Mathematics Preliminary Course
maths
Mathematics Preliminary Course
maths Margaret Grove
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 Every effort has been made to trace and acknowledge copyrighted material. The authors and publishers tender their apologies 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 educational purposes provided that the institution (or the body that administers it) has sent a Statutory Educational notice to Copyright Agency Limited (CAL) and been granted a licence. For details of statutory educational and other copyright licences 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 of this publication may be reproduced, distributed or transmitted in any form or by any means, or stored in a database or retrieval system, without the written permission of McGraw-Hill Australia including, but not limited to, any network or other electronic storage. Enquiries should be made to the publisher via www.mcgraw-hill.com.au National Library of Australia Cataloguing-in-Publication Data Author: Grove, Margaret. Title: Maths in focus: mathematics preliminary course/Margaret Grove. Edition: 2nd ed. ISBN: 9780070278561 (pbk.) Target Audience: For secondary school age. Subjects: Mathematics–Problems, exercises, etc. Mathematics–Textbooks. 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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Contents PREFACE
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ACKNOWLEDGEMENTS
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CREDITS
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FEATURES OF THIS BOOK
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SYLLABUS MATRIX
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STUDY SKILLS
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Chapter 1: Basic Arithmetic
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INTRODUCTION REAL NUMBERS DIRECTED NUMBERS FRACTIONS, DECIMALS AND PERCENTAGES POWERS AND ROOTS ABSOLUTE VALUE TEST YOURSELF 1 CHALLENGE EXERCISE 1 Chapter 2: Algebra and Surds INTRODUCTION SIMPLIFYING EXPRESSIONS BINOMIAL PRODUCTS FACTORISATION COMPLETING THE SQUARE ALGEBRAIC FRACTIONS SUBSTITUTION SURDS TEST YOURSELF 2 CHALLENGE EXERCISE 2 Chapter 3: Equations INTRODUCTION SIMPLE EQUATIONS SUBSTITUTION INEQUATIONS EQUATIONS AND INEQUATIONS INVOLVING ABSOLUTE VALUES EXPONENTIAL EQUATIONS QUADRATIC EQUATIONS QUADRATIC INEQUATIONS SIMULTANEOUS EQUATIONS TEST YOURSELF 3 CHALLENGE EXERCISE 3
3 3 9 12 19 37 41 43 44 45 45 51 55 69 71 73 76 90 93 94 95 95 100 103 107 114 118 125 127 133 134
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Chapter 4: Geometry 1
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INTRODUCTION NOTATION TYPES OF ANGLES PARALLEL LINES TYPES OF TRIANGLES CONGRUENT TRIANGLES SIMILAR TRIANGLES PYTHAGORAS’ THEOREM TYPES OF QUADRILATERALS POLYGONS AREAS TEST YOURSELF 4 CHALLENGE EXERCISE 4
137 137 138 145 149 155 159 167 173 180 184 191 193
Practice Assessment Task Set 1 Chapter 5: Functions and Graphs INTRODUCTION FUNCTIONS GRAPHING TECHNIQUES LINEAR FUNCTION QUADRATIC FUNCTION ABSOLUTE VALUE FUNCTION THE HYPERBOLA CIRCLES AND SEMI-CIRCLES OTHER GRAPHS LIMITS AND CONTINUITY REGIONS TEST YOURSELF 5 CHALLENGE EXERCISE 5 Chapter 6: Trigonometry INTRODUCTION TRIGONOMETRIC RATIOS RIGHT-ANGLED TRIANGLE PROBLEMS APPLICATIONS EXACT RATIOS ANGLES OF ANY MAGNITUDE TRIGONOMETRIC EQUATIONS TRIGONOMETRIC IDENTITIES NON-RIGHT-ANGLED TRIANGLE RESULTS APPLICATIONS AREA TEST YOURSELF 6 CHALLENGE EXERCISE 6 Chapter 7: Linear Functions INTRODUCTION DISTANCE MIDPOINT
195 200 201 201 212 220 224 230 238 242 250 256 260 270 271 274 275 275 283 292 302 306 320 326 331 342 346 349 350 352 353 353 358
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GRADIENT EQUATION OF A STRAIGHT LINE PARALLEL AND PERPENDICULAR LINES INTERSECTION OF LINES PERPENDICULAR DISTANCE TEST YOURSELF 7 CHALLENGE EXERCISE 7 Chapter 8: Introduction to Calculus INTRODUCTION GRADIENT DIFFERENTIATION FROM FIRST PRINCIPLES SHORT METHODS OF DIFFERENTIATION TANGENTS AND NORMALS FURTHER DIFFERENTIATION AND INDICES COMPOSITE FUNCTION RULE PRODUCT RULE QUOTIENT RULE TEST YOURSELF 8 CHALLENGE EXERCISE 8 Practice Assessment Task Set 2 Chapter 9: The Quadratic Function INTRODUCTION GRAPH OF A QUADRATIC FUNCTION QUADRATIC INEQUALITIES THE DISCRIMINANT QUADRATIC IDENTITIES SUM AND PRODUCT OF ROOTS EQUATIONS REDUCIBLE TO QUADRATICS TEST YOURSELF 9 CHALLENGE EXERCISE 9 Chapter 10: Locus and the Parabola INTRODUCTION LOCUS CIRCLE AS A LOCUS PARABOLA AS A LOCUS GENERAL PARABOLA TANGENTS AND NORMALS TEST YOURSELF 10 CHALLENGE EXERCISE 10 Practice Assessment Task Set 3 Answers
360 370 374 379 384 389 390 392 393 394 403 419 425 430 432 436 439 442 443 446 450 451 451 457 461 468 472 477 481 482 484 485 485 493 497 516 531 534 535 536 540
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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. Don’t attempt to do the Challenge exercises until you are confident that you can do the Test Yourself exercises, as these are more difficult 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.
CREDITS Fairfax Photos: p 311 Istockphoto: p 101, p 167 Margaret 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 497 Photolibrary: p 201 Shutterstock: 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 difficult 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 find 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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A syllabus matrix is included to show where each syllabus topic fits 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 arithmetic Chapter 2: Algebra and surds Chapter 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 function Chapter 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 don’t 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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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 wouldn’t go for your driver’s 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 first 24 hours, up to 50% can be forgotten. If it is not practised within 72 hours, up to 85–90% 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 don’t have to study for hours on end!
In the classroom In order to remember, first 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 aren’t giving yourself a chance to remember anything and will have to study harder at home. If you have just had a fight 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 don’t 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 don’t know. Many students feel like they don’t know anything, but it’s 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 don’t understand something ■ Listen for cues from the teacher ■ Look out for what are the main concepts
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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 don’t 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 flexible ■ 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 don’t 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 find 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 confidence 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 night’s sleep before the exam. One of the biggest problems students have with exams is in timing. Make sure you don’t spend too much time on questions you’re unsure about, but work through and find questions you can do first. Divide the time up into smaller chunks for each question and allow some extra time to go back to questions you couldn’t do or finish. 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 finish 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 ■ Don’t 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 finally… Study involves knowing what you don’t 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 can’t do this’, say instead, ‘I can’t do this yet’, and use your teachers, friends, textbooks and other ways of finding 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 find 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!
