© VCAA 2016 – Version 2 – April 2016

VCE Mathematical Methods 2016–2018

Written examinations 1 and 2 – End of year

Examination specifications

Overall conditions There will be two end-of-year examinations for VCE Mathematical Methods – examination 1 and examination 2.

The examinations will be sat at a time and date to be set annually by the Victorian Curriculum and Assessment Authority (VCAA). VCAA examination rules will apply. Details of these rules are published annually in the VCE and VCAL Administrative Handbook.

Examination 1 will have 15 minutes reading time and 1 hour writing time. Students are not permitted to bring into the examination room any technology (calculators or software) or notes of any kind.

Examination 2 will have 15 minutes reading time and 2 hours writing time. Students are permitted to bring into the examination room an approved technology with numerical, graphical, symbolic and statistical functionality, as specified in the VCAA Bulletin and the VCE Exams Navigator. One bound reference may be brought into the examination room. This may be a textbook (which may be annotated), a securely bound lecture pad, a permanently bound student-constructed set of notes without fold-outs or an exercise book. Specifications for the bound reference are published annually in the VCE Exams Navigator.

A formula sheet will be provided with both examinations.

The examinations will be marked by a panel appointed by the VCAA.

Examination 1 will contribute 22 per cent to the study score. Examination 2 will contribute 44 per cent to the study score.

MATHMETH (SPECIFICATIONS)

© VCAA 2016 – Version 2 – April 2016 Page 2

Content The VCE Mathematics Study Design 2016–2018 (‘Mathematical Methods Units 3 and 4’) is the document for the development of the examination. All outcomes in ‘Mathematical Methods Units 3 and 4’ will be examined.

All content from the areas of study, and the key knowledge and skills that underpin the outcomes in Units 3 and 4, are examinable.

Examination 1 will cover all areas of study in relation to Outcome 1. The examination is designed to assess students’ knowledge of mathematical concepts, their skill in carrying out mathematical algorithms without the use of technology, and their ability to apply concepts and skills.

Examination 2 will cover all areas of study in relation to all three outcomes, with an emphasis on Outcome 2. The examination is designed to assess students’ ability to understand and communicate mathematical ideas, and to interpret, analyse and solve both routine and non-routine problems.

Format

Examination 1

The examination will be in the form of a question and answer book.

The examination will consist of short-answer and extended-answer questions. All questions will be compulsory.

The total marks for the examination will be 40.

A formula sheet will be provided with the examination. The formula sheet will be the same for examinations 1 and 2.

All answers are to be recorded in the spaces provided in the question and answer book.

Examination 2

The examination will be in the form of a question and answer book.

The examination will consist of two sections.

Section A will consist of 20 multiple-choice questions worth 1 mark each and will be worth a total of 20 marks.

Section B will consist of short-answer and extended-answer questions, including multi-stage questions of increasing complexity, and will be worth a total of 60 marks.

All questions will be compulsory. The total marks for the examination will be 80.

A formula sheet will be provided with the examination. The formula sheet will be the same for examinations 1 and 2.

Answers to Section A are to be recorded on the answer sheet provided for multiple-choice questions.

Answers to Section B are to be recorded in the spaces provided in the question and answer book.

MATHMETH (SPECIFICATIONS)

© VCAA 2016 – Version 2 – April 2016 Page 3

Approved materials and equipment

Examination 1

• normal stationery requirements (pens, pencils, highlighters, erasers, sharpeners and rulers)

Examination 2

• normal stationery requirements (pens, pencils, highlighters, erasers, sharpeners and rulers) • an approved technology with numerical, graphical, symbolic and statistical functionality • one scientific calculator • one bound reference

Relevant references The following publications should be referred to in relation to the VCE Mathematical Methods examinations:

• VCE Mathematics Study Design 2016–2018 (‘Mathematical Methods Units 3 and 4’) • VCE Mathematical Methods – Advice for teachers 2016–2018 (includes assessment advice) • VCE Exams Navigator • VCAA Bulletin

Advice During the 2016–2018 accreditation period for VCE Mathematical Methods, examinations will be prepared according to the examination specifications above. Each examination will conform to these specifications and will test a representative sample of the key knowledge and skills from all outcomes in Units 3 and 4.

The following sample examinations provide an indication of the types of questions teachers and students can expect until the current accreditation period is over.

