2021 Further Mathematics-nht Written examination 2

FURTHER MATHEMATICSWritten examination 2

Friday 28 May 2021 Reading time: 10.00 am to 10.15 am (15 minutes) Writing time: 10.15 am to 11.45 am (1 hour 30 minutes)

QUESTION AND ANSWER BOOK

Structure of bookSection A – Core Number of

questionsNumber of questions

to be answeredNumber of

marks

8 8 36Section B – Modules Number of

modulesNumber of modules

to be answeredNumber of

marks

4 2 24 Total 60

• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulers,oneboundreference,oneapprovedtechnology(calculatororsoftware)and,ifdesired,onescientificcalculator.CalculatormemoryDOESNOTneedtobecleared.Forapprovedcomputer-basedCAS,fullfunctionalitymaybeused.

• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.

Materials supplied• Questionandanswerbookof35pages• Formulasheet• Workingspaceisprovidedthroughoutthebook.

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

At the end of the examination• Youmaykeeptheformulasheet.

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

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2021

GR

EE

N

ST

RIP

E

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2021

STUDENT NUMBER

Letter

2021FURMATHEXAM2(NHT) 2

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SECTION A – Question 1 – continued

SECTION A – Core

Instructions for Section AAnswerallquestionsinthespacesprovided.Youneednotgivenumericalanswersasdecimalsunlessinstructedtodoso.Alternativeformsmayinclude,forexample,π,surdsorfractions.In‘Recursionandfinancialmodelling’,allanswersshouldberoundedtothenearestcentunlessotherwiseinstructed.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

Data analysis

Question 1 (8marks)Eachyear,manypeopleparticipateinafunrun.ThedatainTable1belowwascollectedfor13participantswhocompetedineitherthe12kmrun (women’sormen’s)orthe6kmrun(women’sormen’s).Thesixvariablesinthetableareasfollows:• number participant’snumber• name participant’sname• event 1=12kmrun,2=6kmrun• gender F=female,M=male• age ageinyears• time timetaken,inminutesandseconds,tocompletetheevent

Table 1

Number Name Event Gender Age Time

2063 M Jane 1 F 34 41:56

1243 HRoz 2 F 27 26:32

4536 JNalin 2 M 19 29:05

3429 KChen 1 M 34 40:58

3657 MFrench 1 F 56 48:12

987 K Morse 1 M 19 44:48

4897 MSharif 1 F 29 49:02

356 WCarey 1 M 39 39:51

234 MChin 1 F 19 55:34

1982 TKhan 1 M 27 46:24

345 RLu 2 F 46 29:32

2390 NGhan 2 F 23 28:13

1965 Z Ali 2 M 20 27:12

3 2021FURMATHEXAM2(NHT)

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SECTION A – Question 1 – continuedTURN OVER

a. Writedownthetwonumericalvariables. 1mark

b. Thevariablenumberisanominalvariable.

HowmanyoftheotherfivevariablesinTable1arenominalvariables? 1mark

c. UsetheinformationinTable1onpage2to

i. determinethemediantime,inminutesandseconds,offemaleparticipantswhocompletedthe6kmrun 1mark

ii. completethetwo-wayfrequencytableshownbelow(Table2). 2marks

Table 2

Gender

Event Female Male

12kmrun

6kmrun

Total


SECTION A – continued

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d. Thefunrunincludeda12kmwalkanda6kmwalkaswell. Thepercentagedsegmentedbarchartbelowshowsthepercentageofmalesandfemaleswho

chosetoparticipateineachofthefourevents(12kmwalk,6kmwalk,12kmrun,6kmrun).

12 km walk

6 km walk

12 km run

6 km run

Key

female malegender

1009080706050403020100

percentage

i. Whatpercentageofmalesparticipatedinawalk? 1mark

ii. Doesthepercentagedsegmentedbarchartsupportthecontentionthat,fortheseparticipants,theeventchosen(12kmwalk,6kmwalk,12kmrun,6kmrun)isassociatedwithgender?Justifyyouranswerbyquotingappropriatepercentages. 2marks


SECTION A – continuedTURN OVER

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Question 2 (5marks)Thehistogramandtheboxplotbelowdisplaytheagedistributionofthefirst80participantstofinishthewomen’s12kmrun.

n = 8018

12

6

0

frequency

5 10 15 20 25 30 35 40age (years)

