2019 further mathematics-nht written examination 2 · writing time: 2.15 pm to 3.45 pm (1 hour 30...
TRANSCRIPT
FURTHER MATHEMATICSWritten examination 2
Thursday 30 May 2019 Reading time: 2.00 pm to 2.15 pm (15 minutes) Writing time: 2.15 pm to 3.45 pm (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
• Studentsaretowriteinblueorblackpen.• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,
sharpeners,rulers,oneboundreference,oneapprovedtechnology(calculatororsoftware)and,ifdesired,onescientificcalculator.CalculatormemoryDOESNOTneedtobecleared.Forapprovedcomputer-basedCAS,fullfunctionalitymaybeused.
• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.
Materials supplied• Questionandanswerbookof34pages• 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.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2019
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2019
STUDENT NUMBER
Letter
2019FURMATHEXAM2(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 (4marks)Thetablebelowdisplaystheaverage sleep time,inhours,forasampleof19typesofmammals.
Type of mammal Average sleep time (hours)
cat 14.5
squirrel 13.8
mouse 13.2
rat 13.2
greywolf 13.0
arcticfox 12.5
raccoon 12.5
gorilla 12.0
jaguar 10.8
baboon 9.8
redfox 9.8
rabbit 8.4
guineapig 8.2
greyseal 6.2
cow 3.9
sheep 3.8
donkey 3.1
horse 2.9
roedeer 2.6
Data:TAllisonandDVCicchetti, ‘SleepinMammals:EcologicalandConstitutionalCorrelates’,
in Science,AmericanAssociationfortheAdvancementof Science,vol.194,no.4266,pp.732–734,12November1976;
accessedfromOzDASL,StatSci.org, <www.statsci.org/data/general/sleep.html>
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SECTION A – continuedTURN OVER
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a. Which of the two variables, type of mammal or average sleep time, is a nominal variable? 1 mark
b. Determine the mean and standard deviation of the variable average sleep time for this sample of mammals.
Write your answers in the boxes provided below. Round your answers to one decimal place. 1 mark
mean = hours standard deviation = hours
c. The average sleep time for a human is eight hours.
What percentage of this sample of mammals has an average sleep time that is less than the average sleep time for a human?
Round your answer to one decimal place. 1 mark
d. The sample is increased in size by adding in the average sleep time of the little brown bat. Its average sleep time is 19.9 hours.
By how many hours will the range for average sleep time increase when the average sleep time for the little brown bat is added to the sample? 1 mark
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SECTION A – continued
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Question 2 (3 marks)The five-number summary below was determined from the sleep time, in hours, of a sample of 59 types of mammals.
Statistic Sleep time (hours)
minimum 2.5
first quartile 8.0
median 10.5
third quartile 13.5
maximum 20.0
a. Show, with calculations, that a boxplot constructed from this five-number summary will not include outliers. 2 marks
b. Construct the boxplot below. 1 mark
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22sleep time (hours)
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SECTION A – continuedTURN OVER
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Question 3 (4marks)Thelife span,inyears,andgestation period,indays,for19typesofmammalsaredisplayedinthetablebelow.
Life span (years) Gestation period (days) 3.20 19 4.70 21 7.60 68 9.00 28 9.80 5213.7 6314.0 6016.2 6317.0 15018.0 3120.0 15122.4 10027.0 18028.0 6330.0 28139.3 25240.0 36541.0 31046.0 336
a. Aleastsquareslinethatenableslife span tobepredictedfromgestation periodisfittedtothisdata.
Nametheexplanatoryvariableintheequationofthisleastsquaresline. 1mark
b. Determinetheequationoftheleastsquareslineintermsofthevariableslife span and gestation period.
