2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued...

27
SPECIALIST MATHEMATICS Written examination 2 Monday 10 November 2014 Reading time: 3.00 pm to 3.15 pm (15 minutes) Writing time: 3.15 pm to 5.15 pm (2 hours) QUESTION AND ANSWER BOOK Structure of book Section Number of questions Number of questions to be answered Number of marks 1 22 22 22 2 5 5 58 Total 80 Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers, a protractor, set-squares, aids for curve sketching, one bound reference, one approved CAS calculator or CAS software and, if desired, one scientific calculator. Calculator memory DOES NOT need to be cleared. Students are NOT permitted to bring into the examination room: blank sheets of paper and/or white out liquid/tape. Materials supplied Question and answer book of 23 pages with a detachable sheet of miscellaneous formulas in the centrefold. Answer sheet for multiple-choice questions. Instructions Detach the formula sheet from the centre of this book during reading time. Write your student number in the space provided above on this page. Check that your name and student number as printed on your answer sheet for multiple-choice questions are correct, and sign your name in the space provided to verify this. All written responses must be in English. At the end of the examination Place the answer sheet for multiple-choice questions inside the front cover of this book. Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room. © VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2014 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Victorian Certificate of Education 2014 STUDENT NUMBER Letter

Transcript of 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued...

Page 1: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

SPECIALIST MATHEMATICSWritten examination 2

Monday 10 November 2014 Reading time: 3.00 pm to 3.15 pm (15 minutes) Writing time: 3.15 pm to 5.15 pm (2 hours)

QUESTION AND ANSWER BOOK

Structure of bookSection Number of

questionsNumber of questions

to be answeredNumber of

marks

1 22 22 222 5 5 58

Total 80

• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulers,aprotractor,set-squares,aidsforcurvesketching,oneboundreference,oneapprovedCAScalculatororCASsoftwareand,ifdesired,onescientificcalculator.CalculatormemoryDOESNOTneedtobecleared.

• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orwhiteoutliquid/tape.

Materials supplied• Questionandanswerbookof23pageswithadetachablesheetofmiscellaneousformulasinthe

centrefold.• Answersheetformultiple-choicequestions.

Instructions• Detachtheformulasheetfromthecentreofthisbookduringreadingtime.• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice

questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.

• AllwrittenresponsesmustbeinEnglish.

At the end of the examination• Placetheanswersheetformultiple-choicequestionsinsidethefrontcoverofthisbook.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2014

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2014

STUDENT NUMBER

Letter

Page 2: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

2014SPECMATHEXAM2 2

SECTION 1–continued

Question 1Theasymptotesofthehyperbolagivenby

x y−( )− =

39 4

12 2

intersectthecoordinateaxesat

A. 0 4 5 0 4 5 3 0, . , , . , ,−( ) ( ) −( )

B. 0 2 0 2 3 0, , , , ,( ) −( ) ( )

C. 0 2 0 2 3 0, , , , ,( ) −( ) −( )

D. 0 4 5 0 4 5 3 0, . , , . , ,−( ) ( ) ( )

E. 2 0 2 0 0 3, , , , ,( ) −( ) −( )

Question 2Theellipsegivenbyx2 – 6x + 2y2 + 8y+16=0hascentre,lengthofhorizontalsemi-axisandlengthofverticalsemi-axisrespectivelyof

A. −( )3 2 1 2, , ,

B. −( )2 3 1 12

, , ,

C. 3 2 12

1, , ,−( )

D. −( )3 2 12

1, , ,

E. 3 2 1 12

, , ,−( )

Question 3Thefeaturesofthegraphofthefunctionwithrule f x

x xx x

( ) = − +− −

2

24 3

6include

A. asymptotesatx=1andx = –2B. asymptotesatx=3andx = –2C. anasymptoteatx=1andapointofdiscontinuityatx=3D. anasymptoteatx=–2andapointofdiscontinuityat x=3E. anasymptoteatx=3andapointofdiscontinuityatx = –2

SECTION 1

Instructions for Section 1Answerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1,anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Taketheacceleration due to gravitytohavemagnitudegm/s2,whereg=9.8.

