MATHEMATICAL METHODSWritten examination 2

Monday 31 May 2021 Reading time: 10.00 am to 10.15 am (15 minutes) Writing time: 10.15 am to 12.15 pm (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• Questionandanswerbookof29pages• Formulasheet• Answersheetformultiple-choicequestions

Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice

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

At the end of the examination• Placetheanswersheetformultiple-choicequestionsinsidethefrontcoverofthisbook.• Youmaykeeptheformulasheet.

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

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2021

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2021

STUDENT NUMBER

Letter

GO

LD

S

TR

IPE

2021MATHMETHEXAM2(NHT) 2

SECTION A – continued

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Question 1Thegraphbelowshowsonecycleofacircularfunction.

x

y

4

2

O

–2

1

TheruleforthefunctioncouldbeA. y=3sin(x)+1

B. y x= −

+3

21sin

π

C. y=−3cos(2π x)+1

D. y=3sin(2π x)−1

E. y=−3sin(2π x)+1

Question 2If3f (x)=f(3x)forx>0,thentheruleforfcouldbe

A. f (x)=3x

B. f x x( ) = 3

C. 3

( ) =3xf x

D. ( ) log3exf x =

E. f (x)=x−3

SECTION A – Multiple-choice questions

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

3 2021MATHMETHEXAM2(NHT)

SECTION A – continuedTURN OVER

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Question 3

Thefunctionf : D → R, f x x x x x( ) = − − +4 3 2

4 39

29 willhaveaninversefunctionfor

A. D = RB. D=(–3,1)C. D=(1,∞)D. D=(–∞,0)E. D=(0,∞)

Question 4Thegraphoff :R → R,f (x)=x3+ax2+bx+chasaturningpointatx=3anday-interceptaty=9.Thevaluesofa,bandccouldbe,respectivelyA. −5,3and9

B. 7,−15and−9

C. − −2 32

9, and

D. 5,−3and−9

E. −1,−3and9

Question 5

Theexpression log3 25

q p

isequivalentto

A. log3(5)–log3(q)–log3(p)

B. 12

5 2 23 3 3log ( ) log ( ) log ( )− −q p

C. 12

5 23 3 3log ( ) log ( ) log ( )− −q p

D. 2 log3(5)–2 log3(q)–log3(p)

E. 2 log3(5)–2 log3(q)–2 log3(p)

Question 6Thesumofthefirstfourpositivesolutionstotheequationtan(2x)−1=0is

A. 32π

B. 52π

C. 2π

D. 72π

E. 4π

2021MATHMETHEXAM2(NHT) 4

SECTION A – continued

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Question 7Partofthegraphofy = f′(x)isshowninthediagrambelow.

3

2

1

–1O 3

2

1–1 4

y

x

Giventhatf(0)=1,thecorrespondingpartofthegraphofy = f(x)couldbe

2

–2

1O 2 3 4

y

x

2

–2

1O 2 3 4

y

x

2

–2

1O 2 3 4

y

x

2

–2

1O 2 3 4

y

x

2

–2

1O 2

3

4

y

x

A. B.

C. D.

E.

1

1

1

1

1

–1–1

–1–1

–1–1

–1–1

–1–1

5 2021MATHMETHEXAM2(NHT)

SECTION A – continuedTURN OVER

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Question 8

Partofthegraphofapolynomialfunctionf isshownbelow.Thisgraphhasturningpointsat − −( )2 2 1, and 2 2 1, −( ).

y

x–4 4O

f(x)isstrictlydecreasingfor

A. x ∈(–∞,–4]∪[4,∞)

B. x ∈[–4,4]

C. x∈ −2 2 2 2,

D. x∈ −∞ −( ∪

, ,2 2 0 2 2

E. x∈ − ∪ ∞

)2 2 0 2 2, ,

Question 9Thecontinuousanddifferentiablefunctionf :R → R hasrootsatx=1andx=6andarepeatedrootatx=4.