Writeyouranswersintheappropriateboxesprovidedbelow. Roundthenumbersrepresentingtheinterceptandslopetothreesignificantfigures. 2marks
= + ×
c. Writethevalueofthecorrelationcoefficientroundedtothreedecimalplaces. 1mark
r =
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SECTION A – Question 4 – continued
Question 4 (8marks)Thescatterplotbelowplotsthevariablelife span,inyears,againstthevariable sleep time,in hours, forasampleof19typesofmammals.
sleep time (hours)
20
25
30
35
40
45
50
15
10
5
010 12 14 16 1886420
life span(years)
Ontheassumptionthattheassociationbetweensleep time and life spanislinear,aleastsquareslineisfittedtothisdatawith sleep timeastheexplanatoryvariable.Theequationofthisleastsquareslineis
life span =42.1–1.90×sleep time
Thecoefficientofdeterminationis0.416
a. Drawthegraphoftheleastsquareslineonthescatterplot above. 1mark
(Answer on the scatterplot above.)
b. Describethelinearassociationbetweenlife span and sleep timeintermsofstrengthanddirection. 2marks
c. Interprettheslopeoftheleastsquareslineintermsoflife span and sleep time. 2marks
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SECTION A – continuedTURN OVER
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d. Interpretthecoefficientofdeterminationintermsoflife span and sleep time. 1mark
e. Thelifespanofthemammalwithasleeptimeof12hoursis39.2years.
Showthat,whentheleastsquareslineisusedtopredictthelifespanofthismammal,theresidualis19.9years. 2marks
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SECTION A – Question 5 – continued
Question 5 (5marks)Arandomsampleof12mammalsdrawnfromapopulationof62typesofmammalswascategorisedaccordingtotwovariables.
likelihood of attack(1=low,2=medium,3=high)exposure to attackduringsleep(1=low,2=medium,3=high)
Thedataisshowninthefollowingtable.
Likelihood of attack 2 2 1 3 2 3 1 3 1 1 3 3
Exposure to attack 3 1 1 1 3 3 1 3 1 1 3 3
a. Usethisdatatocompletethetwo-wayfrequencytablebelow. 1mark
Likelihood of attack Exposure to attack during sleep
low(=1) medium(=2) high(=3)
low(=1) 0 0
medium(=2) 0
high(=3) 0
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SECTION A – continuedTURN OVER
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The following two-way frequency table was formed from the data generated when the entire population of 62 types of mammals was similarly categorised.
Likelihood of attack Exposure to attack during sleep
low medium high
low 31 8 2
medium 2 0 2
high 1 1 15
b. i. How many of these 62 mammals had both a high likelihood of attack and a high exposure to attack during sleep? 1 mark
ii. Of those mammals that had a medium likelihood of attack, what percentage also had a low exposure to attack during sleep? 1 mark
iii. Does the information in the table above support the contention that likelihood of attack is associated with exposure to attack during sleep? Justify your answer by quoting appropriate percentages. It is sufficient to consider only one category of likelihood of attack when justifying your answer. 2 marks
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SECTION A – continued
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Recursion and financial modelling
Question 6 (5marks)Marlonplaysguitarinaband.Hepaid$3264foranewguitar.ThevalueofMarlon’sguitarwillbedepreciatedbyafixedamountforeachconcertthatheplays.After25concerts,thevalueoftheguitarwillhavedecreasedby$200.
a. WhatwillbethevalueofMarlon’sguitarafter25concerts? 1mark
b. WriteacalculationthatshowsthatthevalueofMarlon’sguitarwilldepreciateby $8perconcert. 1mark
c. ThevalueofMarlon’sguitarafternconcerts,Gn,canbedeterminedusingarule.
Completetherulebelowbywritingtheappropriatenumbersintheboxesprovided. 1mark
Gn= − × n
d. Thevalueoftheguitarcontinuestobedepreciatedby$8perconcert.
AfterhowmanyconcertswillthevalueofMarlon’sguitarfirstfallbelow$2500? 2marks
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SECTION A – continuedTURN OVER
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Question 7 (4marks)TishaplaysdrumsinthesamebandasMarlon.Shewouldliketobuyanewdrumkitandhassaved$2500.
a. Tishacouldinvestthismoneyinanaccountthatpaysinterestcompoundingmonthly. Thebalanceofthisinvestmentafternmonths,Tn ,couldbedeterminedusingtherecurrence
relationbelow.
T0 =2500, Tn +1=1.0036×Tn
CalculatethetotalinterestthatwouldbeearnedbyTisha’sinvestmentinthefirstfivemonths. Roundyouranswertothenearestcent. 2marks
Tishacouldinvestthe$2500inadifferentaccountthatpaysinterestattherateof4.08% perannum,compoundingmonthly.Shewouldmakeapaymentof$150intothisaccounteverymonth.
b. LetVnbethevalueofTisha’sinvestmentafternmonths.