Answers to multiple-choice questions are provided at the end of examination 2.

Answers to other questions are not provided.

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

Version2–April2016 3 MATHMETHEXAM1(SAMPLE)

TURN OVER

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

Version2–April2016 5 MATHMETHEXAM1(SAMPLE)

TURN OVER

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

Version2–April2016 7 MATHMETHEXAM1(SAMPLE)

TURN OVER

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

Version2–April2016 9 MATHMETHEXAM1(SAMPLE)

TURN OVER

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

Version2–April2016 11 MATHMETHEXAM1(SAMPLE)

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

Version2–April2016 13 MATHMETHEXAM1(SAMPLE)

END OF QUESTION AND ANSWER BOOK

ii. Findtheminimumtotalshadedareaandthevalueofuforwhichtheareaisaminimum. 2marks

iii. Findthemaximumtotalshadedareaandthevalueofuforwhichtheareaisamaximum. 1mark

S A M P L EMATHEMATICAL METHODS

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 5 5 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• Questionandanswerbookof20pages.• Formulasheet.• Answersheetformultiple-choicequestions.

Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice

questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.• Unlessotherwiseindicated,thediagramsinthisbookarenot drawntoscale.• 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016

Version2–April2016

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education Year

STUDENT NUMBER

Letter

MATHMETHEXAM2(SAMPLE) 2 Version2–April2016

SECTION A – continued

Question 1ThepointP(4,–3)liesonthegraphofafunction f. Thegraphof f istranslatedfourunitsverticallyupandthenreflectedinthey-axis.ThecoordinatesofthefinalimageofPareA. (–4,1)B. (–4,3)C. (0,–3)D. (4,–6)E. (–4,–1)

Question 2Thelinearfunctionf:D → R,f(x)=4–xhasrange[–2,6).ThedomainDofthefunctionisA. [–2,6)B. (–2,2]C. RD. (–2,6]E. [–6,2]

Question 3Thefunctionwithrulef(x)=–3tan(2π x)hasperiod

A. 2π

B. 2

C. 12

D. 14

E. 2π

SECTION A – Multiple-choice questions

Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

Version2–April2016 3 MATHMETHEXAM2(SAMPLE)

SECTION A – continuedTURN OVER

Question 4Ifx + aisafactorof7x3 + 9x2–5ax,wherea R\{0},thenthevalueofaisA. –4B. –2C. –1D. 1E. 2

Question 5

Thefunctiong:[–a,a]→ R,g (x)= sin 26

x −

π hasaninversefunction.Themaximumpossiblevalueofais

A. π12

B. 1

C. π6

D. π4

E. π2

Question 6

If f x dx( ) =∫ 61

4,then 5 2

1

4−( )∫ f x dx( ) isequalto

A. 3B. 4C. 5D. 6E. 16

Question 7ForeventsAandB,Pr (A ∩ B)=p,Pr (A′ ∩ B)= p − 1

8 andPr (A ∩ B′)=35p.

IfAandBareindependent,thenthevalueofpisA. 0

B. 14

C. 38

D. 12

E. 35

MATHMETHEXAM2(SAMPLE) 4 Version2–April2016

SECTION A – continued

Question 8Whichoneofthefollowingfunctionssatisfiesthefunctionalequation f f x x( )( ) = foreveryrealnumberx?A. f x x( ) = 2

B. f x x( ) = 2

C. f x x( ) = 2

D. f x x( ) = − 2

E. f x x( ) = −2

Question 9Abagcontainsfiveredmarblesandfourbluemarbles.Twomarblesaredrawnfromthebag,withoutreplacement,andtheresultsarerecorded.Theprobabilitythatthemarblesaredifferentcoloursis

A. 2081

B. 518

C. 49

D. 4081

E. 59

Question 10

ThetransformationT R R: 2 2→ withrule

Txy

xy

=

−

+ −

1 00 2

12

mapsthelinewithequation x y− =2 3 ontothelinewithequationA. x + y = 0B. x+4y = 0C. –x–y=4D. x+4y=–6E. x–2y=1

Version2–April2016 5 MATHMETHEXAM2(SAMPLE)

SECTION A – continuedTURN OVER

Question 11Ifthetangenttothegraphofy = eax,a≠0,atx = cpassesthroughtheorigin,thencisequaltoA. 0

