victorian certificate of education 2019 · 2019 furmath exam 1 (nht) 6 section a continued question...

FURTHER MATHEMATICSWritten examination 1

Wednesday 29 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)

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• Questionbookof32pages• 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2019

Victorian Certificate of Education 2019

2019FURMATHEXAM1(NHT) 2

THIS PAGE IS BLANK

3 2019FURMATHEXAM1(NHT)

SECTION A – continuedTURN OVER

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 and 2.Thehistogramandboxplotshownbelowbothdisplaythedistributionofthebirth weight,ingrams, of200babies.

percentage

403020100

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500

500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500

birth weight (g)

birth weight (g)

Question 1Theshapeofthedistributionofthebabies’birth weightisbestdescribedasA. positivelyskewedwithnooutliers.B. negativelyskewedwithnooutliers.C. approximatelysymmetricwithnooutliers.D. positivelyskewedwithoutliers.E. approximatelysymmetricwithoutliers.

Question 2Thenumberofbabieswithabirth weightbetween3000gand3500gisclosesttoA. 30B. 32C. 37D. 74E. 80


SECTION A – continued

Question 3Thetotalbirthweightofasampleof12babiesis39.0kg.Themeanbirthweightofthesebabies,inkilograms,isA. 2.50B. 2.75C. 3.00D. 3.25E. 3.50

Question 4Theboxplotanddotplotshownbelowbothdisplaythedistributionofthegestation period,inweeks, for117babygirls.

22 24 26 28 30 32 34 36 38 40 42 44 46

22 24 26 28 30 32 34 36 38 40 42 44 46

n = 117

n = 117

gestation period (weeks)

gestation period (weeks)

Themediangestation period,inweeks,ofthesebabygirlsisA. 38B. 38.5C. 39D. 39.5E. 40



Use the following information to answer Questions 5 to 7.Thebirthweightsofalargepopulationofbabiesareapproximatelynormallydistributedwithameanof3300gandastandarddeviationof550g.

Question 5Ababyselectedatrandomfromthispopulationhasastandardisedweightofz=–0.75Whichoneofthefollowingcalculationswillresultintheactual birth weightofthisbaby?A. actual birth weight=550–0.75×3300

B. actual birth weight=550+0.75×3300

C. actual birth weight=3300–0.75×550

D. actual birth weight=3300+0 75550.

E. actual birth weight = 3300 – 0 75550.

Question 6Usingthe68–95–99.7%rule,thepercentageofbabieswithabirthweightoflessthan1650gisclosesttoA. 0.14%B. 0.15%C. 0.17%D. 0.3%E. 2.5%

Question 7Asampleof600babieswasdrawnatrandomfromthispopulation.Usingthe68–95–99.7%rule,thenumberofthesebabieswithabirthweightbetween2200gand3850gis closesttoA. 111B. 113C. 185D. 408E. 489

Question 8Thevariablesrecovery time after exercise(inminutes)andfitness level(belowaverage,average,aboveaverage)areA. bothnumerical.B. bothcategorical.C. anordinalvariableandanominalvariablerespectively.D. anumericalvariableandanominalvariablerespectively.E. anumericalvariableandanordinalvariablerespectively.



Question 9Aleastsquareslineoftheformy = a+bxisfittedtoasetofbivariatedataforthevariablesxandy.Forthissetofbivariatedata, x s s ay x y= = = = =5 50 5 60 3 03 1 78 3 1. , . , . , . .andTheslopeoftheleastsquaresline,b,isclosesttoA. 0.44B. 0.45C. 0.58D. 0.59E. 0.76

Question 10Astudentusesthedatainthetablebelowtoconstructthescatterplotshownbelow.

x y

0 94

1 91

2 93

3 80

4 74

5 50

6 55

6.5 31

7 9

0 1 2 3 4 5 6 7x

y

0102030405060708090

100110

Asquaredtransformationisappliedtothevariablextolinearisethedata.Aleastsquareslineisfittedtothislineariseddatawithx2astheexplanatoryvariable.TheequationofthisleastsquareslineisclosesttoA. y=94.1–12.3x2

B. y=95.8–1.57x2

C. y=8.76–0.0768x2

D. y = 107 – 111x2

E. y=107–0.0768x2



Question 11TheHuman Development Index(HDI)andthemeannumberofchildrenperwomanfor13countriesarerelated.Thisrelationshipisnon-linear.Tolinearisethedata,alog10transformationisappliedtotheresponsevariablechildren.Aleastsquareslineisthenfittedtothelineariseddata.Theequationofthisleastsquareslineis

log10(children)=1.06–0.00674×HDI

Usingthisequation,themeannumberofchildrenperwomanforacountrywithaHDIof95ispredictedtobeclosesttoA. 0.4B. 1.5C. 2.6D. 2.9E. 3.1

