Cambridge HSC MATHEMATICS GENERAL 2 Second Edition
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Mexico City Cambridge University Press 477 Williamstown Road, Port Melbourne, VIC 3207, Australia www.cambridge.edu.au Information on this title: /9781107654594 The Powers Family Trust 2013 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 2013 Cover design by Sylvia Witte Typeset by Aptara Corp. Printed in China by Print Plus Ltd A Cataloguing-in-Publication entry is available from the catalogue of the National Library of Australia at www.nla.gov.au ISBN 978-1-107-65459-4 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 Introduction vii Acknowledgements xii Chapter 1 Credit and borrowing 1 FM4 1.1 Flat-rate loans 1 1.2 Table of loan repayments 8 1.3 Future value formula 13 1.4 Comparing loans 16 1.5 Credit cards 20 Chapter summary 25 Sample HSC Objective-response questions 26 Sample HSC Short-answer questions 27 Chapter 2 Further applications of area and volume 29 MM4 2.1 Area of circles, annuluses and sectors 29 2.2 Area of composite shapes 35 2.3 Simpson s rule 40 2.4 Surface area of right prisms 45 2.5 Surface area of cylinders and spheres 48 2.6 Volume of pyramids and cones 53 2.7 Volume of composite solids 57 2.8 Errors in measurement 63 Chapter summary 67 Sample HSC Objective-response questions 68 Sample HSC Short-answer questions 69 Chapter 3 Further algebraic skills and techniques 71 AM3 3.1 Adding and subtracting like terms 71 3.2 Index laws 75 3.3 Equations 81 3.4 Solving equations after substitution 88 3.5 Changing the subject of the formula 93 3.6 Simultaneous equations 98 Chapter summary 103 Sample HSC Objective-response questions 104 Sample HSC Short-answer questions 105 iii
iv Contents Chapter 4 Interpreting sets of data 107 DS4 4.1 Grouped frequency tables and histograms 107 4.2 Mean and mode 113 4.3 Median 118 4.4 Measures of spread 122 4.5 Investigating sets of data 128 4.6 Double stem-and-leaf plots 134 4.7 Double box-and-whisker plots 137 4.8 Radar charts and area charts 143 4.9 Comparison of summary statistics 147 4.10 Two-way tables 151 Chapter summary 157 Sample HSC Objective-response questions 158 Sample HSC Short-answer questions 159 Chapter 5 Applications of trigonometry 161 MM5 5.1 Right-angled trigonometry 161 5.2 Bearings 166 5.3 Trigonometry with obtuse angles 171 5.4 The sine rule 174 5.5 Area of a triangle 180 5.6 The cosine rule 183 5.7 Miscellaneous problems 188 5.8 Surveying 193 Chapter summary 199 Sample HSC Objective-response questions 200 Sample HSC Short-answer questions 201 Chapter 6 Multistage events and applications of probability 203 PB2 6.1 Multistage events 203 6.2 Number of arrangements 208 6.3 Ordered selections 213 6.4 Unordered selections 217 6.5 Probability trees: Product rule 221 6.6 Probability trees: Addition rule 226 6.7 Expected outcomes 231 6.8 Expected value 235 Chapter summary 239 Sample HSC Objective-response questions 240 Sample HSC Short-answer questions 241 HSC Practice paper 1 243
Contents v Chapter 7 Annuities and loan repayments 249 FM5 7.1 Future value of an annuity 249 7.2 Present value of an annuity 254 7.3 Loan repayments 258 Chapter summary 265 Sample HSC Objective-response questions 266 Sample HSC Short-answer questions 267 Chapter 8 Normal distribution 269 DS5 8.1 z-scores - 269 8.2 Using z-scores to compare data 274 8.3 Properties of a normal distribution 278 Chapter summary 283 Sample HSC Objective-response questions 284 Sample HSC Short-answer questions 285 Chapter 9 Modelling linear relationships 287 AM4 9.1 Linear functions 287 9.2 Linear functions as models 291 9.3 Intersecting graphs 295 9.4 Break-even analysis 299 Chapter summary 303 Sample HSC Objective-response questions 304 Sample HSC Short-answer questions 305 Chapter 10 Spherical geometry 307 MM6 10.1 Arc length of a circle 307 10.2 Latitude and longitude 311 10.3 Distances along great circles 315 10.4 Time differences and local time 320 10.5 Time zones 325 Chapter summary 331 Sample HSC Objective-response questions 332 Sample HSC Short-answer questions 333 Chapter 11 Sampling and populations 335 DS6 11.1 Samples 335 11.2 The capture recapture technique 339 11.3 Random sampling 343 Chapter summary 347 Sample HSC Objective-response questions 348 Sample HSC Short-answer questions 349
vi Contents Chapter 12 Modelling non-linear relationships 351 AM5 12.1 Quadratic functions 351 12.2 Cubic, exponential and hyperbolic functions 356 12.3 Algebraic modelling 362 12.4 Direct variation 366 12.5 Inverse variation 371 Chapter summary 375 Sample HSC Objective-response questions 376 Sample HSC Short-answer questions 377 Chapter 13 Mathematics and health 379 Focus study 13.1 Body measurements 379 13.2 Correlation coefficient 386 13.3 Line of best fit 391 13.4 Medication 397 13.5 Life expectancy 401 13.6 Predicting life expectancy 404 Chapter summary 407 Sample HSC Objective-response questions 408 Sample HSC Short-answer questions 409 Chapter 14 Mathematics and resources 411 Focus study 14.1 Water availability 411 14.2 Water usage 416 14.3 Dams, land and catchment areas 420 14.4 Energy consumption 426 14.5 Sustainability 431 Chapter summary 435 Sample HSC Objective-response questions 436 Sample HSC Short-answer questions 437 HSC Practice paper 2 439 HSC formula sheet 445 Glossary 447 Answers 457
