SPECIALIST
MATHEMATICS
VCE UNITS 3 & 4
CAMBRIDGE SENIOR MATHEMATICS VCE
SECOND EDITION
MICHAEL EVANS | DAVID TREEBY | KAY LIPSON | JOSIAN ASTRUC
NEIL CRACKNELL | GARETH AINSWORTH | DANIEL MATHEWS
,SPECIALIST
MATHEMATICS
VCE UNITS 3 & 4
CAMBRIDGE SENIOR MATHEMATICS VCE
SECOND EDITION
MICHAEL EVANS | DAVID TREEBY | KAY LIPSON | JOSIAN ASTRUC
NEIL CRACKNELL | GARETH AINSWORTH | DANIEL MATHEWS
,Shaftesbury Road, Cambridge CB2 8EA, 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
103 Penang Road, #05-06/07, Visioncrest Commercial, Singapore 238467
Cambridge University Press is part of Cambridge University Press & Assessment, a department
of the University of Cambridge.
We share the University’s mission to contribute to society through the pursuit of education,
learning and research at the highest international levels of excellence.
www.cambridge.org
c Michael Evans, Neil Cracknell, Josian Astruc, Kay Lipson and Peter Jones 2015
c Michael Evans, David Treeby, Kay Lipson, Josian Astruc, Neil Cracknell, Gareth Ainsworth
and Daniel Mathews 2023
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 & Assessment.
First published 2015
Second Edition 2023
20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Cover design by Denise Lane (Sardine)
Typeset by Jane Pitkethly
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 978-1-009-11057-0 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 12, 66 Goulburn Street
Sydney NSW 2000
Telephone: (02) 9394 7600
Facsimile: (02) 9394 7601
Email: memberservices@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 & Assessment 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 & Assessment does not
guarantee the accuracy of such information thereafter.
Cambridge University Press acknowledges the Australian Aboriginal and Torres Strait Islander
peoples of this nation. We acknowledge the traditional custodians of the lands on which our
company is located and where we conduct our business. We pay our respects to ancestors and
Elders, past and present. Cambridge University Press is committed to honouring Australian
Aboriginal and Torres Strait Islander peoples’ unique cultural and spiritual relationships to the
land, waters and seas and their rich contribution to society.
,Contents
Introduction and overview viii
Acknowledgements xiii
1 Preliminary topics
1A Circular functions . . . . . . . . . . . . . . . . . . . . . . .
1
2
1B The sine and cosine rules . . . . . . . . . . . . . . . . . . . 14
1C Sequences and series . . . . . . . . . . . . . . . . . . . . . 19
1D The modulus function . . . . . . . . . . . . . . . . . . . . . 29
1E Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1F Ellipses and hyperbolas . . . . . . . . . . . . . . . . . . . . 36
1G Parametric equations . . . . . . . . . . . . . . . . . . . . . 43
1H Algorithms and pseudocode . . . . . . . . . . . . . . . . . . 51
Review of Chapter 1 . . . . . . . . . . . . . . . . . . . . . . 57
2 Logic and proof
2A Revision of proof techniques . . . . . . . . . . . . . . . . .
65
66
2B Quantifiers and counterexamples . . . . . . . . . . . . . . . 75
2C Proving inequalities . . . . . . . . . . . . . . . . . . . . . . 79
2D Telescoping series . . . . . . . . . . . . . . . . . . . . . . 82
2E Mathematical induction . . . . . . . . . . . . . . . . . . . . 84
Review of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . 96
,iv Contents
3 Circular functions
3A The reciprocal circular functions . . . . . . . . . . . . . . . 102
101
3B Compound and double angle formulas . . . . . . . . . . . . 109
3C The inverse circular functions . . . . . . . . . . . . . . . . 115
3D Solution of equations . . . . . . . . . . . . . . . . . . . . . 122
3E Sums and products of sines and cosines . . . . . . . . . . . 129
Review of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . 133
4 Vectors
4A Introduction to vectors . . . . . . . . . . . . . . . . . . . . 144
143
4B Resolution of a vector into rectangular components . . . . . . 155
4C Scalar product of vectors . . . . . . . . . . . . . . . . . . . 167
4D Vector projections . . . . . . . . . . . . . . . . . . . . . . 172
4E Collinearity . . . . . . . . . . . . . . . . . . . . . . . . . . 176
4F Geometric proofs . . . . . . . . . . . . . . . . . . . . . . . 179
Review of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . 187
5 Vector equations of lines and planes. . . . . . . . . . . . . . . . . . . 197
5A Vector equations of lines 198
5B Intersection of lines and skew lines . . . . . . . . . . . . . . 206
5C Vector product . . . . . . . . . . . . . . . . . . . . . . . . 212
5D Vector equations of planes . . . . . . . . . . . . . . . . . . 217
5E Distances, angles and intersections . . . . . . . . . . . . . . 223
Review of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . 230
6 Complex numbers
6A Starting to build the complex numbers . . . . . . . . . . . .
