2019 Mathematical Methods-nht Written examination 2€¦ · 5 2019 MATHMETH EXAM 2 (NHT) SECTION A – ot TURN OVER D O N O T W R I T E I N T H I S A R E A D O N O T W R I T E I N

MATHEMATICAL METHODSWritten examination 2

Monday 3 June 2019 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• Questionandanswerbookof26pages• 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2019

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2019

STUDENT NUMBER

Letter

2019MATHMETHEXAM2(NHT) 2

SECTION A – continued

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Question 1Themaximaldomainofthefunctionwithrule f (x)=x2+loge(x)isA. RB. (0,∞)C. [0,∞)D. (−∞,0)E. [1,∞)

Question 2Thediagrambelowshowsonecycleofacircularfunction.

5

4

3

2

1

O–1

–2

–3

1 2 3 4

y

x

Theamplitude,periodandrangeofthisfunctionarerespectivelyA. 3,2and[−2,4]

B. 3,π2 and[−2,4]

C. 4,4and[0,4]

D. 4,π4 and[−2,4]

E. 3,4and[−2,4]

SECTION A – Multiple-choice questions

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

3 2019MATHMETHEXAM2(NHT)

SECTION A – continuedTURN OVER

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Question 3Ifx + aisafactorof8x3–14x2 – a2x,wherea ∈ R\{0},thenthevalueofaisA. 7B. 4C. 1D. –2E. –1

Question 4

Thegraphofthefunction f :D → R, f x xx

( ) = −+

2 34

,whereDisthemaximaldomain,hasasymptotesA. x =–4,y = 2

B. x y= = −32

4,

C. x y= −4 =, 32

D. x y= =32

2,

E. x =2,y =1

Question 5ConsidertheprobabilitydistributionforthediscreterandomvariableXshowninthetablebelow.

x −1 0 1 2 3

Pr(X = x) b b b35

− b 35b

ThevalueofE(X) is

A. 7665

B. 1

C. 0

D. 2

13

E. 8665



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Question 6Let f:[0,∞)→ R, f(x)= x2+1.

Theequation f f x( )( ) = 18516

hasrealsolution(s)

A. x = ± 134

B. x =134

C. x = ± 132

D. x =32

E. x = ± 32

Question 7

Ifmxdx= ∫ 2

1

3,thenthevalueofemis

A. loge(9)

B. −9

C. 19

D. 9

E. −19

Question 8Thesimultaneouslinearequations2y +(m–1)x =2andmy + 3x = khaveinfinitelymanysolutionsforA. m =3andk =−2B. m =3andk = 2C. m =3andk =4D. m =–2andk =−2E. m =−2andk = 3



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Question 9Atthestartofaparticularweek,Kimhasthreeredapplesandtwogreenapples.Sheeatsoneappleeveryday.OnMonday,TuesdayandWednesdayofthatweek,sherandomlyselectsanappletoeat.Inthisthree-dayperiod,theprobabilitythatKimdoesnoteatanappleofthesamecolouronanytwoconsecutivedaysis

A. 1

10

B. 15

C. 3

10

D. 25

E. 625

Question 10

Let f betheprobabilitydensityfunction f R f x kx x x x: , , ( ) ( )( )( ).0 23

2 1 3 2 3 2

→ = + − +

Thevalueofkis

A. 308405

B. −308405

C. −405308

D. 405308

E. 960133

Question 11Thefunction f :D → R, f (x)= 5x3+10x2+1willhaveaninversefunctionforA. D = R

B. D =(−2,∞)

C. D = −∞

, 12

D. D =(–∞,–1]

E. D =[0,∞)

2019 MATHMETH EXAM 2 (NHT) 6


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Question 12The transformation T : R2 → R2, which maps the graph of y x= − + −2 1 3 onto the graph of y x= , has rule

A. Txy

xy

=

−

+

−−

12

0

0 1

13

B. Txy

xy

=

−

+

−

12

0

0 1

13

C. Txy

xy

=

−

+ −

12

0

0 1

13

D. Txy

xy

= −

+ −

2 00 1

13

E. Txy

xy

= −

+

−

2 00 1

13

Question 13The graph of f (x) = x3 – 6(b – 2)x2 + 18x + 6 has exactly two stationary points forA. 1 < b < 2

B. b = 1

C. b = ±4 62

D. 4 62

4 62

−≤ ≤

+b

E. b b<−

>+4 6

24 6

2or



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Question 14If2 loge(x)–loge(x+2)=loge(y),thenxisequalto

A. y y y+ +2 8

2

B. y y y± +2 8

2

C. y y y± −2 8

2

D. − ± −1 4 7

2y

E. − + −1 4 7

2y

Question 15Theareaboundedbythegraphofy=f(x),thelinex=2,thelinex=8andthex-axis,asshadedinthediagrambelow,is3 loge(13).