45 50 55 60 65 70

5 10 15 20 25 30 35 40age (years)

45 50 55 60 65 70

Data:City-BayFunRun,<https://city-bay.org.au/results/>

a. Describetheshapeoftheagedistributionoftheseparticipants,includingthenumberofoutliersifappropriate. 1mark

b. Howmanyoftheseparticipantswereaged50yearsorolder? 1mark

c. Writedownthedifferenceinage,inyears,betweentheyoungestandoldestoftheseparticipants. 1mark

d. i. Showthatthefencesfortheboxplotare7.5yearsand51.5years. 1mark

ii. Usethesefencevaluestoexplainwhythe10-year-oldsinthisgroupofparticipantsarenotshownasanoutlierontheboxplot. 1mark


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Question 3 (7marks)Thetimeseriesplotbelowshowsthewinning time,inminutes,forthewomen’s12kmrunfortheperiod2008to2018.

41

40.5

40

39.5

39winning time

(minutes)

38.5

38

37.5

372007 2008 2009 2010 2011 2012 2013

year2014 2015 2016 2017 2018 2019


a. Inwhichoftheseyearswasthewinningtimethelargest? 1mark

b. Usefive-mediansmoothingtosmooththetimeseriesplot.Markeachsmootheddatapointwithacross(×)onthetime series plot above. 2marks

(Answer on the time series plot above.)



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c. Withwinningtimesconvertedtodecimalnumbers(forexample,39minutesand 45seconds=39.75minutes)theequationoftheleastsquareslineis

winning time=340.22–0.14955×year

Thecorrelationcoefficientisr=–0.690

i. Drawthisleastsquareslineonthetimeseriesplotbelow. 1mark

41

40.5

40

39.5

39winning time

(minutes)

38.5

38

37.5

372007 2008 2009 2010 2011 2012 2013

year2014 2015 2016 2017 2018 2019

ii. Byhowmanyminutesdoestheleastsquareslinepredictthatthewinningtimeforthewomen’s12kmrunwilldecreaseeachyear?

Roundyouranswertotwodecimalplaces. 1mark

iii. Writedownthepercentageofthevariationinwinning timethatisnotexplainedbythevariationinyear. 1mark

iv. Thewinningtimeforthewomen’s12kmrunin2014was38.47minutes.

Determinetheresidual,inminutes,whentheleastsquareslineisusedtopredictthewinningtime.



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Question 4 (4marks)Duringtheperiod2008to2018,thedifference,inminutes,betweenthemen’swinningtimeandthewomen’swinningtimeinthe12kmrundecreased,asshowninthetimeseriesplotbelow.Alsoshownonthetimeseriesplotisaleastsquareslinethatcanbeusedtomodelthisdecreasingtrend.

6

5.5

5

4.5difference(minutes)

4

3.5

32007 2008 2009 2010 2011 2012 2013

year2014 2015 2016 2017 2018 2019


Theequationofthisleastsquareslineis

difference=413.749–0.20327 × year

a. Determinethepredicteddifference,inminutes,betweenthemen’swinningtimeandthewomen’swinningtimeintheyear2021.




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b. Theequationofthisleastsquareslinepredictsthat,sometimeinthefuture,thewomen’swinningtimeinthe12kmrunwillbelowerthanthemen’swinningtimeinthe12kmrun.

i. Inwhichyearisthisfirstpredictedtooccur? 1mark

ii. Byhowmanysecondswillthepredictedwomen’swinningtimebelowerthanthemen’spredictedwinningtimeinthisyear.

Roundyouranswertothenearestsecond. 1mark

iii. Theequationofthisleastsquareslinewascalculatedusingdatafortheperiod2008to2018.

Whatassumptionregardingthisleastsquareslineismadewhenthelineisusedtomakepredictionsforyearsafter2018? 1mark


SECTION A – continued

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Recursion and financial modelling

Question 5 (4marks)Darrellhasareducingbalanceloan.FivelinesoftheamortisationtableforDarrell’sloanareshownbelow.

