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Margaret Grove

Mathematics Preliminary Course

maths

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

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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 preliminary course/Margaret Grove.Edition: 2nd ed.ISBN: 9780070278561 (pbk.)Target Audience: For secondary school age.Subjects: MathematicsProblems, exercises, etc. MathematicsTextbooks.Dewey Number: 510.76

Published in Australia by McGraw-Hill Australia Pty Ltd Level 2, 82 Waterloo Road, North Ryde NSW 2113 Publisher: Eiko Bron Managing Editor: Kathryn Fairfax Production Editor: Natalie Crouch Editorial Assistant: Ivy Chung Art Director: Astred Hicks Cover and Internal Design: Simon Rattray, Squirt Creative Cover Image: Corbis Proofreader: Ron Buck CD-ROM Preparation: Nicole McKenzie Typeset in ITC Stone serif, 10/14 by diacriTech Printed in China on 80 gsm matt art by iBook

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vPREFACE viii

ACKNOWLEDGEMENTS viii

CREDITS viii

FEATURES OF THIS BOOK viii

SYLLABUS MATRIX ix

STUDY SKILLS ix

Chapter 1: Basic Arithmetic 2

INTRODUCTION 3REAL NUMBERS 3DIRECTED NUMBERS 9FRACTIONS, DECIMALS AND PERCENTAGES 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 EQUATIONS 95SUBSTITUTION 100INEQUATIONS 103EQUATIONS AND INEQUATIONS INVOLVING ABSOLUTE VALUES 107EXPONENTIAL EQUATIONS 114QUADRATIC EQUATIONS 118QUADRATIC INEQUATIONS 125SIMULTANEOUS EQUATIONS 127TEST YOURSELF 3 133CHALLENGE EXERCISE 3 134

Contents

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vi

Chapter 4: Geometry 1 136

INTRODUCTION 137NOTATION 137TYPES OF ANGLES 138PARALLEL LINES 145TYPES OF TRIANGLES 149CONGRUENT TRIANGLES 155SIMILAR TRIANGLES 159PYTHAGORAS THEOREM 167TYPES OF QUADRILATERALS 173POLYGONS 180AREAS 184TEST YOURSELF 4 191CHALLENGE EXERCISE 4 193

Practice Assessment Task Set 1 195

Chapter 5: Functions and Graphs 200

INTRODUCTION 201FUNCTIONS 201GRAPHING TECHNIQUES 212LINEAR FUNCTION 220QUADRATIC FUNCTION 224ABSOLUTE VALUE FUNCTION 230THE HYPERBOLA 238CIRCLES AND SEMI-CIRCLES 242OTHER GRAPHS 250LIMITS AND CONTINUITY 256REGIONS 260TEST YOURSELF 5 270CHALLENGE EXERCISE 5 271

Chapter 6: Trigonometry 274

INTRODUCTION 275TRIGONOMETRIC RATIOS 275RIGHT-ANGLED TRIANGLE PROBLEMS 283APPLICATIONS 292EXACT RATIOS 302ANGLES OF ANY MAGNITUDE 306TRIGONOMETRIC EQUATIONS 320TRIGONOMETRIC IDENTITIES 326NON-RIGHT-ANGLED TRIANGLE RESULTS 331APPLICATIONS 342AREA 346TEST YOURSELF 6 349CHALLENGE EXERCISE 6 350

Chapter 7: Linear Functions 352

INTRODUCTION 353DISTANCE 353MIDPOINT 358

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GRADIENT 360EQUATION OF A STRAIGHT LINE 370PARALLEL AND PERPENDICULAR LINES 374INTERSECTION OF LINES 379PERPENDICULAR DISTANCE 384TEST YOURSELF 7 389CHALLENGE EXERCISE 7 390

Chapter 8: Introduction to Calculus 392

INTRODUCTION 393GRADIENT 394DIFFERENTIATION FROM FIRST PRINCIPLES 403SHORT METHODS OF DIFFERENTIATION 419TANGENTS AND NORMALS 425FURTHER DIFFERENTIATION AND INDICES 430COMPOSITE FUNCTION RULE 432PRODUCT RULE 436QUOTIENT RULE 439TEST YOURSELF 8 442CHALLENGE EXERCISE 8 443

