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S A M P L ESPECIALIST MATHEMATICS

Written examination 2Day Date

Reading time: *.** to *.** (15 minutes) Writing time: *.** to *.** (2 hours)

QUESTION AND ANSWER BOOK

Structure of bookSection Number of

questionsNumber of questions

to be answeredNumber of

marks

A 20 20 20B 6 6 60

Total 80

• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulers,aprotractor,setsquares,aidsforcurvesketching,oneboundreference,oneapprovedtechnology(calculatororsoftware)and,ifdesired,onescientificcalculator.CalculatormemoryDOESNOTneedtobecleared.Forapprovedcomputer-basedCAS,fullfunctionalitymaybeused.

• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.

Materials supplied• Questionandanswerbookof23pages.• Formulasheet.• Answersheetformultiple-choicequestions.

Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice

questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.• 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016Version3–July2016

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education Year

STUDENT NUMBER

Letter

SECTION A – continued

SPECMATHEXAM2(SAMPLE) 2 Version3–July2016

Question 1Acirclewithcentre(a,–2)andradius5unitshasequationx2–6x + y2 + 4y = b,whereaandbarerealconstants.ThevaluesofaandbarerespectivelyA. –3and38B. 3and12C. –3and–8D. –3and0E. 3and18

Question 2Themaximaldomainandrangeofthefunctionwithrule f x x( ) = − +−3 4 1

21sin ( ) π arerespectively

A. [–�,2�]and 0, 12

B. 0, 12

and[–�,2�]

C. −

32π π, 3

2and −

12

, 0

D. 0, 12

and[0,3�]

E. −

12

, 0 and[–�,2�]

SECTION A – Multiple-choice questions

Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenot drawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8

SECTION A – continuedTURN OVER

Version3–July2016 3 SPECMATHEXAM2(SAMPLE)

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

Question 4Thealgebraicfraction

7 54 92 2

xx x

−− +( ) ( )

couldbeexpressedinpartialfractionformas

A. A

xB

x−( )+

+4 92 2

B. Ax

Bx

Cx−

+−

++4 3 3

C. A

xBx Cx−( )

+++4 92 2

D. Ax

Bx

Cx Dx−

+−( )

+++4 4 92 2

E. Ax

Bx

Cx−

+−( )

++4 4 92 2

Question 5OnanArganddiagram,asetofpointsthatliesonacircleofradius2centredattheoriginisA. { : }z C zz∈ = 2

B. { : }z C z∈ =2 4

C. { :Re( ) Im( ) }z C z z∈ + =2 2 4

D. { : }z C z z z z∈ +( ) − −( ) =2 2 16

E. { : Re( ) Im( ) }z C z z∈ ( ) + ( ) =2 2 16

Question 6ThepolynomialP(z)hasrealcoefficients.FouroftherootsoftheequationP(z)=0 are z =0,z =1–2i, z =1+2i and z =3i.TheminimumnumberofrootsthattheequationP(z)=0couldhaveisA. 4B. 5C. 6D. 7E. 8



Question 7

–0.5� –0.4� –0.3� –0.2� –0.1� 0.1� 0.2� 0.3� 0.4� 0.5�

1

0.5

–0.5

–1

y

xO

Thedirection(slope)fieldforacertainfirst-orderdifferentialequationisshownabove.Thedifferentialequationcouldbe

A. dydx

x= ( )sin 2

B. dydx

x= ( )cos 2

C. dydx

y=

cos 1

2

D. dydx

y=

sin 1

2

E. dydx

x=

cos 1

2



Question 8Let f :[–π,2π] → R,wheref (x)=sin3(x).Usingthesubstitutionu=cos(x),theareaboundedbythegraphoffandthex-axiscouldbefoundbyevaluating

A. − −( )−∫ 1 22

u duπ

π

B. 3 1 21

1−( )

−∫ u du

C. − −( )∫3 1 20

u duπ

D. 3 1 21

1−( )

−

∫ u du

E. − −( )−∫ 1 2

1

1u du

Question 9

Letdydx

xx x

=+

+ +2

2 12 and(x0 ,y0)=(0,2).

