FURTHER MATHEMATICSWritten examination 1

Friday 2 June 2017 Reading time: 2.00 pm to 2.15 pm (15 minutes) Writing time: 2.15 pm to 3.45 pm (1 hour 30 minutes)

MULTIPLE-CHOICE QUESTION BOOK

Structure of bookSection Number of

questionsNumber of questions

to be answeredNumber of modules

Number of modulesto be answered

Number of marks

A – Core 24 24 24B – Modules 32 16 4 2 16

Total 40

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

• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.

Materials supplied• Questionbookof33pages.• Formulasheet.• Answersheetformultiple-choicequestions.• Workingspaceisprovidedthroughoutthebook.

Instructions• Checkthatyourname and student numberasprintedonyouranswersheetformultiple-choice

questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

At the end of the examination• Youmaykeepthisquestionbookandtheformulasheet.

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

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2017

Victorian Certificate of Education 2017

2017FURMATHEXAM1(NHT) 2

THIS PAgE IS BLANK

3 2017FURMATHEXAM1(NHT)

TURN OVER

THIS PAgE IS BLANK

2017FURMATHEXAM1(NHT) 4

SECTION A – continued

SECTION A – Core

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

Data analysis

Use the following information to answer Questions 1–3.Thehistogrambelowshowsthedistributionofmarksobtainedby52studentsonatest.

10

5

0

frequency

0 5 10 15 20 25 30 35 40 45 50marks

n = 52

Question 1TheshapeofthedistributionofmarksisA. symmetric.B. bell-shaped.C. positivelyskewed.D. negativelyskewed.E. approximatelynormal.

Question 2Thepassmarkforthetestwassetat25.ThepercentageofstudentswhopassedthistestwasclosesttoA. 14%B. 28%C. 32%D. 52%E. 73%

5 2017FURMATHEXAM1(NHT)

SECTION A – continuedTURN OVER

Question 3ThemedianmarkforthetestwasA. greaterthan20butlessthan25.B. greaterthan25butlessthan30.C. greaterthan30butlessthan35.D. greaterthan35butlessthan40.E. greaterthan40butlessthan45.

Question 4Theassociationbetweentheamount of solar energycapturedbyasolarpanel,inmegajoules,andthe capital city(Melbourne,Adelaide,etc.)inwhichitiscapturedwillbeinvestigated.Thevariablesamount of solar energyandcapital city areA. bothnumericalvariables.B. bothcategoricalvariables.C. anominalvariableandanumericalvariablerespectively.D. anumericalvariableandanominalvariablerespectively.E. anumericalvariableandanordinalvariablerespectively.

Question 5Thehistogrambelowdisplaysthedistributionofpopulation density,inpeoplepersquarekilometre,for53countries.Thehorizontalscaleofthehistogramislog(population density).

25

20

15

10frequency

5

0–1 0 1 2

log(population density)3 4

n = 53

Data:UnitedNations,DepartmentofEconomicandSocialAffairs, PopulationDivision(2015),‘WorldPopulationProspects:The2015Revision’(DVDedition)

Basedonthehistogram,howmanycountrieshaveapopulationdensitythatislessthan10peoplepersquarekilometre?A. 2B. 5C. 7D. 29E. 46

2017 FURMATH EXAM 1 (NHT) 6

SECTION A – continued

Question 6Which of the following could be used to identify and describe the association between the variables height (short, medium, tall) and hat size (small, medium, large)?A. a histogramB. a scatterplotC. parallel boxplotsD. a segmented bar chartE. a back-to-back stem plot

Question 7A least squares line fitted to a scatterplot will always A. maximise the number of data points lying on the line.B. equalise the number of data points on either side of the line.C. minimise the sum of the squares of the vertical distance from each data point to the line.D. minimise the sum of the squares of the shortest distance from each data point to the line.E. minimise the sum of the squares of the horizontal distance from each data point to the line.

