�,;,,# NELSONt - CENGAGE Learning·

Nelson Senior Maths Specialist 11

1st Edition

Stephen Swift

Ross Brodie

Jim Green

Sue Garner

Janet Hunter

Allason McNamara

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National Library of Australia Cataloguing-in-Publication Data

Swift, Stephen, author.

Nelson Maths Specialist 11 Student Book / Stephen Swift,

Ross Brodie, Jim Green, Sue Garner,

Janet Hunter and Allason McNamara.

9780170250276 (paperback)

Includes index.

For secondary school age.

Mathematics--Australia--Textbooks.

Mathematics··Study and teaching (Secondary)--Australia.

510

Cengage learning Australia

Level 7, 80 Dorcas Street

South Melbourne, Victoria Australia 3205

Cengage learning New Zealand

Unit 48 Rosedale Office Park

331 Rosedale Road, Albany, North Shore 0632, NZ

For learning solutions, visit cengage.com.au

Printed in China by China Translation & Printing Services.

2 3 4 5 6 7 8 20 19 1817 16

ABO.UT THIS SERIES

There are 6 books for this series currently. These cover the subj�cts General lv;Iathematics; Mathematical Methods and Specialist Mathematics. An additional 2 books are to follow, which will cover the Essential Mathematics course.

These books have all been written specifically, from scratch, by a national author team for the Senior Australian Curriculum.

There are worked solutions available for purchase, which accompany each of the Student Books.

There are also Exam View question banks which can be purchased separately. These wil� contain a large number of multiple choice questions covering each course: General, Methods and Specialist. An Exam View test generator is also provided, so that t_eachers cari construct-their own tests for eachtopic.

Accompanying each printed textbook is a digital textbook called the N els<;mNetBook and a NelsonNet website.

Go to www.nelsonnet.com.au to log in.

For each chapter the resources are listed. Simply click on the required resource.

This is a web:based eBook which ca� be customised to suit your own learning needs.

The icons with the blue NelsonNet logo are 'hotspots'. Click on the icon and the resource will open. : � '

.

The tools on the vertical tool bar allow you to personalise pages in a variety of ways, including voice reco_rdings, drawings and links to favourite websites. You are also able to zoom in and out.

The tools on the horizontal tbolbar allow you to navigate around your eBook and change settings.