Giventhat4

1( )f x dx a=∫ and

6

4( )f x dx b=∫ ,wherea,b ∈ R, f x dx( ) +( )∫ 1

1

6isequalto

A. a+b+1B. a–b+1C. a+b+5D. a–b–5E. a–b

Question 10Considerthegraphoff:R → R,f (x)=– x2–4x+5.Thetangenttothegraphoff isparalleltothelineconnectingthenegativex-interceptandthey-interceptoffwhenxisequaltoA. −3

B. −52

C. −32

D. −1

E. −12

2021MATHMETHEXAM2(NHT) 6

SECTION A – continued

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Question 11Asurveyofalargerandomsampleofpeoplefoundthatanapproximate95%confidenceintervalfortheproportionofpeoplewhoownedayellowrubberduckwas(0.6299,0.6699).ThenumberofpeopleintherandomsampleisclosesttoA. 569B. 1793C. 2108D. 2179E. 2185

Question 12ThetransformationT :R2 → R2mapsthegraphofy = x3−xontothegraphofy=2(x −1)3−2(x −1)+4.ThetransformationTcouldbegivenby

A. Txy

xy

=

+

1 00 2

14

B. Txy

xy

=

+

1 0

0 12

14

C. Txy

xy

=

+

2 00 1

12

D. Txy

xy

=

+

12

0

0 1

12

E. Txy

xy

=

+

1 00 2

12

Question 13

Forthefunctionp (x)=ke−k x,wherex≥0andk>0,thevalueof aforwhich p a p( ) ( )=12

0 is

A. 1 1

2k elog

B. 1 2k elog ( )

C. kloge (2)

D. k elog 12

E. kkelog 1

2

7 2021MATHMETHEXAM2(NHT)

SECTION A – continuedTURN OVER

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Question 14Acontinuousrandomvariable,X,hastheprobabilitydensityfunction

f xe x

x

x

( ). .

=≥<

−0 2 00 0

0 2

ThevarianceofXisA. 25B. 12.5C. 6.25D. 3.125E. 0

Question 15Thegraphsoffunctionsfandgareshownbelow.Bothfunctionshavethesamedomainof[0,b],where b>0,andthesameaveragevalue.

O

2b

f

y

xb O

g

y

xb

8

ThevalueofbisA. 1B. 2C. 4D. 8E. 16

Question 16Inaparticularcity,itisknownthat70%ofalladultsgettheirhaircuteverymonth.Arandomsampleof 720adultsfromthiscityisselected.Fromthissample,theprobabilitythattheproportionofadultswhogettheirhaircuteverymonthisgreaterthan0.72isA. 0.2104B. 0.1359C. 0.1187D. 0.0847E. 0.0392

2021MATHMETHEXAM2(NHT) 8

SECTION A – continued

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Question 17Partofthegraphofthefunctionf isshownbelow.Thesmallestpositivex-interceptofthegraphoccursatx = a.Thehorizontallineisatangenttofatthelocalminimum b f b, ( )( ).Theshadedareaistheareaboundedbythegraphoff,thex-axis,they-axisandthegraphofy = f (b).

( ), ( )b f b

(a, 0)

y

Ox

Theareaoftheshadedregionis

A. a f b f x dxa

b( ) ( )+ ∫

B. a f b f x dxa

b( ) ( )− ∫

C. f x dx b f ba

b( ) ( )+∫

D. b f b f x dxa

b( ) ( )− ∫

E. f x dx b f ba

b( ) ( )−∫

Question 18

Giventhatd x x

dxx x x

cos( )cos( ) sin( )( )

= − , sin( )x x dx∫ isequalto

A. cos(x)−xcos(x)

B. cos cos( ) ( )x x x dx+ ∫C. x x x dxcos( ) cos( )− ∫D. cos cos( ) ( )x dx x x−∫E.

−x xx

cos( )cos( )

9 2021MATHMETHEXAM2(NHT)

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END OF SECTION ATURN OVER

Question 19Acubicpolynomialfunctionf :R → Rhasrootsatx=1andx=3onlyanditsgraphhasay-interceptat y =3.Whichoneofthefollowingstatementsmustbetrueaboutthefunctiong,whereg x f x( ) ( )= ?A. Thefunctionghasalocalmaximumatx = 2B. g(2)=1C. Thedomainofgdoesnotincludetheinterval(1,3)D. Thedomainofgincludestheinterval(1,3)E. Thedomainofgdoesnotincludetheinterval(3,∞)

Question 20TheprobabilitydistributionforthediscreterandomvariableX,whereb ∈ R,isshowninthetablebelow.

x 0 1 2 3

Pr(X = x) 45

110

3− b 15 250

2b − 9 550b +

ThevalueofbisA. –0.4B. –0.3C. –0.2D. 0.2E. 0.5

2021MATHMETHEXAM2(NHT) 10

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SECTION B – Question 1–continued

Question 1 (11marks)Let f R R f x x x: , ( ) cos( ) cos( )→ = − +( )2 4 andg :R → R,g (x)=2cos(x).

a. Statetheperiodandtheamplitudeofg. 1mark

b. Findthevalueofcforwhichf(c)=0,wherec∈

0

2, π . 1mark

c. Findtheminimumvalueoff. 1mark

SECTION B

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

11 2021MATHMETHEXAM2(NHT)

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SECTION B – Question 1–continuedTURN OVER

Partsofthegraphsofthefunctionsoffandgareshownbelow.