Use the following information to answer Questions 8 and 9.The beak length of small birds in a large population is approximately normally distributed with a mean of 9.5 mm and a standard deviation of 0.50 mm.

Question 8Which one of the following statements relating to this population of birds is not true?A. No bird will have a beak length that is less than 8.0 mm.B. More than 99% of the birds will have a beak length that is less than 11 mm.C. Approximately half of the birds will have a beak length that is less than 9.5 mm.D. Approximately 2.5% of the birds will have a beak length that is greater than 10.5 mm.E. Approximately 34% of the birds will have a beak length that is between 9.5 mm and 10.0 mm.

Question 9A random sample of 250 of these birds is captured and the beak length of each bird is measured.The expected number of these captured birds with beak lengths that are greater than 9 mm is closest to A. 6B. 13C. 170D. 210E. 244

7 2017FURMATHEXAM1(NHT)

SECTION A – continuedTURN OVER

Use the following information to answer Questions 10 and 11.Thescatterplotbelowdisplaysthebeaklengthandbeakwidthof68birdsofthesamespecies.Aleastsquareslinehasbeenfittedtothedata.

n = 68r = 0.8616

10

9

width (mm)

8

77 8 9

length (mm)10 11

Data:HowardHughesMedicalInstitute

Thecorrelationcoefficientisr=0.8616Theleastsquareslinehasbeenfittedtothescatterplotusingbeaklengthastheexplanatoryvariable.

Question 10TheequationofthislineisclosesttoA. width=0.56+3.5×lengthB. width=3.5+0.56×lengthC. width=7.4+0.56×lengthD. length=7.4+0.56×widthE. length=3.5+0.56×width

Question 11Whichoneofthefollowingstatementsisnottrue?A. Theslopeoftheleastsquareslineispositive.B. Birdswithlongerbeakstendtohavewiderbeaks.C. Thereisastrongpositivelinearassociationbetweenbeakwidthandbeaklengthforthesebirds.D. Approximately74%ofthevariationinbeakwidthisexplainedbythevariationinbeaklength.E. Usingtheleastsquareslinetopredictthebeakwidthofabirdwithabeaklengthof8.1mmwouldbe

anexampleofextrapolation.

2017FURMATHEXAM1(NHT) 8

SECTION A – continued

Question 12Thetablebelowshowsthespeed,inkilometresperhour,andthebraking distance,inmetres,ofacartravellingateightdifferentspeeds.Ascatterplothasbeenconstructedfromthisdata.

Speed (km/h) Distance (m) 70

60

50

40

30

distance (m)

20

10

00 20 40 60

speed (km/h)80 100 120

40 9

50 14

60 20

70 27

80 36

90 45

100 56

110 67

Data:©TheStateofQueensland(DepartmentofTransportandMainRoads)2010–2016

Thescatterplotshowsthattheassociationbetweendistanceandspeedisnon-linear.Asquaredtransformationisappliedtothevariablespeed tolinearisethedata.Aleastsquareslineisthenfittedtothetransformeddatawithdistanceastheresponsevariable.TheequationofthisleastsquareslineisclosesttoA. distance=–15.6+180×speed 2

B. distance=0.0056+0.092×speed 2

C. distance=0.092+0.0056×speed 2

D. speed 2=180–15.6×distanceE. speed 2=0.0056+0.092×distance2

9 2017FURMATHEXAM1(NHT)

SECTION A – continuedTURN OVER

Use the following information to answer Questions 13 and 14.Thetimeseriesplotbelowchartsthequarterlysalesfigures,inmillionsofdollars,ofasmallmanufacturingbusinessoveraperiodoffouryears.

16

14

12

10

8sales

($ millions)

6

4

2

00 4 8

quarter number12 16

Question 13ThetimeseriesplotisbestdescribedashavingA. novariability.B. seasonalityonly.C. irregularvariationonly.D. adecreasingtrendwithseasonality.E. anincreasingtrendwithseasonality.