Question 12Whichoneofthefollowingstatementscouldbetruewhenwrittenaspartoftheresultsofastatisticalinvestigation?A. Thecorrelationcoefficientbetweenheight(incentimetres)andfoot length(incentimetres)was

foundtober=1.24B. Thecorrelationcoefficientbetweenheight (belowaverage,average,aboveaverage)and

arm span(incentimetres)wasfoundtober=0.64C. Thecorrelationcoefficientbetweenblood pressure(low,normal,high)andage

(under25,25–49,over50)wasfoundtober=0.74D. Thecorrelationcoefficientbetweentheheightofstudentsofthesameage(incentimetres)and

themoneytheyspentonsnackfood(indollars)wasfoundtober=0.22E. Thecorrelationcoefficientbetweenheight of wheat(incentimetres)andgrain yield (intonnes)was

foundtober=–0.40andthecoefficientofdeterminationwasfoundtober 2=–0.16

Question 13Theassociationbetweenamount of protein consumed(ingrams/day)andfamily income(indollars)isbestdisplayedusingA. ascatterplot.B. atimeseriesplot.C. parallelboxplots.D. back-to-backstemplots.E. atwo-wayfrequencytable.



Question 14Thetimeseriesplotbelowshowsthedailynumberofvisitorstoahistoricalsiteoveratwo-weekperiod.

0

50

100

150

200

250

numberof visitors

0 1 2 3 4 5 6 7 8day

9 10 11 12 13 14 15

Thistimeseriesplotistobesmoothedusingseven-mediansmoothing.Thesmoothednumberofvisitorsonday4isclosesttoA. 120B. 140C. 145D. 150E. 160

Question 15Thetablebelowshowsthelong-termmonthlyaverageheatingcost,indollars,forasmalloffice.

Month Jan. Feb. Mar. Apr. May June Jul. Aug. Sep. Oct. Nov. Dec.

Long-term average heating cost 78 83 73 109 156 130 146 168 159 140 136 122

TheseasonalindexforAprilisclosesttoA. 0.80B. 0.87C. 0.96D. 1.25E. 1.56



Question 16AsmallcafeisopeneverydayoftheweekexceptTuesday.Thetablebelowshowsthedailyseasonalindicesforlabourcostsatthiscafe.

Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Seasonal index 1.47 closed 1.24 1.01 0.77 0.76 0.75

Excludingthedaythatthecafeisclosed,thelong-termaverageweeklylabourcostis$9786.TheexpecteddailylabourcostonaWednesdayisclosesttoA. $1127B. $1315C. $1631D. $1733E. $2022



Recursion and financial modelling

Question 17Asequenceofnumbersisgeneratedbytherecurrencerelationshownbelow.

P0=2, Pn+1 = 3 Pn – 1

WhatisthevalueofP3?A. 2B. 5C. 11D. 41E. 122

Question 18Atruckwaspurchasedfor$134000.Usingthereducingbalancemethod,thevalueofthetruckisdepreciatedby8.5%eachyear.Whichoneofthefollowingrecurrencerelationscouldbeusedtodeterminethevalueofthetruckafter nyears,Vn?A. V0=134000, Vn+1=0.915×Vn B. V0=134000, Vn+1=1.085×VnC. V0=134000, Vn+1 = Vn–11390D. V0=134000, Vn+1=0.915×Vn–8576E. V0=134000, Vn+1=1.085×Vn–8576

Question 19Considertherecurrencerelationshownbelow.

V0=125000, Vn+1=1.013Vn – 2000

ThisrecurrencerelationcouldbeusedtodeterminethevalueofA. aperpetuitywithapaymentof$2000perquarter.B. anannuitywithwithdrawalsof$2000perquarter.C. anannuityinvestmentwithadditionalpaymentsof$2000perquarter.D. anitemdepreciatingat aflatrateof1.3%ofthepurchasepriceperquarter.E. acompoundinterestinvestmentearninginterestattherateof1.3%perannum.



Question 20Martyhasbeendepreciatingthevalueofhiscareachyearusingflatratedepreciation.Afterthreeyearsofownership,thevalueofthecarwashalvedduetoanaccident.Martycontinuedtodepreciatethevalueofhiscarbythesameamounteachyearaftertheaccident.WhichoneofthefollowinggraphscouldshowthevalueofMarty’scarafternyears,Cn?

Cn Cn

Cn Cn

Cn

n n

n n

n

0 1 2 3 4 5 6 0 1 2 3 4 5 6

0 1 2 3 4 5 6

0 1 2 3 4 5 6

0 1 2 3 4 5 6

A. B.

C. D.

E.



Question 21Yazhenhasareducingbalanceloan.SixlinesoftheamortisationtableforYazhen’sloanareshownbelow.