B. 1a

C. 1

D. a

E. −1a

Question 12Thesimultaneouslinearequations ax–3y=5 and 3x–ay=8–a haveno solution forA. a=3B. a=–3C. botha=3anda=–3D. a ∈ R\{3}E. a ∈ R\[–3,3]

Question 13Itisknownthat26%ofthe19-year-oldsinaregiondonothaveadriver’slicence.Ifarandomsampleoften19-year-oldsfromtheregionistaken,theprobability,correcttofourdecimalplaces,thatmorethanhalfofthemwillnothaveadriver’slicenceisA. 0.0239B. 0.0904C. 0.2600D. 0.9096E. 0.9761

Question 14Anopinionpollsterreportedthatforarandomsampleof574votersinatown,76%indicatedapreferenceforretainingthecurrentcouncil.Anapproximate90%confidenceintervalfortheproportionofthetotalvotingpopulationwithapreferenceforretainingthecurrentcouncilcanbefoundbyevaluating

A. 0 76 0 76 0 24574

0 76 0 24574

. . . , . .−

× ×

0.76 +

B. 0 76 1 65 0 76 0 24574

0 76 65 0 76 0 24574

. . . . , . . .−

× ×

+1.

C. 0 76 2 58 0 76 0 24574

0 76 0 24574

. . . . , . .−

× ×

0.76 + 2.58

D. 436 1 96 0 76 0 24 574 0 76 0 24 574− × × × ×( ). . . , . .436 +1.96

E. 0 76 2 0 76 0 24 574 0 76 0 24 574. . . , . .− × × × ×( )0.76 + 2

MATHMETHEXAM2(SAMPLE) 6 Version2–April2016

SECTION A – continued

Question 15Thecubicfunctionf :R → R,f(x)=ax3–bx2 + cx,wherea,bandcarepositiveconstants,hasnostationarypointswhen

A. c ba

>2

4

B. c ba

<2

4

C. c<4b2a

D. c ba

>2

3

E. c ba

<2

3

Question 16Apartofthegraphofg :R → R,g (x)=x2–4isshownbelow.

A

B

y

xaO

TheareaoftheregionlabelledAisthesameastheareaoftheregionlabelledB.TheexactvalueofaisA. 0

B. 6

C. 6

D. 12E. 2 3

Version2–April2016 7 MATHMETHEXAM2(SAMPLE)

SECTION A – continuedTURN OVER

Question 17Theequationx3–9x2+15x + w=0hasonlyonesolutionforxwhenA. –7<w < 25B. w≤–7C. w≥25D. w<–7orw > 25E. w>1

Question 18Acubicfunctionhastheruley = f(x).Thegraphofthederivativefunction f ʹcrossesthex-axisat(2,0)and(–3,0).Themaximumvalueofthederivativefunctionis10.Thevalueofxforwhichthegraphofy = f(x)hasalocalmaximumisA. –2B. 2C. –3D. 3

E. -12

Question 19ButterfliesofaparticularspeciesdieTdaysafterhatching,whereTisanormallydistributedrandomvariablewithameanof120daysandastandarddeviationofdays.If,fromapopulationof2000newlyhatchedbutterflies,150areexpectedtodieinthefirst90days,thenthevalueofisclosesttoA. 7days.B. 13days.C. 17days.D. 21days.E. 37days.

MATHMETHEXAM2(SAMPLE) 8 Version2–April2016

END OF SECTION A

Question 20Thegraphsof y = f (x)andy = g (x)areshownbelow.

y

O x

y = f (x)

y = g (x)

Thegraphofy = f (g (x))isbestrepresentedby

y

Ox

y

O x

y

O x

y

O x

y

O x

A. B.

C. D.

E.

Version2–April2016 9 MATHMETHEXAM2(SAMPLE)

SECTION B – continuedTURN OVER

Question 1 (7marks)Thepopulationofwombatsinaparticularlocationvariesaccordingtotherule

n t t( ) = +

1200 400

3cos π ,wherenisthenumberofwombatsandtisthenumberofmonthsafter

1March2013.

a. Findtheperiodandamplitudeofthefunctionn. 2marks

b. Findthemaximumandminimumpopulationsofwombatsinthislocation. 2marks

c. Findn(10). 1mark

d. Overthe12monthsfrom1March2013,findthefractionoftimewhenthepopulationofwombatsinthislocationwaslessthann(10). 2marks

SECTION B

Instructions for Section BAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

MATHMETHEXAM2(SAMPLE) 10 Version2–April2016

SECTION B – Question 2–continued

Question 2 (10marks)Asolidblockintheshapeofarectangularprismhasabaseofwidthxcentimetres.Thelengthofthebaseistwo-and-a-halftimesthewidthofthebase.