Question 14Thesalesfiguresusedtogeneratethistimeseriesplotaredisplayedinthetablebelow.

Year Quarter 1 Quarter 2 Quarter 3 Quarter 4

2013 6.5 13.4 7.4 3.8

2014 10.2 11.8 7.4 4.5

2015 9.6 14.5 8.6 5.3

2016 10.3 14.2 7.5 4.9

Thefour-meansmoothedsaleswithcentringforQuarter3in2015,inmillionsofdollars,wasclosesttoA. 8.6B. 9.3C. 9.5D. 9.6E. 9.7

2017FURMATHEXAM1(NHT) 10

SECTION A – continued

Use the following information to answer Questions 15 and 16.Theseasonalindices(SI)forthedailyearningsofacafeinatouristtown,fromMondaytoSaturday,areshowninthetablebelow.TheseasonalindexforSundayisnotshown.

Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday

SI 0.65 0.60 0.74 0.82 1.12 1.45

Question 15LastSunday,thecafeearned$3839.ThedeseasonalisedearningsforthisdaywereclosesttoA. $2370B. $2500C. $2650D. $5570E. $6220

Question 16TheseasonalindexforWednesdayis0.74ThistellsusthatWednesdayearningstendtobeA. 26%lessthantheaveragedailyearnings.B. 26%morethantheaveragedailyearnings.C. 35%lessthantheaveragedailyearnings.D. 35%morethantheaveragedailyearnings.E. 74%lessthantheaveragedailyearnings.

11 2017FURMATHEXAM1(NHT)

SECTION A – continuedTURN OVER

Recursion and financial modelling

Question 17Asequenceisgeneratedbytherecurrencerelationbelow.

A0=2, An+1 = 3An – 3

ThesequenceisA. 2,1,0,–3…B. 2,3,0,–3…C. 2,3,2,3…D. 2,3,3,3…E. 2,3,6,15…

Question 18Andredeposited$20000intoasavingsaccountearningcompoundinterestattherateof3.1%perannum,compoundingannually.Whichoneofthefollowingrecurrencerelationscanbeusedtodeterminetheamountinthesavingsaccount,Sn ,afternyears?

A. S0=20000, Sn+1 = Sn+620

B. S0=20000, Sn+1=1.031×Sn

C. S0=20000, Sn+1=620×Sn

D. S0=20000, Sn+1=3.1×Sn+620

E. S0=20000, Sn+1 = Sn+3.1×620

Question 19Considertherecurrencerelationbelow.

L0=2000, Ln +1 = Ln+80

ThisrecurrencerelationcouldbeusedtomodelaA. simpleinterestinvestmentof$2000withanannualinterestrateof4%.B. simpleinterestinvestmentof$2000withanannualinterestrateof8%.C. simpleinterestinvestmentof$2000withanannualinterestrateof40%.D. compoundinterestinvestmentof$2000withanannualinterestrateof4%.E. compoundinterestinvestmentof$2000withanannualinterestrateof8%.

Question 20Amusicschoolhas$80000toinvestinaperpetuity.Theinterestearnedfromthisperpetuitywillprovideanannualprizeof$3000toatalentedmusicianfromtheschool.Whatannualinterestratewouldberequiredforthisinvestment?A. 0.3125%B. 3.75%C. 3.90%D. 41.92%E. 45.00%

2017 FURMATH EXAM 1 (NHT) 12

SECTION A – continued

Question 21The amortisation table below shows the repayment, interest, principal reduction and balance of a reducing balance loan after the first repayment.

