Prelims

Mathematics Extension 1 HSC Course

maths

Margaret Grove

Mathematics Extension 1 HSC Course

maths

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National Library of Australia Cataloguing-in-Publication DataAuthor: Grove, Margaret.Title: Maths in focus: mathematics extension 1 HSC course/Margaret Grove.Edition: 2nd ed.ISBN: 9780070278592 (pbk.)Target Audience: For secondary school age.Subjects: Mathematics. Mathematics–Problems, exercises, etc.Dewey Number: 510.76

Published in Australia byMcGraw-Hill Australia Pty LtdLevel 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: CorbisProofreaders: 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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ContentsPREFACE viii

ACKNOWLEDGEMENTS viii

CREDITS viii

FEATURES OF THIS BOOK viii

SYLLABUS MATRIX ix

STUDY SKILLS x

Chapter 1: Geometry 2 2

INTRODUCTION 3PlaNe FIgURe geOmeTRy 3SURFaCe aReaS aND VOlUmeS 16COORDINaTe meThODS IN geOmeTRy 21CIRCLE PROPERTIES 25

TeST yOURSelF 1 43ChalleNge exeRCISe 1 45

Chapter 2: Geometrical Applications of Calculus 50

INTRODUCTION 51gRaDIeNT OF a CURVe 51TyPeS OF STaTIONaRy POINTS 57hIgheR DeRIVaTIVeS 61SIgN OF The SeCOND DeRIVaTIVe 62DeTeRmININg TyPeS OF STaTIONaRy POINTS 70CURVe SkeTChINg 73FURTHER CURvE SKETCHING 77

maxImUm aND mINImUm ValUeS 79PROblemS INVOlVINg maxIma aND mINIma 83PRImITIVe FUNCTIONS 95TeST yOURSelF 2 100ChalleNge exeRCISe 2 102

Chapter 3: Integration 104

INTRODUCTION 105aPPROxImaTION meThODS 105INTegRaTION aND The PRImITIVe FUNCTION 117DeFINITe INTegRalS 120INDeFINITe INTegRalS 123aReaS eNClOSeD by The x-axIS 128aReaS eNClOSeD by The y-axIS 133SUmS aND DIFFeReNCeS OF aReaS 136VOlUmeS 138INTEGRATION USING SUBSTITUTION 145

TeST yOURSelF 3 150ChalleNge exeRCISe 3 151

Practice Assessment Task Set 1 153

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Chapter 4: Exponential and Logarithmic Functions 160

INTRODUCTION 161DIFFERENTIATION OF EXPONENTIAL FUNCTIONS 161INTEGRATION OF EXPONENTIAL FUNCTIONS 169LOGARITHMS 172DERIVATIVE OF THE LOGARITHMIC FUNCTION 183INTEGRATION AND THE LOGARITHMIC FUNCTION 187TEST YOURSELF 4 190CHALLENGE EXERCISE 4 191

Chapter 5: Trigonometric Functions 194

INTRODUCTION 195CIRCULAR MEASURE 195TRIGONOMETRIC RESULTS 199FURTHER TRIGONOMETRIC EQUATIONS 204

CIRCLE RESULTS 209SMALL ANGLES 218TRIGONOMETRIC GRAPHS 222DIFFERENTIATION OF TRIGONOMETRIC FUNCTIONS 236INTEGRATION OF TRIGONOMETRIC FUNCTIONS 240INTEGRATION OF SIN2 X AND COS2 X 244

TEST YOURSELF 5 247CHALLENGE EXERCISE 5 248

Chapter 6: Applications of Calculus to the Physical World 250

INTRODUCTION 251RATES OF CHANGE 251RATES INVOLVING TWO OR MORE VARIABLES 255

EXPONENTIAL GROWTH AND DECAY 260A MORE COMPLEX FORMULA FOR GROWTH AND DECAY 269

MOTION OF A PARTICLE IN A STRAIGHT LINE 275MOTION AND DIFFERENTIATION 283MOTION AND INTEGRATION 290VELOCITY AND ACCELERATION IN TERMS OF X 294SIMPLE HARMONIC MOTION 302PROJECTILES 313

TEST YOURSELF 6 324CHALLENGE EXERCISE 6 326

Practice Assessment Task Set 2 329

Chapter 7: Inverse Functions 334

INTRODUCTION 335INVERSE FUNCTIONS 335GRAPH OF INVERSE FUNCTIONS 339INVERSE TRIGONOMETRIC FUNCTIONS 352DIFFERENTIATION OF INVERSE TRIGONOMETRIC FUNCTIONS 363INTEGRATION OF INVERSE TRIGONOMETRIC FUNCTIONS 370TEST YOURSELF 7 373CHALLENGE EXERCISE 7 374

prelims.indd viprelims.indd vi 6/30/09 11:32:58 AM6/30/09 11:32:58 AM

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Chapter 8: Series 376

INTRODUCTION 377geNeRal SeRIeS 377SIgma NOTaTION 383aRIThmeTIC SeRIeS 385geOmeTRIC SeRIeS 394aPPlICaTIONS OF SeRIeS 410PROOF BY MATHEMATICAL INDUCTION 431