Writedownarecurrencerelation,intermsofV0,Vn and Vn+1,thatwouldmodelthechangeinthevalueofthisinvestment. 1mark
c. Tishawouldliketohaveabalanceof$4500,tothenearestdollar,after12months.
WhatannualinterestratewouldTisharequire? Roundyouranswertotwodecimalplaces. 1mark
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END OF SECTION A
Question 8 (3marks)Arecordproducergavetheband$50000towriteandrecordanalbumofsongs.This$50000wasinvestedinanannuitythatprovidesamonthlypaymenttotheband.Theannuitypaysinterestattherateof3.12%perannum,compoundingmonthly.Aftersixmonthsofwritingandrecording,thebandhas$32667.68remainingintheannuity.
a. Whatisthevalue,indollars,ofthemonthlypaymenttotheband? 1mark
b. Aftersixmonthsofwritingandrecording,thebanddecidedthatitneedsmoretimetofinishthealbum.
Toextendthetimethattheannuitywilllast,thebandwillworkforthreemoremonthswithoutwithdrawingapayment.
Afterthis,thebandwillreceivemonthlypaymentsof$3800foraslongaspossible. Theannuitywillendwithonefinalmonthlypaymentthatwillbesmallerthanallofthe
others.
Calculatethetotalnumberofmonthsthatthisannuitywilllast. 2marks
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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..........................................................................................19
Module3–Geometryandmeasurement....................................................................................................... 24
Module4–Graphsandrelations................................................................................................................... 30
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SECTION B – Module 1 – Question 1 – continued
Module 1 – Matrices
Question 1 (5marks)AtotalofsixresidentsfromtwotownswillbecompetingattheInternationalGames.MatrixA,shownbelow,containsthenumberofmale(M)andthenumberoffemale(F)athletescompetingfromthetownsofGillen(G)andHaldaw(H).
M F
AGH
=2 21 1
a. HowmanyoftheseathletesareresidentsofHaldaw? 1mark
Eachofthesixathleteswillcompeteinoneevent:tabletennis,runningorbasketball.Matrices T and R,shownbelow,containthenumberofmaleandfemaleathletesfromeachtownwhowillcompeteintabletennisandrunningrespectively.
Table tennis Running
M F
TGH
=0 11 0
M F
RGH
=1 10 0
b. MatrixBcontainsthenumberofmaleandfemaleathletesfromeachtownwhowillcompeteinbasketball.
CompletematrixBbelow. 1mark
BG
H=
M F
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SECTION B – Module 1 – continued TURN OVER
MatrixCcontainsthecostofoneuniform,indollars,foreachofthethreeevents:tabletennis(T),running(R)andbasketball(B).
CTRB
=515550580
c. i. Forwhicheventwillthetotalcostofuniformsfortheathletesbe$1030? 1mark
ii. Writeamatrixcalculation,thatincludesmatrixC,toshowthatthetotalcostofuniformsfortheeventnamedinpart c.i.iscontainedinthematrixanswerof[1030]. 1mark
d. MatrixVandmatrixQaretwonewmatriceswhereV = Q × C and:• matrixQ isa4×3matrix• element v11 =totalcostofuniformsforallfemaleathletesfromGillen• element v21 =totalcostofuniformsforallfemaleathletesfromHaldaw• element v31 =totalcostofuniformsforallmaleathletesfromGillen• element v41 =totalcostofuniformsforallmaleathletesfromHaldaw
• CTRB
=515550580
CompletematrixQwiththemissingvalues. 1mark
Q =
10 0 10 1 1
0
2019 FURMATH EXAM 2 (NHT) 16
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SECTION B – Module 1 – continued
Question 2 (2 marks)Three television channels, C1, C2 and C3, will broadcast the International Games in Gillen.Gillen’s 2000 residents are expected to change television channels from hour to hour as shown in the transition matrix T below. The option for residents not to watch television (NoTV) at that time is also indicated in the transition matrix.
this hourC1 C2 C3 NoTV
T =
0.50 0.05 0.10 0.200.10 0.60 0.20 0.200.255 0.10 0.50 0.100.15 0.25 0.20 0.50
C1
C2
C3
NoTV
next hour
The state matrix G0 below lists the number of Gillen residents who are expected to watch the games on each of the channels at the start of a particular day (9.00 am).Also shown is the number of Gillen residents who are not expected to watch television at that time.