UsingEuler’smethodwithastepsizeof0.1,thevalueofy1,correcttotwodecimalplaces,isA. 0.17B. 0.20C. 1.70D. 2.17E. 2.20



Question 10

Thecurvegivenbyy=sin–1(2x),where0≤x≤ 12,isrotatedaboutthey-axistoformasolidofrevolution.

Thevolumeofthesolidmaybefoundbyevaluating

A. π

π

41 2

0

2

−( )∫ cos( )y dy

B. π8

1 20

12

−( )∫ cos( )y dy

C. π

π

81 2

0

2

−( )∫ cos( )y dy

D. 18

1 20

2

−( )∫ cos( )y dy

π

E. π

π

π

81 2

2

2

−( )−

∫ cos( )y dy

Question 11Theanglebetweenthevectors3 6 2 2 2

i j k and i j k+ − − + ,correcttothenearesttenthofadegree,isA. 2.0°B. 91.0°C. 112.4°D. 121.3°E. 124.9°

Question 12Thescalarresoluteof

a i k= −3 inthedirectionof

b i j k= − −2 2 is

A. 810

B. 89

2 2

i j k− −( )

C. 8

D. 45

3

i k−( )

E. 83



Question 13

Thepositionvectorofaparticleattimetseconds,t ≥0,isgivenby

r i j + 5k( ) ( )t t t= − −3 6 .Thedirectionofmotionoftheparticlewhent=9is

A. − −6

i 18j + 5k

B. − −

i j

C. − −6

i j

D. − − +

i j k5

E. − − +13 5 108 45.

i j k

Question 14Thediagrambelowshowsarhombus,spannedbythetwovectors

a and

b .

�a

�b

ItfollowsthatA.

a b. = 0

B.

a = b

C.

a b a b+( ) −( ) =. 0

D.

a b a b+ = −

E. 2 2 0

a b+ =

Question 15A12kgmassmovesinastraightlineundertheactionofavariableforceF,sothatitsvelocityvms–1whenitisxmetresfromtheoriginisgivenby v x x= − +3 162 3 .TheforceFactingonthemassisgivenby

A. 12 3 32

2x x−

B. 12 3 162 3x x− +( )

C. 12 6 3 2x x−( )

D. 12 3 162 3x x− +

E. 12 3 3−( )x



Question 16

Theacceleration,a ms–2,ofaparticlemovinginastraightlineisgivenby a vve

=log ( )

,wherevisthe

velocityoftheparticleinms–1attimetseconds.Theinitialvelocityoftheparticlewas5ms–1.Thevelocityoftheparticle,intermsoft, isgivenbyA. v = e2t

B. v = e2t + 4

C. v e t e= +2 5log ( )

D. v e t e= +2 5 2(log )

E. v e t e= − +2 5 2(log )

Question 17A12kgmassissuspendedinequilibriumfromahorizontalceilingbytwoidenticallightstrings.Eachstringmakesanangleof60°withtheceiling,asshown.

60° 60°

12 kg

Themagnitude,innewtons,ofthetensionineachstringisequaltoA. 6 g

B. 12g

C. 24 g

D. 4 3 g

E. 8 3 g

Question 18GiventhatXisanormalrandomvariablewithmean10andstandarddeviation8,andthatYisanormalrandomvariablewithmean3andstandarddeviation2,andXandYareindependentrandomvariables,therandomvariableZdefinedbyZ = X–3YwillhavemeanμandstandarddeviationσgivenbyA. μ=1,σ = 28B. μ=19,σ = 2

C. μ=1,σ = 2 7D. μ=19,σ=14E. μ=1,σ=10


END OF SECTION ATURN OVER

Question 19ThemeanstudyscoreforalargeVCEstudyis30withastandarddeviationof7.Aclassof20studentsmaybeconsideredasarandomsampledrawnfromthiscohort.Theprobabilitythattheclassmeanforthegroupof20exceeds32isA. 0.1007B. 0.3875C. 0.3993D. 0.6125E. 0.8993

Question 20AtypeIerrorwouldoccurinastatisticaltestwhereA. H0isacceptedwhenH0isfalse.B. H1isacceptedwhenH1istrue.C. H0 isrejectedwhenH0istrue.D. H1isrejectedwhenH1istrue.E. H0 isacceptedwhenH0istrue.


