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: Copyright Agency Limited Level 15, 233 Castlereagh Street Sydney NSW 2000 Telephone: (02) 9394 7600 Email: [email protected] 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