Introduction Cambridge HSC Mathematics General 2 Second Edition has been completely revised for the stage 6 Mathematics General syllabus to be implemented in 2013 14. This resourcee closely follows the syllabus and is divided into strands and focus studies. The focus studiess are designed to be integrated across the strands. Teachers can decide on the integration depending on the ability and knowledge of their students. A teaching program is provided that outlines one method of integration. Key features HSC course divided into smaller manageable topics to assist teaching Clear design with appropriate diagrams and tables Each section and exercise is easily accessible at the top of the page Syllabus topic and content listed at the beginning of each chapter Suggested teaching program for the new syllabus can be downloaded from Cambridge GO (www.cambridge.edu.au/go) Each focus study contained in a single chapter to provide easy access Step-by-step worked examples with precise explanations to encourage independent learning Extensive exercises divided into foundation, development and challenge questions that cater for students at different levels Important concepts highlighted for easy reference Excel spreadsheet activities integrated throughout the text Graphics calculator explanations and problems integrated into the text Chapter reviews containing a summary as well as sample HSC objective-response (multiple-choice) and short-answer questions The sample HSC objective-response questions can also be answered by students in a self-marking Quiz Me format for web browsers and smartphones Two complete HSC Practice Papers Companion website on Cambridge GO with a downloadable digital version and an online interactive version of the textbook with links to additional resources Comprehensive glossary and HSC Formula sheet. Also available to support and extend this text is the Cambridge HSC Mathematics General Teacher s Resource Package on Cambridge GO. It contains literacy worksheets, PowerPoint presentations, spreadsheet skills, spreadsheet files, topic tests and a copy of the teaching program. vii
viii Introduction Guide to the icons used in the textbook Identifies challenge questions in the exercises. Challenge questions 1 (placed at the end of each chapter) A PowerPoint file or Word worksheet containing extra challenge questions is available on Cambridge GO. 1.1 Integrated HOTmaths course available (access by teacher account or student subscription). 1B Spreadsheet file available on Cambridge GO. 14.1 Used in Chapters 1 12 to indicate where the teaching program suggests that a Focus Study section be done next. 14A An alternative worksheet format is available for the exercise on Cambridge GO. Study guide 1 (placed on the Chapter Summary bar) A PowerPoint file containing a study guide is available on Cambridge GO. Additional Resources in the Teacher s Resource package on Cambridge GO Lesson Notes a new resource: PowerPoint files containing comprehensive lesson notes and additional examples that can be used in class or given to students as tutorials Chapter tests as worksheets, with worked solutions Literacy worksheets activities to help with mathematical terminology Spreadsheet skills worksheets to use with spreadsheet files provided Copies of the teaching programs and scope and sequence charts
Introduction ix About the author Greg Powers is currently the Head of Mathematics at Cabramatta High School and the coordinator of the Mathematics Head Teacher Western Network. He is an experienced classroom teacher, having taught for over 30 years at a range of different schools. Greg has been a senior marker for the HSC, an educational consultant for the Metropolitan South West Region and has presented at numerous MANSW inservice days. He has also enjoyed several curriculum roles with the Department of Education and Training. Greg is an experienced author who has written numerous texts on mathematics and technology.
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Acknowledgements The author and publisher wish to thank the following sources for permission to reproduce material: Images: Shutterstock.com/muzsy, p.111 / Laszlo Szirtesi, p. 345 / Gheorghe Bunescu Bogdan Mircea, p.401; Courtesy of Sydney Water, p.418; Google Earth, p.420, 421, 423, 424, 425; Commonwealth of Australia (Department of Climate Change and Energy Efficiency). This file is licensed under the Creative Commons Attribution 3.0 Australia License, p.427; UIG via Getty Images, p.429; Courtesy of the NSW Government Department of Planning and Infrastructure. Reproduced by permission, p.431; All other images 2013 used under license from Shutterstock.com. 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. xii