–10 –8 –6 –4 –2

O 2 4 6 8 10

y

y = f (x)x

Thevalueof 3 24

10f x dx( )−∫ is

A. 3 loge(13)B. 9 loge(13)C. 6 loge(39)D. loge(13)E. 9 loge(11)



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Question 16Partsofthegraphsofy = f (x)andy = g(x)areshownbelow.

O

y = g(x)

y = f (x)y

x

Thecorrespondingpartofthegraphofy = g(f (x))isbestrepresentedby

O

y

xO

y

x

O

y

x

O

y

x

O

y

x

A. B.

C. D.

E.



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Question 17Thegraphofthefunctiongisobtainedfromthegraphofthefunction f withrule f x x( ) cos( )= −

38bya

dilationoffactor4πfromthey-axis,adilationoffactor

43fromthex-axis,areflectioninthey-axisanda

translationof 32unitsinthepositiveydirection,inthatorder.

Therangeandtheperiodofgarerespectively

A. −

13

73

2, and

B. −

13

73

, and 8

C. −

73

13

, and 2

D. −

73

13

, and 8

E. −

43

43 2

2, and π

Question 18Partofthegraphofthefunction f,where f (x)= 8 – 2x−1,isshownbelow.ItintersectstheaxesatthepointsAandB. ThelinesegmentjoiningAandBisalsoshownonthegraph.

y

A

xBO

y = f (x)

Theareaoftheshadedregionis

A. 16 −152log ( )e

B. 17 −15

2 2log ( )e

C. 72

15916log ( )e

−

D. 16 152 2

−log ( )e

E. 17

2 215

log ( )e−


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END OF SECTION A

Question 19Arandomsampleofcomputeruserswassurveyedaboutwhethertheusershadplayedaparticularcomputergame.Anapproximate95%confidenceintervalfortheproportionofcomputeruserswhohadplayedthisgamewascalculatedfromthisrandomsampletobe(0.6668,0.8147).ThenumberofcomputerusersinthesampleisclosesttoA. 5B. 33C. 135D. 150E. 180

Question 20Let f (x)=(ax + b)5andletgbetheinversefunctionof f.Giventhat f (0)=1,whatisthevalueofg ′(1)?

A. 5a

B. 1

C. 1

5a

D. 5a(a+1)4

E. 0


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TURN OVER

CONTINUES OVER PAGE


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SECTION B – Question 1–continued

Question 1 (9marks)Partsofthegraphsof f (x)=(x–1)3(x+2)3andg(x)=(x–1)2(x+2)3areshownontheaxesbelow.

y

x

y = f (x)

y = g(x)

8

O–2 –1 1 2

–8

Thetwographsintersectatthreepoints,(–2,0),(1,0)and(c,d).Thepoint(c,d)isnotshowninthediagramabove.

a. Findthevaluesofc andd. 2marks

b. Findthevaluesofxsuchthat f (x)>g(x). 1mark

SECTION B

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


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SECTION B – continuedTURN OVER

c. Statethevaluesofxforwhich

i. f ′(x)>0 1mark

ii. g′(x)>0 1mark

d. Showthat f (1+m)=f (–2–m)forallm. 1mark

e. Findthevaluesofhsuchthatg(x + h)=0hasexactlyonenegativesolution. 2marks

f. Findthevaluesofksuchthat f (x)+k=0hasnosolutions. 1mark


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Question 2 (14marks)Thewindspeedataweathermonitoringstationvariesaccordingtothefunction

v t t( ) sin= +

20 16

14π

wherevisthespeedofthewind,inkilometresperhour(km/h),andtisthetime,inminutes,after 9am.

a. Whatistheamplitudeandtheperiodofv(t)? 2marks

b. Whatarethemaximumandminimumwindspeedsattheweathermonitoringstation? 1mark

c. Findv(60),correcttofourdecimalplaces. 1mark

d. Findtheaveragevalueofv(t)forthefirst60minutes,correcttotwodecimalplaces. 2marks


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SECTION B – Question 2–continuedTURN OVER

Asuddenwindchangeoccursat10am.Fromthatpointintime,thewindspeedvariesaccordingtothenewfunction

v t t k1 28 18

7( ) sin ( )

= +−

π

wherev1isthespeedofthewind,inkilometresperhour,tisthetime,inminutes,after9amand k ∈ R+.Thewindspeedafter9amisshowninthediagrambelow.