9780170250276 About this series I iii

About this series iii 3.07 Applications of counting

About this book vi and permutations 104

About the authors ix Chapter summary 107

Symbols and abbreviations X Chapter review 108

1 Basic vectors 2 Mixed revision for Chapters 1-3 110

1.01 Two-dimensional vectors 4 4 Applications of vectors 112

1.02 Addition of vectors 12 4.01 The scalar product 114

1.03 Component and polar forms 4.02 The component form 117

of vectors 18 4.03 Properties of the scalar product 122

1.04 Multiplication by scalars 23 4.04 Parallel and perpendicular vectors 124

1.05 Unit vectors 25 4.05 Projection of a vector 127

1.06 Using components 29 4.06 Resolution of forces 130

1.07 Vector properties 32 4.07 Applications of the scalar product 136

1.08 Applications of vectors 35 4.08 Application of vectors to navigation 140

Chapter summary 39 Chapter summary 145

Chapter review 40 Chapter review 147

2 Mathematical proof 44 5 Counting methods and combinations 150

2.01 Mathematical proof 46 5.01 Simple combinations 152

2.02 Counterexamples 50 5.02 Using combinations 156

2.03 Converse 53 5.03 Pascal's triangle 160

2.04 Contrapositive statements 55 5.04 The inclusion-exclusion principle 166

2.05 Euclidean geometry 58 5.05 Simple applications to probability 169

2.06 Geometric proofs using vectors 66 5.06 General use of counting methods 171

2.07 Quantifiers and Proof 5.07 General applications to probability 176

by Contradiction 71 Chapter summary 179

Chapter summary 73 Chapter review 180

Chapter review 74 6 Circle geometry 18.2

3 Counting methods and 6.01 Angles at the centre of circles 184

permutations 78 6.02 Angles at the circumference

3.01 The multiplication principle 80 of circles 191 3.02 The addition principle 85 6.03 Semicircle angle 194

3.03 The pigeonhole principle 90 6.04 Arcs and chords 199

3.04 Simple permutations 93 6.05 Intersecting chords 204

3.05 Restricted permutations 98 6.06 Tangents and secants 207

3.06 Permutations with repetition 102 6.07 Figures in circles 214

Mixed circle problems 220 10.03 Complex conjugates

Chapter summary 225 10.04 Operations with complex numbers 385

Chapter review 228 10.05 The complex plane 389

Mixed revision for Chapters 4-6 235 10.06 The modulus of a complex number 393

7 Real numbers and proofs 238 10.07 Operations in the Argand plane 396

7.01 Integers and subsets 240 10.08 Properties of complex numbers 400

7.02 Simple proofs involving integers 246 10.09 Quadratic equations 404

7.03 Decimal representation 250 Chapter summary 409

7.04 Rational numbers 257 Chapter review 411

7.05 Irrational numbers 261 11 Transformations in the plane 414

7.06 Real numbers 265 11.01 Translations 416

7.07 The principle of mathematical 11.02 Linear transformations 421

induction 270 11.03 Dilations 426

Chapter summary 275 11.04 Rotations 432

Chapter review 277 11.05 Reflections 438

Matrix arithmetic 280 11.06 Composition of transformations 446

8.01 Matrices 282 11.07 Inverse transformations 455

8.02 Scalar multiplication of matrices 287 Determinants and geometry 460 11.08

8.03 Addition and subtraction of matrices 291 468 Chapter summary

8.04 Matrix multiplication 297 470

8.05 Identities and inverses 304 Chapter review

311 12 Trigonometric functions

8.06 Matrix equations and graphs 472

Chapter summary 316 12.01 Period, amplitude and phase shift 474

Chapter review 318 12.02 General trigonometric functions 478

Trigonometric identities 320 12.03 Approximate solution of

9.01 Reciprocal trigonometric functions 322 trigonometric equations 483

9.02 The Pythagorean identities 334 12.04 Exact solution of trigonometric

9.03 Angle sum and difference identities 337 equations 487

9.04 Double angle formulas 343 12.05 Reciprocal trigonometric functions 493

9.05 Finding and using exact values 346 12.06 a sin[x] + b cos[x] 498

9.06 Products to sums 350 12.07 Using a cos[x] + b sin[x] 500

9.07 Sums to products 354 12.08 Modelling periodic motion 503

9.08 Trigonometric identities 360 Chapter summary 508

Chapter summary 364 Chapter review 509

Chapter review 366 Mixed revision for Chapters 10-12 512

Mixed revision for Chapters 7-9 368 Answers 514

10 Complex numbers 370 Glossary 560

10.01 Imaginary numbers 372 Index 568

Complex numbers 375

ABOUT THIS BOOK

A clear outline of chapter contents is provided.

Links to curriculum content descriptions are included.

A Prior learning worksheet is provided, enabling teachers to assess pre-requisite understanding.

·@ Example 1

Decide whether these proofs are deductive, inductive or by contradiction.

a All sheep arc black.

This is a sheep.

Therefore it is black.

b In triangle A, the angles add up to 180°.

In triangle B, the angles add up to 180°.

Therefore, in all triangles the angles add up to 180°.

c Assume that pentagons are squares.

Squares have 4 sides.

Therefore pentagons have 4 sides.

But pentagons have 5 sides, so the assumption cannot be true.

Therefore pentagons are not squares.

Solution

a Proof a begins with some statements and a

conclusion is drawn.

Proof a is deductive.

b Proofb begins with some specific statements Proofb is inductive.

and a general conclusion is drawn.

c Proof c begins with an assumption and a Proof c is by contradiction.

contradiction is shown.

Examples are clearly set out with reasoning (in black) and writing (in blue) in separate columns.

Examples are sequenced in a logical order.

There are generally three examples leading to an exercise.