Repayment number

Repayment Interest Principal reduction

Balance of loan

0 0.00 0.00 0.00 180 000.00

1 850.00 720.00 130.00 179 870.00

2 850.00

What amount of interest is paid with Repayment number 2?A. $608.56B. $609.44C. $717.12D. $719.48E. $720.00

Question 22Vusa has invested $420 000 in an annuity that pays interest at the rate of 3.6% per annum, compounding monthly.After the interest has been added each month, Vusa immediately receives a payment from the annuity.The value of Vusa’s investment is $372 934.71 after three years.The monthly payment that Vusa receives from the annuity is closest toA. $1260B. $1310C. $2500D. $15 120E. $16 900

13 2017FURMATHEXAM1(NHT)

END OF SECTION ATURN OVER

Question 23Thevalueofapianoisdepreciatedusingthereducingbalancemethod.Thegraphbelowshowsthevalueofthepianoasitdepreciatesoveraperiodof10years.

8000

7000

6000

5000

4000

3000

2000

1000

0 1 2 3 4 5 6 7 8 9 10

value ($)

n (years)

(3, 6859)(4, 6516.05)

LetPnbethevalueofthepianoafternyears.ArecurrencerelationthatcouldbeusedtodeterminePnisA. P0=8000, Pn+1=0.95×Pn

B. P0=8000, Pn+1=342.95×Pn

C. P0=8000, Pn+1=1.05×Pn – 3

D. P0=8000, Pn+1=0.95×Pn–342.95

E. P0=8000, Pn+1 = Pn–342.95

Question 24Geoffhasacompoundinterestinvestmentthatearnsinterestcompoundingmonthly.ThebalanceofGeoff’scompoundinterestinvestmentwas$4418.80aftersixmonths.ThebalanceofGeoff’scompoundinterestinvestmentwas$4862.80aftertwoyears.TheamountofmoneythatGeoffinitiallyinvestedisclosesttoA. $4000B. $4015C. $4280D. $4370E. $4715

2017FURMATHEXAM1(NHT) 14

SECTION B –continued

SECTION B – Modules

Instructions for Section BSelecttwomodulesandanswerallquestionswithintheselectedmodulesinpencilontheanswersheetprovidedformultiple-choicequestions.Showthemodulesyouareansweringbyshadingthematchingboxesonyourmultiple-choiceanswersheetandwritingthenameofthemoduleintheboxprovided.Choosetheresponsethatiscorrectforthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

Contents Page

Module1–Matrices......................................................................................................................................15

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

Module3–Geometryandmeasurement.......................................................................................................25

Module4–Graphsandrelations................................................................................................................... 29

15 2017FURMATHEXAM1(NHT)

SECTION B – Module 1 – continuedTURN OVER

Question 14 21 5

34 21 5

+

isequalto

A. 8 42 10

B. 11 15 13

C. 16 84 20

D. 24 126 30

E. 38 3431 40

Question 2Thecostoffruitatastall,indollarsperkilogram,isshowninthetablebelow.

Apples $2.50

Pears $3.20

Bananas $1.90

Seanwantstobuy2kgofapples,1kgofpearsand3kgofbananas.WhichoneofthefollowingmatrixproductswillresultinamatrixthatcontainsthetotalcostofSean’sfruitpurchase,indollars?

A. [ ]...

2 1 32 503 201 90

B. [ ][ . . . ]2 1 3 2 50 3 20 1 90

C. 213

2 503 201 90

.

.

.D.

2 0 00 1 00 0 3

2 503 201 90

.

.

.

E. 2 503 201 90

2 1 3...

[ ]

Module 1 – Matrices

Beforeansweringthesequestions,youmustshadethe‘Matrices’boxontheanswersheetfor multiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

2017FURMATHEXAM1(NHT) 16

SECTION B – Module 1 – continued

Question 3Peter(P),Sally(S)andBen(B)aremanagersinabusiness.Thediagrambelowshowsthedirectcommunicationthatispossiblebetweenthesemanagersandaworker,Whitney(W).

P

W

B

S

Forexample,thearrowfromS to W indicatesthatSallyisabletocommunicatedirectlywithWhitney.AcommunicationmatrixcanrepresentthedirectcommunicationthatispossiblebetweenthemanagersandWhitney.Theelementsinthematrixaresuchthat:• ‘1’indicatesthatdirectcommunicationfromonepersontoanotherispossible• ‘0’indicatesthatdirectcommunicationisnotpossible.

