482QUOTIENT RULE 485ANGLE BETWEEN 2 CURVES 487



Chapter 9: Properties of the Circle 498

INTRODUCTION 499PARTS OF A CIRCLE 499ARCS, ANGLES AND CHORDS 500CHORD PROPERTIES 512CONCYCLIC POINTS 519TANGENT PROPERTIES 525TEST YOURSELF 9 537CHALLENGE EXERCISE 9 539

Chapter 10: The Quadratic Function 542

INTRODUCTION 543GRAPH OF A QUADRA TIC FUNCTION 543QUADRATIC INEQUALITIES 549THE DISCRIMINANT 555QUADRATIC IDENTITIES 562SUM AND PRODUCT OF ROOTS 566EQUATIONS REDUCIBLE TO QUADRATICS 571TEST YOURSELF 10 575CHALLENGE EXERCISE 10 576


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Chapter 11: Locus and the Parabola 578

INTRODUCTION 579LOCUS 579CIRCLE AS A LOCUS 587PARABOLA AS A LOCUS 591GENERAL PARABOLA 610TANGENTS AND NORMALS 625PARAMETRIC EQUATIONS OF THE PARABOLA 627CHORDS, TANGENTS AND NORMALS 634PROPERTIES OF THE PARABOLA 643LOCUS PROBLEMS 648



Chapter 12: Polynomials 1 662

INTRODUCTION 663DEFINITION OF A POLYNOMIAL 663DIVISION OF POLYNOMIALS 667REMAINDER AND FACTOR THEOREMS 672GRAPH OF A POLYNOMIAL 681ROOTS AND COEFFICIENTS OF POLYNOMIAL EQUATIONS 706TEST YOURSELF 12 713CHALLENGE EXERCISE 12 714

Chapter 13: Permutations and Combinations 716

INTRODUCTION 717FUNDAMENTAL COUNTING PRINCIPLE 717

PERMUTATIONS 730COMBINATIONS 740TEST YOURSELF 13 746CHALLENGE EXERCISE 13 747


Answers 756


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PREFACEThis book covers the Preliminary syllabus for Mathematics and Extension 1. The extension materialis easy to see as it has green headings and there is green shading next to all extension question andanswers. The syllabus is available through the NSW Board of Studies website on www.boardofstudies.nsw.edu.au . You can also access resources, study techniques, examination technique, sample andpast examination papers through other websites such as www.math.nsw.edu.au and www.csu.edu.au . Searching the Internet generally will pick up many websites supporting the work in this course.

Each chapter has comprehensive fully worked examples and explanations as well as ample setsof graded exercises. The theory follows a logical order, although some topics may be learned in anyorder. Each chapter contains Test Yourself and Challenge exercises, and there are several practiceassessment tasks throughout the book.

If you have trouble doing the Test Yourself exercises at the end of a chapter, you will need togo back into the chapter and revise it before trying them again. Dont attempt to do the Challengeexercises until you are con dent that you can do the Test Yourself exercises, as these are more dif cultand are designed to test the more able students who understand the topic really well.

ACKNOWLEDGEMENTSThanks go to my family, especially my husband Geoff, for supporting me in writing this book.

CREDITSFairfax Photos: p 327Istockphoto: p 101, p 171Margaret Grove: p 37, p 163, p 206, p 246, p 260, p 291, p308 (bottom), p 310, p 311, p 313, p 316,p 391, p 499, p 543, p 591, p 717, p 719, p 726, p 729, p 730, p 739Photolibrary: p 205Shutterstock: p 74, p 164, p 229, p 308 (top), p 580

FEATURES OF THIS BOOKThis second edition retains all the features of previous Maths in Focus books while adding in newimprovements.

The main feature of Maths in Focus is in its readability, its plentiful worked examples andstraightforward language so that students can understand it and use it in self-paced learning. Thelogical progression of topics, the comprehensive fully worked examples and graded exercises are stillmajor features.

