##### Document Text Contents

Page 1

prelims.indd

Mathematics

Preliminary Course

maths

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Chapter 4: Geometry 1 136

INTRODUCTION 137

NOTATION 137

TYPES OF ANGLES 138

PARALLEL LINES 145

TYPES OF TRIANGLES 149

CONGRUENT TRIANGLES 155

SIMILAR TRIANGLES 159

PYTHAGORAS’ THEOREM 167

TYPES OF QUADRILATERALS 173

POLYGONS 180

AREAS 184

TEST YOURSELF 4 191

CHALLENGE EXERCISE 4 193

Practice Assessment Task Set 1 195

Chapter 5: Functions and Graphs 200

INTRODUCTION 201

FUNCTIONS 201

GRAPHING TECHNIQUES 212

LINEAR FUNCTION 220

QUADRATIC FUNCTION 224

ABSOLUTE VALUE FUNCTION 230

THE HYPERBOLA 238

CIRCLES AND SEMI-CIRCLES 242

OTHER GRAPHS 250

LIMITS AND CONTINUITY 256

REGIONS 260

TEST YOURSELF 5 270

CHALLENGE EXERCISE 5 271

Chapter 6: Trigonometry 274

INTRODUCTION 275

TRIGONOMETRIC RATIOS 275

RIGHT-ANGLED TRIANGLE PROBLEMS 283

APPLICATIONS 292

EXACT RATIOS 302

ANGLES OF ANY MAGNITUDE 306

TRIGONOMETRIC EQUATIONS 320

TRIGONOMETRIC IDENTITIES 326

NON-RIGHT-ANGLED TRIANGLE RESULTS 331

APPLICATIONS 342

AREA 346

TEST YOURSELF 6 349

CHALLENGE EXERCISE 6 350

Chapter 7: Linear Functions 352

INTRODUCTION 353

DISTANCE 353

MIDPOINT 358

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GRADIENT 360

EQUATION OF A STRAIGHT LINE 370

PARALLEL AND PERPENDICULAR LINES 374

INTERSECTION OF LINES 379

PERPENDICULAR DISTANCE 384

TEST YOURSELF 7 389

CHALLENGE EXERCISE 7 390

Chapter 8: Introduction to Calculus 392

INTRODUCTION 393

GRADIENT 394

DIFFERENTIATION FROM FIRST PRINCIPLES 403

SHORT METHODS OF DIFFERENTIATION 419

TANGENTS AND NORMALS 425

FURTHER DIFFERENTIATION AND INDICES 430

COMPOSITE FUNCTION RULE 432

PRODUCT RULE 436

QUOTIENT RULE 439

TEST YOURSELF 8 442

CHALLENGE EXERCISE 8 443

Practice Assessment Task Set 2 446

Chapter 9: The Quadratic Function 450

INTRODUCTION 451

GRAPH OF A QUADRATIC FUNCTION 451

QUADRATIC INEQUALITIES 457

THE DISCRIMINANT 461

QUADRATIC IDENTITIES 468

SUM AND PRODUCT OF ROOTS 472

EQUATIONS REDUCIBLE TO QUADRATICS 477

TEST YOURSELF 9 481

CHALLENGE EXERCISE 9 482

Chapter 10: Locus and the Parabola 484

INTRODUCTION 485

LOCUS 485

CIRCLE AS A LOCUS 493

PARABOLA AS A LOCUS 497

GENERAL PARABOLA 516

TANGENTS AND NORMALS 531

TEST YOURSELF 10 534

CHALLENGE EXERCISE 10 535

Practice Assessment Task Set 3 536

Answers 540

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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 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 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 couldn’t 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 ■

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 fi nally…

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 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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prelims.indd

Mathematics

Preliminary Course

maths

prelims.indd i 8/4/09 2:39:29 PM

Page 2

prelims.indd

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Page 6

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vi

Chapter 4: Geometry 1 136

INTRODUCTION 137

NOTATION 137

TYPES OF ANGLES 138

PARALLEL LINES 145

TYPES OF TRIANGLES 149

CONGRUENT TRIANGLES 155

SIMILAR TRIANGLES 159

PYTHAGORAS’ THEOREM 167

TYPES OF QUADRILATERALS 173

POLYGONS 180

AREAS 184

TEST YOURSELF 4 191

CHALLENGE EXERCISE 4 193

Practice Assessment Task Set 1 195

Chapter 5: Functions and Graphs 200

INTRODUCTION 201

FUNCTIONS 201

GRAPHING TECHNIQUES 212

LINEAR FUNCTION 220

QUADRATIC FUNCTION 224

ABSOLUTE VALUE FUNCTION 230

THE HYPERBOLA 238

CIRCLES AND SEMI-CIRCLES 242

OTHER GRAPHS 250

LIMITS AND CONTINUITY 256

REGIONS 260

TEST YOURSELF 5 270

CHALLENGE EXERCISE 5 271

Chapter 6: Trigonometry 274

INTRODUCTION 275

TRIGONOMETRIC RATIOS 275

RIGHT-ANGLED TRIANGLE PROBLEMS 283

APPLICATIONS 292

EXACT RATIOS 302

ANGLES OF ANY MAGNITUDE 306

TRIGONOMETRIC EQUATIONS 320

TRIGONOMETRIC IDENTITIES 326

NON-RIGHT-ANGLED TRIANGLE RESULTS 331

APPLICATIONS 342

AREA 346

TEST YOURSELF 6 349

CHALLENGE EXERCISE 6 350

Chapter 7: Linear Functions 352

INTRODUCTION 353

DISTANCE 353

MIDPOINT 358

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Page 7

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vii

GRADIENT 360

EQUATION OF A STRAIGHT LINE 370

PARALLEL AND PERPENDICULAR LINES 374

INTERSECTION OF LINES 379

PERPENDICULAR DISTANCE 384

TEST YOURSELF 7 389

CHALLENGE EXERCISE 7 390

Chapter 8: Introduction to Calculus 392

INTRODUCTION 393

GRADIENT 394

DIFFERENTIATION FROM FIRST PRINCIPLES 403

SHORT METHODS OF DIFFERENTIATION 419

TANGENTS AND NORMALS 425

FURTHER DIFFERENTIATION AND INDICES 430

COMPOSITE FUNCTION RULE 432

PRODUCT RULE 436

QUOTIENT RULE 439

TEST YOURSELF 8 442

CHALLENGE EXERCISE 8 443

Practice Assessment Task Set 2 446

Chapter 9: The Quadratic Function 450

INTRODUCTION 451

GRAPH OF A QUADRATIC FUNCTION 451

QUADRATIC INEQUALITIES 457

THE DISCRIMINANT 461

QUADRATIC IDENTITIES 468

SUM AND PRODUCT OF ROOTS 472

EQUATIONS REDUCIBLE TO QUADRATICS 477

TEST YOURSELF 9 481

CHALLENGE EXERCISE 9 482

Chapter 10: Locus and the Parabola 484

INTRODUCTION 485

LOCUS 485

CIRCLE AS A LOCUS 493

PARABOLA AS A LOCUS 497

GENERAL PARABOLA 516

TANGENTS AND NORMALS 531

TEST YOURSELF 10 534

CHALLENGE EXERCISE 10 535

Practice Assessment Task Set 3 536

Answers 540

prelims.indd vii 8/4/09 2:39:44 PM

Page 12

prelims.indd

xii

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 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 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 couldn’t 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 ■

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 fi nally…

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 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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