Payment number

Payment ($) Interest ($) Principal reduction ($)

Balance ($)

0 0.00 0.00 0.00 240000.00

1 1500.00 800.00 700.00 239300.00

2 1500.00 797.67 702.33 238597.67

3 1500.00 795.33 704.67 237893.00

4 1500.00 P Q R

a. WhatamountdidDarrelloriginallyborrow? 1mark

InterestiscalculatedmonthlyandDarrellmakesmonthlypayments.

b. Showthattheinterestrateforthisloanis4%perannum. 1mark

c. WritedownthevaluesofP, Q and R, roundedtothenearestcent,intheboxesprovidedbelow. 2marks

P= Q= R=



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Question 6 (3marks)Darrellspent$120000ofthemoneyheborrowedonmachinery.Thevalueofthemachinerywillbedepreciatedusingtheunitcostmethodby$3.50perhourofuse.Therecurrencerelationbelowcanbeusedtomodelthevalueofthemachinery,Vn,afternyears.

V0=120000 Vn+1=Vn–10920

a. Userecursiontoshowthatthevalueofthemachineryaftertwoyearsis$98160. 1mark

b. Themachineryisusedall52weeksoftheyearandforthesamenumberofhourseachweek.

Forhowmanyhourseachweekisthemachineryused? 1mark

c. Therecurrencerelationabovecouldalsomodeltheyear-to-yearvalueofthemachineryusingflatratedepreciation.

Whatannualpercentageflatrateofdepreciationisrepresented? 1mark


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END OF SECTION A

Question 7 (3marks)Darrelltakesoutanewreducingbalanceloanof$220000.Theinterestratefortheloanis4.4% perannum,compoundingfortnightly.Thisloanistoberepaidfortnightlyover10years.

a. Inwhichofthe10yearswillDarrellpaythemostinterest? 1mark

Thescheduledrepaymentsare$1046.62perfortnight.However,Darrellfindsthathecanaffordtopay$1200perfortnightanddecidestodosoforthedurationoftheloan.

b. HowmanyofDarrell’srepaymentswillbeexactly$1200? 1mark

c. Afterfiveyearsofrepayments,Darrellreceivesaninheritanceof$100000andwishestoimmediatelypayofftheremainingbalanceoftheloan.

WillDarrellhaveenoughmoneytopayofftheloaninfull? Justifyyouranswerwitharelevantcalculation. 1mark

Question 8 (2marks)TofundhisretirementDarrellinvests$600000inaperpetuity.Theperpetuityearnsinterestattherateof3.8%perannum.Interestiscalculatedandpaidquarterly.LetVnbethevalueofDarrell’sinvestmentafternquarters.

Writedownarecurrencerelation,intermsofV0 ,Vn+1 and Vn ,thatwouldmodelthevalueofthisinvestmentovertime.


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SECTION B – continuedTURN OVER

SECTION B – Modules

Instructions for Section BSelect twomodulesandanswerallquestionswithintheselectedmodules.Youneednotgivenumericalanswersasdecimalsunlessinstructedtodoso.Alternativeformsmayinclude,forexample,π,surdsorfractions.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

Contents Page

Module1–Matrices......................................................................................................................................14

Module2–Networksanddecisionmathematics.......................................................................................... 20

Module3–Geometryandmeasurement....................................................................................................... 24

Module4–Graphsandrelations...................................................................................................................31


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SECTION B – Module 1 – continued

Module 1 – Matrices

Question 1 (4marks)MatrixN liststhenumberofstudentsenrolledinYear7,Year8andYear9ataschool.

N ��

�

��

�

�

��

240260300

Year 7Year 8Year 9

a. WritedowntheorderofmatrixN. 1mark

Studentsattheseyearlevelscanbeawardedagradeofdistinction(D),credit(C)orpass(P)attheendofthefirstsemester.MatrixE liststheproportionofstudentsawardedeach gradeineachyearlevelforEnglish.

D C PE � � �0 25 0 55 0 20. . .

b. LetthematrixR =N × E.

i. DeterminematrixR. 1mark

ii. Explainwhatthematrixelementr32represents. 1mark

c. Theschoolwantstopresentacertificatetoeachstudentwhoachievesadistinction(D)inEnglishattheendofthefirstsemester.Theprintingcostwillbe$0.25foreachYear7certificate,$0.28foreachYear8certificateand$0.30foreachYear9certificate.