Thiscommunicationmatrixcouldbe

A.

toP S B W

from

PSBW

0 1 1 10 0 1 11 1 0 00 1 0 0

B.

toP S B W

from

PSBW

0 1 2 11 0 2 22 2 0 11 2 1 0

C.

toP S B W

from

PSBW

0 1 1 10 0 0 11 1 0 00 1 0 0

D.

toP S B W

from

PSBW

0 1 1 11 0 1 11 1 0 11 1 1 0

E.

toP S B W

from

PSBW

0 0 1 01 0 1 11 1 0 01 1 0 0

17 2017FURMATHEXAM1(NHT)

SECTION B – Module 1 – continuedTURN OVER

Question 4 TheelementinrowiandcolumnjofmatrixMismi j.TheelementsinmatrixMaredeterminedusingtherulemi j = 2i+j.MatrixMcouldnotbeA. [3] B. [ ]3 4 5

C. 3 45 6

D.

3456

E. 3 4 55 6 77 8 9

Question 5Yvetteneedstobuyatotalofninepensandmarkers.Fivepensandfourmarkerswillcost$31.00Fourpensandfivemarkerswillcost$32.00Letpbethecostofapen.Letmbethecostofamarker.Considerthefollowingmatrixequations.

pm

=

−5 44 5

3132

1 pm

=

−3231

4 55 4

1

pm

= −

−−

19

4 55 4

3231

pm

=

−

−

59

49

49

59

3231

HowmanyofthematrixequationsabovecouldYvettesolvetogetamatrixthatcontainsthepriceofapenandthepriceofamarker?A. 0B. 1C. 2D. 3E. 4

2017FURMATHEXAM1(NHT) 18

SECTION B – Module 1 – continued

Question 6Studentsataschoolchooseanafternoonactivityeveryweek.Theycanchooseeithersport(S),art(A)ormusic(M).Thetablebelowshowsthenumberofstudentswhochosesport,artandmusicinWeek1oftheschoolterm.

Sport (S) Art (A) Music (M)

150 85 35

ThestudentsareexpectedtochangetheactivitytheychoosefromweektoweekasshowninthetransitionmatrixPbelow.

this weekS A M

P =

0 80 0 20 0 050 10 0 70 0 150 10 0 10 0 80

. . .

. . .

. . .

SAM

next week

Whichoneofthefollowingstatementsistrueforthissituation?A. 30%ofthestudentswillneverchooseart.B. Thenumberofstudentswhochoosemusicwilldecreaseeveryweek.C. Thenumberofstudentswhochoosesportinoneweekwillalwaysbe20%lessthanthenumberof

studentswhochosesportinthepreviousweek.D. InWeek3oftheschoolterm,thenumberofstudentswhochoosemusicwillbelessthanhalfthe

numberofstudentswhochooseart.E. Inthelongterm,thenumberofstudentswhochoosemusicwillbemorethanthenumberofstudents

whochooseart.

Question 7Abadmintoncompetitionisheldbetweenfourplayers,Amanda(A),Ben(B),Carlos(C)andDarius(D).Inthecompetition,eachplayercompetesinonegamewitheachoftheotherthreeplayers.ThematrixS 2belowshowsthetwo-stepdominancethateachplayerhasovertheotherplayers.

loserA B C D

S winner

ABCD

2

0 2 1 00 0 0 00 0 0 00 1 0 0

=

Fromthematrixabove,whichoneofthefollowingeventsmusthaveoccurredduringthecompetition?A. BenbeatAmanda.B. BenbeatCarlos.C. CarlosbeatAmanda.D. DariusbeatAmanda.E. DariusbeatCarlos.