G
CCCNoTV
0 =
1004001001400
1
2
3
a. Complete the calculation below to show that 835 Gillen residents are not expected to watch television (NoTV) at 10.00 am that day. 1 mark
× 100 + × 400 + × 100 + × 1400 = 835
b. Determine the number of residents expected to watch the games on C3 at 11.00 am that day. 1 mark
17 2019FURMATHEXAM2(NHT)
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SECTION B – Module 1 – continued TURN OVER
Question 3 (2marks)ThebasketballfinalswillbetelevisedonC3from12.00noonuntil4.00pm.Itisexpectedthat600GillenresidentswillbewatchingC3atanytimefrom12.00noonuntil4.00pm.Theremaining1400GillenresidentswillnotbewatchingC3from12.00noonuntil4.00pm(represented byNotC3).ThetransitionmatrixP belowshowshowthe2000GillenresidentsareexpectedtochangetheirviewinghabitseachhourbetweenwatchingC3 andnotwatchingC3 from12.00noonuntil4.00pm.
this hourC NotC
Pv w
xnext hour
3 3
=0.35
C3
NotC3
Writedownthevaluesofv,w and xintheboxesprovidedbelow.
v= w= x=
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End of Module 1 – SECTION B – continued
Question 4 (3marks)After5.00pm,touristswillstarttoarriveinGillenandtheywillstayovernight.Asaresult,thenumberofpeopleinGillenwillincreaseandthetelevisionviewinghabitsofthetouristswillalsobemonitored.Assumethat50touristsarriveeveryhour.Itisexpectedthat80%ofarrivingtouristswillwatchonlyC2duringthehourthattheyarrive.Theremaining20%ofarrivingtouristswillnotwatchtelevisionduringthehourthattheyarrive.LetWmbethestatematrixthatshowsthenumberofpeopleineachcategorymhoursafter5.00pmonthisday.TherecurrencerelationthatmodelsthechangeinthetelevisionviewinghabitsofthisincreasingnumberofpeopleinGillenmhoursafter5.00pmonthisdayisshownbelow.
Wm+1 = T Wm + V
where
this hourC C C NoTV
T
1 2 30 50 0 05 0 10 0 200 10 0 60 0 20 0 200 25 0
=
. . . .
. . . .
. .110 0 50 0 100 15 0 25 0 20 0 50
1
2
3. .. . . .
CCCNoTV
next hour, and W0
400600300700
=
a. WritedownmatrixV. 1mark
b. HowmanypeopleinGillenareexpectedtowatchC2at7.00pmonthisday? 2marks
19 2019FURMATHEXAM2(NHT)
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SECTION B – Module 2 – Question 1 – continued TURN OVER
Module 2 – Networks and decision mathematics
Question 1 (5marks)Azoohasanentrance,acafeandnineanimalexhibits:bears(B),elephants(E),giraffes(G), lions(L),monkeys(M),penguins(P),seals(S),tigers(T)andzebras(Z).Theedgesonthegraphbelowrepresentthepathsbetweentheentrance,thecafeandtheanimalexhibits.Thenumbersoneachedgerepresentthelength,inmetres,alongthatpath.Visitorstothezoocanuseonlythesepathstotravelaroundthezoo.
entrance cafe
P
Z
SB E
G
L
T
M
2520
30
40
40
40
2525
35
35
5515
4550
30
60
a. Whatistheshortestdistance,inmetres,betweentheentranceandthesealexhibit(S)? 1mark
b. Freddyisavisitortothezoo.Hewishestovisitthecafeandeachanimalexhibitjustonce,startingandendingattheentrance.
i. Whatisthemathematicaltermusedtodescribethisroute? 1mark
ii. DrawonepossibleroutethatFreddymaytakeonthegraphbelow. 1mark
entrance cafe
P
Z
SB E
G
L
T
M
2520
30
40
40
40
2525
35
35
5515
4550
30
60
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SECTION B – Module 2 – continued
Areptileexhibit(R) willbeaddedtothezoo.Anewpathoflength20mwillbebuiltbetweenthereptileexhibit(R) andthegiraffeexhibit(G).Asecondnewpath,oflength35m,willbebuiltbetweenthereptileexhibit(R) andthecafe.
c. Completethegraphbelowwiththenewreptileexhibitandthetwonewpathsadded.LabelthenewvertexRandwritethedistancesonthenewedges. 1mark
entrance cafe
P
Z
SB E
G
L
T
M
2520
30
40
40
40
2525
35
35
5515
4550
30
60
d. Thenewpathsreducetheminimumdistancethatvisitorshavetowalkbetweenthegiraffeexhibit(G)andthecafe.