52x cm

x cm

h cm

Theblockhasatotalsurfaceareaof6480cm2.

a. Showthatiftheheightoftheblockishcentimetres,h = 6480 57

2− xx

. 2marks

Version2–April2016 11 MATHMETHEXAM2(SAMPLE)

SECTION B – continuedTURN OVER

b. Thevolume,Vcubiccentimetres,oftheblockisgivenbyV(x)= 5 6480 514

2x x( ).−

GiventhatV (x)>0andx >0,findthepossiblevaluesofx. 3marks

c. Find dVdx

,expressingyouranswerintheform dVdx

= ax2 + b,whereaandbarerealnumbers. 3marks

d. Findtheexactvaluesofxandhiftheblockistohavemaximumvolume. 2marks

MATHMETHEXAM2(SAMPLE) 12 Version2–April2016

SECTION B – Question 3–continued

Question 3 (20marks)FullyFitisaninternationalcompanythatownsandoperatesmanyfitnesscentres(gyms)inseveralcountries.Ithasmorethan100000membersworldwide.AteveryoneofFullyFit’sgyms,eachmemberagreestohavetheirfitnessassessedeverymonthbyundertakingasetofexercisescalledS.IfsomeonecompletesSinlessthanthreeminutes,theyareconsideredfit.

a. IthasbeenfoundthattheprobabilitythatanymemberofFullyFitwillcompleteSinlessthan

threeminutesis58.Thisisindependentofanyothermember.Arandomsampleof20FullyFit

membersistaken.Forasampleof20members,letXbetherandomvariablethatrepresentsthenumberofmemberswhocompleteSinlessthanthreeminutes.

i. FindPr(X≥10)correcttofourdecimalplaces. 2marks

ii. FindPr(X≥15|X≥10)correcttothreedecimalplaces. 3marks

Version2–April2016 13 MATHMETHEXAM2(SAMPLE)

SECTION B – Question 3–continuedTURN OVER

Forsamplesof20members,P̂istherandomvariableofthedistributionofsampleproportionsofpeoplewhocompleteSinlessthanthreeminutes.

iii. FindtheexpectedvalueandvarianceofP̂. 3marks

iv. Findtheprobabilitythatasampleproportionlieswithintwostandarddeviationsof58.

Giveyouranswercorrecttothreedecimalplaces.Donotuseanormalapproximation. 3marks

v. FindPr(P̂≥34 |P̂≥

58 ).Giveyouranswercorrecttothreedecimalplaces.Donotuse

anormalapproximation. 2marks

MATHMETHEXAM2(SAMPLE) 14 Version2–April2016

SECTION B – Question 3–continued

b. PaulaisamemberofFullyFit’sgyminSanFrancisco.ShecompletesSeverymonthasrequired,butotherwisedoesnotattendregularlyandsoherfitnesslevelvariesovermanymonths.Paulafindsthatifsheisfitonemonth,theprobabilitythatsheisfitthenextmonth

is34 ,andifsheisnotfitonemonth,theprobabilitythatsheisnotfitthenextmonthis

12 .

IfPaulaisnotfitinoneparticularmonth,whatistheprobabilitythatsheisfitinexactlytwoofthenextthreemonths? 2marks

c. WhenFullyFitsurveyedallitsgymsthroughouttheworld,itwasfoundthatthetimetakenbymemberstocompleteanotherexerciseroutine,T,isacontinuousrandomvariableWwithaprobabilitydensityfunctiong,asdefinedbelow.

g w

w w

w w( )

( )

=

− +≤ ≤

+< ≤

3 64256

3

29128

3 5

0

3 1

elsewhere

i. FindE(W)correcttofourdecimalplaces. 2marks

ii. Inarandomsampleof200FullyFitmembers,howmanymemberswouldbeexpectedtotakemorethanfourminutestocompleteT?Giveyouranswertothenearestinteger. 2marks