Thegraphsoffandgtouch,butdonotcross,atthepoints(p, q)and π3

, v

.

y

xO

g

(p, q)

f

,3

vπ

d. Findthevaluesofpandq. 2marks

e. i. Findthevalueofthederivativeoffandthevalueofthederivativeofg at x = π3. 2marks

ii. Findtheequationofthetangenttothegraphsoffandg at x = π3. 1mark

iii. Findtheequationofthelineperpendiculartothegraphsoffandg at x = π3. 1mark

2021MATHMETHEXAM2(NHT) 12

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SECTION B – continued

f. Theareaboundedbythegraphsoffandgisshadedinthediagrambelow.

y

xO

g

f

Findtheareaoftheshadedregion. 2marks

13 2021MATHMETHEXAM2(NHT)

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SECTION B – continuedTURN OVER

CONTINUES OVER PAGE

2021MATHMETHEXAM2(NHT) 14

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SECTION B – Question 2–continued

Question 2 (10marks)

Thefunctionh xx

x xe( ) log , [ , ]=−( )

−

∈

32005

54

10 502 where ,modelstherateatwhichheatislost

fromthewaterinahot-waterpipewithinsulation,whereh(x) istherateatwhichunitsofheatarelostfromthewaterandx istheradiusofthehot-waterpipewithitsinsulation,inmillimetres.Thediagrambelowshowsacross-sectionofthepipewithitsinsulation.

radius of pipewith insulation

radius of pipe

O

x

Theradiusofthepipewithoutitsinsulationis10mm.Thegraphoftherateofheatlostfromthewateroverthegivendomainisshownbelow.

O

40

30

20

10

10 20 30 40 50x

h

a. Findtherateatwhichheatislostfromthewaterinapipewithnoinsulation,correcttothreedecimalplaces. 1mark

15 2021MATHMETHEXAM2(NHT)

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SECTION B – Question 2–continuedTURN OVER

b. i. Statethederivativeofh (x). 1mark

ii. Findthemaximumrateatwhichheatislostfromthewater,correcttothreedecimalplaces. 1mark

c. Aparticularinsulatedpipehasthesamerateofheatlostfromthewaterasapipewithnoinsulation.

Findthethicknessofinsulationforthispipe,inmillimetres,correcttothreedecimalplaces. 1mark

d. i. Ifboththeradiusofthepipewithoutinsulationandtheradiusofthepipewithinsulation,asshowninthediagramonpage14,aredoubled,showthattherateofheatlostfromthewater,h1,isnowgivenby

h x xx e1 212 800 10

810( ) log=

−

−( )

andstatethedomainofh1. 2marks

ii. Describethetransformationthatmapsthegraphofhtothegraphofh1. 1mark

2021MATHMETHEXAM2(NHT) 16

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SECTION B – continued

e. i. Findtheareabetweenthegraphofh1andthehorizontalaxisoveritsdomain.Giveyouranswercorrecttothreedecimalplaces. 2marks

ii. Lettheareafoundinpart e.i.beA.

Determinetheareabetweenthegraphofh andthehorizontalaxisoveritsdomain,intermsofA. 1mark

17 2021MATHMETHEXAM2(NHT)

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SECTION B – continuedTURN OVER

CONTINUES OVER PAGE

2021MATHMETHEXAM2(NHT) 18

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SECTION B – Question 3–continued

Question 3 (10marks)Theparabolicarchofatunnelismodelledbythefunctionf:[−c,c] → R,f(x)=ax2+b,where a < 0,b ∈ Randc>0.Letxbethehorizontaldistance,inmetres,fromtheoriginandletybetheverticaldistance, inmetres,abovethebaseofthearch.Thegraphoff isshownbelow,wherethecoordinatesofthey-interceptare(0,k)andthecoordinatesofthex-interceptsare(−c,0)and(c,0).

y

c–c O

k

x

f

a. Expressa andb intermsofc andk. 2marks

19 2021MATHMETHEXAM2(NHT)

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SECTION B – Question 3–continuedTURN OVER

Aparticulartunnelhasanarchmodelledbyf.Ithasaheightof6matthecentreandawidthof 8matthebase.

b. i. Findtheruleforthisarch. 1mark

ii. Atruckthathasaheightof3.7mandawidthof2.7mwillfitthroughthearchwiththefunctionf foundinpart b.i.

y

x

k

–c cO

d

truck

f

Assumingthatthetruckdrivesdirectlythroughthemiddleofthearch,letdbetheminimumdistancebetweenthearchandthetopcornerofthetruck.

