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
SECTION A – continued
SPECMATHEXAM2(SAMPLE) 4 Version3–July2016
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
SECTION A – continuedTURN OVER
Version3–July2016 5 SPECMATHEXAM2(SAMPLE)
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
SECTION A – continued
SPECMATHEXAM2(SAMPLE) 6 Version3–July2016
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
SECTION A – continuedTURN OVER
Version3–July2016 7 SPECMATHEXAM2(SAMPLE)
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
SECTION A – continued
SPECMATHEXAM2(SAMPLE) 8 Version3–July2016
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
Version3–July2016 9 SPECMATHEXAM2(SAMPLE)
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.
SPECMATHEXAM2(SAMPLE) 10 Version3–July2016
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
Version3–July2016 11 SPECMATHEXAM2(SAMPLE)
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
SPECMATHEXAM2(SAMPLE) 12 Version3–July2016
SECTION B – Question 2–continued
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
Version3–July2016 13 SPECMATHEXAM2(SAMPLE)
SECTION B – continuedTURN OVER
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
SPECMATHEXAM2(SAMPLE) 14 Version3–July2016
SECTION B – Question 3–continued
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
Version3–July2016 15 SPECMATHEXAM2(SAMPLE)
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
SPECMATHEXAM2(SAMPLE) 16 Version3–July2016
SECTION B – continued
e. SketchthegraphofNasafunctionoftontheaxesbelowfor0≤t≤15. 2marks
N
t
400
300
200
100
5 10 15O
Version3–July2016 17 SPECMATHEXAM2(SAMPLE)
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
SPECMATHEXAM2(SAMPLE) 18 Version3–July2016
SECTION B – Question 4–continued
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
Version3–July2016 19 SPECMATHEXAM2(SAMPLE)
SECTION B – continuedTURN OVER
d. Atwhatspeed,inmetrespersecond,doestheskierlandonthedown-slope?Giveyouranswercorrecttoonedecimalplace. 3marks
SPECMATHEXAM2(SAMPLE) 20 Version3–July2016
SECTION B – Question 5–continued
Question 5 (10marks)Thediagrambelowshowsparticlesofmass1kgand3kgconnectedbyalightinextensiblestringpassingoverasmoothpulley.ThetensioninthestringisT1newtons.
T1 T1
g 3 g
a. Letams–2betheaccelerationofthe3kgmassdownwards.
Findthevalueofa. 2marks
b. FindthevalueofT1. 1mark
Version3–July2016 21 SPECMATHEXAM2(SAMPLE)
SECTION B – continuedTURN OVER
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
SPECMATHEXAM2(SAMPLE) 22 Version3–July2016
SECTION B – Question 6–continued
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
Version3–July2016 23 SPECMATHEXAM2(SAMPLE)
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