Byhowmanymetreswillthesenewpathsreducetheminimumdistancebetweenthegiraffeexhibit(G)andthecafe? 1mark
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SECTION B – Module 2 – continued TURN OVER
Question 2 (3marks)Theconstructionofthenewreptileexhibitisaprojectinvolvingnineactivities,A to I.Thedirectednetworkbelowshowstheseactivitiesandtheircompletiontimesinweeks.
start finish
A, 3
E, 7
F, 3C, 4
D, 6
B, 5 H, 7
G, 5I, 4
a. Whichactivitieshavemorethanoneimmediatepredecessor? 1mark
b. Writedownthecriticalpathforthisproject. 1mark
c. Whatisthelateststarttime,inweeks,foractivityB? 1mark
2019 FURMATH EXAM 2 (NHT) 22
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SECTION B – Module 2 – Question 3 – continued
Question 3 (4 marks)The zoo’s management requests quotes for parts of the new building works.Four businesses each submit quotes for four different tasks. Each business will be offered only one task.The quoted cost, in $100 000, of providing the work is shown in Table 1 below.
Table 1
Task 1Constructing the pathways
Task 2Constructing
the new reptile exhibit
Task 3Heating and lighting the new exhibit
Task 4Landscaping
the surrounding grounds
Business 1 12 12 5 7
Business 2 10 11 4 7
Business 3 12 8 8 6
Business 4 13 11 9 8
The zoo’s management wants to complete the new building works at minimum cost.The Hungarian algorithm is used to determine the allocation of tasks to businesses.The first step of the Hungarian algorithm involves row reduction; that is, subtracting the smallest element in each row of Table 1 from each of the elements in that row. The result of the first step is shown in Table 2 below.
Table 2
Task 1Constructing the pathways
Task 2Constructing
the new reptile exhibit
Task 3Heating and lighting the new exhibit
Task 4Landscaping
the surrounding grounds
Business 1 7 7 0 2
Business 2 6 7 0 3
Business 3 6 2 2 0
Business 4 5 3 1 0
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End of Module 2 – SECTION B – continued TURN OVER
The second step of the Hungarian algorithm involves column reduction; that is, subtracting the smallest element in each column of Table 2 from each of the elements in that column.The results of the second step of the Hungarian algorithm are shown in Table 3 below. The values of Task 1 are given as A, B, C and D.
Table 3
Task 1Constructing the pathways
Task 2Constructing
the new reptile exhibit
Task 3Heating and lighting the new exhibit
Task 4Landscaping
the surrounding grounds
Business 1 A 5 0 2
Business 2 B 5 0 3
Business 3 C 0 2 0
Business 4 D 1 1 0
a. Write down the values of A, B, C and D. 1 mark
A = B = C = D =
b. The next step of the Hungarian algorithm involves covering all the zero elements with horizontal or vertical lines. The minimum number of lines required to cover the zeros is three.
Draw these three lines on Table 3 above. 1 mark
(Answer on Table 3 above.)
c. An allocation for minimum cost is not yet possible. When all steps of the Hungarian algorithm are complete, a bipartite graph can show the
allocation for minimum cost.
Complete the bipartite graph below to show this allocation for minimum cost. 1 mark
Business 1 • • Task 1
Business 2 • • Task 2
Business 3 • • Task 3
Business 4 • • Task 4
d. Business 4 has changed its quote for the construction of the pathways. The new cost is $1 000 000. The overall minimum cost of the building works is now reduced by reallocating the tasks.
How much is this reduction? 1 mark
2019 FURMATH EXAM 2 (NHT) 24
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SECTION B – Module 3 – Question 1 – continued
Module 3 – Geometry and measurement
Question 1 (3 marks)An engineering conference is being held in Rome. Rome is located at latitude 42° N and longitude 12° E.Four presenters from different cities will travel to Rome to attend the conference. The presenters and the coordinates (latitude and longitude) of the city each presenter lives in are shown below.