Writedownamatrixcalculationthatdeterminesthetotalprintingcostfordistinction(D)certificates. 1mark


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SECTION B – Module 1 – continued TURN OVER

Question 2 (2marks) TheyearlevelcoordinatorsforYears7to11attheschoolareAmy(A),Brian(B),Cleo(C), David(D)andEllie(E).Afaultytelephonesystemmeansthatsomeofthesecoordinatorscannot directlycallothercoordinators.Thecommunicationmatrix,M,belowshowswhichofthesecoordinatorscandirectlycallanothercoordinator.

receiverA B C D E

M caller

ABCDE

�

�

�

��

0 1 0 1 01 0 1 0 00 0 0 1 10 0 1 0 10 1 1 0 0��

�

�

��

The‘0’inrowD,columnAofmatrixMindicatesthatDavidcannotdirectlycallAmy.The‘1’inrowD,columnCofmatrixMindicatesthatDavidcandirectlycallCleo.

a. WritethenamesofthecoordinatorswhocancallBriandirectly. 1mark

b. CleowantstosendamessagetoAmyusingtheleastnumberofothercoordinators.

Write,inorder,thenamesofthecoordinatorsCleomustusetosendthismessagetoAmy. 1mark


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SECTION B – Module 1 – Question 3 – continued

Question 3 (4marks)StudentsattheschoolwhoarestudyingmathematicsinYears7to10canreceiveoneofthreegradesattheendoftheyear:distinction(D),credit(C)orpass(P).AregulartransitionmatrixTthatrepresentshowstudents’gradeschangefromyeartoyearisgivenbelow.Fourofthevaluesarelistedasj,k,l and m.

this yearD C P

Tl

j km

DCP

next year��

�

��

�

�

��

0 65 0 290 38

0 03 0 09

. ..

. .

a. Explainthemeaningofthevalue0.29inthetransitionmatrixT. 1mark

b. Showthatthevalueofk is0.62 1mark


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c. LetS0bethestatematrixthatshowsthenumberofstudentsachievingD,C or PinYear7GeneralMathematicsattheendof2017.

SDCP

0

8010050

=

AllofthesestudentscompletedYear8in2018,withatotalof82studentsreceiving adistinction.

i. Usingthisinformation,determinethevalueofl intransitionmatrix T. 1mark

ii. StudentsmustachieveadistinctioninGeneralMathematicsattheendofYear10toqualifyforYear11AdvancedMathematics.AllstudentswhocompletedYear7in2017alsocompletedYear10in2020.

HowmanyoftheYear7studentsfrom2017werepredictedtoqualifyforYear11 AdvancedMathematicsin2021?

Roundyouranswertothenearestwholenumber. 1mark


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End of Module 1 – SECTION B – continued

Question 4 (2marks)Ateachersavesexaminationquestionsinafile.Therearetwotypesofquestions:multiplechoice(M)andproblemsolving(P).Eachmonthshechangessomequestionsfrommultiplechoicetoproblemsolvingorfromproblemsolvingtomultiplechoice.Shealsoaddssevennewmultiple-choicequestionsandtwonewproblem-solvingquestionseachmonth.LetQnbethestatematrixthatshowsthenumberofmultiple-choiceandproblem-solvingquestionsinthefileattheendofthenthmonth.LetTbethetransitionmatrixthatrepresentshowthequestiontypesofexistingquestionsareexpectedtochangefrommonthtomonth.LetBbethematrixthatshowsthenumberofnewquestionsofeachtypeaddedtothefileeachmonth.Thematrixrecurrencerelationbelowcanbeusedtopredicttheexpectednumberofquestionsinthefileattheendofaparticularmonth.

Qn+1=TQn + B

where

this monthM P

TMP

next month� ��

��

0 90 0 200 10 0 80. .. .

B � ��

72

ThestatematrixforthenumberofquestionsinthefileattheendofthefourthmonthisQ43814

� ��

��.

Howmanymultiple-choicequestionswereinthefileattheendofthesecondmonth?


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SECTION B – continuedTURN OVER

CONTINUES OVER PAGE


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Module 2 – Networks and decision mathematics

Question 1 (4marks)ThediagrambelowshowsanetworkofroadsfromLeah’shometotheairport.Herhomeandtheairportarelabelledasverticesonthenetwork.TheverticesP, Q, R and Srepresentroadintersections.Thenumberoneachedgerepresentsthedistance,inkilometres,alongthatsectionofroad.

50

20

home15

25

P

10

Q

R 12

35

S 8 airport

drop-offzone

1

a. WhatisthedegreeofvertexP? 1mark

b. Whatistheshortestdistance,inkilometres,betweenLeah’shomeandtheairport? 1mark

c. Whatisthemathematicalnameoftheedgethatformsthedrop-offzone? 1mark


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d. Anincompleteadjacencymatrixforthenetworkonpage20isshownbelow.

