2019 further mathematics-nht written examination 2 · writing time: 2.15 pm to 3.45 pm (1 hour 30...

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>

3 2019 FURMATH EXAM 2 (NHT)

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

2019 FURMATH EXAM 2 (NHT) 4

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)

5 2019FURMATHEXAM2(NHT)


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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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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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d. Interpretthecoefficientofdeterminationintermsoflife span and sleep time. 1mark

e. Thelifespanofthemammalwithasleeptimeof12hoursis39.2years.

Showthat,whentheleastsquareslineisusedtopredictthelifespanofthismammal,theresidualis19.9years. 2marks


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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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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


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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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


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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


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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


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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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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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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


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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 3 12 8 8 6


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 2Constructing



Task 4Landscaping


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 2Constructing



Task 4Landscaping


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




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


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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


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c. Thekeyspeaker,Michael,willtravelfromAucklandtoRometoattendtheconference. MichaelleftAucklandat9.40amonaMonday. ThejourneytoRometook30hours. ThetimedifferencebetweenAuckland(37°S,175°E)andRome(42°N,12°E)is10hours.

OnwhatdayandatwhattimedidMichaelarriveinRome? 1mark


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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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Thetopofthehemisphericaldomewillbemadeofglass,asshownshadedinthediagrambelow.

h

d

20 m

Theglasssectionhasabasediameterofd metresandaheightofhmetres.Theareaofthebaseoftheglasssectionisonequarteroftheareaofthebaseofthedome.

c. i. Show,withcalculations,thatthebasediameteroftheglasssection,d,is10m. 1mark

ii. Calculatetheheight,h,inmetres,oftheglasssectionofthedome. Roundyouranswertotwodecimalplaces. 1mark


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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


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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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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


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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

2019 further mathematics-nht written examination 2 · writing time: 2.15 pm to 3.45 pm (1 hour 30...

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