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3 2014SPECMATHEXAM2

SECTION 1–continuedTURN OVER

Question 4Thedomainof arcsin 2 1x −( ) is

A. [ , ]−1 1

B. [ , ]−1 0

C. 0 1,[ ]

D. −

12

12

,

E. 0 12

,

Question 5Ifthecomplexnumberzhasmodulus 2 2 andargument

34π,thenz2isequalto

A. –8i

B. 4i

C. −2 2i

D. 2 2i

E. –4i

Question 6Giventhati n = pandi 2=–1,theni 2n+3intermsofpisequaltoA. p2 – iB. p2 + iC. –p2

D. –ip2

E. ip2

Question 7Thesumoftherootsofz3 – 5z2 + 11z–7=0,wherez C∈ ,is

A. 1 2 3+ i

B. 5i

C. 4 2 3− i

D. 2 3i

E. 5

Page 4: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

2014SPECMATHEXAM2 4

SECTION 1–continued

Question 8Theprincipalargumentof

− −+

3 2 62 2

ii is

A. −13

12π

B. 712π

C. 1112π

D. 1312π

E. −11

12π

Question 9Thecircle z i− − =3 2 2 isintersectedexactlytwicebythelinegivenby

A. z i z− = +1

B. z i z− − = −3 2 5

C. z i z i− − = −3 2 10

D. Im(z) = 0

E. Re(z) = 5

Question 10Alargetankinitiallyholds1500Lofwaterinwhich100kgofsaltisdissolved.Asolutioncontaining2kgofsaltperlitreflowsintothetankatarateof8Lperminute.Themixtureisstirredcontinuouslyandflowsoutofthetankthroughaholeatarateof10Lperminute.ThedifferentialequationforQ,thenumberofkilogramsofsaltinthetankaftert minutes,isgivenby

A. dQdt

Qt

= −−

16 5750

B. dQdt

Qt

= −+

16 5750

C. dQdt

Qt

= +−

16 5750

D. dQdt

Qt

=−

100750

E. dQdt

Qt

= −−

81500 2

Page 5: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

5 2014SPECMATHEXAM2

SECTION 1–continuedTURN OVER

Question 11

Let dydx

x xy y x= − = =3 2 1and when .

UsingEuler’smethodwithastepsizeof0.1,theapproximationtoy whenx =1.1isA. 0.9B. 1.0C. 1.1D. 1.9E. 2.1