Writethemissingfourelementsinthespacesprovidedinthematrix. 1mark

0

0

1

0

1

1

1

0

0

1

0

1

0

1

1

0

1

0

0

1

0

1

0

1

0

1

1

0

0

0

1

0

P Q R Shome

P

Q

R

S

airport

home airport


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Question 2 (4marks)Leahmustbookalast-minuteflightfromMelbourne(M)toLondon(L).Therearenodirectflights.However,TravelsafeAirlineshasflightsavailablefromMelbourne toLondonviaBrisbane(B),HongKong(HK),AbuDhabi(AD),Amsterdam(A)and LosAngeles(LA).Thenetworkbelowshowsthemaximumnumberofseatsstillavailableontheseflights.Acut,labelledCut1,isalsoshownonthenetwork.

B

5

M

22

12

20

AD

HK 8

10

9

A

14 LA Cut 1

12

L

14

27

a. HowmanydifferentflightroutesareavailablefromMelbourne(M)toLondon(L)? 1mark

Whenconsideringthepossibleflowthroughthisnetwork,differentcutscanbemade.

b. WhatisthecapacityofCut1? 1mark

c. WhatisthemaximumnumberofseatsstillavailabletotravelfromMelbourne(M)to London(L)onthisday? 1mark

d. TravelsafeAirlinescanaddeightextraseatstooneofitsflightsinordertoalloweightmorepassengerstotravelfromMelbourne(M)toLondon(L)onthisday.

Nametwocitiesbetweenwhichtheseextraseatscouldnowbeavailable. 1mark


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End of Module 2 – SECTION B – continued TURN OVER

Question 3 (4marks)TravelsafeAirlinesisplanningtorenovateapassengerterminal.Thisprojectwillinvolve12activities:A to L.Thedirectednetworkbelowshowstheseactivitiesandtheircompletiontimes,inweeks.Thecompletiontimeforactivity Jislabelledx.

E, 3

B, 4

A, 5

L, 3

D, 3

F, 7

G, 8C, 6

I, 7

K, 1

H, 6

J, x

finish

start

Theminimumcompletiontimeforthisprojectis20weeks.

a. Whatisthevalueofx? 1mark

b. Completethefollowingsentencebyfillingintheboxesprovided. 1mark

Activity hasthelongestfloattimeof weeks.

c. Howmanyactivitiescouldhavetheircompletiontimeincreasedbytwoweekswithoutalteringtheminimumcompletiontime? 1mark

d. TravelsafeAirlineshasemployedmorestafftoworkontherenovationproject. Thishasensuredthatnoactivityonthenetworkabovewillhaveanincreaseincompletion

time.However,thecompletiontimesofactivitiesG and H areeachreducedbytwoweeks.

Howwillthisaffecttheoverallcompletiontimeoftheproject? 1mark


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Module 3 – Geometry and measurement

Question 1 (5marks)Shonaisinchargeofdecorationsforanevent.Shewantstohangthedecorationsfromtheceiling.Theceilingistriangularinshape,asshowninthediagrambelow.Allthreesidesoftheceilingare23mlong.

23 m

a. Whatistheperimeter,inmetres,oftheceiling? 1mark

ThedecorationsaretobehungfromabeamABthatrunsacrossthecentreoftheceiling,asshowninthediagrambelow.

23 m

beam

A

B

b. WriteacalculationthatshowsthatthelengthofthisbeamAB,roundedtoonedecimalplace,is19.9m. 1mark


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SECTION B – Module 3 – Question 1 – continued TURN OVER

c. Whatisthearea,insquaremetres,oftheceiling? Roundyouranswertothenearestwholenumber. 1mark

Shonawantstohangspheresfromthebeam.Eachspherehasaradiusof18cm,asshowninthediagrambelow.

18 cm

d. Whatisthevolume,incubiccentimetres,ofonesphere? Roundyouranswertothenearestwholenumber. 1mark


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e. Eachspherewillcontainalightglobe. Thelightglobewillbesuspendedfromthebeambyacable. Thelowerendofthecablemusthangabovethecentreofthesphereandbe12cmfromeach

sideofthesphere,asshowninthediagrambelow. Thetopofthespheremustbe15cmfromthebeam.