Presenter Coordinates
Yvette (3° S, 12° E)
Anthony (17° S, 12° E)
Rany (54° N, 12° E)
Shana (65° N, 12° E)
a. Which presenter lives closest to Rome? 1 mark
Rome has a latitude of 42° N.The diagram below shows the small circle of Earth with latitude 42° N.The radius of this circle is labelled r.Assume that the radius of Earth is 6400 km.
42°6400 km
rnorth
b. Calculate the radius of the small circle of Earth with latitude 42° N. Round your answer to the nearest kilometre. 1 mark
25 2019FURMATHEXAM2(NHT)
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SECTION B – Module 3 – continued TURN OVER
c. Thekeyspeaker,Michael,willtravelfromAucklandtoRometoattendtheconference. MichaelleftAucklandat9.40amonaMonday. ThejourneytoRometook30hours. ThetimedifferencebetweenAuckland(37°S,175°E)andRome(42°N,12°E)is10hours.
OnwhatdayandatwhattimedidMichaelarriveinRome? 1mark
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SECTION B – Module 3 – Question 2 – continued
Question 2 (5marks)Michaelwillpresentabuildingdesignattheconference.Thehorizontalcross-sectionofthebuildingisintheshapeofaregularhexagon.Eachsideofthehexagonis12mlong.TriangleABC isshownshadedbelow.
A
B
C
12 m
a. i. Show,withcalculations,thattheareaoftriangleABC,roundedtoonedecimalplace, is62.4m2. 1mark
ii. Calculatetheareaofthehexagon. Roundyouranswertothenearestsquaremetre. 1mark
Ahemisphericaldomewillsitontheroofofthebuilding.Thedomehasadiameterof20m.Theviewofthebuildingfromaboveisshowninthediagrambelow.
20 m 12 m
b. Theareaoftheroofthatsurroundsthedomeisshadedinthediagramabove.
Calculatethisshadedarea. Roundyouranswertothenearestsquaremetre. 1mark
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SECTION B – Module 3 – continued TURN OVER
Thetopofthehemisphericaldomewillbemadeofglass,asshownshadedinthediagrambelow.
h
d
20 m
Theglasssectionhasabasediameterofd metresandaheightofhmetres.Theareaofthebaseoftheglasssectionisonequarteroftheareaofthebaseofthedome.
c. i. Show,withcalculations,thatthebasediameteroftheglasssection,d,is10m. 1mark
ii. Calculatetheheight,h,inmetres,oftheglasssectionofthedome. Roundyouranswertotwodecimalplaces. 1mark
2019FURMATHEXAM2(NHT) 28
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SECTION B – Module 3 – Question 3 – continued
Question 3 (4marks)Someofthepresentersorganisedahelicoptertourofnearbylandmarks.ThetourstartedinRomeandtravelleddirectlytoGenzanodiRoma,asshowninthediagrambelow.
Rome
Genzano di Roma
north
GenzanodiRomais37kmfromRomeonabearingof153°.
a. HowfarsouthofRomeisGenzanodiRoma? Roundyouranswertothenearestkilometre. 1mark
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End of Module 3 – SECTION B – continued TURN OVER
FromGenzanodiRoma,thehelicopterthenfollowedaroutetoTivoli,CalcataandthenbacktoRome,asshowninthediagrambelow.
Rome
Genzano di Roma
north
Tivoli
Calcata
Tivoliis25kmfromGenzanodiRomaonabearingof016°.Calcatais42kmfromTivolionabearingof309°.
b. CalculatethedistancebetweenGenzanodiRomaandCalcata. Roundyouranswertothenearestkilometre. 2marks
c. Calcatais40.3kmnorthand11.8kmwestofRome.