237
238
6B Modulus, conjugate and division . . . . . . . . . . . . . . . 246
6C Polar form of a complex number . . . . . . . . . . . . . . . 251
6D Basic operations on complex numbers in polar form . . . . . 255
6E Solving quadratic equations over the complex numbers . . . . 262
6F Solving polynomial equations over the complex numbers . . . 266
6G Using De Moivre’s theorem to solve equations . . . . . . . . . 273
6H Sketching subsets of the complex plane . . . . . . . . . . . . 277
Review of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . 281
7 Revision of Chapters 1–6
7A Technology-free questions . . . . . . . . . . . . . . . . . .
289
289
7B Multiple-choice questions . . . . . . . . . . . . . . . . . . . 295
7C Extended-response questions . . . . . . . . . . . . . . . . . 305
7D Algorithms and pseudocode . . . . . . . . . . . . . . . . . . 314
, Contents v
8 Differentiation and rational
8A
functions
. . . . . . . . . . . . . . . . . . . . . . . . .
Differentiation
316
317
8B Derivatives of x = f(y) . . . . . . . . . . . . . . . . . . . . . 322
8C Derivatives of inverse circular functions . . . . . . . . . . . 326
8D Second derivatives . . . . . . . . . . . . . . . . . . . . . . 331
8E Points of inflection . . . . . . . . . . . . . . . . . . . . . . 333
8F Related rates . . . . . . . . . . . . . . . . . . . . . . . . . 346
8G Rational functions . . . . . . . . . . . . . . . . . . . . . . . 354
8H A summary of differentiation . . . . . . . . . . . . . . . . . 363
8I Implicit differentiation . . . . . . . . . . . . . . . . . . . . . 365
Review of Chapter 8 . . . . . . . . . . . . . . . . . . . . . . 371
9 Techniques of integration . . . . . . . . . . . . . . . . . . . . . . . 381
9A Antidifferentiation 382
9B Antiderivatives involving inverse circular functions . . . . . . 390
9C Integration by substitution . . . . . . . . . . . . . . . . . . 392
9D Definite integrals by substitution . . . . . . . . . . . . . . . 398
9E Use of trigonometric identities for integration . . . . . . . . . 400
9F Further substitution . . . . . . . . . . . . . . . . . . . . . . 404
9G Partial fractions . . . . . . . . . . . . . . . . . . . . . . . 407
9H Integration by parts . . . . . . . . . . . . . . . . . . . . . . 415
9I Further techniques and miscellaneous exercises . . . . . . . 420
Review of Chapter 9 . . . . . . . . . . . . . . . . . . . . . . 424
10 Applications of integration
10A The fundamental theorem of calculus . . . . . . . . . . . . . 430
429
10B Area of a region between two curves . . . . . . . . . . . . . 436
10C Integration using a CAS calculator . . . . . . . . . . . . . . 443
10D Volumes of solids of revolution . . . . . . . . . . . . . . . . 449
10E Lengths of curves in the plane . . . . . . . . . . . . . . . . . 458
10F Areas of surfaces of revolution . . . . . . . . . . . . . . . . 462
Review of Chapter 10 . . . . . . . . . . . . . . . . . . . . . 467
11 Differential equations
11A An introduction to differential equations . . . . . . . . . . . 478
477
11B Differential equations involving a function of the
independent variable . . . . . . . . . . . . . . . . . . . . . 482
11C Differential equations involving a function of the
dependent variable . . . . . . . . . . . . . . . . . . . . . . 490
11D Applications of differential equations . . . . . . . . . . . . . 493
11E . . . . . . . . . . . . . . . 504
The logistic differential equation
11F Separation of variables . . . . . . . . . . . . . . . . . . . . 507
,vi Contents
11G Differential equations with related rates . . . . . . . . . . . 511
11H Using a definite integral to solve a differential equation . . . . 516
11I Using Euler’s method to solve a differential equation . . . . . 518
11J Slope field for a differential equation . . . . . . . . . . . . . 525
Review of Chapter 11 . . . . . . . . . . . . . . . . . . . . . 528
12 Kinematics
12A Position, velocity and acceleration . . . . . . . . . . . . . . 538
537
12B Constant acceleration . . . . . . . . . . . . . . . . . . . . . 553
12C Velocity–time graphs . . . . . . . . . . . . . . . . . . . . . 558
12D Differential equations of the form v = f(x) and a = f(v) . . . . . 565
12E Other expressions for acceleration . . . . . . . . . . . . . . 569
Review of Chapter 12 . . . . . . . . . . . . . . . . . . . . . 574
13 Vector functions and vector. .calculus
13A Vector functions . . . . . . . . . . . . . . . . . . . . .
583
584
13B Position vectors as a function of time . . . . . . . . . . . . . 588
13C Vector calculus . . . . . . . . . . . . . . . . . . . . . . . . 594
13D Velocity and acceleration for motion along a curve . . . . . . 600
Review of Chapter 13 . . . . . . . . . . . . . . . . . . . . . 608
14 Revision of Chapters 8–13
14A Technology-free questions . . . . . . . . . . . . . . . . . .