12 cm12 cm

cable

beam

15 cm

Whatisthelengthofthecable,incentimetres,thatwillberequiredfromthebeamtothelightglobe?



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SECTION B – Module 3 – Question 2 – continued TURN OVER

Question 2 (4marks)Shonawillplacecylindricalbowlsoneachtableattheeventasacentrepiece.Eachcylindricalbowlhasaradiusof6cm,asshowninthediagrambelow.

h

6 cm

Eachbowlhasavolumeof1244cm3.

a. Writeacalculationthatshowsthattheheight,h,ofonecylindricalbowl,roundedtothenearestwholenumber,is11cm. 1mark

b. Acandle,alsointheshapeofacylinder,istobeplaceduprightinsideeachbowlsothatittouchesthebaseofthebowl.

Thecandlehasaradiusof3cmandaheightof18cm. Oncethecandlehasbeenplacedinsidethebowl,theremainingvolumeofthebowlwillbe

filledwithsand.

Whatvolumeofsand,incubiccentimetres,isrequiredtofillthecylindricalbowloncethecandleisplacedinsideit?



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c. Eachbowlistobeplacedonacircularplate. Belowisadiagramofabowlontopoftheplate,asseenfromabove. Theareaoftheplatethatisnotcoveredbythebowlisshaded.

bowl

Theratiooftheareaofthebaseofthebowltotheshadedareais4:5

Whatisthearea,insquarecentimetres,ofoneplate? Roundyouranswertoonedecimalplace. 2marks


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End of Module 3 – SECTION B – continued TURN OVER

Question 3 (3marks)IshawillflyfromBrisbanetoAdelaidefortheevent.ShecantakeadirectflightfromBrisbanetoAdelaide.Brisbaneis918kmnorthand1310kmeastofAdelaide.

a. Show,withcalculations,thatthebearingofAdelaidefromBrisbane,roundedtothenearestdegree,is235°. 1mark

b. IshacouldalsotakeaflightfromBrisbanetoAdelaideviaMelbourne,asshownonthediagrambelow.

Brisbane

Melbourne

Adelaide

north

Melbourneis1370kmfromBrisbaneonabearingof211°.

Whatisthedistance,inkilometres,betweenMelbourneandAdelaide? Roundyouranswertothenearestwholenumber. 2marks


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SECTION B – continued

THIS PAGE IS BLANK


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Module 4 – Graphs and relations

Question 1 (4marks)Inanannualevent,athletescompeteagainstavintagetrainina15kmrace.Aroadrunsalongsidearailwaytrackfortheentirelengthoftherace.Therearethreerailcrossings,atpointsA,B and C.Graph1belowshowstheheight above sea level,inmetres,alongthisroadandtherailwaytrack.

Graph 1

start

A

B

Cfinish

180175170165160155150145140135130125120115110

height abovesea level

(m)

0 1 2 3 4 5 6 7distance from the start (km)

8 9 10 11 12 13 14 15

a. Whatisthedifferenceinheight,inmetres,betweenthestartingpointoftheraceandrailcrossingB? 1mark

b. Betweenthestartoftheraceandthefinish,whatlengthofroad,inkilometres,isdownhill? 1mark

c. ConsiderthesectionofroadbetweenrailcrossingsA and B.

i. Writeacalculationthatshowsthattheaverageslopeofthissectionofroadis12.5 1mark

ii. Whatistheunitinwhichthisaverageslopeof12.5ismeasured? 1mark


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Question 2 (5marks)Onraceday,Jasminestartedtheraceatthesametimeasthetrain.ThegraphshowingthetimetakenbyJasminetocompletetheraceisshownonGraph2below.

Graph 2

0

10

20

30

time(minutes)

40

50

60

70

0 1 2 3 4 5 6 7 8distance (km)

9 10 11 12 13 14 15

ThetrainrunsataconstantspeedbetweenthestartandpointB(pointBisonGraph1onpage31)andthenalsobetweenpointBandthefinish.Thetablebelowshowsthetimesatwhichthetrainisatthestart,pointB andthefinishoftherace.

Location Distance travelled (km) Time (minutes)

start 0 0

pointB 5 35

finish 15 60

a. OnGraph 2 above, addthegraphthatshowsthetrain’stimeinthisrace. 1mark

(Answer on Graph 2 above.)