Examples show solutions and steps which guide students through the use of the TI-Nspire CX and

the CASIO ClassPad CAS calculators.

vi I NELSON SENIOR MATHS Specialist 11 9780170250276

D«lok"·ht1Mror no11M11mmcntNlowl1t rurbychttkinglM1ruth ofthrconu,f'Oliliw: lfoq....JrilJmaJ/r.JJfo11rritltlon,.'ilJ.tht11i1/10,'f""U.

Solution

finJthrcon1,,ro,i1i,.,,. h ii tn,d No, sinu a r«l.,,gl< h not• sq1mt but ltl1Ufourrigh 1 ,ngk1.

n..con1upo>ith"U.: lfi1i,,.._,,0Jq1W11"flw110.,,....,ril,afm,/""'1M lw-,.,f,x,,rijltl� n..u,n1upotilh.,,>htrmcntbnocmx. :.Thr origin.i,uttmrnt i1NOTtruc,.

NitfiJil(ffijj[ iii Mathematical proof in real·life

lmY:n< (1.>nctioo1 rriyon ..-htth u or not thfcon,·uK iltruc, for• 1t1trmcnl P� Q .

for init�r.ifx•·J,IMnx'•9,butthuonwn<linotn«ffS•rilrt""'·lfx'•9,1Mn x• tl. Thi, me-an, thlt 1hr htl'f:IK fur,,ctionofsquuingli not n«eC'U-lri l y 1dlngth,r tq11U< tool, unl fu\Moriginal munbcr...-uroi,ith'I'. $M.iriryonlhrinmnrtrelit1onK<Urrin1Y:11<functions,1lon1"ithco.kbm.k.inginG(ncrd Find ou t a, much as )'UllUn about 1hr Enlsm.r .IIJ<Afrw1nJ Alan Turing.. Whr1tli Ble tchlcy P.ul<?

1§3§;@�ifJ•GI Contrapositive statements

Concepts and techniques

\\'ritr lhr conl r•f'(Hith·• ,t,te mcnt for uch ,t,trmtnt Nlow. a lf)'Ullli,.,,inCoobrrl'Ny,thtniooli,.,,unJuground. b lf)'Ullca ndriw,cu,lhrn)""h""''k•nc•. C [f)'Ulllr<ol J.lhrn)'Ollnc..Jhr1ri"1liJS. dlfx•l,thrnx'•-1. e Ac,nr10.Jiun 1mrt,ibu1L I Allmrnurmort,L

2 Giwn tM coneuroo�i,., ,mernrnt, brio..-. writrdown rach origin,l ,t,trmtnL 1 [f tht1l')'i1no1 blue,thenitil11011unny. b lfa 1tudrn!dor1notm>Jy.thrnMdounotra,1hi1narru. c lf )oodo notli,.,,inAlk�Spring s . thrn )""do notlh.,,inthrd<1<rt. d lfa numbcri1no lf'OlilhY:,thrnit!1no1.1counting numbu. e !fawhidri1no11nr .1Mni1dor1not ha,.,,four..-httl1. I lf )'Ulldono1h1w1job.1he n)'OU..-illnothi, ·rm onry.

CHAPIEll21l.\o�1t,01;..,c,I P'oofl 57

Exercise questions are clearly divided into two categories: concepts and techniques and reasoning

and communication.

Where appropriate, worksheets are provided for additional practice and consolidation of key

concepts.

Each chapter contains at least one investigation, providing students with the opportunity to apply

their understanding to a practical application.

2CHAPTER REVIEW

MATHEMATICAL PROOF

Muttip!echoic� lbt�1olA•Jlo

AJ•A IA•i -�· .. --"""'-­

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J �:,,.=.·�., .. -11i.11� ... ....,_·,o_..,,...,,.lofa!M....:l.ll>t

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"

• v .. 1.3:, .... .i:�11-...... �-,,,-o. e VuQ.V1.w•l,..l,lhll1•; .. knw•O. D "••0.3:,,w•1"-tllhll1•;•t.r,..,•O.

