Maths in focus. Year 12 : mathematics extension 2

390 Pages·36.966 MB·English

by  Janet Hunter, Jim Green

Jim Green
12
YEAR
Janet Hunter
Series editor: Robert Yen
Maths in Focus 12 Mathematics Extension 2 © 2019 Cengage Learning Australia Pty Limited
1st Edition
Jim Green Copyright Notice
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PREFACE
Maths in Focus 12 Mathematics Extension 2 is The theory presented in this book follows a
written for the new Mathematics Extension 2 logical order, although some topics may be
syllabus (2017). Although this is a new book, learned in any order. We have endeavoured to
students and teachers will find that it contains produce a practical text that captures the spirit
the familiar features that have made Maths in of the course, providing relevant and meaningful
Focus a leading senior mathematics series, such applications of mathematics.
as clear and abundant worked examples in plain
The NelsonNet teacher website contains
English, comprehensive sets of graded exercises,
additional resources such as worksheets,
chapter Test yourself exercises and practice sets of
ExamView quizzes and questionbank, topic tests
mixed revision and exam-style questions.
and worked solutions (see page viii). We wish
The Mathematics Extension 2 course is designed all teachers and students using this book every
for students who intend to study mathematics success in embracing the new Mathematics
at university, possibly majoring in the subject. Extension 2 course.
AB UT THE AUTHORS
Jim Green is Head of Mathematics at Reflections, and recently co-authored Nelson
Trinity Catholic College, Lismore, where he Senior Maths 11-12 Specialist Mathematics for the
has spent most of his teaching career of over Australian Curriculum.
35 years. He has taught pre-service teachers
at the Southern Cross University, written
CONTRIBUTING
HSC examinations and syllabus writing
drafts, composed questions for the Australian AUTHORS
Mathematics Competition and recently
Jim Green and Janet Hunter also wrote the
co-authored Nelson Senior Maths 11-12 Specialist
topic tests.
Mathematics for the Australian Curriculum.
Roger Walter wrote the ExamView questions.
Janet Hunter is Head of Mathematics at
Ascham School, Edgecliff, and has worked in the Shane Scott wrote the worked solutions to all
finance sector and in tertiary/adult education as exercise sets.
a lecturer. She has been a senior HSC examiner
and judge, an HSC Advice Line adviser, served
on the editorial team for the MANSW journal
ISBN 9780170413435 Preface iii
C NTENTS
3
PREFACE ......................................................iii
SYLLABUS REFERENCE GRID ............................vi
3D VECTORS 78
ABOUT THIS BOOK .....................................viii
V1.1 3.01 Review of 2D vectors ..................80
MATHEMATICAL VERBS.................................xiii
V1.1 3.02 3D vectors ................................86
V1.2 3.03 Angle between vectors ................89
V1.2 3.04 Geometry proofs using vectors ......93
1
V1.3 3.05 3D space .................................97
V1.3 3.06 Vector equation of a curve .........100
COMPLEX NUMBERS 2
V1.3 3.07 Vector equation of a
N1.1 1.01 Complex numbers ........................4
straight line .............................106
N1.1 1.02 Square root of a complex
V1.3 3.08 Parallel and perpendicular
number ....................................14
lines.......................................110
N1.2 1.03 The Argand diagram ..................16
Test yourself 3 ............................................114
N1.2 1.04 Modulus and argument ...............23
N1.2 1.05 Properties of moduli and
arguments ................................29
4
N1.3 1.06 Euler’s formula ...........................36
N1.3 1.07 Applying Euler’s formula ..............39
APPLYING COMPLEX NUMBERS 116
Test yourself 1 ..............................................42
N2.1 4.01 De Moivre’s theorem ................118
N2.1 4.02 Quadratic equations with
complex coefficients .................126
2
N2.1 4.03 Polynomial equations ................132
N2.2 4.04 Operations on the complex
MATHEMATICAL PROOF 46
plane ....................................137
P1 2.01 The language of proof ....................48
N2.2 4.05 Roots of unity ..........................149
P1 2.02 Proof by contradiction .....................59
N2.1 4.06 Roots of complex numbers .........158
P1 2.03 Proof by counterexample .................61
