Mastering HSC Mathematics - Year 12 Extension 2 Mathematics

299 Pages·32.942 MB·English

by  Anne Joshua, Jonathan Le

Mathematical Association of NSW
Mathematical Association of NSW
67 /73 St Hilliers Road, Auburn, NSW 2144, Australia
Email: [email protected]
Website: www.masteringhscmathematics.com.au
National Library of Australia
Mastering HSC Mathematics Extension 2 Year 12
ISBN: 978-0-9923745-4-9 Paperback
Published in Australia by the Mathematical Association of NSW
© Jonathan Le 2020
1s t Edition published September 2020
Cover and interior design by Daisy Chen.
Printing by Round Printing.
Typeset in To\'IEX- by Jonathan Le, Elio A. Farina, Jose L. Leon
Copyright:
All rights reserved. Except as permitted under the Australian Copyright Act 1968, no part of this publication
may be reproduced, stored in a retrieval system, or transmitted in any form or by any means ( electronic,
mechanical, photocopying, recording, or otherwise) without the prior permission of the authors. The moral
rights of the authors have been asserted. Enquiries are to be made to the authors of this book.
Copyright for educational purposes:
Where copies of part or the entirety of the book are made, the law requires that the educational institution, or
the body that administers it, provide a remuneration notice to the Copyright Agency Limited, who may be
contacted as follows:
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Errors:
Whilst the authors have taken utmost care with the editing and checking of this book, the authors cannot
guarantee that all errors have been found and corrected. If you identify an error, please contact the authors
directly at [email protected]
Disclaimer:
The authors have taken all possible and reasonable measures to acknowledge and credit any sources of
material. We apologise for any accidental infringement of copyright and welcome information that would
redress this situation.
Features of this book
This book is suitable for all students studying the RSC Mathematics courses. It has been designed in
a thoroughly organised manner to help students master each syllabus topic in the new Stage 6 RSC
Mathematics Extension 2 course. This book will teach, consolidate, test and challenge students. It is
an essential resource for all students and teachers.
In flavour with the new course, this book has the following features:
• Interpretation questions.
• Modelling and application problems.
• Verification questions.
Within each chapter, there are subsections divided as follows.
Fundamentals
The carefully constructed fundamentals section appears before the main body of questions. The
purpose of this section is to
• test all key formulae, definitions, concepts and theory.
• test essential mathematical terms and language through doze-passages.
• ensure that the student has knowledge of the essential prerequisites.
• provide a summary of basic requirements for the topic.
Questions
This is the main body of questions with the following features.
• Step-by-step questions to assist the student with more difficult problems.
• Carefully graded exercises.
• "Show"-type questions, both guides the student, and offers good exam preparation.
• Proofs and explanations to strengthen understanding and develop problem-solving skills.
• Application questions to demonstrate future uses of learned theory.
• Technology-based questions to teach and reinforce concepts.
Challenge
These are more difficult questions that provide
• a challenge for students wishing to test their mastery of the topic.
• rigour and higher-order thinking skills.
• extension and more in-depth treatment of the unit of work.
Chapter Review
This section appears at the end of every chapter, and offers the following.
• Revision and consolidation of the previous exercises.
• Questions that require a combination of ideas from previous exercises.
Investigations
These tasks are potential assignments and research projects. Teachers may use and adapt these to
cover the new NESA requirements on investigative assessment tasks. This section provides for the
student
• application and modelling scenarios.
• research tasks involving data collection and analysis.
• scaffolding of learning tasks.
• open-ended style problems for discussions.
• opportunity to use appropriate technology effectively in a range of contexts.
• opportunity for students to demonstrate critical thinking.
Answers
• Quick answers to questions.
• "Show" and "prove" answers can be found in the full worked solutions.
Full worked Solutions
• Can be found online for free, or a full-colour hard copy purchased for convenience.
• Provide complete worked solutions to all questions, except investigative tasks to maintain the
open-ended nature of the tasks.
• Includes several alternative solutions to problems, where possible.
Jonathan Le
B.Sc. Pure & Applied Mathematics (8yd)
. I
I
Acknowledgements
From Jonathan:
Thank you to my family and friends for your support and encouragement.