A wide variety of questions is maintained, with more comprehensive and more dif cult questionsincluded in each topic. At the end of each chapter is a consolidation set of exercises (Test yourself)in no particular order that will test whether the student has grasped the concepts contained in thechapter. There is also a challenge set for the more able students.

The four practice assessment tasks provide a comprehensive variety of mixed questions fromvarious chapters. These have been extended to contain questions in the form of sample examinationquestions, including short answer, free response and multiple-choice questions that students mayencounter in assessments.

The second edition also features a short summary of general study skills that students will nduseful, both in the classroom and when doing assessment tasks and examinations. These study skillsare also repeated in the HSC book.


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A syllabus matrix is included to show where each syllabus topic ts into the book. Topics aregenerally arranged in a logical order. For example, arithmetic and algebra are needed in most, if notall other topics, so these are treated at the beginning of the book.

Some teachers like to introduce particular topics before others, e.g. linear functions before moregeneral functions. However, part of the work on gradient requires some knowledge of trigonometry

and the topic of angles of any magnitude in trigonometry needs some knowledge of functions. Sothe order of most chapters in the book have been carefully thought out. Some chapters, however,could be covered in a different order, such as geometry which is covered in Chapter 4, and quadraticfunctions and locus, which are near the end of the book.

SYLLABUS MATRIXThis matrix shows how the syllabus is organised in the chapters of this book.

Mathematics (2 Unit)

Basic arithmetic and algebra (1.1 1.4) Chapter 1: Basic arithmetic

Chapter 2: Algebra and surdsChapter 3: Equations

Real functions (4.1 4.4) Chapter 5: Functions and graphs

Trigonometric ratios (5.1 5.5) Chapter 6: Trigonometry

Linear functions (6.1 6.5, 6.7) Chapter 7: Linear functions

The quadratic polynomial and the parabola (9.1 9.5) Chapter 10: The quadratic functionChapter 11: Locus and the parabola

Plane geometry (2.1 2.4) Chapter 4: Geometry 1

Tangent to a curve and derivative of a function (8.1 8.9) Chapter 8: Introduction to calculus

Extension 1

Other inequalities (1.4E) Chapter 3: Equations

Circle geometry (2.6 2.10E) Chapter 9: Properties of the circle

Further trigonometry (5.6 5.9E) Chapter 6: Trigonometry

Angles between two lines (6.6E) Chapter 7: Linear functions


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Internal and external division of lines into given ratios (6.7E) Chapter 7: Linear functions

Parametric representation (9.6E) Chapter 11: Locus and the parabola

Permutations and combinations(18.1E)

Chapter 13: Permutations and combinations

Polynomials (16.1 16.3E) Chapter 12: Polynomials 1

STUDY SKILLSYou may have coasted through previous stages without needing to rely on regular study, but in thiscourse many of the topics are new and you will need to systematically revise in order to build up yourskills and to remember them.

The Preliminary course introduces the basics of topics such as calculus that are then applied inthe HSC course. You will struggle in the HSC if you dont set yourself up to revise the preliminarytopics as you learn new HSC topics.

Your teachers will be able to help you build up and manage good study habits. Here are a fewhints to get you started.

There is no right or wrong way to learn. Different styles of learning suit different people. Thereis also no magical number of hours a week that you should study, as this will be different for everystudent. But just listening in class and taking notes is not enough, especially when learning materialthat is totally new.

You wouldnt go for your drivers licence after just one trip in the car, or enter a dance competitionafter learning a dance routine once. These skills take a lot of practice. Studying mathematics is just

the same.If a skill is not practised within the rst 24 hours, up to 50% can be forgotten. If it is not practised

within 72 hours, up to 8590% can be forgotten! So it is really important that whatever your studytimetable, new work must be looked at soon after it is presented to you.

With a continual succession of new work to learn and retain, this is a challenge. But the goodnews is that you dont have to study for hours on end!

In the classroom

In order to remember, rst you need to focus on what is being said and done.According to an ancient proverb:

I hear and I forget

I see and I remember

I do and I understand

If you chat to friends and just take notes without really paying attention, you arent giving yourself achance to remember anything and will have to study harder at home.