N2.2 4.07 Curves and regions on the
P1 2.04 Proofs involving numbers .................64 complex plane ........................163
P1 2.05 Proofs involving inequalities .............68 Test yourself 4 ............................................174
Test yourself 2 ..............................................74
PRACTICE SET 1 ........................................179
iv MATHS IN FOCUS 12. Mathematics Extension 2 ISBN 9780170413435
5 7
FURTHER MATHEMATICAL MECHANICS 254
INDUCTION 188 M1.2 7.01 Velocity and acceleration in
P2 5.01 Review of mathematical induction ...190 terms of x ...............................256
P2 5.02 Further mathematical induction .......194 M1.1 7.02 Simple harmonic motion ...........260
P2 5.03 Series and sigma notation .............197 M1.4 7.03 Projectile motion ......................268
P2 5.04 Applications of mathematical M1.2 7.04 Forces and equations of
induction ....................................202 motion ...................................278
P2 5.05 Recursive formula proofs ................208 M1.3 7.05 Resisted horizontal motion .........287
P1 5.06 Proofs involving inequalities and M1.3 7.06 Resisted vertical motion .............292
graphs .......................................211 M1.4 7.07 Resisted projectile motion ..........302
Test yourself 5 ............................................217 Test yourself 7 ............................................316
PRACTICE SET 2 ........................................318
6
ANSWERS ...............................................322
FURTHER INTEGRATION 220
C1 6.01 Integration by substitution ..............222 INDEX ......................................................374
C1 6.02 Rational functions with quadratic
denominators .............................229
C1 6.03 Partial fractions ...........................233
C1 6.04 Integration by parts .....................244
C1 6.05 Recurrence relations .....................249
Test yourself 6 ............................................253
ISBN 9780170413435 Contents v
SYLLABUS REFERENCE GRID
Topic and subtopic Maths in Focus 12
Mathematics Extension 2 chapter
PROOF
MEX-P1 The nature of proof 2 Mathematical proof
5 Further mathematical induction
MEX-P2 Further proof by mathematical induction 5 Further mathematical induction
VECTORS
MEX-V1 Further work with vectors 3 3D vectors
V1.1 Introduction to three-dimensional vectors
V1.2 Further operations with three-dimensional vectors
V1.3 Vectors and vector equations of lines
COMPLEX NUMBERS
MEX-N1 Introduction to complex numbers 1 Complex numbers
N1.1 Arithmetic of complex numbers
N1.2 Geometric representation of a complex number
N1.3 Other representations of complex numbers
MEX-N2 Using complex numbers 4 Applying complex numbers
N2.1 Solving equations with complex numbers
N2.2 Geometric implications of complex numbers
CALCULUS
MEX-C1 Further integration 6 Further integration
MECHANICS
MEX-M1 Applications of calculus to mechanics 7 Mechanics
M1.1 Simple harmonic motion
M1.2 Modelling motion without resistance
M1.3 Resisted motion
M1.4 Projectiles and resisted motion
vi MATHS IN FOCUS 12. Mathematics Extension 2 ISBN 9780170413435
MATHS IN FOCUS AND NEW CENTURY MATHS 11–12
MATHS IN FOCUSADVACNEDMATHEMATICS MATHS IN FOCUSEXTENSION 1MATHEMATICS (PATHWAY 1)NEW CENTURY MATHSSTANDARDMATHEMATICS (PATHWAY 2)NEW CENTURY MATHSSTANDARDMATHEMATICS
Margaret Grove Margaret Grove Judy BinnsSue Thomson Robert YenMargaret Willard Sarah Hamper Klaas Bootsma
EDITION3RD EDITION3RD EDITION2ND EDITION4TH Klaas Bootsma
YEAR 11 M3RaD rEgDaITrIOeNt Grove YEAR 11 M3RaD rEgDaITrIOeNt Grove YEAR 11 SJ2uNuDde Ey TD BIhTiIoOnmNnsson YEAR 11 SMR4TaoaHrb raEgeDhaIr TtHrI OeYaNtem Wnpiellra rd
MATHS IN FOCUSADVACNEDMATHEMATICS MATHS IN FOCUSEXTENSION 1MATHEMATICS NEW CENTURY MATHSSTANDARD 1MATHEMATICS NEW CENTURY MATHSSTANDARD 2MATHEMATICS
Margaret Grove Margaret Grove Judy BinnsSue Thomson Robert YenMargaret Willard Sarah Hamper Klaas Bootsma
EDITION3RD EDITION3RD EDITION2ND EDITION4TH Klaas Bootsma
YEAR 12 M3RaD rEgDaITrIOeNt Grove YEAR 12 M3RaD rEgDaITrIOeNt Grove YEAR 12 SJ2uNuDde Ey TD BIhTiIoOnmNnsson YEAR 12 SMR4TaoaHrb raEgeDhaIr TtHrI OeYaNtem Wnpiellra rd
M
ATHS IN FO
CUS
EXTENSION 2MATHEMATICS
Janet Hunter Jim Green
YEAR 12 JJaimne Gt rHeuennt er
ISBN 9780170413435 Syllabus reference grid vii
ABOUT
THIS B K
AT THE BEGINNING OF EACH CHAPTER
• Each chapter begins on a
double-page spread showing PROOF
2.