Finddandthevalueofxforwhichthisoccurs,correcttothreedecimalplaces. 3marks

2021MATHMETHEXAM2(NHT) 20

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SECTION B – Question 3–continued

Adifferenttunnelhasasemicirculararch.Thisarchcanbemodelledbythefunctiong R g x r x: [ , ] , ( )− → = −6 6 2 2 ,wherer > 0.Thegraphofgisshownbelow.

6

x6O–6

y

g

c. Statethevalueofr. 1mark

21 2021MATHMETHEXAM2(NHT)

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SECTION B – continuedTURN OVER

d. Twolightshavebeenplacedonthearchtolighttheentranceofthetunnel.Thepositionsofthelightsare −( ) ( )11 5 11 5, ,and .Theareathatislitbytheselightsisshadedinthe

diagrambelow.

6

x6O–6

y

( )11, 5− ( )11, 5

Determinetheproportionofthecross-sectionofthetunnelentrancethatislitbythelights.Giveyouranswerasapercentage,correcttothenearestinteger. 3marks

2021MATHMETHEXAM2(NHT) 22

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SECTION B – Question 4–continued

Question 4 (17marks)Aparticularpetrolstationhastwoairpumps,AandB,toinflatetyres.Eachinflationofatyreisindependentofanyotherinflationofatyre.WhenpumpAissetto320kilopascals(kPa),thepressurethatthetyreswillbeinflatedtofollows anormaldistributionwithameanof320kPaandastandarddeviationof10kPa.

a. Findtheprobabilitythatatyrewillbeinflatedtoapressuregreaterthan330kPawheninflatedbypumpA,correcttofourdecimalplaces. 1mark

b. TheprobabilitythatatyreisinflatedbypumpA toapressuregreaterthanais0.9

Findthevalueofa,correcttothenearestkilopascal. 2marks

WhenpumpBissetto320kPa,thepressurethatthetyreswillbeinflatedtoismodelledbythefollowingprobabilitydensityfunction.

b xx x x

( )( ) ( )

=− − ≤ ≤

340000

310 330 310 330

0

2

elsewhere

c. DeterminethemeantyrepressurefortyresinflatedbypumpB. 2marks

d. ArandomlyselectedtyreisinflatedbypumpB.

FindtheprobabilitythatthistyrewillbeinflatedtoapressuregreaterthanthemeantyrepressureoftyresinflatedbypumpB,correcttofourdecimalplaces. 2marks

23 2021MATHMETHEXAM2(NHT)

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SECTION B – Question 4–continuedTURN OVER

e. TheprobabilitythatatyreisinflatedbypumpB toapressurelessthankis0.95

Findthevalueofk,correcttothenearestkilopascal. 2marks

f. AmotoristisequallylikelytouseeitherpumpAorpumpBtoinflateoneoftheircar’styres.

FindtheprobabilitythatthemotoristhasusedpumpAgiventhatthetyreisinflatedtoapressuregreaterthan325kPa.Giveyouranswercorrecttofourdecimalplaces. 2marks

Thecompanythatmanufacturesthepumpstestsallofitspumpsandremovesthosethataredefective.Theprobabilitythatarandomlyselectedpumpisdefective,fromallofthepumpstested,is0.08

g. Findtheprobabilitythatfourpumpsaredefectivefromasampleof25randomlyselectedpumps,correcttofourdecimalplaces. 2marks

h. Forrandomsamplesof25pumps,P̂ istherandomvariablethatrepresentstheproportionof pumpsthataredefective.

FindtheprobabilitythatP̂ isgreaterthan15%,correcttofourdecimalplaces. 2marks

2021MATHMETHEXAM2(NHT) 24

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SECTION B – continued

i. Forrandomsamplesofnpumps,P̂ nistherandomvariablethatrepresentstheproportionofpumpsthataredefective.

Findtheleastvalueofnsuchthat 1ˆPr 0.15nPn

< <

2marks

25 2021MATHMETHEXAM2(NHT)

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SECTION B – continuedTURN OVER

CONTINUES OVER PAGE

2021MATHMETHEXAM2(NHT) 26

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SECTION B – Question 5–continued

Question 5 (12marks)

Let f R R f xx

g R R g x x: , ( ) : , ( )+ → = − → = −1

565

452 and .