SECTION B – Question 1–continued

Question 1 (10marks)Considerthefunctionf :[0,3)→ R,wheref (x)=–2+2sec

π x6

.

a. Evaluatef (2). 1mark

Letf –1betheinversefunctionoff.

b. Ontheaxesbelow,sketchthegraphsoffand f –1,showingtheirpointsofintersection. 2marks

1

O 1 2 3 4

2

3

4

y

x

SECTION B

Instructions for Section BAnswerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8


SECTION B – continuedTURN OVER

c. Theruleforf –1canbewrittenasf –1(x)=karccos2

2x +

.

Findtheexactvalueofk. 2marks

LetAbethemagnitudeoftheareaenclosedbythegraphsoffandf –1.

d. WriteadefiniteintegralexpressionforAandevaluateitcorrecttothreedecimalplaces. 2marks

e. i. Writedownadefiniteintegralintermsofxthatgivesthearclengthofthegraphof ffromx = 0 to x =2. 2marks

ii. Evaluatethisdefiniteintegralcorrecttothreedecimalplaces. 1mark



Question 2 (9marks)

Letu = 12

32

+ i .

a. i. Expressuinpolarform. 2marks

ii. Henceshowthatu6=1. 1mark

iii. Plotallrootsofz6–1=0ontheArganddiagrambelow,labellinguandwwhere w =–u. 2marks

Im(z)

Re(z)1 2 3–3 –2 –1

2

1

–1

–2

O



b. i. Drawandlabelthesubsetofthecomplexplanegivenby S z z= ={ }: 1 ontheArganddiagrambelow. 1mark

Im(z)

Re(z)1 2 3–3 –2 –1

2

1

–1

–2

O

ii. DrawandlabelthesubsetofthecomplexplanegivenbyT z z u z u= − = +{ }: ontheArganddiagramabove. 1mark

iii. FindthecoordinatesofthepointsofintersectionofSandT. 2marks



Question 3 (11marks)Thenumberofmobilephones,N,ownedinacertaincommunityaftertyearsmaybemodelledbyloge(N)=6–3e–0.4t,t≥0.

a. Verifybysubstitutionthatloge(N)=6–3e–0.4tsatisfiesthedifferentialequation

1 0 4 2 4 0NdNdt

Ne+ − =. log ( ) . 2marks

b. Findtheinitialnumberofmobilephonesownedinthecommunity.Giveyouranswercorrecttothenearestinteger. 1mark

c. Usingthismathematicalmodel,findthelimitingnumberofmobilephonesthatwouldeventuallybeownedinthecommunity.Giveyouranswercorrecttothenearestinteger. 2marks


SECTION B – Question 3–continuedTURN OVER

Thedifferentialequationinpart a.canalsobewrittenintheformdNdt =0.4N(6–loge(N )).

d. i. Findd Ndt

2

2 intermsofNandloge(N ). 2marks

ii. ThegraphofNasafunctionofthasapointofinflection.

Findthevaluesofthecoordinatesofthispoint.GivethevalueoftcorrecttoonedecimalplaceandthevalueofNcorrecttothenearestinteger. 2marks


SECTION B – continued

e. SketchthegraphofNasafunctionoftontheaxesbelowfor0≤t≤15. 2marks

N

t

400

300

200

100

5 10 15O


SECTION B – Question 4–continuedTURN OVER

Question 4 (10marks)Askieracceleratesdownaslopeandthenskisupashortskijump,asshownbelow.Theskierleavesthejumpataspeedof12ms–1andatanangleof60°tothehorizontal.Theskierperformsvariousgymnastictwistsandlandsonastraight-linesectionofthe45°down-slopeTsecondsafterleavingthejump.LettheoriginOofacartesiancoordinatesystembeatthepointwheretheskierleavesthejump,with

i aunitvectorinthepositivexdirectionand

j aunitvectorinthepositiveydirection.Displacementsaremeasuredinmetresandtimeinseconds.