50

40

30

20

10

O 60 120

wind speed (km/h)

time (min)

e. Findthesmallestvalueofk,correcttofourdecimalplaces,suchthatv(t)andv1(t)areequalandarebothincreasingat10am. 2marks


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SECTION B – continued

f. Anotherpossiblevalueofk wasfoundtobe31.4358 Usingthisvalueofk,theweathermonitoringstationsendsasignalwhenthewindspeedis

greaterthan38km/h.

i. Findthevalueoftatwhichasignalisfirstsent,correcttotwodecimalplaces. 1mark

ii. Findtheproportionofonecycle,tothenearestwholepercent,forwhichv1>38. 2marks

g. Let f x x( ) sin= +

20 16

14π and g x x k( ) sin ( ) .= +

−

28 18

7π

ThetransformationTxy

ab

xy

cd

=

+

00

mapsthegraphof f ontothegraphofg.

Statethevaluesofa,b,candd,intermsofkwhereappropriate. 3marks


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Question 3 (14marks)ConcertsattheMathslandConcertHallbeginLminutesafterthescheduledstartingtime.Lisarandomvariablethatisnormallydistributedwithameanof10minutesandastandarddeviationoffourminutes.

a. Whatproportionofconcertsbeginbeforethescheduledstartingtime,correcttofour decimalplaces? 1mark

b. Findtheprobabilitythataconcertbeginsmorethan15minutesafterthescheduled startingtime,correcttofourdecimalplaces. 1mark

Ifaconcertbeginsmorethan15minutesafterthescheduledstartingtime,thecleanerisgivenanextrapaymentof$200.Ifaconcertbeginsupto15minutesafterthescheduledstartingtime,thecleanerisgivenanextrapaymentof$100.Ifaconcertbeginsatorbeforethescheduledstartingtime,thereisnoextrapaymentforthecleaner.LetCbetherandomvariablethatrepresentstheextrapaymentforthecleaner,indollars.

c. i. Usingyourresponsesfrompart a.andpart b.,completethefollowingtable,correcttothreedecimalplaces. 1mark

c 0 100 200

Pr(C=c)

ii. Calculatetheexpectedvalueoftheextrapaymentforthecleaner,tothenearestdollar. 1mark

iii. CalculatethestandarddeviationofC,correcttothenearestdollar. 1mark


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TheownersoftheMathslandConcertHalldecidetoreviewtheiroperation.Theystudyinformationfrom1000concertsatothersimilarvenues,collectedasasimplerandomsample.Thesamplevalueforthenumberofconcertsthatstartmorethan15minutesafterthescheduledstartingtimeis43.

d. i. Findthe95%confidenceintervalfortheproportionofconcertsthatbeginmorethan 15minutesafterthescheduledstartingtime.Givevaluescorrecttothreedecimalplaces. 1mark

ii. Explainwhythisconfidenceintervalsuggeststhattheproportionofconcertsthatbeginmorethan15minutesafterthescheduledstartingtimeattheMathslandConcertHallisdifferentfromtheproportionatthevenuesinthesample. 1mark

19 2019 MATHMETH EXAM 2 (NHT)

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The owners of the Mathsland Concert Hall decide that concerts must not begin before the scheduled starting time. They also make changes to reduce the number of concerts that begin after the scheduled starting time. Following these changes, M is the random variable that represents the number of minutes after the scheduled starting time that concerts begin. The probability density function for M is

f x xx

x( ) ( )= +

≥

<

82

0

0

0

3

where x is the time, in minutes, after the scheduled starting time.