Question 12

If dydx

x= +( )2 16 andy=5whenx=1,thenthevalueofy whenx =4isgivenby

A. ( )2 161

4x dx+ +( )∫ 5

B. ( )2 161

4x dx+∫

C. ( )2 161

4x dx+ +∫ 5

D. ( )2 161

4x dx+ −∫ 5

E. ( )2 161

4x dx+ −( )∫ 5

Question 13Usingthesubstitutionu x= +1 then

dxx x( )+ +∫ 2 10

2canbeexpressedas

A. 1121

3

u udu

( )+∫

B. 2120

2

udu

+∫

C. 111

3

u udu

( )+∫

D. 14

112 20

2

u udu

( )+∫

E. 2 1121

3

udu

+∫

Page 6: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

2014SPECMATHEXAM2 6

SECTION 1–continued

Question 14

y

x

Thedifferentialequationthatisbestrepresentedbytheabovedirectionfieldis

A. dydx x y

=−1

B. dydx

y x= −

C. dydx y x

=−1

D. dydx

x y= −

E. dydx y x

=+1

Page 7: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

7 2014SPECMATHEXAM2

SECTION 1–continuedTURN OVER

Question 15Ifθistheanglebetween a i j k~ ~ ~ ~

= + −3 4 and b i j k~ ~ ~ ~= − +4 3 ,then cos 2θ( ) is

A. −45

B. 725

C. − 725

D. 1425

E. − 2425

Question 16Twovectorsaregivenby a i j k~ ~ ~

= + −4 3~ m and b i j k~ ~ ~ ~= − + −2 n ,wherem n R, .∈ +

If a = 10~

and a~ isperpendicularto b~ ,thenmandnrespectivelyare

A. 5 3 33

,

B. 5 3 3,

C. −5 3 3,

D. 93 5 9393

,

E. 5,1

Question 17Theaccelerationvectorofaparticlethatstartsfromrestisgivenby

a ( i 0 ) j k , where~ ~ ~~ ) sin( ) cos( .t t t e tt= − + − ≥−4 2 2 2 20 02

Thevelocityvectoroftheparticle,v(~ ),t isgivenby

A. − − + −8 2 40 2 40 2cos( ) sin(t t e ti ) j k~ ~ ~

B. 2 2 1 2 10 2cos( ) sin(t t e ti 0 ) j k~ ~ ~+ + −

C. 8 8 2 40 2 40 402− ( )( ) − ( ) + ( − )−cos sint t e ti j k~ ~ ~

D. 2 2 2 1 2 10 102cos sint t e t( ) −( ) + ( ) + ( − )−i 0 j k~ ~ ~

E. 4 2 4 20 2 20 20 2cos sint t e t( ) −( ) + ( ) + ( − )−i j k~ ~ ~

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2014SPECMATHEXAM2 8

SECTION 1–continued

Question 18Abodyonahorizontalsmoothplaneisacteduponbyfourforces, F F F and F~ ~ ~ ~, , ,1 2 3 4 asshown.

Theforce F~1actsinanortherlydirectionandtheforce F~4

actsinawesterlydirection.

60°

60°150°

N

EW

S

F4~

F1~F2~

F3~

Giventhat F = 1, F = 2, F = 4 and F = 5,~1 ~2 ~3 ~4 themotionofthebodyissuchthatit

A. isinequilibrium.B. movestothewest.C. movestothenorth.D. movesinthedirection30°southofwest.E. movestotheeast.

Question 19Thevelocityvectorofa5kgmassmovinginthecartesianplaneisgivenbyv i j~ ~ ~

sin cos ,t t t( ) = ( ) + ( )3 2 4 2 wherevelocitycomponentsaremeasuredinm/s.Duringitsmotion,themaximummagnitudeofthenetforce,innewtons,actingonthemassisA. 8B. 30C. 40D. 50E. 70

Page 9: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

9 2014SPECMATHEXAM2

SECTION 1–continuedTURN OVER

Question 20

3 kg

5 kg

Particlesofmass3kgand5kgareattachedtotheendsofalightinextensiblestringthatpassesoverafixedsmoothpulley,asshownabove.Thesystemisreleasedfromrest.Assumingthesystemremainsconnected,thespeedofthe5kgmassaftertwosecondsisA. 4.0m/sB. 4.9m/sC. 9.8m/sD. 10.0m/sE. 19.6m/s

Question 21Theacceleration,inms–2,ofaparticlemovinginastraightlineisgivenby–4x,wherex metresisitsdisplacementfromafixedoriginO.Iftheparticleisatrestwherex=5,thespeedoftheparticle,inms–1,wherex=3isA. 8

B. 8 2

C. 12

D. 4 2

E. 2 34

Page 10: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

2014SPECMATHEXAM2 10

END OF SECTION 1

Question 22Thevelocity–timegraphbelowshowsthemotionofabodytravellinginastraightline,wherevms–1isitsvelocityafter tseconds.

O

10987654321

–1–2–3–4–5–6

v

t1 2 3 4 5 6 7 8 9 10

Thevelocityofthebodyoverthetimeinterval t∈[ ]4 9, isgivenby v t t( ) = − −( ) +9

164 92 .

Thedistance,inmetres,travelledbythebodyoverninesecondsisclosesttoA. 45.6B. 47.5C. 48.6D. 51.0E. 53.4

Page 11: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

11 2014SPECMATHEXAM2

TURN OVER

Workingspace

Page 12: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

2014SPECMATHEXAM2 12

SECTION 2 – Question 1–continued

Question 1 (11marks)Considerthefunction f withrule ( )( )

9( )2 4

f xx x

=+ −

overitsmaximaldomain.

a. Findthecoordinatesofthestationarypoint(s). 3marks

b. Statetheequationsofallasymptotesofthegraphof f. 2marks

SECTION 2

Instructions for Section 2Answerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegm/s2,whereg=9.8.