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74IULMllllUIIO.�IIIJ�lo,'__.,11

Each chapter concludes with a detailed chapter summary and a comprehensive review of concepts.

At the end of every three chapters a comprehensive mixed revision set is provided.

9780170250276 About this book I vii

ICONS IN THE TEXTBOOK

HHnHM Link from question to worked

example

Prior learning

Prior learning

- CAS exercise and example

Worksheet

Worksheets

Weblink

Interactive spreadsheet

Spreodsheet

Practice quiz

Practice quiz

GREEK ALPHABET

A,

B,

r,

Ll.,

E,

z,

H,

0,

GREEK ALPHABET

a alpha I,

� beta K,

y gamma A,

() delta M,

E epsilon N,

I; zeta �,

Tj eta 0,

8 theta n,

viii I NELSON SENIOR MATHS Specialist 11

K

"A

µ

V

s

0

7t

iota P, p rho

kappa I:, Ci sigma

lambda T, 'C tau

mu Y, 1) upsilon

nu <I>, <I> phi

xi X, X chi

omicron \Jl, \jf psi

pi Q, (I) omega

9780170250276

ABOUT THE AUTHORS

Lead author: Stephen Swift

Stephen Swift was the lead author of the Nelson Think Maths series. He has also written as a

member of an author team on successful senior mathematics textbooks. Stephen started teaching

Mathematics, Science and Computing in 1973 and has taught at all levels from Year 7 to 12 in

several states, in urban and country schools until retiring in 2010 from the role of Mathematics

Head of Department at Wellington Point State High School in Brisbane.

Ross Brodie

Ross has worked as a classroom teacher and Mathematics Head of Department at a number of

regional and metropolitan secondary schools. He has taught Mathematics at Years 8 to 12. Ross

brings a wide range of experience from education and other sectors to the writing of mathematics

student books.

Jim Green

Jim started his teaching career in Sydney over 30 years ago. He moved to Lismore in Northern NSW

in 1985. He has been a Head of Mathematics for over 20 years. Jim has been involved in various

mathematics associations for most of his teaching career. Jim has been a writer for Mathematics

competitions, NSW School Certificate Examinations and HSC Trial Examination Papers. Jim has

written articles for teaching journals and has presented at local, state and national professional

development conferences. Jim is very interested in the use of technology to teach Mathematics.

Sue Garner

Sue Garner is assistant Head of VCE Mathematics at Ballarat Grammar in Victoria, where she has

taught for twenty years, specialising in both Mathematical Methods (CAS) and Specialist

Mathematics. Sue has been involved in the assessment of senior examinations in both of these

subject areas. Sue has taught in secondary schools in Melbourne Ballarat and the United Kingdom.

She has lectured at the University of Ballarat, La Trobe University and the University of Melbourne

and has published extensively.

Janet Hunter

Janet has taught at a number of independent schools and is currently the Head of Mathematics at

Ascham School in eastern Sydney. Her extensive experience in secondary, tertiary and adult

education was broadened by four years spent in the finance sector. She has been a NSW Higher

School Certificate senior examiner and judge, written HSC review courses and worked on

curriculum development committees for the NSW Board of Studies. Janet has lectured at

universities and international conferences, and her interests include financial mathematics,

multiple intelligences and resilience in education. She currently convenes the regional Maths

Enrichment Group to foster mathematical talent and is active in promoting teacher expertise

through professional associations such as ACARA and NSWIT.

Allason McNamara

Allason is the Head of Mathematics at Mount Scopus Memorial College. She has a Masters in

Mathematical Modelling and Data Analysis and has been a co-author of several mathematics texts.

Allason is currently the President of the MAV and is also a member of the AAMT council.

The Digital Resources Team includes Margaret Denham, Issam Kanaan, Jason Warwick and

Roger Walter.