19 2017FURMATHEXAM1(NHT)

End of Module 1 – SECTION B–continuedTURN OVER

Question 8MatrixAisann×nmatrixwheren>1.MatrixRisarowmatrix.MatrixCisacolumnmatrix.Whichoneofthematrixproductsbelowcouldresultina1×1matrix?A. ACRB. ARCC. CARD. RACE. RCA

2017FURMATHEXAM1(NHT) 20

SECTION B – Module 2 – continued

Question 1Hunterrideshisbiketoschooleachday.TheedgesofthenetworkbelowrepresenttheroadsthatHuntercanusetoridetoschool.Thenumbersontheedgesgivethedistance,inkilometres,alongeachroad.

2

1 4

4

2

354

25

3

home

school

WhatistheshortestdistancethatHuntercanridebetweenhomeandschool?A. 10kmB. 11kmC. 12kmD. 14kmE. 23km

Module 2 – Networks and decision mathematics

Beforeansweringthesequestions,youmustshadethe‘Networksanddecisionmathematics’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

21 2017FURMATHEXAM1(NHT)

SECTION B – Module 2 – continuedTURN OVER

Question 2Considerthetwographsbelow.

Whichoneofthefollowingstatementsisnottrue?A. Eachgraphisconnected.B. EachgraphcontainsanEuleriantrail.C. EachgraphcontainsaHamiltoniancycle.D. Eachgraphhasatleastonevertexofdegreetwo.E. Thesumofthedegreesoftheverticesforeachgraphiseven.

Question 3Thegraphshownbelowisplanar.

Howmanyfacesdoesthisgraphhave?A. 5B. 6C. 7D. 8E. 9

2017FURMATHEXAM1(NHT) 22

SECTION B – Module 2 – continued

Question 4ThematrixbelowshowsthenumberofroadconnectionsbetweentownsF,G,HandI.

F G H IFGHI

0 0 1 10 1 1 21 1 0 01 2 0 0

Whichoneofthefollowinggraphsshowsalloftheseroadconnections?

F

G

H

I

I

H

G

F

I

G

H

F

H

G

I

F

FG H

I

A. B.

C. D.

E.

23 2017 FURMATH EXAM 1 (NHT)

SECTION B – Module 2 – continuedTURN OVER

Use the following information to answer Questions 5 and 6.The activity network below shows the sequence of activities required to complete a project. The number next to each activity in the network is the time it takes to complete that activity, in days.

D, 5 H, 3

E, 1

N, 1

K, 4

I, 4

L, 5

M, 5 P, 2

J, 3

A, 2

O, 2F, 2C, 2

B, 3

G, 6

finishstart

Question 5Beginning with Activity C, the number of paths from start to finish isA. 1B. 2C. 3D. 4E. 5

Question 6What is the latest starting time for Activity I, in days, so that the project is completed in the shortest time possible?A. 3B. 4C. 5D. 6E. 7

2017FURMATHEXAM1(NHT) 24

End of Module 2 – SECTION B–continued

Question 7Considertheweightedgraphshownbelow.

6

6

6

6

10

10

6

6

Howmanydifferentminimalspanningtreesarepossible?A. 2B. 3C. 4D. 5E. 6

Question 8Abbey,Barb,CathalandDinharefourworkersatabusiness.Eachworkerwillperformoneduty.Thetimeforeachworkertocompleteduties1,2,3and4,inminutes,isshowninthetablebelow.

Duty 1 Duty 2 Duty 3 Duty 4

Abbey 6 5 6 8

Barb 9 12 10 6

Cathal 8 7 4 8

Dinh 5 3 6 4

Theminimumtotaltimeforalldutiesis19minutes,withDinhperformingDuty2.Beforethedutiesareperformed,itisfoundthatDinhwillrequire7minutesforDuty2ratherthan3minutes.Ifthedutiesareallocatedagain,theminimumtotaltimeforalldutieswillA. remainthesame.B. increaseby1minute.C. increaseby2minutes.D. increaseby3minutes.E. increaseby4minutes.

25 2017FURMATHEXAM1(NHT)

SECTION B – Module 3 – continuedTURN OVER

Question 1Asectoriscutfromacircleofradius12mm.Thissectorisshownshadedinthediagrambelow.

