i MATHEMATICAL METHODS UNITS 1 & 2 CAMBRIDGE SENIOR MATHEMATICS FOR QUEENSLAND MICHAEL EVANS KAY LIPSON DOUGLAS WALLACE DAVID GREENWOOD Consultants: Chicri Maksoud Steve Sisson Trevor Redmond Ray Minns
University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314 321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi 110025, India 79 Anson Road, #06 04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108451611 Michael Evans, Kay Lipson, Douglas Wallace and David Greenwood 2018 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2018 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Cover designed by Sardine Design Text designed by Jane Pitkethly Typeset by Jane Pitkethly and diacritech Printed in China by C & C Offset Printing Co. Ltd. A catalogue record for this book is available from the National Library of Australia at www.nla.gov.au ISBN 978-1-108-45161-1 Paperback Additional resources for this publication at www.cambridge.edu.au/go Reproduction and communication for educational purposes The Australian Copyright Act 1968 (the Act) allows a maximum of one chapter or 10% of the pages of this publication, whichever is the greater, to be reproduced and/or communicated by any educational institution for its educational purposes provided that the educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL) under the Act. For details of the CAL licence for educational institutions contact: Copyright Agency Limited Level 15, 233 Castlereagh Street Sydney NSW 2000 Telephone: (02) 9394 7600 Facsimile: (02) 9394 7601 Email: info@copyright.com.au Reproduction and communication for other purposes Except as permitted under the Act (for example a fair dealing for the purposes of study, research, criticism or review) no part of this publication may be reproduced, stored in a retrieval system, communicated or transmitted in any form or by any means without prior written permission. All inquiries should be made to the publisher at the address above. Cambridge University Press has no responsibility for the persistence or accuracy of URLS for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.
Contents About the lead author and consultants Introduction and overview Acknowledgements 1 Reviewing linear equations 1 1A Linear equations......................... 2 1B Constructing linear equations.................. 7 1C Simultaneous equations..................... 10 1D Constructing simultaneous linear equations........... 16 1E Solving linear inequalities..................... 19 1F Using and transposing formulas................. 23 Review of Chapter 1....................... 28 2 Coordinate geometry and linear relations 33 2A Distance and midpoints...................... 34 2B The gradient of a straight line.................. 37 2C The equation of a straight line.................. 43 2D Graphing straight lines...................... 51 2E Parallel and perpendicular lines.................. 54 2F Families of straight lines..................... 58 2G Linear models........................... 61 2H Simultaneous linear equations.................. 64 Review of Chapter 2....................... 71 ix x xv
iv Contents 3 Quadratics 80 3A Expanding and collecting like terms............... 81 3B Factorising............................ 85 3C Quadratic equations....................... 90 3D Graphing quadratics....................... 95 3E Completing the square and turning points............ 100 3F Graphing quadratics in polynomial form............. 105 3G Solving quadratic inequalities................... 109 3H The general quadratic formula.................. 111 3I The discriminant......................... 114 3J Solving simultaneous linear and quadratic equations...... 118 3K Families of quadratic polynomial functions........... 121 3L Quadratic models......................... 130 Review of Chapter 3....................... 134 4 Arithmetic sequences 142 4A Introduction to sequences.................... 143 4B Arithmetic sequences....................... 150 4C Arithmetic series......................... 155 Review of Chapter 4....................... 161 5 A gallery of graphs 165 5A Rectangular hyperbolas...................... 166 5B The graph of y 2 = x........................ 170 5C The graph of y = x....................... 171 5D Circles............................... 174 5E Determining rules......................... 180 Review of Chapter 5....................... 184 6 Variation 189 6A Direct variation.......................... 190 6B Inverse variation......................... 194 6C Fitting data............................ 198 Review of Chapter 6....................... 205 7 Functions, relations and transformations 210 7A Set notation and sets of numbers................ 211 7B Relations, domain and range................... 215 7C Functions............................. 221