Thank you to Jack Tiger Lam, Matthew Cash, and my lovely students for helping to proof-read the
original manuscripts of this text.
Thank you to Anne for giving me the opportunity to create this book.
Images:
Alexandre Godreau, Alvaro Pinot, Danuel Straub, David Jorre, Gerson Repreza, Jason Leung, Joakim
Honksal, Lucas Gallone, Ricardo Gomez, Sylvie Tittel, Sam Wermut
The authors have taken all possible and reasonable measures to acknowledge and credit any sources
of material. We apologise for any accidental infringement of copyright and welcome information that
would redress this situation.
· 1
Table of Contents
Chapter 1: 2K Euler's Formula ................ 94
The Nature of Proof Review Chapter 2 ........ ............ 96
■ Investigation Task ................... 102
Exercise
Investigation Task ................... 103
1A Language of Proof ........ ..... 2
Investigation Task ................... 104
1B Direct Proof .................... 7 Investigation Task ................... 105
1C Contrapositive .................. 9
Chapter 3:
1D Proof by Contradiction ......... 11
3D Vectors
1E Examples and
■
Counter-examples .............. 13 Exercise
1F Algebraic Inequalities .......... 17 3A Introduction to 3D Vectors ..... 108
1G Mathematical Induction ........ 23 3B Proofs using 3D Vectors ....... 111
1H Inequalities using 3C Vector Equation of a Line ...... 120
Differentiation .................. 32
3D Parameterising 3D Curves ..... 125
11 Inequalities using Integration ... 35
3E Spheres and Circles ........... 134
Review Chapter 1 .................... 41
Review Chapter 3 .................... 137
Investigation Task ................... 45
Investigation Task ................... 140
Investigation Task ................... 46
Investigation Task ................... 141
Chapter 2: Investigation Task ................... 142
Complex Numbers
Chapter 4:
■
Exercise Further Integration
■
2A Arithmetic of Complex
Exercise
Numbers ....................... 50
4A Integration by Substitution ..... 146
2B Solving and Factorising
4B Trigonometric Integrals ......... 150
Quadratics .. ...... ..... ........ 53
4C Trigonometric Substitutions .... 152
2C Polar Form and the Argand
Diagram ........................ 55 4D Harder Standard Integrals ..... 154
2D Vector Representation ......... 59 4E Partial Fractions ................ 157
2E Locus ........................... 67 4F t-formula Substitutions ......... 162
2F De Moivre's Theorem .......... 71 4G Integration by Parts ............ 164
-2G Applications of de Moivre's 4H Reduction Formulae ........... 166
Theorem ...... .. .. ..... ........ 75 41 Further Substitutions ......... .. 171
2H Roots of Unity .................. 81 Review Chapter 4 .................... 173
21 Applications of Roots of Unity .. 85 Investigation Task ................... 176
2J Solving Polynomials ............ 89 Investigation Task ................... 177
Investigation Task . . . . . . . . . . . . . . . . . . . 178 5D Resisted Horizontal Motion . . . . 202
Investigation Task . . . . . . . . . . . . . . . . . . . 179
5E Resisted Vertical Motion . . . . . . . 205
Chapter 5: 5F Resisted Projectile Motion ..... 211
Mechanics 5G Inclined Planes and Pulleys .... 215
■
Exercise Review Chapter 5 . . . . . . . . . . . . . . . . . . . . 225
5A Velocity-Displacement Investigation Task . . . . . . . . . . . . . . . . . . . 229
Equations . . . . . . . . . . . . . . . . . . . . . . 182 Investigation Task . . . . . . . . . . . . . . . . . . . 230
58 Simple Harmonic Motion . . . . . . . 186 Investigation Task . . . . . . . . . . . . . . . . . . . 231
5C Projectile Motion . . . . . . . . . . . . . . . 192 Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
THE NATURE OF PROOF
Language of Proof
■
Direct Proof
■
Contrapositive
■
Proof by Contradiction
■
Examples and Counter-examples
■
Algebraic Inequalities
■
Mathematical Induction
■
Inequalities using Differentiation
■
Inequalities using Integration
■
2 Chapter 1: The Nature of Proof
Exercise 1A
Language of Proof
~ Fundamentals
Fundamentals 1
(a) Expressions like 'The cat is black' or 'It will rain today' are called P-----· They are
denoted by p and q.
(b) If we have
then it means that
'if p, then q'.
This is called a logical i _____ or a c _____ statement.
(c) The c _ __ of the statement is that
q ⇒ _.
( d) If both directions are true, then we say that the statement is an 'iff' statement, which is
short for _____ _______
Fundamentals 2
(a) The negation of a statement is the 'o ____ , of the statement.
(b) Complete the following negation.
Statement: All cats are cute.
Negation: __ all cats are cute.
( c) The negation of a proposition p is denoted by __ .
Fundamentals 3
You can also have a negation of a conditional statement. The negation of a conditional statement
is the directly contradictory statement that 'proves it wrong'.
Statement: If I study, I will do well in my exam.
Negation: I studied, but _ _ __________
The negation of a conditional statement is important because that's how we disprove statements.
MASTERING MATHEMATICS