12 mm58°

Theangleatthepointofthesectoris58°.Theareaofthesector,insquaremillimetres,isclosesttoA. 1B. 6C. 46D. 73E. 146

Question 2Averticaltreeofheight20mstandsonhorizontalground.Thetreeis100mawayfrompointP,asshowninthediagrambelow.

100 m P

20 m

x°

Thevalueoftheanglex,showninthediagramabove,isclosesttoA. 3°B. 11°C. 12°D. 78°E. 79°

Module 3 – geometry and measurement

Beforeansweringthesequestions,youmustshadethe‘Geometryandmeasurement’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

2017FURMATHEXAM1(NHT) 26

SECTION B – Module 3 – continued

Question 3Whichofthefollowingpairsofcitieshasthelargesttimedifferencebetweenthem?A. Belfast(55°N,6°W)andJohannesburg(26°S,28°E)B. Havana(23°N,82°W)andLagos(7°N,3°E)C. Helsinki(60°N,25°E)andManila(15°N,121°E)D. Malang(8°S,113°E)andRockhampton(23°S,151°E)E. Moscow(56°N,38°E)andAtlanta(34°N,84°W)

Question 4Paulahasbuiltamodelhouseusingatriangularprismontopofarectangularbox.Thedimensionsofthemodelhouseareshownonthediagrambelow.

52 cm 52 cm

75 cm96 cm80 cm

Paulawillpainttheoutsidewallsandtheroofofthemodelhouse.Theareathatwillbepainted,insquarecentimetres,isclosesttoA. 12 600B. 26 400C. 36 400D. 37700E. 39 000

27 2017FURMATHEXAM1(NHT)

SECTION B – Module 3 – continuedTURN OVER

Question 5Atrianglehasonesideoflength8cmandanothersideoflength12cm,asshowninthediagrambelow.Theanglesxandyarealsoshown.

x y

12 cm8 cm

Theanglexissuchthat sin( )x =25.

Whatisthevalueofsin(y)?

A. 35

B. 45

C. 53

D. 415

E. 715

Question 6CityAhaslatitude25°Nandlongitude50°E.CityBhaslatitude35°Sandlongitude50°E.AssumethattheradiusofEarthis6400km.TheshortestdistancealongthemeridianbetweenAandB,inkilometres,isclosesttoA. 1117B. 2793C. 3910D. 6400E. 6702

2017FURMATHEXAM1(NHT) 28

End of Module 3 – SECTION B–continued

Question 7Thelatitudeandlongitudeoftwocities,AandB,areshowninthetablebelow.

City Latitude Longitude

A 50°N 85°E

B 50°N 112°W

AssumethattheradiusofEarthis6400km.TheshortestdistancealongtheparallelbetweenAandB,inkilometres,canbefoundfromwhichoneofthefollowingcalculations?

A. 163360

2 6400 40× × × ×π sin

B. 197360

2 6400 40× × × ×π sin

C. 197360

6400 40× × ×π sin

D. 197360

2 6400 50× × × ×π sin

E. 163360

2 6400 50× × × ×π sin

Question 8AtrianglehasverticesP,SandR.PointRliesdueeastofpointP.PointQliesonsidePR.ThelengthofsideSPisthesameasthelengthofsideSQ,asshowninthediagrambelow.

80 m

P Q R

S

north

100 m

ThelengthofsideSRis80m.ThelengthofsidePRis100m.ThebearingofQfromSis120°.ThedistancebetweenQandR,inmetres,isclosesttoA. 26B. 43C. 58D. 74E. 100

29 2017FURMATHEXAM1(NHT)

SECTION B – Module 4 – continuedTURN OVER

Question 1Thegraphbelowshowsthetemperature,indegreesCelsius,overthecourseofaday.

temperature (°C)