Contents v 7D Piecewise-defined functions................... 231 7E Applying function notation.................... 234 7F Functions and modelling exercises................ 236 7G Translations of functions..................... 240 7H Dilations and reflections..................... 244 7I Combinations of transformations................ 248 7J Determining transformations................... 249 Review of Chapter 7....................... 252 8 Polynomials 259 8A The language of polynomials................... 260 8B Division of polynomials...................... 265 8C Factorisation of polynomials................... 270 8D Solving cubic equations...................... 276 8E Cubic functions of the form f(x) = a(x h) 3 + k......... 280 8F Graphs of factorised cubic functions............... 282 8G Families of cubic polynomial functions.............. 288 8H Quartic and other polynomial functions............. 292 8I Applications of polynomial functions............... 297 Review of Chapter 8....................... 302 9 Revision of Chapters 2 8 309 9A Technology-free questions.................... 309 10 9B Multiple-choice questions.................... 311 9C Extended-response questions.................. 316 9D Problem-solving and modelling questions............ 322 9E Problem-solving and modelling investigations.......... 325 9F Degree-of-difficulty classified questions............. 328 Probability 332 10A Sets and Venn diagrams..................... 333 10B Sample spaces and probability.................. 338 10C Estimating probabilities...................... 346 10D Multi-stage experiments..................... 352 10E The addition rule......................... 357 10F Probability tables......................... 360 10G Conditional probability...................... 364 10H Independent events....................... 373 10I Solving probability problems using simulation.......... 379 Review of Chapter 10...................... 383
vi Contents 11 12 13 14 Counting methods and binomial expansions 390 11A Addition and multiplication principles.............. 391 11B Arrangements........................... 394 11C Selections............................. 400 11D Applications to probability.................... 405 11E Pascal s triangle and the binomial theorem........... 407 Review of Chapter 11...................... 412 Revision of Chapters 10 11 416 12A Technology-free questions.................... 416 12B Multiple-choice questions.................... 418 12C Extended-response questions.................. 421 12D Problem-solving and modelling questions............ 424 12E Problem-solving and modelling investigations.......... 426 12F Degree-of-difficulty classified questions............. 428 Exponential functions and logarithms 434 13A The index laws.......................... 435 13B Rational indices.......................... 442 13C Standard form.......................... 444 13D Graphs of exponential functions................. 447 13E Solving exponential equations and inequalities......... 453 13F Logarithms............................ 457 13G Using logarithms to solve exponential equations and inequalities.......................... 461 13H Graphs of logarithm functions.................. 464 13I Exponential models and applications............... 468 Review of Chapter 13...................... 476 Geometric sequences 481 14A Geometric sequences....................... 482 15 14B Geometric series......................... 487 14C Applications of geometric sequences............... 490 14D Zeno s paradox and infinite geometric series........... 497 Review of Chapter 14...................... 500 Trigonometric functions 504 15A Measuring angles in degrees and radians............ 505 15B Defining trigonometric functions: sine and cosine........ 508 15C Another trigonometric function: tangent............ 510 15D Reviewing trigonometric ratios.................. 511 15E Symmetry properties of trigonometric functions......... 512 15F Exact values of trigonometric functions............. 515
Contents vii 16 17 18 19 15G Graphs of sine and cosine.................... 517 15H Solution of trigonometric equations............... 524 15I Sketch graphs of y = a sin n(t ± ε) and y = a cos n(t ± ε)..... 529 15J Sketch graphs of y = a sin n(t ± ε) ± b and y = a cos n(t ± ε) ± b. 531 15K The tangent function....................... 533 15L Further symmetry properties and the Pythagorean identity... 537 15M Circle mensuration........................ 539 15N Applications of trigonometric functions............. 544 Review of Chapter 15...................... 547 Revision of Chapters 13 15 554 16A Technology-free questions.................... 554 16B Multiple-choice questions.................... 556 16C Extended-response questions.................. 559 16D Problem-solving and modelling questions............ 563 16E Problem-solving and modelling investigations.......... 566 16F Degree-of-difiiculty classified questions............. 568 Rates of change 572 17A Recognising relationships..................... 573 17B Constant rate of change..................... 577 17C Average rate of change...................... 580 17D Instantaneous rate of change.................. 585 17E Position and average velocity.................. 591 Review of Chapter 17...................... 598 Differentiation of polynomials 604 18A The tangent to a curve at a point................ 605 18B The derivative........................... 609 18C Rules for differentiation...................... 612 18D Graphs of the derivative function................ 620 18E Limits and continuity....................... 628 18F When is a function differentiable?................ 634 Review of Chapter 18...................... 638 Applications of differentiation of polynomials 642 19A Tangents and normals...................... 643 19B Rates of change.......................... 646 19C Stationary points......................... 650 19D Types of stationary points.................... 653 19E Applications to maximum and minimum problems....... 657 19F Applications of differentiation to kinematics........... 664 Review of Chapter 19...................... 672
