S A M P L EMATHEMATICAL METHODS

Written examination 1

Day Date Reading time: *.** to *.** (15 minutes) Writing time: *.** to *.** (1 hour)

QUESTION AND ANSWER BOOK

Structure of bookNumber of questions

Number of questions to be answered

Number of marks

10 10 40

• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpenersandrulers.

• StudentsareNOTpermittedtobringintotheexaminationroom:anytechnology(calculatorsorsoftware),notesofanykind,blanksheetsofpaperand/orcorrectionfluid/tape.

Materials supplied• Questionandanswerbookof13pages.• Formulasheet.• Workingspaceisprovidedthroughoutthebook.

Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.• AllwrittenresponsesmustbeinEnglish.

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

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016

Version2–April2016

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education Year

STUDENT NUMBER

Letter

MATHMETHEXAM1(SAMPLE) 2 Version2–April2016

THIS PAGE IS BLANK

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Question 1 (3marks)a. Differentiate 4 − x withrespecttox. 1mark

b. If f x xx

( )sin ( )

= ,find f ′ π2

. 2marks

InstructionsAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegiven,unlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmustbeshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

MATHMETHEXAM1(SAMPLE) 4 Version2–April2016

Question 2 (3marks)Ontheaxesbelow,sketchthegraphof f :R\{–1}→R, f(x)= 2

41

−+x

.

Labeleachaxisinterceptwithitscoordinates.Labeleachasymptotewithitsequation.

y

Ox

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Question 3 (4marks)

a. Findanantiderivativeof 12 1 3x −( )

withrespecttox.2marks

b. Thefunctionwithrule g (x)hasderivativeg′(x)=sin(2πx).

Giventhatg (1)=1π,findg (x). 2marks

MATHMETHEXAM1(SAMPLE) 6 Version2–April2016

Question 4 (3marks)LetXbetherandomvariablethatrepresentsthenumberoftelephonecallsthatDanielreceivesonanygivendaywithprobabilitydistributiongivenbythetablebelow.

x 0 1 2 3

Pr(X=x) 0.2 0.2 0.5 0.1

a. FindthemeanofX. 2marks

b. WhatistheprobabilitythatDanielreceivesonlyonetelephonecalloneachofthreeconsecutivedays? 1mark

Question 5 (3marks)Thegraphsof y=cos(x)and y=asin(x),whereaisarealconstant,haveapointofintersection

at x = π3.

a. Findthevalueofa. 2marks

b. Ifx∈[0,2π],findthex-coordinateoftheotherpointofintersectionofthetwographs. 1mark

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Question 6 (5marks)a. Solvetheequation2log3(5)–log3(2)+log3(x)=2forx. 2marks

b. Solve3e t=5+8e–tfort. 3marks

MATHMETHEXAM1(SAMPLE) 8 Version2–April2016

Question 7 (4marks)Astudentperformsanexperimentinwhichacomputerisusedtosimulatedrawingarandomsampleofsizenfromalargepopulation.Theproportionofthepopulationwiththecharacteristicofinteresttothestudentisp.

a. LettherandomvariableP̂representthesampleproportionobservedintheexperiment.

If p = 15,findthesmallestintegervalueofthesamplesizesuchthatthestandarddeviationof

P̂islessthanorequalto1

100 . 2marks

Eachof23studentsinaclassindependentlyperformstheexperimentdescribedaboveandeachstudentcalculatesanapproximate95%confidenceintervalforpusingthesampleproportionsfortheirsample.Itissubsequentlyfoundthatexactlyoneofthe23confidenceintervalscalculatedbytheclassdoesnotcontainthevalueofp.

b. Twooftheconfidenceintervalscalculatedbytheclassareselectedatrandomwithoutreplacement.

Findtheprobabilitythatexactlyoneoftheselectedconfidenceintervalsdoesnotcontainthevalueofp. 2marks

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Question 8 (4marks)Acontinuousrandomvariable,X,hasaprobabilitydensityfunctiongivenby

f xe x

x

x

( ) = ≥

<

−15

0

0 0

5

ThemedianofXism.

a. Determinethevalueofm. 2marks

b. Thevalueofmisanumbergreaterthan1.

Find Pr .X X m< ≤( )1 2marks

MATHMETHEXAM1(SAMPLE) 10 Version2–April2016

Question 9 (4marks)Partofthegraphof f :R+→R,f (x)=xloge(x)isshownbelow.

1 3O

y

x

a. Findthederivativeofx2loge(x). 1mark

b. Useyouranswertopart a.tofindtheareaoftheshadedregionintheformaloge(b)+c,wherea,bandcarenon-zerorealconstants. 3marks

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Question 10–continuedTURN OVER

Question 10 (7marks)ThelinecontainingthepointsMandNintersectsthex-axisatthepointMwithcoordinates(6,0).Thelineisalsoatangenttothegraphofy=ax2+bxatthepointQwithcoordinates(2,4),asshownbelow.

O

N

Q (2, 4)

M (6, 0)

y

x

a. Ifaandbarenon-zerorealnumbers,findthevaluesofaandb. 3marks

MATHMETHEXAM1(SAMPLE) 12 Version2–April2016

Question 10–continued

b. ThelinecontainingthepointsUandVintersectsthecoordinateaxesatthepointsUandVwithcoordinates(u,0)and(0,v),respectively,whereuandvarepositiverealnumbersand52 ≤u≤6,asshownbelow.

TherectangleOPQRhasavertexatQontheline.ThecoordinatesofQare(2,4),asshownbelow.

O

R

P

V (0, v)

Q (2, 4)

U (u, 0)

y

x

i. Findanexpressionforvintermsofu. 1mark

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END OF QUESTION AND ANSWER BOOK

ii. Findtheminimumtotalshadedareaandthevalueofuforwhichtheareaisaminimum. 2marks

iii. Findthemaximumtotalshadedareaandthevalueofuforwhichtheareaisamaximum. 1mark