45°60°

y

xskijump

down-slope

O

a. Showthattheinitialvelocityoftheskierwhenleavingthejumpis6 6 3

i j+ . 1mark



b. Theaccelerationoftheskier, tsecondsafterleavingtheskijump,isgivenby

��

� �r( ) i jt t g t= − − −( )0 1 0 1. . ,0≤t≤T

Showthatthepositionvectoroftheskier, tsecondsafterleavingthejump,isgivenby

r i jt t t t gt t( ) = −

+ − +

6 1

606 3 1

2160

3 2 3 ,0≤t≤T 3marks

c. ShowthatT g= +( )12 3 1 . 3marks



d. Atwhatspeed,inmetrespersecond,doestheskierlandonthedown-slope?Giveyouranswercorrecttoonedecimalplace. 3marks



Question 5 (10marks)Thediagrambelowshowsparticlesofmass1kgand3kgconnectedbyalightinextensiblestringpassingoverasmoothpulley.ThetensioninthestringisT1newtons.

T1 T1

g 3 g

a. Letams–2betheaccelerationofthe3kgmassdownwards.

Findthevalueofa. 2marks

b. FindthevalueofT1. 1mark



The3kgmassisplacedonasmoothplaneinclinedatanangleof °tothehorizontal.ThetensioninthestringisnowT2newtons.

3 gT2

T2

g

° θ

c. When °=30°,theaccelerationofthe1kgmassupwardsisbms–2.

Findthevalueofb. 3marks

d. Forwhatangle °willthe3kgmassbeatrestontheplane?Giveyouranswercorrecttoonedecimalplace. 2marks

e. Whatangle °willcausethe3kgmasstoaccelerateuptheplaneatg4

1 32

2−

−ms ? 2marks



Question 6 (10marks)Acertaintypeofcomputer,oncefullycharged,isclaimedbythemanufacturertohaveμ=10hourslifetimebeforearechargeisneeded.Whenchecked,arandomsampleofn=25suchcomputersisfoundtohaveanaveragelifetimeofx =9.7hoursandastandarddeviationofs=1hour.Todecidewhethertheinformationgainedfromthesampleisconsistentwiththeclaimμ=10,astatisticaltestistobecarriedout.Assumethatthedistributionoflifetimesisnormalandthats isasufficientlyaccurateestimateofthepopulation(oflifetimes)standarddeviationσ.

a. WritedownsuitablehypothesesH0andH1totestwhetherthemeanlifetimeislessthanthatclaimedbythemanufacturer. 2marks

b. Findthepvalueforthistest,correcttothreedecimalplaces. 2marks

c. StatewithareasonwhetherH0shouldberejectedornotrejectedatthe5%levelofsignificance. 1mark

LettherandomvariableX denotethemeanlifetimeofarandomsampleof25computers,assumingμ=10.

d. FindC *suchthatPr * .X C< =( ) =µ 10 0 05.Giveyouranswercorrecttothreedecimalplaces. 2marks


e. i. Ifthemeanlifetimeofallcomputersisinfactμ =9.5hours,findPr * .X C> =( )µ 9 5 ,givingyouranswercorrecttothreedecimalplaces,whereC*isyouranswertopart d. 2marks

ii. Doestheresultinpart e.i.indicateatypeIortypeIIerror?Explainyouranswer. 1mark

END OF QUESTION AND ANSWER BOOK

SPECMATH EXAM 2 (SAMPLE – ANSWERS)

© VCAA 2016 – Version 3 – July 2016

Answers to multiple-choice questions

Question Answer

1 B

2 B

3 D

4 D

5 D

6 B

7 B

8 B

9 E

10 C

11 C

12 E

13 B

14 C

15 A

16 D

17 D

18 E

19 A

20 C