Version2–April2016 15 MATHMETHEXAM2(SAMPLE)

SECTION B – continuedTURN OVER

d. Fromarandomsampleof100members,itwasfoundthatthesampleproportionofpeoplewhospentmorethantwohoursperweekinthegymwas0.6

Findanapproximate95%confidenceintervalforthepopulationproportioncorrespondingtothissampleproportion.Givevaluescorrecttothreedecimalplaces. 1mark

MATHMETHEXAM2(SAMPLE) 16 Version2–April2016

SECTION B – Question 4–continued

Question 4 (8marks)TheshadedregioninthediagrambelowistheplanofaminesitefortheBlackPossumminingcompany.Alldistancesareinkilometres.Twooftheboundariesoftheminesiteareintheshapeofthegraphsofthefollowingfunctions.

f :R → R,f (x)=ex g:R+ → R,g (x)=loge(x)

y

x–3 –2 –1 1 2 3

–3

–2

–1

O

1

2

3

x = 1

y = f (x) y = g (x)

y = –2

a. i. Evaluate f x dx( ) .−∫ 2

0

1mark

ii. Hence,orotherwise,findtheareaoftheregionboundedbythegraphofg,thex-axisandy-axis,andtheliney =–2. 1mark

Version2–April2016 17 MATHMETHEXAM2(SAMPLE)

SECTION B – Question 4–continuedTURN OVER

iii. Findthetotalareaoftheshadedregion. 1mark

b. Theminingengineer,Victoria,decidesthatabettersiteforthemineistheregionboundedbythegraphofgandthatofanewfunctionk:(–∞,a)→ R,k (x)=–loge(a – x),whereaisapositiverealnumber.

i. Find,intermsofa,thex-coordinatesofthepointsofintersectionofthegraphsof gandk. 2marks

ii. Hence,findthesetofvaluesofaforwhichthegraphsofgandkhavetwodistinctpointsofintersection. 1mark

MATHMETHEXAM2(SAMPLE) 18 Version2–April2016

SECTION B – continued

c. Forthenewminesite,thegraphsofgandkintersectattwodistinctpoints,AandB.ItisproposedtostartminingoperationsalongthelinesegmentAB,whichjoinsthetwopointsofintersection.

Victoriadecidesthatthegraphofkwillbesuchthatthex-coordinateofthemidpointofAB is 2 .

Findthevalueofainthiscase. 2marks

Version2–April2016 19 MATHMETHEXAM2(SAMPLE)

SECTION B – Question 5–continuedTURN OVER

Question 5 (15marks)Let f:R → R, f (x)=(x–3)(x–1)(x 2+3)andg:R → R,g (x)=x 4–8x.

a. Expressx4–8x intheformx x a x b c−( ) + +( )( )2 . 2marks

b. Describethetranslationthatmapsthegraphofy = f(x)ontothegraphofy = g(x). 2marks

c. Findthevaluesofdsuchthatthegraphofy = f(x + d )has i. onepositivex-axisintercept 1mark

ii. twopositivex-axisintercepts. 1mark

d. Findthevalueofn forwhichtheequationg (x)= nhasonesolution. 1mark

MATHMETHEXAM2(SAMPLE) 20 Version2–April2016

END OF QUESTION AND ANSWER BOOK

e. Atthepoint u g u, ( ) ,( ) thegradientofy = g (x)ismandatthepoint v g v, ( ) ,( ) thegradientis–m,wheremisapositiverealnumber.

i. Findthevalueof u3 + v3. 2marks

ii. Finduandvif u + v=1. 1mark

f. i. Findtheequationofthetangenttothegraphofy = g(x)atthepoint p g p, ( )( ). 2marks

ii. Findtheequationsofthetangentstothegraphofy = g(x)thatpassthroughthepoint

withcoordinates 32

12, −

. 3marks

MATHMETH EXAM 2 (SAMPLE – ANSWERS)

© VCAA 2016 – Version 2 – April 2016

Answers to multiple-choice questions

Question Answer

1 A

2 D

3 C

4 E

5 A

6 A

7 C

8 E

9 E

10 C

11 B

12 B

13 A

14 B

15 D

16 E

17 D

18 B

19 D

20 E