TeST yOURSelF 8 436ChalleNge exeRCISe 8 438

Practice Assessment Task Set 3 440

Chapter 9: Polynomials 2 446

INTRODUCTION 447ESTIMATION OF ROOTS 447TEST YOURSELF 9 464CHALLENGE EXERCISE 9 465

Chapter 10: The Binomial Theorem 466

INTRODUCTION 467COMBINATIONS 467BINOMIAL THEOREM 474FURTHER WORK WITH COEFFICIENTS 483TEST YOURSELF 10 494CHALLENGE EXERCISE 10 495

Chapter 11: Probability 496

INTRODUCTION 497SImPle PRObabIlITy 497mUlTI-STage eVeNTS 509COUNTING TECHNIqUES 521BINOMIAL PROBABILITY DISTRIBUTION 533

TeST yOURSelF 11 543ChalleNge exeRCISe 11 545

Practice Assessment Task Set 4 548

Sample Examination Papers 552

Answers 562

viii

PREFACEThis book covers the HSC syllabus for Mathematics and Extension 1. It follows the same style as the Year 11 Preliminary course, and provides a thorough coverage of the HSC syllabus. The extension material is easy to see as it has purple headings and there is purple shading next to all extension questions and answers.

The syllabus is available through the NSW Board of Studies website at 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 Istockphoto: p 105Margaret Grove: p 3, p 25, p 92, p 94, p 144, p 195, p 231, p 234, p 235, p 260, p 265, p 266, p 267, p 268, p 269, p 274, p 322, p 414, p 415, p416, p 497, p 508, p 519, p 520, p 535, p 541Shutterstock: p 21, p 251, p 301, p 335, p 514

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 four 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 HSC 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.

A syllabus matrix is included to show where each syllabus topic fi ts into the book. Topics are generally arranged in a logical order but there is room for some topics to be done in a different way. For example, probability can be done at any time.

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

Mathematics (2 Unit)

Coordinate methods in geometry (6.8) Chapter 1: Geometry 2

Applications of geometrical properties (2.5) Chapter 1: Geometry 2

Geometrical applications of differentiation (10.1 – 10.8) Chapter 2: Geometrical applications of calculus

Integration (11.1 – 11.4) Chapter 3: Integration

Trigonometric functions (13.1 – 13.6, 13.7) Chapter 5: Trigonometric functions

Logarithms and exponential functions (12.1 – 12.5) Chapter 4: Exponential and logarithmic functions

Applications of calculus to the physical world (14.1 – 14.3) Chapter 6: Applications of calculus to the physical world

Probability (3.1 – 3.3) Chapter 11: Probability

Series (7.1 – 7.3) and Series applications (7.5) Chapter 8: Series

Extension 1

Methods of integration (11.5E) Chapter 3: Integration

Primitive of sin 2x and cos 2x (13.6E) Chapter 5: Trigonometric functions

Equation dNdt

= k(N - P) (14.2E)Chapter 6: Applications of calculus to the physical world

Velocity and acceleration as a function of x (14.3E) Chapter 6: Applications of calculus to the physical world

x

Projectile motion (14.3E) Chapter 6: Applications of calculus to the physical world

Simple harmonic motion (14.4E) Chapter 6: Applications of calculus to the physical world

Inverse functions and inverse trigonometric functions (15.1 – 15.5E)

Chapter 7: Inverse functions

Induction (7.4E) Chapter 8: Series

Binomial theorem (17.1 – 17.3E) Chapter 10: The binomial theorem

Further probability (18.2E) Chapter 11: Probability

Iterative methods for numerical estimation of the roots of a polynomial equation (16.4E)

Chapter 9: Polynomials 2

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.

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 fi rst 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!

xi

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 of 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 Q

Clear your head of other issues in the classroom Q

Active, not passive learning is more memorable Q

Ask questions if you don’t understand something Q

Listen for cues from the teacher Q

Look out for what are the main concepts Q

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 Q

Use different colours to highlight important points or formulae Q

Make notes in textbooks (using pencil if you don’t own the textbook) Q

Use highlighter pens to point out important points Q

Summarise the main points Q

If notes are scribbled, rewrite them at home Q

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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 Q

Have a realistic timetable and be flexible Q

Summarise the main points Q

Revise when you are fresh and energetic Q

Divide study time into smaller rather than longer chunks Q

Study in a quiet environment Q

Have a balanced life and don’t forget to have fun! Q

If you are given exercises out of a textbook to do for homework, consider asking the teacher if you can leave some of them until 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 Q

Use colourful posters Q

Use mindmaps and diagrams Q

Discuss work with a group of friends Q

Read notes out aloud Q

Make up songs and rhymes Q

Do exercises regularly Q

Role play teaching someone else Q

Assessment tasks and exams

Many of the assessment tasks for maths are closed book examinations.You will cope better in exams if you have practiced 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.

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Here are some general guidelines for doing exams:

Read through and ensure you know how many questions there are Q

Divide your time between questions with extra time at the end Q

Don’t spend too much time on one question Q

Read each question carefully, underlining key words Q

Show all working out, including diagrams and formulae Q

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

Write legibly Q

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 Q

tomorrow Q

in a week Q

in a month Q

Some people hardly ever find time to study while others give up their outside lives to devote all 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!