viii Contents 20 21 22 23 Further differentiation 681 20A Differentiating x n where n is a negative integer......... 682 20B The chain rule........................... 686 20C Differentiating rational powers.................. 691 20D Sketch graphs........................... 694 20E The product rule......................... 698 20F The quotient rule......................... 702 Review of Chapter 20...................... 706 Revision of Chapters 17 20 710 21A Technology-free questions.................... 710 21B Multiple-choice questions.................... 711 21C Extended-response questions.................. 717 21D Problem-solving and modelling questions............ 720 21E Problem-solving and modelling investigations.......... 723 21F Degree-of-difficulty classified questions............. 725 Discrete probability distributions 729 22A Discrete random variables.................... 730 22B Determining discrete probability distributions.......... 736 22C Expected value, variance and standard deviation........ 742 Review of Chapter 22...................... 749 Revision of Chapters 1 22 755 23A Technology-free questions.................... 755 23B Multiple-choice questions.................... 759 23C Extended-response questions.................. 764 23D Degree-of-difficulty classified questions............. 768 Glossary 772 Answers 781 Online Appendices: Guides to using technology These online guides are accessed through the Interactive Textbook or PDF Textbook A1A A2A Online guide to the TI-Nspire CX Non-CAS graphics calculator Online guide to the TI-84 Plus CE graphics calculator A3A Online guide to the Casio fxcg20au and Casio fxcg50au graphics calculators A4A Online guide to the Desmos graphing calculator
About the lead author and consultants About the lead author Michael Evans was a consultant to ACARA on the writing of the Australian Curriculum on which the new Queensland syllabus is based. He is a consultant with the Australian Mathematical Sciences Institute, and is coordinating author of the ICE-EM 7 10 series also published by Cambridge. He has also been active in the Australian Mathematics Trust, being involved with the writing of enrichment material and competition questions. He has many years experience as a Chief Examiner and Chairperson of examination panels. About the consultants Chicri Maksoud is Senior Mathematics Teacher at Brisbane Boys College Steve Sisson is Curriculum Leader Mathematics at Redeemer Lutheran College, Rochedale, QLD Trevor Redmond is Head of Mathematics at Somerville House, South Brisbane Ray Minns is Head of Mathematics at Northpine Christian College, Dakabin, QLD
Introduction and overview Cambridge Senior Mathematics for Queensland Mathematical Methods Units 1 & 2 provides complete and close coverage of the QCAA syllabus to be implemented in Year 11 from 2019. Its four components the print book, downloadable PDF textbook, online Interactive Textbook (ITB) and Online Teaching Resource (OTS), both powered by the HOTmaths platform contain a huge range of resources, including worked solutions and revision of Year 10 material, available to schools in a single package at one convenient price (the OTS is included with class adoptions, conditions apply). There are no extra subscriptions or per-student charges to pay. Review of Years 9 & 10: Chapter 1: Reviewing linear equations and Chapter 2: Coordinate geometry and linear relations can be considered as a review of material covered in years 9 and 10. The topics covered in these chapters are essential knowledge before proceeding with the rest of the book. You may choose to complete these chapters prior to the beginning of Year 11. In addition, some diagnostic tests and HOTmaths Year 10 lessons can be accessed in the ITB for revision of prerequisite knowledge by clicking on the Knowledge check icon. Degree of difficulty classification of questions: in the exercises, questions are classified as simple familiar SF, complex familiar CF, and complex unfamiliar CU questions. The revision chapters described below also contain model questions for each of these categories, and tests are also provided in the teacher resources, made up of such categorised model questions. Five revision chapters of material covered in the course: The first four of these chapters contain sections on Technology-free questions, Multiple choice questions, Extended-response questions, Modelling and problem-solving questions, Modelling and problem-solving investigations, and degree of difficulty classification of questions. The last revision chapter contains questions revising the whole book. Problem-solving and modelling: The problem-solving and modelling questions are multipart questions where the students undertake modelling or problem solving with an indication of the path to be taken. These can be used in introducing the techniques of problem-solving and modelling. The Investigation section is less structured and the student will have to make decisions about the pathway they will follow. Answers for them are not given in the textbook, but model solutions are in the Online Teaching Suite. See the QCAA Mathematical Methods syllabus for further guidelines for such tasks. Additional problem-solving and modelling investigations are provided in the ITB, linked via an icon from the review section of each chapter.