615
615
14B Multiple-choice questions . . . . . . . . . . . . . . . . . . . 619
14C Extended-response questions . . . . . . . . . . . . . . . . . 630
14D Algorithms and pseudocode . . . . . . . . . . . . . . . . . . 642
15 Linear combinations of random variables and. the. . sample
15A Linear functions of a random variable
mean
. . . . . . . . . .
645
646
15B Linear combinations of random variables . . . . . . . . . . . 651
15C Linear combinations of normal random variables . . . . . . . 661
15D The sample mean of a normal random variable . . . . . . . . 663
15E Investigating the distribution of the sample mean
using simulation . . . . . . . . . . . . . . . . . . . . . . . 666
15F The distribution of the sample mean . . . . . . . . . . . . . . 671
Review of Chapter 15 . . . . . . . . . . . . . . . . . . . . . 677
, Contents vii
16 Confidence intervals and hypothesis testing for the mean
16A . . . . . . . . .
Confidence intervals for the population mean
682
683
16B Hypothesis testing for the mean . . . . . . . . . . . . . . . . 692
16C One-tail and two-tail tests . . . . . . . . . . . . . . . . . . . 702
16D Two-tail tests revisited . . . . . . . . . . . . . . . . . . . . 708
16E Errors in hypothesis testing . . . . . . . . . . . . . . . . . . 712
Review of Chapter 16 . . . . . . . . . . . . . . . . . . . . . 717
17 Revision of Chapters 15–16
17A Technology-free questions
726
. . . . . . . . . . . . . . . . . . 726
17B Multiple-choice questions . . . . . . . . . . . . . . . . . . . 728
17C Extended-response questions . . . . . . . . . . . . . . . . . 730
17D Algorithms and pseudocode . . . . . . . . . . . . . . . . . . 732
18 Revision of Chapters 1–17
18A Technology-free questions
734
. . . . . . . . . . . . . . . . . . 734
18B Multiple-choice questions . . . . . . . . . . . . . . . . . . . 738
18C Extended-response questions . . . . . . . . . . . . . . . . . 743
Glossary 752
Answers 764
Online appendices accessed through the Interactive Textbook or PDF Textbook
Appendix A Guide to the TI-Nspire CAS calculator in VCE mathematics
Appendix B Guide to the Casio ClassPad II CAS calculator in VCE mathematics
Appendix C Introduction to coding using Python
Appendix D Introduction to coding using the TI-Nspire
Appendix E Introduction to coding using the Casio ClassPad
, Introduction and
overview
Cambridge Specialist Mathematics VCE Units 3&4 Second Edition provides a complete
teaching and learning resource for the VCE Study Design to be first implemented in 2023.
It has been written with understanding as its chief aim, and with ample practice offered
through the worked examples and exercises. The work has been trialled in the classroom,
and the approaches offered are based on classroom experience and the helpful feedback of
teachers to earlier editions.
Specialist Mathematics Units 3 and 4 provide a study of elementary functions, algebra,
calculus, and probability and statistics and their applications in a variety of practical and
theoretical contexts. This book has been carefully prepared to meet the requirements of the
new Study Design.
The book begins with a review of some topics from Specialist Mathematics Units 1 and 2,
including algorithms and pseudocode, circular functions and proof.
The concept of proof now features more strongly throughout the course. To account for
this, we have a specially written Proof chapter that involves topics such as divisibility;
inequalities; graph theory; combinatorics; sequences and series, including partial sums and
partial products and related notations; complex numbers; matrices; vectors and calculus.
Other chapters also feature exercises aimed to further develop your students’ skills in
mathematical reasoning.
In addition to the online appendices on the general use of calculators, there are three online
appendices for using both the programming language Python and the inbuilt capabilities
of students’ CAS calculators.
The four revision chapters provide technology-free, multiple-choice and extended-response
questions. Each of the first three revision chapters contain a section on algorithms and
pseudocode.
The TI-Nspire calculator examples and instructions have been completed by Peter Flynn,
and those for the Casio ClassPad by Mark Jelinek, and we thank them for their helpful
contributions.
, Overview of the print book
1 Graded step-by-step worked examples with precise explanations (and video versions)
encourage independent learning, and are linked to exercise questions.
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 Questions that suit the use of a CAS calculator to solve them are identified within
exercises.
5 Chapter reviews contain a chapter summary and technology-free, multiple-choice, and
extended-response questions.
6 Revision chapters provide comprehensive revision and preparation for assessment,
including new practice Investigations.
7 The glossary includes page numbers of the main explanation of each term.
8 In addition to coverage within chapters, print and online appendices provide additional
support for learning and applying algorithms and pseudocode, including the use of Python
and TI-Nspire and Casio ClassPad for coding.
Numbers refer to descriptions above.