CalculatethebearingofRomefromCalcata. Roundyouranswertothenearestdegree. 1mark
2019FURMATHEXAM2(NHT) 30
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SECTION B – Module 4 – continued
Module 4 – Graphs and relations
Question 1 (2marks)Rumihasasmallclothesshop.Hesellsmen’stiesandcravats.Alltiesarethesamepriceandallcravatsarethesameprice.Lettrepresentthesellingpriceofeachtieandletcrepresentthesellingpriceofeachcravat.Onecustomer,Andy,purchasedsixtiesandfourcravatsandpaid$230.AlinearrelationrepresentingAndy’spurchaseis6t + 4c=230.
a. Asecondcustomer,Barry,purchasedfourtiesandeightcravatsandpaid$260.
WritedownalinearrelationrepresentingBarry’spurchase. 1mark
b. ThesellingpriceofeachcravatinRumi’sshopis$20.
Whatisthesellingpriceofeachtie? 1mark
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SECTION B – Module 4 – continued TURN OVER
Question 2 (3marks)CustomerswhovisitRumi’sshopparktheircarsinanundergroundcarpark.Thegraphbelowshowsthetotalchargeforparking,indollars,accordingtothenumberofhoursparkedinonevisit.
charge ($)
151413121110987654321
1 2 3 4 5hours
O
a. OnMonday,LeonparkedintheundergroundcarparkfortwohoursandLucyparkedfor one-and-a-halfhours.
WhatwasthetotalcombinedchargeforLeonandLucy? 1mark
b. Customerswhoparkformorethanfourhoursarecharged$11.50iftheydonotstaylongerthanfivehours.
Addthisinformationtothegraph above. 1mark
(Answer on the graph above.)
c. Anothercustomer,Pam,parkedintheundergroundcarparkonbothTuesdayandWednesday.Eachdaysheparkedformorethantwohoursbutlessthanfourhours.
Pamwaschargedlessthan$20intotalforthesetwodays.
WritedownthetwopossibleamountsthatPamcouldhavebeencharged. 1mark
2019 FURMATH EXAM 2 (NHT) 32
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SECTION B – Module 4 – continued
Question 3 (2 marks)Rumi’s web designers, Judy and Geoff, charge fees that consist of a fixed amount and a charge per hour. The graph of the fee charged for the number of hours worked by Judy is shown below.
1 2 3 4 5
20406080
100120140160180200220240260280300320340360380400420440460480500
fee ($)
hours
a. Complete the following sentence. 1 mark
Judy charges a fixed amount of $ and a charge per hour of $ .
b. Judy and Geoff both charge a fee of $320 for three hours worked. Geoff’s charge per hour is $100.
Draw the linear relation representing the fee charged by Geoff for the number of hours worked on the graph above. 1 mark
(Answer on the graph above.)
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SECTION B – Module 4 – Question 4 – continued TURN OVER
Question 4 (5marks)Rumialsosellsjacketsformenandwomen.LetxbethenumberofjacketsformenthatRumisellsinoneweek.LetybethenumberofjacketsforwomenthatRumisellsinoneweek.Salesrecordssuggestconstraintsonthesaleofjacketseachweek.TheseconstraintsarerepresentedbyInequalities1to4.
Inequality1 0≤x≤75Inequality2 0≤y≤80Inequality3 x + y≥40Inequality4 x + y ≤ 100
a. ExplainwhatInequality3andInequality4tellusaboutthenumberofjacketsthatRumisellseachweek. 1mark
ThegraphbelowshowsthelinesthatrepresenttheboundariesofInequalities1to4.Thefeasibleregionhasbeenshaded.
x
405060708090
100
302010
050 60 70 80 90403020100 100
y
Rumimakesaprofitof$30onthesaleofeachjacketformen.Rumimakesaprofitof$50onthesaleofeachjacketforwomen.
b. DeterminethemaximumprofitthatRumicanmakeonallsalesofjacketsinoneweek. 1mark
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END OF QUESTION AND ANSWER BOOK
c. InaparticularweekRumisoldexactly30jacketsformen.
DeterminethemaximumprofitthatRumicouldmakeonallsalesofjacketsinthatweek. 1mark
d. Later,Rumifindsthattheprofithemakesonthesaleofeachjacketforwomenhaschanged. Theprofitonthesaleofeachjacketformenremainsas$30. Themaximumprofithecanmakeinoneweekwillnowbeadifferentamount. LastweekRumiwasabletomakethismaximumprofitbyselling65jacketsformenand
35jacketsforwomen.
WhatprofitdidRumimakelastweek? 2marks
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 2019
Victorian Certificate of Education 2019
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