Page 13: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

13 2014SPECMATHEXAM2

SECTION 2–continuedTURN OVER

c. Sketchthegraphof f for x∈ −[ ]6 6, ontheaxesbelow,showingasymptotes,thevaluesofthecoordinatesofanyinterceptswiththeaxes,andthestationarypoint(s). 3marks

–6 –4 –2 O 2 4 6

6

4

2

–2

–4

–6

y

x

Theregionboundedbythecoordinateaxes,thegraphof f andthelinex=3,isrotatedaboutthex-axistoformasolidofrevolution.

d. i. Writedownadefiniteintegralintermsofxthatgivesthevolumeofthissolidofrevolution. 2marks

ii. Findthevolumeofthissolid,correcttotwodecimalplaces. 1mark

Page 14: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

2014SPECMATHEXAM2 14

SECTION 2 – Question 2–continued

Question 2 (13marks)Considerthecomplexnumber z i1 3 3= − .

a. i. Expressz1inpolarform. 2marks

ii. FindArg z14( ). 1mark

iii. Giventhat z i1 3 3= − isonerootoftheequation z3 24 3 0+ = , findtheothertworoots,expressingyouranswersincartesianform. 2marks

b. i. Findthevalueof z i z i1 12 2+( ) −( ), where z i1 3 3= − . 1mark

ii. Showthattherelation z i z i+( ) −( ) =2 2 4 canbeexpressedincartesianformas x y2 22 4+ +( ) = . 2marks

Page 15: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

15 2014SPECMATHEXAM2

SECTION 2–continuedTURN OVER

iii. Sketch{ : }z z i z i+( ) −( ) =2 2 4 ontheaxesbelow. 2marks

Im(z)

Re(z)–6 –4 –2 O 2 4 6

6

4

2

–2

–4

–6

c. Thelinejoiningthepointscorrespondingtok – 2iand − +( )2 k i ,wherek<0,istangentto thecurvegivenby{ : }z z i z i+( ) −( ) =2 2 4 .

Findthevalueofk. 3marks

Page 16: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

2014SPECMATHEXAM2 16

SECTION 2 – Question 3–continued

Question 3 (10marks)Let a i j k and b i j k~ ~ ~ ~ ~ ~ ~ ~ .= + + = − −3 2 2 2

a. Express a~ asthesumoftwovectorresolutes,oneofwhichisparallelto b~ andtheotherofwhichisperpendicularto b~ .Identifyclearlytheparallelvectorresoluteandtheperpendicularvectorresolute. 5marks

Page 17: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

17 2014SPECMATHEXAM2

SECTION 2–continuedTURN OVER

OABC isaparallelogramwhereDisthemidpointofCB .

OB and AD intersectatpointP.

LetOA→

= a~ andOC→

= c.~

C D B

O

P

A

b. i. GiventhatAP AD→ →

=α , writeanexpressionfor AP→

intermsofα, a~ and c.~ 2marks

ii. GiventhatOP OB→ →

= β ,writeanotherexpressionfor AP→

intermsofβ, a~ and c.~ 1mark

iii. Hencededucethevaluesofαandβ. 2marks

Page 18: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

2014SPECMATHEXAM2 18

SECTION 2 – Question 4–continued

Question 4 (12marks)Atawaterfunpark,aconicaltankofradius0.5mandheight1misfillingwithwater.Atthesametime,somewaterflowsoutfromthevertex,wettingthoseunderneath.Whenthetankeventuallyfills,ittipsoverandthewaterfallsout,drenchingallthoseunderneath.Thetankthenreturnstoitsoriginalpositionandbeginstorefill.