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If you have just had a ght with a friend, have been chatting about weekend activities or myriadother conversations outside the classroom, it helps if you can check these at the door and dont keepchatting about them once the lesson starts.

If you are unsure of something that the teacher has said, the chances are that others are also notsure. Asking questions and clarifying things will ultimately help you gain better results, especially

in a subject like mathematics where much of the knowledge and skills depends on being able tounderstand the basics.Learning is all about knowing what you know and what you dont know. Many students feel like

they dont know anything, but its surprising just how much they know already. Picking up the mainconcepts in class and not worrying too much about other less important parts can really help. Theteacher can guide you on this.

Here are some pointers to get the best out of classroom learning:

Take control and be responsible for your own learning

Clear your head of other issues in the classroom

Active, not passive, learning is more memorable

Ask questions if you dont understand something

Listen for cues from the teacher

Look out for what are the main concepts

Note taking varies from class to class, but there are some general guidelines that will help when youcome to read over your notes later on at home:

Write legibly

Use different colours to highlight important points or formulae

Make notes in textbooks (using pencil if you dont own the textbook) Use highlighter pens to point out important points

Summarise the main points

If notes are scribbled, rewrite them at home

At home

You are responsible for your own learning and nobody else can tell you how best to study. Somepeople need more revision time than others, some study better in the mornings while others do betterat night, and some can work at home while others prefer a library.

There are some general guidelines for studying at home:

Revise both new and older topics regularly

Have a realistic timetable and be exible

Summarise the main points

Revise when you are fresh and energetic

Divide study time into smaller rather than longer chunks


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Study in a quiet environment

Have a balanced life and dont forget to have fun!

If you are given exercises out of a textbook to do for homework, consider asking the teacher if youcan leave some of them till later and use these for revision. It is not necessar y to do every exercise at

one sitting, and you learn better if you can spread these over time.People use different learning styles to help them study. The more variety the better, and you will

nd some that help you more than others. Some people (around 35%) learn best visually, some (25%)learn best by hearing and others (40%) learn by doing.

Here are some ideas to give you a variety of ways to study:

Summarise on cue cards or in a small notebook

Use colourful posters

Use mindmaps and diagrams

Discuss work with a group of friends

Read notes out aloud

Make up songs and rhymes

Do exercises regularly

Role play teaching someone else

Assessment tasks and exams

Many of the assessment tasks for maths are closed book examinations.You will cope better in exams if you have practised doing sample exams under exam conditions.

Regular revision will give you con dence and if you feel well prepared, this will help get rid of nerves

in the exam. You will also cope better if you have had a reasonable nights sleep before the exam.One of the biggest problems students have with exams is in timing. Make sure you dont spend toomuch time on questions youre unsure about, but work through and nd questions you can do rst.

Divide the time up into smaller chunks for each question and allow some extra time to go backto questions you couldnt do or nish. For example, in a 2 hour exam with 6 questions, allow around15 minutes for each question. This will give an extra half hour at the end to tidy up and nish offquestions.

Here are some general guidelines for doing exams:

Read through and ensure you know how many questions there are

Divide your time between questions with extra time at the end

Dont spend too much time on one question

Read each question carefully, underlining key words

Show all working out, including diagrams and formulae

Cross out mistakes with a single line so it can still be read

Write legibly


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And nally

Study involves knowing what you dont know, and putting in a lot of time into concentrating onthese areas. This is a positive way to learn. Rather than just saying, I cant do this, say instead, I cantdo this yet, and use your teachers, friends, textbooks and other ways of nding out.

With the parts of the course that you do know, make sure you can remember these easily underexam pressure by putting in lots of practice.

Remember to look at new work

today

tomorrow

in a week

in a month

Some people hardly ever nd time to study while others give up their outside lives to devote theirtime to study. The ideal situation is to balance study with other aspects of your life, including goingout with friends, working and keeping up with sport and other activities that you enjoy.

Good luck with your studies!


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