e. Calculate the expected value of M. 2 marks

f. i. Find the probability that a concert now begins more than 15 minutes after the scheduled starting time. 1 mark

ii. Find the probability that each of the next nine concerts begins no more than 15 minutes after the scheduled starting time and the 10th concert begins more than 15 minutes after the scheduled starting time. Give your answer correct to four decimal places. 2 marks

iii. Find the probability that a concert begins up to 20 minutes after the scheduled starting time, given that it begins more than 15 minutes after the scheduled starting time. Give your answer correct to three decimal places. 2 marks


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SECTION B – Question 4 – continued

Question 4 (13 marks)A mining company has found deposits of gold between two points, A and B, that are located on a straight fence line that separates Ms Pot’s property and Mr Neg’s property. The distance between A and B is 4 units.The mining company believes that the gold could be found on both Ms Pot’s property and Mr Neg’s property.The mining company initially models the boundary of its proposed mining area using the fence line and the graph of

f : [0, 4] → R, f (x) = x(x – 2)(x – 4)

where x is the number of units from point A in the direction of point B and y is the number of units perpendicular to the fence line, with the positive direction towards Ms Pot’s property. The mining company will only mine from the boundary curve to the fence line, as indicated by the shaded area below.

y

xA

B

42

Ms Pot’s property

Mr Neg’s property

fence line

4

2

O

–2

–4

a. Determine the total number of square units that will be mined according to this model. 2 marks

The mining company offers to pay Mr Neg $100 000 per square unit of his land mined and Ms Pot $120 000 per square unit of her land mined.

b. Determine the total amount of money that the mining company offers to pay. 1 mark


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Theminingcompanyreviewsitsmodeltousethefencelineandthegraphof

p R p x x xa

x:[ , ] , ( ) ( )0 4 4 41

4→ = − ++

−

wherea>0.

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

d. Solvep′(x)=0forxintermsofa. 2marks

MrNegdoesnotwanthispropertytobeminedfurtherthan4unitsmeasuredperpendicularfromthefenceline.

e. Findthesmallestvalueofa,correcttothreedecimalplaces,forthisconditiontobemet. 2marks


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SECTION B – continued

f. Find the value of a for which the total area of land mined is a minimum. 3 marks

g. The mining company offers to pay Ms Pot $120 000 per square unit of her land mined and Mr Neg $100 000 per square unit of his land mined.

Determine the value of a that will minimise the total cost of the land purchase for the mining company. Give your answer correct to three decimal places. 2 marks


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CONTINUES OVER PAGE


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Question 5 (10marks)

Let f R R f x e g R R g x xx

e: , ( ) : , ( ) log ( ).→ = → =

+2 2and

a. Findg−1(x). 1mark

b. FindthecoordinatesofpointA,wherethetangenttothegraphof f at Aisparalleltothegraphofy=x. 2marks

c. Showthattheequationofthelinethatisperpendiculartothegraphofy=xandgoesthrough pointAisy=−x+2 loge (2)+2. 1mark

LetBbethepointofintersectionofthegraphsofgandy = −x+2 loge (2)+2,asshowninthediagrambelow.

y = f (x) y = x

y = g(x)

B

A

y

x

y = –x + 2loge(2) + 2O

d. DeterminethecoordinatesofpointB. 1mark


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e. Theshadedregionbelowisenclosedbytheaxes,thegraphsof f andg,andtheline y = −x+2 loge (2)+2.

y = f (x) y = x

y = g(x)

B

y

x

y = –x + 2loge(2) + 2O

A

Findtheareaoftheshadedregion. 2marks


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END OF QUESTION AND ANSWER BOOK

Let p R R p x e q R R q xk

xkxe: , ( ) : , ( ) log ( ).→ = → =+and 1

f. Thegraphsofp,qandy = xareshowninthediagrambelow.Thegraphsofpandqtouchbutdonotcross.

y = p(x) y = x

y = q(x)

y

xO

Findthevalueofk. 2marks

g. Findthevalueofk, k >0,forwhichthetangenttothegraphofpatitsy-interceptandthetangenttothegraphofqatitsx-interceptareparallel. 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 2019

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2019

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 ( )( ) = 11 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

Documents

2019 Mathematical Methods-nht Written examination 2€¦ · 5 2019 MATHMETH EXAM 2 (NHT) SECTION A – ot TURN OVER D O N O T W R I T E I N T H I S A R E A D O N O T W R I T E I N