1 m

0.5 m

h

Waterflowsinataconstantrateof0.02πm3/minandflowsoutatavariablerateof

0 01. π h m3/min,wherehmetresisthedepthofthewateratanyinstant.

a. Showthatthevolume, Vcubicmetres,ofwaterintheconewhenitisfilledtoadepthof h metresisgivenbyV h=

π12

3. 1mark

b. Findtherate,inm/min,atwhichthedepthofthewaterinthetankisincreasingwhenthedepthis0.25m. 4marks

Page 19: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

19 2014SPECMATHEXAM2

SECTION 2 – Question 4–continuedTURN OVER

Thetankisemptyattimet=0minutes.

c. Byusinganappropriatedefiniteintegral, findthetimeittakesforthetanktofill.Giveyouranswerinminutes,correcttoonedecimalplace. 2marks

Page 20: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

2014SPECMATHEXAM2 20

SECTION 2–continued

Anotherwatertank,shownbelow,hastheshapeofalargebucket(partofacone)withthedimensionsgiven.Waterfillsthetankatarateof0.05πm3/min,butnowaterleaksout.

0.75 m

0.5 m

x

1 m

Whenfilledtoadepthofxmetres,thevolumeofwater,V cubicmetres,inthetankisgivenby

V x x x= + +( )π48

6 123 2

d. Giventhatthetankisinitiallyempty, findthedepth,xmetres,asafunctionoftimet. 5marks

Page 21: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

21 2014SPECMATHEXAM2

SECTION 2 – Question 5–continuedTURN OVER

Question 5 (12marks)Thediagrambelowshowsblocksofmass3kgand5kgonasmoothplaneofinclinationθdegreestothehorizontal,connectedbyatautlightinextensiblerope.The5kgblockisconnectedbyanotherlightinextensibleropeviaasmoothpulleyatthetopoftheinclinedplanetoablockofmass2kg,hangingvertically.The2kgblockhasaccelerationupwardsofams–2.Theforcesoneachofthethreeblocksareshownonthediagram.

N

RT1

T2T2

5g

3g

T1

2gθ

a. i. Writedownanequationofmotionforthe2kgblock. 1mark

ii. Byresolvingforcesactingparalleltotheplaneontheothertwoblocks,writedownanequationofmotionforeachofthe3kgand5kgblocks,usingthesymbolsdefinedintheabovediagram. 2marks

Page 22: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

2014SPECMATHEXAM2 22

SECTION 2 – Question 5–continued

iii. Showthatag

=−( )4 1

5sin( )

1mark

iv. Findtheangleθforthesystemtobeinequilibrium.Giveyouranswerindegrees,correcttoonedecimalplace. 1mark

Nowconsideradifferent situation wherethe3kgand5kgblocksareplacedatrestonarough plane.Theplaneisinclinedat30°tothehorizontal,asshown,andtherearenostringsattachedtotheblocks.Thecoefficientoffrictionbetweenthe3kgblockandtheplaneis0.11andthecoefficientoffrictionbetweenthe5kgblockandtheplaneis0.01.Initially,thedistancebetweenthetwoclosestfacesoftheblocksis3m.

b. i. Onthediagrambelow,show andlabeltheforcesactingoneachofthetwoblocks. 2marks

3 m

3 kg

5 kg

30°

Page 23: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

23 2014SPECMATHEXAM2

ii. Calculatetheaccelerationofeachblockdowntheplane.Giveyouranswersinm/s2, correcttothenearest0.01m/s2. 2marks

iii. Calculatethetimetakenforthe5kgblocktocollidewiththe3kgblock.Giveyouranswerinseconds,correcttothenearest0.01s. 3marks

END OF QUESTION AND ANSWER BOOK

Page 24: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

SPECIALIST MATHEMATICS

Written examinations 1 and 2

FORMULA SHEET

Directions to students

Detach this formula sheet during reading time.

This formula sheet is provided for your reference.