Introduction and overview xi Calculator guidance: Throughout the book there is guidance for the use of the TI-Nspire CX non-cas and the Casio fxcg20au and fxcg50au graphics calculators for the solution of problems. Guidance on the TI-84Plus CE is included in the interactive textbook, accessed via icons next to the TI-Nspire boxes. There are also online guides for the general use of each of these calculators. The online graphing calculator from Desmos.com is also embedded in the interactive textbook, as blank screens that students and teachers can use for their own calculations, or as widgets which have been set up for a variety of activities. The new Desmos geometry tool is also embedded in the interactive textbook, and activities and widgets using the tool will be added as they are developed. Assessment: examination practice questions and assessment tasks are provided in the revision chapters and the Online Teaching Suite. Check the updates there for developments as more guidance is published closer to implementation of the new assessment. Overview of the print book (shown below) 1 Graded step-by-step worked examples with precise explanations (and video versions) encourage independent learning, and are linked to exercises. 2 Section summaries provide important concepts in boxes for easy reference. 3 Additional linked resources in the Interactive Textbook are indicated by icons, such as skillsheets and video versions of examples. 4 Degree of difficulty categories are indicated in exercises (similar familiar, complex familiar and complex unfamiliar). 5 Chapter reviews contain a chapter summary and technology-free, multiple-choice, and extended-response questions. The latter are classified by degree of difficulty. 6 The glossary includes page numbers of the main explanation of each term. 3 1 Numbers refer to descriptions above. 2 3 2 5 6 4 1
xii Introduction and overview Overview of the downloadable PDF textbook 7 The convenience of a downloadable PDF textbook has been retained for times when users cannot go online. 8 PDF annotation and search features are enabled. Overview of the Interactive Textbook (shown on the page opposite) The Interactive Textbook (ITB) is an online HTML version of the print textbook powered by the HOTmaths platform, included with the print book or available as a separate purchase. 9 The material is formatted for on screen use with a convenient and easy-to-use navigation system and links to all resources. 10 The new Workspaces enable students to enter working and answers online and to save them. Input is by typing, with the help of a symbol palette, handwriting and drawing on tablets, or by uploading images of writing or drawing. 11 The new self-assessment tools enable students to check answers, mark their own work, and rate their confidence level in their work. This helps develop responsibility for learning and communicates progress and performance to the teacher. Student accounts can be linked to the learning management system used by the teacher in the Online Teaching Suite, so that teachers can review student self-assessment and provide feedback or adjust marks. 12 Examples have video versions to encourage independent learning. 13 Worked solutions are included and can be enabled or disabled in the student ITB accounts by the teacher. 14 Interactive Desmos widgets and activities based on embedded graphics calculator and geometry tool windows demonstrate key concepts and enable students to visualise the mathematics. 15 The Desmos graphics calculator, scientific calculator, and geometry tool are also embedded for students to use for their own calculations and exploration. 16 Revision of prior knowledge is provided with links to diagnostic tests and Year 10 HOTmaths lessons. 17 Quick quizzes containing automarked multiple choice questions enable students to check their understanding. 18 Definitions pop up for key terms in the text, and are also provided in a dictionary. 19 Messages from the teacher assign tasks and tests. 20 Practice exam-style questions are provided in downloadable PDF and Word files. 21 Additional problem-solving and modelling questions and investigations are provided as downloadable PDFs and editable Word documents. 8 8
Introduction and overview xiii INTERACTIVE TEXTBOOK POWERED BY THE HOTmaths PLATFORM A selection of features is shown. Numbers refer to the descriptions on the opposite page. HOTmaths platform features are updated regularly 14 9 17 15 9 18 12 WORKSPACES AND SELF-ASSESSMENT 10 11 19 13
xiv Introduction and overview Overview of the Online Teaching Suite powered by the HOTmaths platform (shown below) The Online Teaching Suite is automatically enabled with a teacher account and is integrated with the teacher s copy of the Interactive Textbook. All the teacher resources are in one place for easy access. The features include: 22 The HOTmaths learning management system with class and student analytics and reports, and communication tools. 23 Teacher s view of a student s working and self-assessment which enables them to modify the student s self-assessed marks, and respond where students flag that they had difficulty. 24 A HOTmaths-style test generator. 25 Chapter test worksheets and additional assessment items. 26 Editable curriculum grids and teaching programs. 24 22 22 23 25, 26
Acknowledgements The author and publishers wish to thank Russell Brown and Maria Schaffner for their work in preparing calculator guides. The author and publisher wish to thank the following sources for permission to reproduce material: Cover: Getty Images / DuxX Images: Shutterstock.com / ILeysen, Chapter 1, 10 Opener / Mauro Fabbro, Chapter 2 Opener / LizaLutik, Chapter 3 Opener / Vanillamilk, Chapter 3, 13, 15 Opener / Melamory, Chapter 5 Opener / Curly Pat, Chapter 6, 7, 8, 10 Opener / Melanory, Chapter 5 Opener / Redcollegiya, Chapter 8 Opener / Bellenixi, Chapter 11 Opener / Attitude, Chapter 17 Opener / Hakki Arslan, Chapter 18 Opener / Shutterstock, Chapter 19 Opener / Kittikorn Phonok, Chapter 20 Opener / metrue, Chapter 21 Opener / kentoh, Chapter 22 Opener / Shutterstock, Chapter 23 Opener / 100 words, 1F (1) / 2H (1) / wavebreakmedia, 3L (1); Alamy / RTImages, 10G (1). Every effort has been made to trace and acknowledge copyright. The publisher apologises for any accidental infringement and welcomes information that would redress this situation.