9780170250276 About the authors I ix

SYMBOLS AND

ABBREVIATIONS

"" approximately equal to RIO or R/{0} means the real numbers except 0 equal to c,� the set of complex numbers

- identically equal to, congruent E is a member of "# not equal to � is not a member of < less than p,p the vector p

> greater than IPl,P the magnitude (norm) of the vector p

:::; less than or equal to AB displacement vector from point A;c: greater than or equal to to point B

plus or minus OA, a, OA, OA the position vector of the -1 square root (r, 8), ( �)

point A* a general operation polar form of the vector with I: summation magnitude r and direction 8 � if ... then, implies, so P � Q means (x, y), (;) component (Cartesian) form of

if P then Q or P implies Q vector ¢:::>, iff if and only if, so P <=> Q means P -a additive inverse of the vector a

if and only if Q A unit vector in the direction of the p

Q,-,Q, Q' not Q vector p

Ill, - similar i, j unit vectors in the x and yII parallel directions .l perpendicular tw change in the vector v RTP required to prove a-b the scalar product (dot product) of QED quod erat demonstrandum, the vectors a and b

indicating a proof is complete 080° bearing of 80°

V for all 0 degree 3 there exists L angle LHS left-hand side 6 triangle RHS right-hand side sss side, side, side test of triangle N,N the set of natural numbers, congruency

{l, 2, 3, 4, .... } SAS side, angle, side test of triangle Z, Z or /the set of integers,{ ... -3, -2, -1, 0, congruency

1, 2, 3, ... } AAS angle, angle, side test of triangle Q,(Q) the set of rational numbers congruency Q'orQ the set of irrational numbers RHS right angle, hypotenuse, side test of Q+ the set of positive rational numbers triangle congruency R,� the set of real numbers factorial, so 5! = 5 x 4 x 3 x 2 x 1

t,, the nth term of a sequence

X I NELSON SENIOR MATHS Specialist 11 9780170250276

"Pr the number of permutations of r things from n things (;} "C,. the number of combinations of r objects from n objects Pa element r in row n of Pascal's '11,r triangle A = {2, 4, 6, 8} the set A consisting of the numbers 2, 4, 6 and 8 A= {x: 0 < x < IO and x an even number} the set A of elements x such that 0 < x < IO and xis an even number � is a subset of C is a proper subset of n intersection u union {}, 0 null set (empty set) u universal set A,A' the complement of the set or event A

IAI the cardinality of the set A; for finite sets, the number of elements n(A) for finite sets, the number of elements in the set A

P(A) the probability of the event A

<p the golden ratio P(n) a proposition associated with the number n

A, A the matrix A

A:(a;}2x3 = (}4

�2 �)=(}4

�2 �)the 2 x 3 matrix A with elements aij

-A the additive inverse of the matrix A

l11 the n X n identity matrix

8ij

the Kronecker delta function A-

1 the multiplicative inverse of the (square) matrix A

<let A, IAI, A the determinant of the matrix A T: (x, y) � (x', y') the transformation that changes the point (x, y) to (x', y')

9780170250276

D a dilation A

p ,½ the dilation factors in the x and y directions for a dilation

Ra a rotation around the origin through and angle a M a reflection across a line through the origin SoT the composition of the transformations S and TT1 the inverse of the transformation T7t (pi) about 3.14159 ... sin (8) the sine of the angle 8 cos (8) the cosine of the angle 8 tan (8) the tangent of the angle 8 cot (8) the cotangent of the angle 8 sec (8) the secant of the angle 8 cosec (8) the cosecant of the angle 8 x � 00 x becomes large and positive

(x approaches infinity) x � -00 x becomes large and negative

(x approaches negative infinity) 7t 7t arcsin (a), sin-1(a) theangle--:::::;8 :::;-such 2 2 that sin (8) = a

arccos (a), cos- 1(a) the angle O :s; 8 :s; 7t such that cos (8) = a 7t 1l arctan (a),tan- 1(a) the angle--:::::;8:s;-2 2 such that tan (8) = a

the imaginary number (ri_) Re(z) the real part of the complex number z

Im(z) the imaginary part of the complex number z

z the complex conjugate of zArg(z) the argument of z

lzl, mod(z) the modulus of the complex number z

Symbols and abbreviations I xi