the Chapter contents and a
list of chapter outcomes
MATHEMATICAL PROOF IN THIS CHAPTER YOU WILL:
Mbmaaastthihse eommf aamttiicacatahll e apmrrgaoutoimcf sei.sn Iutnss etohdfi s a toc th ydapepetet etrhrm,a iytn oiesu cwwohimlel mtehexoarn m ainsins neeur ttmihoebn esn ra,a trauelrg etre uobefr apo rrao fnoadfl s aegn.e dPo rmdoeoevtfrsey l.fooprm r igthoer ous •••• lusuetssaaeetre ntp h trtehohe oep fcr fooboorynmf t rccaaoolp nnloatcrsaneitgdpivuitcesat igoaoefnn di maa ncnpdodli nc scvayoetmiruosbnento, e olrsnef e oxagaf asmpttaripootlenoem,f, eeiqnnctuliuvdailnegnc ∀e ,a ∃n,d ⇒ eq, u⇔al,it y¬P, iff and ∈
• prove results involving numbers and inequalities
• prove further results involving inequalities based on previous results
CHAPTER OUTLINE
2.01 The language of proof
2.02 Proof by contradiction
2.03 Proof by counterexample
2.04 Proofs involving numbers
2.05 Proofs involving inequalities
Test yourself 2
iStock.com/Radachynskyi
• Terminology is a chapter
TERMINOLOGY Implication
gkwelioyths wsianor yrthd tesh aacnth dpap rpethevrrieawsess tfhroe m ccceimoooqPnsQswcPnnuupttao vtanar il⇒in⇒vrtiimettttc aeearePwt ammppl esQrPe)trnleo.eieeini :o eIt((sxnn PceiifsTianff ttte Pm PQii PQh:¬ifsv s fe ⇒)tPep⇔ Qttan Qh: lch t⇔e o Teo )⇒QeeQ:t.nn n m h AIt. Q vi fQer¬sPne eu Pct))rnP( ee .rP soit ⇒.uexsi n:sia e ¬fttP m,a orQ a Qtla ⇒pnshP poa lde⇒ oen ⇒n Qts odrt i thnut ¬ (QhQiaeilvPfyet. e ⇒ ( P sic(isffih o tf t QoPa Pnhntw t e,eott rnahhsmatl eeptsQQehnonon, a s tQwtotih t rt)eaieo v n i es tpQsnrtreiaEttaa‘woagdthhnsnoDeraesaadiguftmmttt: i ltm tboiteqeoehshyn unp nneiee no: tcnst i ¬didTtet,oo rhaPqpiennahr eru,rltet a eroieoPract s gdatn ptl′ dri f id eotooacuaeyigmlarspsecs: ali p erPt otot aie.onixiorr.ooq ss gn tnfniu rtau+: i ael ormoA sn e foreyd ed fpa pu(n ’ bra. msrt≥o ut esia ,ottm s t L af xenu titam eossh+tmeetiae dn:ybtne A otfnat o,o st tPs∀hs r seiu )s ihn.xms ot t, nreewyuon s∈ te ct Phe  e, . I‘TEW iffX h PrPeiA ta tIennhMf o tdeIht n PadQe to QLi fosnEo’tn ’l atal 1 nosPs dwtP u⇒ i fdn⇒o ygrQ, Qtts hhct.aaeetn ne2 ma Ipl sewaonri tltbsl i efnoa rf im ela.anadt ihafe-st m‘hPae tinmic asptll ainetoes tmQat’ei.onnt,: we can write the statement
Solution
HCoomWWnevweSSrosrek 2Afpcooa rprn. trat0iorlclaou d1cflaai icsrn te ic somTa. nsTahe to.hh rTeie scmh iosealu rdtaneiic tfaesfnre rieres ge xdmnaimtfuof fesprtarol eeomn.gf tTt e atehnnye p ae yenxo saa armorfefg p bpulpaermso wereodnhfo ,te os rtnuoeh c rathfeth aaesshs so odtnawet,ds ea uma cw estatinvaytte eom, mfin ateydhn uibtnc ekot iitrvnr epug er aae notmdtnr liipybsre uof otioesrfd tab r uy e D ThefeinnPQ e:w: IPIe d waconainlndl’ t fwQ asirt laiu.tsde f:yo.llows:
CHonotmraepwoosritkive mPloatsot oafntden A trois tthoetl eC.lassical Greek mathematicians and philosophers such as Euclid, Socrates, P implies Q, or P ⇒ Q.
WS In mathematics, deductive proof and proof by contradiction are often used, whereas inductive Converse
HQoumaenwtifoierrks proof is common in science. Proof by counterexample is used to show a statement is not true. To find the converse of a statement, we reverse the implication.
SAbo tsatthat)te.emmeenntt, ,p rporpoopsiotisoint ioorn p roerm pisree ism a isseentence that is either true or false (but not CF‘Poo irmn thvpelei esrtssa Qtee’m ise ‘nQt ‘iImf pPl itehse Pn’ .Q’, the converse is ‘If Q then P’, or the converse of
‘It is raining’ is a statement. ‘Is it raining?’ is not a statement but a question. We denote a In symbols, the converse of P ⇒ Q is Q ⇒ P.
WsAQS1ato a <siemtst ea w1baemt t eitwliemrlmn uhneteeeo ns nbsw tyt t ahac d atoe>e e uc mtfbalrid npuei seitnb th ttaesr .ol ou mmlfee a ateott shentterelay;rm tm feioafmi rtna ieec oanxalntalo; dm gdf yoebp p rula eeers,neex Pd dab:ms oi Inoptt hn llieo s pt ,g hroQiaecsi: ian ctFliio nvponegrrt o. ebaoxulltft ..r iFeta oilsr x ni,n odsdttx atrn(uxce2e ,+f toh2rex a )sllt= ant2uexmm+be2ne.rt st.hat TbFPsCuuyoohmtntrhe vt e ahoecxgrofeaos nertmoha:v rspIeei’f glr stestihq,hen eub eaooaol rfs rets qemahtusa t :Ptao reIruyfnme tet hh oaesa nfetr ga titoogt hmethrehmat aels-yoe’ra nn nto2htggr es emlmieosddtarae ye systm ir. odnir eaoa n mnotg dfbal eaeyi, t ttnstrr hoicuaetoen nb.sgqvelue etr arisrusee ee a.oq rEfue qta thlur etuao lehl .yty,hp teoh tseeu ncmou nsoevf e itsrh seeeq smuqaauly at robe est htorefu e
the other 2 sides, then the triangle is right angled.
Implication or ‘if-then’ statement However, consider the statement ‘If you are a cow, then you eat grass’. This is a true
One of the most common arguments is the if-then statement, also known as the statement.
implication statement or conditional statement. This can be written using symbols. Tarhe eo cthoenrv earnsiem iasl ‘sI ft hyaotu e eaat tg grarasss.s, then you are a cow’. The converse is not true because there
48 MATHS IN FOCUS 12. Mathematics Extension 2 ISBN 9780170413435 ISBN 9780170413435 2. Mathematical proof 49
viii MATHS IN FOCUS 12. Mathematics Extension 2 ISBN 9780170413435
IN EACH CHAPTER
• Important facts and formulas are highlighted
INVESTIGATION
in a shaded box. 1 = –1?