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b. Thetimetakenbythetrain,tminutes,totraveldkilometresisgivenby

td

ad bdd

��

� ��

��

7 0 55 15

i. Usingonlythenumbersgiveninthetableonpage32,writeacalculationthatshowsthat a=2.5 1mark

ii. Usinga=2.5andthenumbersgiveninthetableonpage32,writeacalculationthatshowsthatb=22.5 1mark

TheequationforJasmine’sexpectedracecompletiontimeis

tdd

dd

��

��

� ��

54 5

0 55 15

c. Inkilometresperhour,whatisJasmine’sspeedforthefirst5kmoftherace? 1mark

d. HowlongafterthestartoftheracedoesittakeforthetraintocatchuptoJasmine? Writeyouranswerinminutesandseconds. 1mark


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Question 3 (3marks)Theorganisingcommitteemustbudgetfortheracesothatitdoesnotmakealoss.Letxbethecostofrunningthetrainonraceday.Letybethecostofotherracedayoperations.Thegraphbelowshowstheshadedfeasibleregionforinequalitiesthatconstrainthevaluesof x and y.Thefeasibleregionincludesthefiveboundariesshown.Fourofthefiveinequalitiesthatdefinethefeasibleregionare:• Inequality1 y≥4x• Inequality2 y≤5x• Inequality3 x≥8000• Inequality4 x + y≥45000

J K

LMN

y

xO

Thecoordinatesofthefivepointsshownonthegrapharegiveninthetablebelow.

J K L M N

(8000,40000) (10000,40000) (10000,35000) (8000,32000) (7500,37500)

a. Inequality5completesthedefinitionofthefeasibleregion.

WritedownInequality5. 1mark

Anadditional5%ofthecostofotherracedayoperations(y)isaddedasacostforpromotions.Thetotalcost,$C,oforganisingandrunningtheraceisgivenbyC=x+1.05y.

b. Determinetheminimumtotalcost. 1mark


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END OF QUESTION AND ANSWER BOOK

c. Theraceorganisingcommitteehadarevenueof$75000fromsponsorshipsandentryfeesfortherace.

Allcostswillbepaidfromthisrevenue.

Determinetheminimumprofitthattheorganisingcommitteecanmakefromthisrace. 1mark

FURTHER MATHEMATICS

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 CURRICULUM AND ASSESSMENT AUTHORITY 2021

Victorian Certificate of Education 2021

FURMATH EXAM 2

Further Mathematics formulas

Core – Data analysis

standardised score z x xsx

=−

lower and upper fence in a boxplot lower Q1 – 1.5 × IQR upper Q3 + 1.5 × IQR

least squares line of best fit y = a + bx, where b rss

y

x= and a y bx= −

residual value residual value = actual value – predicted value

seasonal index seasonal index = actual figuredeseasonalised figure

Core – Recursion and financial modelling

first-order linear recurrence relation u0 = a, un + 1 = bun + c

effective rate of interest for a compound interest loan or investment

r rneffective

n= +

−

×1

1001 100%

Module 1 – Matrices

determinant of a 2 × 2 matrix A a bc d=

, det A

ac

bd ad bc= = −

inverse of a 2 × 2 matrix AA

d bc a

− =−

−

1 1det

, where det A ≠ 0

recurrence relation S0 = initial state, Sn + 1 = T Sn + B

Module 2 – Networks and decision mathematics

Euler’s formula v + f = e + 2

3 FURMATH EXAM

END OF FORMULA SHEET

Module 3 – Geometry and measurement

area of a triangle A bc=12

sin ( )θ

Heron’s formula A s s a s b s c= − − −( )( )( ), where s a b c= + +12

( )

sine rulea

Ab

Bc

Csin ( ) sin ( ) sin ( )= =

cosine rule a2 = b2 + c2 – 2bc cos (A)

circumference of a circle 2π r

length of an arc r × × °π

θ180

area of a circle π r2

area of a sector πθr2

360×

°

volume of a sphere43π r 3

surface area of a sphere 4π r2

volume of a cone13π r 2h

volume of a prism area of base × height

volume of a pyramid13

× area of base × height

Module 4 – Graphs and relations

gradient (slope) of a straight line m y y

x x=

−−

2 1

2 1

equation of a straight line y = mx + c

Documents

2021 Further Mathematics-nht Written examination 2