Partsofthegraphsoffandgareshownbelow.

A

B

y

O

–1

–2

1 2x

f

g

a. Findthecoordinatesofthepointsofintersectionofthegraphsoff andg,labelledAandB inthediagramabove. 2marks

b. DeterminetheareaboundedbythegraphsoffandgbetweenAandB.Giveyouranswerin

theform r s tu+ ,wherer,s,tanduareintegers. 2marks

27 2021MATHMETHEXAM2(NHT)

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SECTION B – Question 5–continuedTURN OVER

Let 21 1 (1 ): , ( ) and : , ( ) for 1a a xp R R p x q R R q x a

ax a a+ + −→ = − → = > .

c. Findthevalueofaforwhichp (x)=f(x)andq (x)=g (x)forallx. 1mark

d. Findthepositivex-interceptofpintermsofa. 1mark

PointMliesonthegraphofy = p (x).Thetangenttop at Misparalleltoq.

e. Findthex-coordinateofMintermsofa. 2marks

2021MATHMETHEXAM2(NHT) 28

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SECTION B – Question 5–continued

f. i. Findthey-interceptofthetangenttop at Mintermsofa. 1mark

ii. Giventhat 23

2 113

23x x≥ −( ) forx>1,showthatthetangenttop paralleltoqwillhave

anegativey-interceptforalla>1. 1mark

iii. Thetangenttopparalleltoqhasanegativey-intercept.

Explainwhythisimpliespandqwillalwaysenclosearegionboundedbybothgraphsforalla>1. 1mark

29 2021MATHMETHEXAM2(NHT)

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END OF QUESTION AND ANSWER BOOK

g. Partsofthegraphsofpandqareshownbelowforwhena=100.

y

O

–1

1x

p

q

Theshadedareaisboundedbythegraphsofpandq.

Findthesmallestvalue,b,suchthattheshadedareaislessthanbforalla≥100. 1mark

MATHEMATICAL METHODS

Written examination 2

FORMULA SHEET

Instructions

This formula sheet is provided for your reference.A question and answer book is provided with this formula sheet.

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

Victorian Certificate of Education 2021

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2021

MATHMETH EXAM 2

Mathematical Methods formulas

Mensuration

area of a trapezium 12a b h+( ) volume of a pyramid 1

3Ah

curved surface area of a cylinder 2π  rh volume of a sphere

43

3π r

volume of a cylinder π r 2h area of a triangle12bc Asin ( )

volume of a cone13

2π r h

Calculus

ddx

x nxn n( ) = −1 x dxn

x c nn n=+

+ ≠ −+∫ 11

11 ,

ddx

ax b an ax bn n( )+( ) = +( ) −1 ( )( )

( ) ,ax b dxa n

ax b c nn n+ =+

+ + ≠ −+∫ 11

11

ddxe aeax ax( ) = e dx a e cax ax= +∫ 1

ddx

x xelog ( )( ) = 1 1 0x dx x c xe= + >∫ log ( ) ,

ddx

ax a axsin ( ) cos( )( ) = sin ( ) cos( )ax dx a ax c= − +∫ 1

ddx

ax a axcos( )( ) −= sin ( ) cos( ) sin ( )ax dx a ax c= +∫ 1

ddx

ax aax

a axtan ( )( )

( ) ==cos

sec ( )22

product ruleddxuv u dv

dxv dudx

( ) = + quotient ruleddx

uv

v dudx

u dvdx

v

=

−

2

chain ruledydx

dydududx

=

3 MATHMETH EXAM

END OF FORMULA SHEET

Probability

Pr(A) = 1 – Pr(A′) Pr(A ∪ B) = Pr(A) + Pr(B) – Pr(A ∩ B)

Pr(A|B) = Pr

PrA BB∩( )( )

mean µ = E(X) variance var(X) = σ 2 = E((X – µ)2) = E(X 2) – µ2

Probability distribution Mean Variance

discrete Pr(X = x) = p(x) µ = ∑ x p(x) σ 2 = ∑ (x – µ)2 p(x)

continuous Pr( ) ( )a X b f x dxa

b< < = ∫ µ =

−∞

∞

∫ x f x dx( ) σ µ2 2= −−∞

∞

∫ ( ) ( )x f x dx

Sample proportions

P Xn

=̂ mean E(P̂ ) = p

standard deviation

sd P p pn

(ˆ ) ( )=

−1 approximate confidence interval

,p zp p

np z

p pn

−−( )

+−( )

1 1ˆ ˆ ˆˆˆ ˆ