Mathematics is based on definitions and proof. It is very important to be accurate
with the definitions when we set up a proof and to use rigour when applying the
• Important words and phrases are printed mathematical algorithms so that the conclusions are valid.
Study the proof that 1 = -1 below and see if you can find the error. There must be a flaw
in red and listed in the Terminology somewhere because 1 ≠ -1!
Proof that 1 = –1
chapter glossary. Consider the complex number i= −1.
Then we know that:
i=i
• Graded exercises include exam-style problems −1= −1
and realistic applications. −11= −11
1= −1
−1 1
• Worked solutions to all exercise questions Then cross-multiplying we have:
1× 1= −1× −1
are provided on the NelsonNet teacher ∴ 1 = -1 QED
website.
2.04 Proofs involving numbers
In this section we will develop some techniques for proving common properties of numbers.
• Investigations explore the syllabus in more
Properties of positive integers
detail, providing ideas for modelling activities An even number can be described by the formula 2n where n ∈ .
An odd number can be described by the formula 2n - 1 where n ∈ .
and assessment tasks. A square number can be described by the formula n2 where n ∈ .
A number, X (X ∈ ), is divisible by another number, p (p ∈ ), if ∃ Y (Y ∈ )
such that X = pY.
• Did you know? contains interesting facts
Note that in general, properties such as even, odd, multiples, factors, refer to the positive
and applications of the mathematics learned sinqtueagree rnsu omnblye.r F.or instance, we would not usually say -7 is an odd number or that 49 is a
in the chapter.
64 MATHS IN FOCUS 12. Mathematics Extension 2 ISBN 9780170413435
3 Decide whether each statement is true or false. If it is false, provide a counterexample.
EXAMPLE 11 a If a quadrilateral has diagonals that are perpendicular, then it is a square.
Determine whether this statement is true: b If p ≤ 3, then 1p≥13
All camels have one hump. c (x + y)2 ≥ x2 + y2 for all x, y ∈ 
Solution d If pq = rq then r = p
e All rectangles are similar.
Most camels have one hump, but the Bactrian camel has two humps.
4 In each case, determine whether the counterexample shows that the statement is false.
Therefore the statement is false. a Statement: x2 = 3x - 2 for x ∈ 
Counterexample: For x = 3, LHS = 32 = 9
EXAMPLE 12 RHS = 3(3) - 2 = 7 ∴ LHS ≠ RHS
The statement is not true.
Explain what is wrong with the following argument. b Statement: All dogs are domesticated.
Teacher: Smoking is bad for you. Counterexample: Cats are domesticated.
Student: I have a counterexample: social media is bad for you. The statement is not true.
5 Decide whether the statement below is true or false. If it is false, find a counterexample.
STshohoelwu s tttuhiodaten nsmt hoaksi nfogu ins dn oatn boathde fro hr ayboiut .that is bad for you rather than a counterexample to 6 IIsf ait >a lbw tahyes ntr u1ae< th1ba.t (x−1)1(x−2)+(x−2)1(x−3)=(x−1)(2xx−−24)(x−3) for x = 4, 5, 6, …?
Exercise 2.03 Proof by counterexample 7 Is it always true that n1<n1−1?
1 Find a counterexample to prove that each statement is not true. 8 Are all squares rhombuses? Are all rhombuses squares?
a ∀ n ∈ , n2 ≥ n 9 A circle can always be drawn through the four vertices of a rectangle. Is this true for all
b ∀ n ∈ , n2 + n ≥ 0 quadrilaterals?
ecd ∀A∀l lxx ,p ∈yr i ∈m ,e x n a+unmx1d bn≥e ∈2rs ar, ex no d+d y.n = (x + y)(xn – 1 - xn – 2y + xn – 3y2 - … x2yn – 2 + yn – 1) 111012 IDIss otith taehl ewa ndagyialseg t osrunumeal ts oh ofa fta al l x kp i-ot elyy ag l=wo n aysy -sw ixint ht ?e nrs esicdte isn Ssind =e t1h8e0 °k(inte -? 2)?