30

25

20

15

10

5

O 6.00 am 9.00 am 12.00noon

3.00 pm 6.00 pmtime

Thetimethatthetemperaturefirstreached25°CwasclosesttoA. 12.00noonB. 12.30pmC. 1.00pmD. 1.30pmE. 2.00pm

Question 2Playersatafootballclubpayafeeof$130eachyear.Theyalsopayafeeof$12foreverygametheyplayinthatyear.Lastyear,Jennypaidatotalof$262infeesatthisfootballclub.HowmanygamesdidJennyplaylastyear?A. 10B. 11C. 12D. 13E. 14

Module 4 – graphs and relations

Beforeansweringthesequestions,youmustshadethe‘Graphsandrelations’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

2017FURMATHEXAM1(NHT) 30

SECTION B – Module 4 – continued

Question 3Thegraphbelowshowsthecostofsendingapackagebypriorityairmail,indollars,accordingtotheweightofthepackage.

200

150

100

50

10 20 30 40 50

cost ($)

weight (kg)O

Manushipsthreeseparatepackages:• one15kgpackage• one35kgpackage• one48kgpackage

Whatisthetotalcostofsendingallthreepackages?A. $120B. $230C. $360D. $520E. $560

Question 4Thegraphbelowshowsastraightlinethattouchesthex-axisatthepoint(p,0).

(3, 10)(0, 12)

O (p, 0)

y

x

Whatisthevalueofp?A. 12B. 14C. 16D. 18E. 22

31 2017FURMATHEXAM1(NHT)

SECTION B – Module 4 – continuedTURN OVER

Question 5Theshadedregiononthegraphbelowshowsthefeasibleregiondefinedbyfiveinequalities.

y

x

10

8

6

4

2

2 4 6 8 10O

OneoftheinequalitiesusedtodefinethisfeasibleregioncouldbeA. 4x+3y≤24B. 4x+3y≥24C. 3x – y≤9D. 3x – y≥9E. 2≤x≤6

Question 6Abookstoreishavingasale.Eachpaperbackbookcosts$8andeachhardbackbookcosts$14.Janebought12booksforatotalof$126.ThenumberofhardbackbooksthatJaneboughtisA. 5B. 7C. 8D. 12E. 14

2017FURMATHEXAM1(NHT) 32

SECTION B – Module 4 – continued

Question 7Theshadedareainthegraphbelowshowsthefeasibleregionforalinearprogrammingproblemdefinedbythefollowinginequalities.

Inequality1 y≤0.8x+2Inequality2 y≥−0.7x+6Inequality3 x≤7

y

x

A

B

C

O

A,BandCarethecornerpointsofthisregion.AllofthepointsalongthelineABgivethemaximumvalueofanobjectivefunctionZ.ThisobjectivefunctioncouldhavetheequationA. Z = −56x +80yB. Z = −52x +65yC. Z = 52x +65yD. Z = 56x – 80yE. Z = 56x +80y

33 2017FURMATHEXAM1(NHT)

END OF MULTIPLE-CHOICE QUESTION BOOK

Question 8Atankisbeingfilledwithwaterbytwohoses.After15minutes,oneofthehosesisremoved.Thegraphbelowshowsthevolumeofwater,Vlitres,inthetankattimetminutes.

V (litres)

600

400

200

Ot (minutes)

(a, 600)

(15, 400)

Thetankhasacapacityof600litresandisfullafteraminutes.Ifbothhoseshadcontinuedtofillthetank,itwouldhavereacheditscapacitynineminutesearlierthanthis.Therateatwhichthetankisfilling,inlitresperminute,overthetimeintervalfrom15minutestoaminutes,isclosesttoA. 12B. 13C. 22D. 23E. 27

FURTHER MATHEMATICS

Written examination 1

FORMULA SHEET

Instructions

This formula sheet is provided for your reference.A multiple-choice question 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 2017

Victorian Certificate of Education 2017

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 rssy

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

acbd ad bc= = −

inverse of a 2 × 2 matrix AAd 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 ruleaA

bB

cCsin ( ) 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