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2014

Page 25: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

SPECMATH 2

Specialist Mathematics formulas

Mensuration

area of a trapezium: 12 a b h+( )

curved surface area of a cylinder: 2π rh

volume of a cylinder: π r2h

volume of a cone: 13π r2h

volume of a pyramid: 13 Ah

volume of a sphere: 43 π r

3

area of a triangle: 12 bc Asin

sine rule: aA

bB

cCsin sin sin

= =

cosine rule: c2 = a2 + b2 – 2ab cos C

Coordinate geometry

ellipse: x ha

y kb

−( )+

−( )=

2

2

2

2 1 hyperbola: x ha

y kb

−( )−

−( )=

2

2

2

2 1

Circular (trigonometric) functionscos2(x) + sin2(x) = 1

1 + tan2(x) = sec2(x) cot2(x) + 1 = cosec2(x)

sin(x + y) = sin(x) cos(y) + cos(x) sin(y) sin(x – y) = sin(x) cos(y) – cos(x) sin(y)

cos(x + y) = cos(x) cos(y) – sin(x) sin(y) cos(x – y) = cos(x) cos(y) + sin(x) sin(y)

tan( ) tan( ) tan( )tan( ) tan( )

x y x yx y

+ =+

−1 tan( ) tan( ) tan( )tan( ) tan( )

x y x yx y

− =−

+1

cos(2x) = cos2(x) – sin2(x) = 2 cos2(x) – 1 = 1 – 2 sin2(x)

sin(2x) = 2 sin(x) cos(x) tan( ) tan( )tan ( )

2 21 2x x

x=

function sin–1 cos–1 tan–1

domain [–1, 1] [–1, 1] R

range −

π π2 2

, [0, �] −

π π2 2

,

Page 26: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

3 SPECMATH

Algebra (complex numbers)z = x + yi = r(cos θ + i sin θ) = r cis θ

z x y r= + =2 2 –π < Arg z ≤ π

z1z2 = r1r2 cis(θ1 + θ2) zz

rr

1

2

1

21 2= −( )cis θ θ

zn = rn cis(nθ) (de Moivre’s theorem)

Calculusddx

x nxn n( ) = −1

∫ =

++ ≠ −+x dx

nx c nn n1

111 ,

ddxe aeax ax( ) =

∫ = +e dx

ae cax ax1

ddx

xxelog ( )( )= 1

∫ = +1xdx x celog

ddx

ax a axsin( ) cos( )( )=

∫ = − +sin( ) cos( )ax dxa

ax c1

ddx

ax a axcos( ) sin( )( )= −

∫ = +cos( ) sin( )ax dxa

ax c1

ddx

ax a axtan( ) sec ( )( )= 2

∫ = +sec ( ) tan( )2 1ax dx

aax c

ddx

xx

sin−( ) =−

12

1

1( )

−=

+ >−1 0

2 21

a xdx x

a c asin ,

ddx

xx

cos−( ) = −

12

1

1( )

−=

+ >−1 0

2 21

a xdx x

a c acos ,

ddx

xx

tan−( ) =+

12

11

( )

∫+

=

+

−aa x

dx xa c2 2

1tan

product rule: ddxuv u dv

dxv dudx

( ) = +

quotient rule: ddx

uv

v dudx

u dvdx

v

=

2

chain rule: dydx

dydududx

=

Euler’s method: If dydx

f x= ( ), x0 = a and y0 = b, then xn + 1 = xn + h and yn + 1 = yn + h f (xn)

acceleration: a d xdt

dvdt

v dvdx

ddx

v= = = =

2

221

2

constant (uniform) acceleration: v = u + at s = ut +12

at2 v2 = u2 + 2as s = 12

(u + v)t

TURN OVER

Page 27: 2014 Specialist Mathematics Written examination 22014 SPECMATH EXAM 2 4 SECTION 1 – continued Question 8 The principal argument of 32 6 22 i i is A. −13 12 π B. 7 12 π C. 11

SPECMATH 4

END OF FORMULA SHEET

Vectors in two and three dimensions

r i j k~ ~ ~ ~= + +x y z

| r~ | = x y z r2 2 2+ + = r

~ 1. r~ 2 = r1r2 cos θ = x1x2 + y1y2 + z1z2

Mechanics

momentum: p v~ ~= m

equation of motion: R a~ ~= m

friction: F ≤ µN