Maths in focus : mathematics extension preliminary course

861 Pages·2009·21.416 MB·English

by  Margaret Grove

Mathematics Extension 1
Preliminary Course
maths
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Mathematics Extension 1 Preliminary Course
maths
Margaret Grove
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Text © 2010 Grove and Associates Pty Ltd
Illustrations and design © 2010 McGraw-Hill Australia Pty Ltd
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National Library of Australia Cataloguing-in-Publication Data
Author: 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: 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
Proofreaders: Terence Townsend and 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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v
Contents
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 3
DIRECTED NUMBERS 9
FRACTIONS, DECIMALS AND PERCENTAGES 12
POWERS AND ROOTS 19
ABSOLUTE VALUE 37
TEST YOURSELF 1 41
CHALLENGE EXERCISE 1 43
Chapter 2: Algebra and Surds 44
INTRODUCTION 45
SIMPLIFYING EXPRESSIONS 45
BINOMIAL PRODUCTS 51
FACTORISATION 55
COMPLETING THE SQUARE 69
ALGEBRAIC FRACTIONS 71
SUBSTITUTION 73
SURDS 76
TEST YOURSELF 2 90
CHALLENGE EXERCISE 2 93
Chapter 3: Equations 94
INTRODUCTION 95
SIMPLE EQUATIONS 95
SUBSTITUTION 100
INEQUATIONS 103
EQUATIONS AND INEQUATIONS INVOLVING ABSOLUTE V ALUES 107
EXPONENTIAL EQUATIONS 114
QUADRATIC EQUATIONS 118
FURTHER INEQUATIONS 125
QUADRATIC INEQUATIONS 129
SIMULTANEOUS EQUATIONS 132
TEST YOURSELF 3 138
CHALLENGE EXERCISE 3 139
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vi
Chapter 4: Geometry 1 140
INTRODUCTION 141
NOTATION 141
TYPES OF ANGLES 142
PARALLEL LINES 149
TYPES OF TRIANGLES 153
CONGRUENT TRIANGLES 159
SIMILAR TRIANGLES 163
PYTHAGORAS’ THEOREM 171
TYPES OF QUADRILATERALS 177
POLYGONS 184
AREAS 188
TEST YOURSELF 4 195
CHALLENGE EXERCISE 4 197
Practice Assessment Task Set 1 199
Chapter 5: Functions and Graphs 204
INTRODUCTION 205
FUNCTIONS 205
GRAPHING TECHNIQUES 216
LINEAR FUNCTION 224
QUADRATIC FUNCTION 228
ABSOLUTE VALUE FUNCTION 234
THE HYPERBOLA 242
CIRCLES AND SEMI-CIRCLES 246
OTHER GRAPHS 254
LIMITS AND CONTINUITY 260
FURTHER GRAPHS 264
REGIONS 277
TEST YOURSELF 5 287
CHALLENGE EXERCISE 5 288
Chapter 6: Trigonometry 290
INTRODUCTION 291
TRIGONOMETRIC RATIOS 291
RIGHT-ANGLED TRIANGLE PROBLEMS 299
APPLICATIONS 308
EXACT RATIOS 318
ANGLES OF ANY MAGNITUDE 322
TRIGONOMETRIC EQUATIONS 336
TRIGONOMETRIC IDENTITIES 342
NON-RIGHT-ANGLED TRIANGLE RESULTS 347
APPLICATIONS 358
AREA 362
TRIGONOMETRY IN THREE DIMENSIONS 365
SUMS AND DIFFERENCES OF ANGLES 367
FURTHER TRIGONOMETRIC EQUATIONS 374
TEST YOURSELF 6 385
CHALLENGE EXERCISE 6 387
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vii
Chapter 7: Linear Functions 390
INTRODUCTION 391
DISTANCE 391
MIDPOINT 396
GRADIENT 398
EQUATION OF A STRAIGHT LINE 408
PARALLEL AND PERPENDICULAR LINES 412
INTERSECTION OF LINES 417
PERPENDICULAR DISTANCE 422
ANGLE BETWEEN TWO LINES 426
RATIOS 430
TEST YOURSELF 7 434
CHALLENGE EXERCISE 7 435
Chapter 8: Introduction to Calculus 438
INTRODUCTION 439
GRADIENT 440
DIFFERENTIATION FROM FIRST PRINCIPLES 449
SHORT METHODS OF DIFFERENTIATION 465
TANGENTS AND NORMALS 471
FURTHER DIFFERENTIATION AND INDICES 476
COMPOSITE FUNCTION RULE 478
PRODUCT RULE 482
QUOTIENT RULE 485
ANGLE BETWEEN 2 CURVES 487
TEST YOURSELF 8 490
CHALLENGE EXERCISE 8 491
Practice Assessment Task Set 2 494
Chapter 9: Properties of the Circle 498
INTRODUCTION 499
PARTS OF A CIRCLE 499
ARCS, ANGLES AND CHORDS 500
CHORD PROPERTIES 512
CONCYCLIC POINTS 519
TANGENT PROPERTIES 525
TEST YOURSELF 9 537
CHALLENGE EXERCISE 9 539
Chapter 10: The Quadratic Function 542
INTRODUCTION 543
GRAPH OF A QUADRATIC FUNCTION 543
QUADRATIC INEQUALITIES 549
THE DISCRIMINANT 555
QUADRATIC IDENTITIES 562
SUM AND PRODUCT OF ROOTS 566
EQUATIONS REDUCIBLE TO QUADRATICS 571
TEST YOURSELF 10 575
CHALLENGE EXERCISE 10 576
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viii
Chapter 11: Locus and the Parabola 578
INTRODUCTION 579
LOCUS 579
CIRCLE AS A LOCUS 587
PARABOLA AS A LOCUS 591
GENERAL PARABOLA 610
TANGENTS AND NORMALS 625
PARAMETRIC EQUATIONS OF THE PARABOLA 627
CHORDS, TANGENTS AND NORMALS 634
PROPERTIES OF THE PARABOLA 643
LOCUS PROBLEMS 648
TEST YOURSELF 11 652
CHALLENGE EXERCISE 11 653
Practice Assessment Task Set 3 655
Chapter 12: Polynomials 1 662
INTRODUCTION 663
DEFINITION OF A POLYNOMIAL 663
DIVISION OF POLYNOMIALS 667
REMAINDER AND FACTOR THEOREMS 672
GRAPH OF A POLYNOMIAL 681
ROOTS AND COEFFICIENTS OF POLYNOMIAL EQUATIONS 706
TEST YOURSELF 12 713
CHALLENGE EXERCISE 12 714
Chapter 13: Permutations and Combinations 716
INTRODUCTION 717
FUNDAMENTAL COUNTING PRINCIPLE 717
PERMUTATIONS 730
COMBINATIONS 740
TEST YOURSELF 13 746
CHALLENGE EXERCISE 13 747
Practice Assessment Task Set 4 749
Answers 756
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ix
PREFACE
This book covers the Preliminary syllabus for Mathematics and Extension 1. The extension material
is easy to see as it has green headings and there is green shading next to all extension question and
answers. 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 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.
CREDITS
Fairfax Photos: p 327
Istockphoto: p 101, p 171
Margaret 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 739
Photolibrary: p 205
Shutterstock: p 74, p 164, p 229, p 308 (top), p 580
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 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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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 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 10: The quadratic function
Chapter 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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