Practice Assessment Task Set 2 446

Chapter 9: The Quadratic Function 450

INTRODUCTION 451GRAPH OF A QUADRATIC FUNCTION 451QUADRATIC INEQUALITIES 457THE DISCRIMINANT 461QUADRATIC IDENTITIES 468SUM AND PRODUCT OF ROOTS 472EQUATIONS REDUCIBLE TO QUADRATICS 477TEST YOURSELF 9 481CHALLENGE EXERCISE 9 482

Chapter 10: Locus and the Parabola 484

INTRODUCTION 485LOCUS 485CIRCLE AS A LOCUS 493PARABOLA AS A LOCUS 497GENERAL PARABOLA 516TANGENTS AND NORMALS 531TEST YOURSELF 10 534CHALLENGE EXERCISE 10 535

Practice Assessment Task Set 3 536

Answers 540

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viii

PREFACE This book covers the Preliminary syllabus for Mathematics. 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 and past 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 sets of graded exercises. The theory follows a logical order, although some topics may be learned in any order. Each chapter contains Test Yourself and Challenge exercises, and there are several practice assessment tasks throughout the book.

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

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

CREDITSFairfax Photos: p 311Istockphoto: p 101, p 167Margaret Grove: p 18, p 37, p 159, p 202, p 242, p 256, p 275, p 292 (bottom), p 294, p 295, p 297, p 300, p 353, p 497Photolibrary: p 201Shutterstock: p 74, p 160, p 225, p 292 (top), p 486

FEATURES OF THIS BOOK This second edition retains all the features of previous Maths in Focus books while adding in new improvements.

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

A wide variety of questions is maintained, with more comprehensive and more diffi cult questions included 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 the chapter. There is also a challenge set for the more able students.

The three practice assessment tasks provide a comprehensive variety of mixed questions from various chapters. These have been extended to contain questions in the form of sample examination questions, including short answer, free response and multiple-choice questions that students may encounter in assessments.

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

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ix

A syllabus matrix is included to show where each syllabus topic fi ts into the book. Topics are generally arranged in a logical order. For example, arithmetic and algebra are needed in most, if not all 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 more general 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. So the 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 quadratic functions and locus, which are near the end of the book.

SYLLABUS MATRIX This 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 arithmeticChapter 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 9: The quadratic functionChapter 10: 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

STUDY SKILLS You may have coasted through previous stages without needing to rely on regular study, but in this course many of the topics are new and you will need to systematically revise in order to build up your skills and to remember them.

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

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

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

You wouldnt go for your drivers licence after just one trip in the car, or enter a dance competition after 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 fi 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 study timetable, 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 good news is that you dont have to study for hours on end!

In the classroom

In order to remember, fi 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 a chance to remember anything and will have to study harder at home.

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

If you are unsure of something that the teacher has said, the chances are that others are also not sure. 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 to understand 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 main concepts in class and not worrying too much about other less important parts can really help. The teacher 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

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xi

Note taking varies from class to class, but there are some general guidelines that will help when you come 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. Some people need more revision time than others, some study better in the mornings while others do better at 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 fl exible Summarise the main points Revise when you are fresh and energetic Divide study time into smaller rather than longer chunks 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 you can leave some of them till later and use these for revision. It is not necessary 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 fi 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

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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 confi 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 too much time on questions youre unsure about, but work through and fi nd questions you can do fi rst.

Divide the time up into smaller chunks for each question and allow some extra time to go back to questions you couldnt do or fi nish. For example, in a 2 hour exam with 6 questions, allow around 15 minutes for each question. This will give an extra half hour at the end to tidy up and fi nish off questions.

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

And fi nally

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

With the parts of the course that you do know, make sure you can remember these easily under exam 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 fi nd time to study while others give up their outside lives to devote their time to study. The ideal situation is to balance study with other aspects of your life, including going out with friends, working and keeping up with sport and other activities that you enjoy.

Good luck with your studies!

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Maths in FocusMaths in Focus: Mathematics Preliminary CourseCopyright PageContentsPrefaceAcknowledgementsCreditsFeatures of this BookSyllabus MatrixStudy Skills