2 Find a counterexample for each statement to demonstrate it is false.
a If n2 = 100, then n = 10. 13 Is it always true that if n > m then nk > mk?
b The statement x3 - 6x2 + 11x - 6 = 0 is true for x = 1, 2, 3, …
c All lines that never meet are parallel. DID YOU KNOW?
d If an animal lays eggs, then it is a bird. Bertrand Russell
Bertrand Russell was a philosopher and mathematician who studied logic. He is famous
for inventing a paradox in 1901 that is understandably called Russell’s Paradox. It was a
proof by contradiction for an idea in set theory.
62 MATHS IN FOCUS 12. Mathematics Extension 2 ISBN 9780170413435 ISBN 9780170413435 2. Mathematical proof 63
ISBN 9780170413435 About this book ix
AT THE END OF EACH CHAPTER
• Test yourself contains chapter revision exercises.
• Practice sets (after several chapters) provide a comprehensive variety of mixed exam-style
questions from various chapters, including short-answer, free-response and multiple-choice
questions.
2.
TEST YOURSELF
Practice set 1
1 Write each if-then statement in the form P ⇒ Q and state P and Q. In Questions 1 to 10, select the correct answer A, B, C or D.
a If I get a lot of sleep, then I am healthy.
2 Wabc ritAIIeff t⇒athh pee oB ctl oeyangcvohener srhe ia sos nf5 ie cbsaei cd, het hss¬,et taPnht eeI⇒m nw eiQitln l i tsl,e aa rpne.ntagco n.N ⇒ ¬M d ¬B ⇒ ¬F 1 JHzza1koz2ew2= l i=mzst⋅aezzn1d y tz ha2 ree s ec ocorarmregcpz tz1l?ez1xz2 2=n = uz m1azrb2g e zr1 r ×u laersg i nz 2a summary. zz12=zz12
e If I can save money, then I can buy a car. A all B one C some D none
fg IIff ma =y bc othmepnu ate3 r= i sb 3b.roken, then I am bored. 2 (Asi n θs i−n in cθo −s θi )cno =s nθ B sin (−nθ) + i cos (−nθ)
3 Define ‘iff’. Give an example. C (−i)n(cos nθ + i sin nθ) D in(cos nθ + i sin nθ)
4 For each statement, find the converse and determine if the statement is an equivalence. 3 If ω is a complex root of z3 = 1, which statement is false?
If it is an equivalence, then write an iff statement. A ω2 + ω + 1 = 0 B ω2 = ω
ab IIff ax q>u 1a,d trhielant ex1ra<l h1.as equal diagonals, then it is a square. 4 CC onsωid4 e=r ωth e statement: ‘If there is a stationarDy poiωn−t 2a =t xω1 = 3 then f ′(3) = 0.’
c If I pass my exams, then I study hard. Which of the following is false?
d If a = 3, then a2 = 9. A The converse B The contrapositive
e If a triangle has 2 equal angles, then it is isosceles. C The negation D The proposition
5 Write the negation of each statement. 5 Which inequality always holds for a > b?
ac IKt oisa lraasi narineg c.u te. db TSohme ea pppeloep ilse naoret rsiepxeis.t. A 1a<1b B a12<b12 C a2 > b2 D a3 > b3
eg Tp ∈he y are all correct. f x ≤ 4 6 Consider the vectors u−31, v62, w−39 and z−31.
6 Joe was asked what was the negation of the statement ‘There were more than 10’. Which vectors are parallel?
He said, ‘There were less than 10.’ Is he correct? A u and v B u and w
7 Write the contrapositive of each statement. C v and w D w and z
a A ⇒ B b ¬P ⇒ Q c N ⇒ ¬M d ¬B ⇒ ¬F 7 What do the set of equations x = cos t, y = sin t, z = a, where 0 ≤ a ≤ 5 describe?
e If the boy has red hair, then he has blue eyes. A A circle B A sphere C A cylinder D A helix
f If the country is rich, then the citizens have money.
g If a quadrilateral is a kite, then the adjacent sides are equal in length.
h If x = y, then x2 = y2
i If a ∈ , then a ∈ 
74 MATHS IN FOCUS 12. Mathematics Extension 2 ISBN 9780170413435 ISBN 9780170413435 Practice set 1 179
AT THE END OF THE BOOK
Answers and Index (worked solutions on the teacher website).
x MATHS IN FOCUS 12. Mathematics Extension 2 ISBN 9780170413435