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
2021FURMATHEXAM2(NHT) 4
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
5 2021FURMATHEXAM2(NHT)
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
2021FURMATHEXAM2(NHT) 6
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SECTION A – Question 3 – continued
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
Data:City-BayFunRun,<https://city-bay.org.au/results/>
a. Inwhichoftheseyearswasthewinningtimethelargest? 1mark
b. Usefive-mediansmoothingtosmooththetimeseriesplot.Markeachsmootheddatapointwithacross(×)onthetime series plot above. 2marks
(Answer on the time series plot above.)
7 2021FURMATHEXAM2(NHT)
SECTION A – continuedTURN OVER
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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.
Roundyouranswertotwodecimalplaces. 1mark
2021FURMATHEXAM2(NHT) 8
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SECTION A – Question 4 – continued
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
Data:City-BayFunRun,<https://city-bay.org.au/results/>
Theequationofthisleastsquareslineis
difference=413.749–0.20327 × year
a. Determinethepredicteddifference,inminutes,betweenthemen’swinningtimeandthewomen’swinningtimeintheyear2021.
Roundyouranswertotwodecimalplaces. 1mark
9 2021FURMATHEXAM2(NHT)
SECTION A – continuedTURN OVER
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b. Theequationofthisleastsquareslinepredictsthat,sometimeinthefuture,thewomen’swinningtimeinthe12kmrunwillbelowerthanthemen’swinningtimeinthe12kmrun.
i. Inwhichyearisthisfirstpredictedtooccur? 1mark
ii. Byhowmanysecondswillthepredictedwomen’swinningtimebelowerthanthemen’spredictedwinningtimeinthisyear.
Roundyouranswertothenearestsecond. 1mark
iii. Theequationofthisleastsquareslinewascalculatedusingdatafortheperiod2008to2018.
Whatassumptionregardingthisleastsquareslineismadewhenthelineisusedtomakepredictionsforyearsafter2018? 1mark
2021FURMATHEXAM2(NHT) 10
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=
11 2021FURMATHEXAM2(NHT)
SECTION A – continuedTURN OVER
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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
2021FURMATHEXAM2(NHT) 12
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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.
13 2021FURMATHEXAM2(NHT)
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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
2021FURMATHEXAM2(NHT) 14
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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
15 2021FURMATHEXAM2(NHT)
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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
2021FURMATHEXAM2(NHT) 16
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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
17 2021FURMATHEXAM2(NHT)
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SECTION B – Module 1 – continued TURN OVER
c. LetS0bethestatematrixthatshowsthenumberofstudentsachievingD,C or PinYear7GeneralMathematicsattheendof2017.
SDCP
0
8010050
=
AllofthesestudentscompletedYear8in2018,withatotalof82studentsreceiving adistinction.
i. Usingthisinformation,determinethevalueofl intransitionmatrix T. 1mark
ii. StudentsmustachieveadistinctioninGeneralMathematicsattheendofYear10toqualifyforYear11AdvancedMathematics.AllstudentswhocompletedYear7in2017alsocompletedYear10in2020.
HowmanyoftheYear7studentsfrom2017werepredictedtoqualifyforYear11 AdvancedMathematicsin2021?
Roundyouranswertothenearestwholenumber. 1mark
2021FURMATHEXAM2(NHT) 18
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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?
19 2021FURMATHEXAM2(NHT)
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SECTION B – continuedTURN OVER
CONTINUES OVER PAGE
2021FURMATHEXAM2(NHT) 20
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SECTION B – Module 2 – Question 1 – continued
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
21 2021FURMATHEXAM2(NHT)
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SECTION B – Module 2 – continued TURN OVER
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
2021FURMATHEXAM2(NHT) 22
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SECTION B – Module 2 – continued
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
23 2021FURMATHEXAM2(NHT)
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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
2021FURMATHEXAM2(NHT) 24
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SECTION B – Module 3 – Question 1 – continued
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
25 2021FURMATHEXAM2(NHT)
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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
2021FURMATHEXAM2(NHT) 26
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SECTION B – Module 3 – continued
e. Eachspherewillcontainalightglobe. Thelightglobewillbesuspendedfromthebeambyacable. Thelowerendofthecablemusthangabovethecentreofthesphereandbe12cmfromeach
sideofthesphere,asshowninthediagrambelow. Thetopofthespheremustbe15cmfromthebeam.
12 cm12 cm
cable
beam
15 cm
Whatisthelengthofthecable,incentimetres,thatwillberequiredfromthebeamtothelightglobe?
Roundyouranswertothenearestwholenumber. 1mark
27 2021FURMATHEXAM2(NHT)
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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?
Roundyouranswertothenearestwholenumber. 1mark
2021FURMATHEXAM2(NHT) 28
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SECTION B – Module 3 – continued
c. Eachbowlistobeplacedonacircularplate. Belowisadiagramofabowlontopoftheplate,asseenfromabove. Theareaoftheplatethatisnotcoveredbythebowlisshaded.
bowl
Theratiooftheareaofthebaseofthebowltotheshadedareais4:5
Whatisthearea,insquarecentimetres,ofoneplate? Roundyouranswertoonedecimalplace. 2marks
29 2021FURMATHEXAM2(NHT)
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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
2021FURMATHEXAM2(NHT) 30
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SECTION B – continued
THIS PAGE IS BLANK
31 2021FURMATHEXAM2(NHT)
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SECTION B – Module 4 – continued TURN OVER
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
2021FURMATHEXAM2(NHT) 32
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SECTION B – Module 4 – Question 2 – continued
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.)
33 2021FURMATHEXAM2(NHT)
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SECTION B – Module 4 – continued TURN OVER
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
2021FURMATHEXAM2(NHT) 34
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SECTION B – Module 4 – Question 3 – continued
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