Repayment number

Repayment Interest Principal reduction

Balance of loan

15 1000.00 349.50 650.50 34299.50

16 1000.00 343.00 657.00 33642.50

17 1000.00 403.71 596.29 33046.21

18 1000.00 396.55 603.45 32442.76

19 1000.00 389.31 610.69 31832.07

20 1000.00 381.98 618.02 31214.05

TheinterestrateforYazhen’sloanincreasedafteroneofthesesixrepaymentshadbeenmade.ThefirstrepaymentmadeatthehigherinterestratewasrepaymentnumberA. 15B. 16C. 17D. 18E. 19


END OF SECTION ATURN OVER

Use the following information to answer Questions 22 and 23.Armandborrowed$12000topayforaholiday.Hewillbechargedinterestattherateof6.12%perannum,compoundingmonthly.Thisloanwillberepaidwithmonthlyrepaymentsof$500.

Question 22Afterfourmonths,thetotalinterestthatArmandwillhavepaidisclosesttoA. $231B. $245C. $255D. $734E. $1796

Question 23Aftereightrepayments,Armanddecidedtoincreasethevalueofhismonthlyrepayments.Hewillmakeanumberofmonthlyrepaymentsof$850andthenonefinalrepaymentthatwillhaveasmallervalue.ThisfinalrepaymenthasavalueclosesttoA. $168B. $169C. $180D. $586E. $681

Question 24Robynhasacurrentbalanceof$347283.45inhersuperannuationaccount.Robyn’semployerdeposits$350intothisaccounteveryfortnight.Thisaccountearnsinterestattherateof2.5%perannum,compoundingfortnightly.Robynwillstopworkafter15yearsandwillnolongerreceivedepositsfromheremployer.Thebalanceofhersuperannuationaccountatthistimewillbeinvestedinanannuitythatwillpayinterestattherateof3.6%perannum,compoundingmonthly.After234monthlypaymentstherewillbenomoneyleftinRobyn’sannuity.ThevalueofRobyn’smonthlypaymentwillbeclosesttoA. $3993B. $5088C. $6664D. $8051E. $9045


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


SECTION B – Module 1 – continuedTURN OVER

Question 1ThenumberofindividualpointsscoredbyRhianna(R),Suzy(S),Tina(T),Ursula(U)andVicki(V)infivebasketballmatches(F,G,H,I,J)isshowninmatrixPbelow.

matchF G H I J

P

RSTUV

=

2 0 3 1 84 7 2 5 36 4 0 0 51 6 1 4 50 5 3 2 0

pplayer

Whoscoredthehighestnumberofpointsandinwhichmatch?A. SuzyinmatchIB. TinainmatchHC. VickiinmatchFD. UrsulainmatchGE. RhiannainmatchJ

Question 2Fourteams,blue(B),green(G),orange(O)andpink(P),playedeachotheronceinacompetition.Therewerenodrawsinthiscompetition.Theresultsofthecompetitionareshowninthematrixbelow.

Thelettersv,wandxeachhaveavalueof‘0’or‘1’.A‘1’inthematrixshowsthattheteamnamedinthatrowdefeatedtheteamnamedinthatcolumn.A‘0’inthematrixshowsthattheteamnamedinthatrowwasdefeatedbytheteamnamedinthatcolumn.Adash(–)inthematrixshowsthatnogamewasplayed.Thevaluesofv,wandx areA. v=0,w=1,x = 0B. v=0,w=1,x = 1C. v=1,w=0,x = 1D. v=1,w=1,x = 0E. v=1,w=1,x = 1

Module 1 – Matrices

Beforeansweringthesequestions,youmustshadethe‘Matrices’boxontheanswersheetfor multiple-choicequestionsandwritethenameofthemoduleintheboxprovided.


SECTION B – Module 1 – continued

Question 3InmatrixA,theelementa21=7.TheorderofmatrixAis3×2.MatrixBisthetransposeofmatrixA.MatrixBcouldbeA. 3 9 4

2 7 1

B. 4 7 21 0 5

C. 4 1 37 8 0

D. 5 67 14 2

E. 9 73 40 2

Question 4Inagamebetweentwoteams,HillsideandRovers,eachteamcanscorepointsintwoways.TheteammayhitaFullortheteammayhitaBit.MorepointsarescoredforhittingaFullthanforhittingaBit.Ateam’stotalpointscoreisthesumofthepointsscoredfromhittingFullsandBits.Thetablebelowshowsthescoresattheendofthegame.

Full Bit Total

Hillside 4 8 52

Rovers 5 2 49

Let f bethenumberofpointsscoredbyhittingoneFull.LetbbethenumberofpointsscoredbyhittingoneBit.Whichoneofthefollowingmatrixproductscanbeevaluatedtofindthematrix

fb

?

A. 4 85 2

×5249

–1 B. 4 85 2

×5249

C.52 49

4 85 2

[ ]

D.52 49

4 85 2

–1[ ]

E. 4 8 525 2 49

–1



Question 5Apopulationofbirdsfeedsattwodifferentlocations,AandB,onanisland.ThechangeinthepercentageofthebirdsateachlocationfromyeartoyearcanbedeterminedfromthetransitionmatrixTshownbelow.

this yearA B

TABnext year=

0.8 0.40.2 0.6

In2018,55%ofthebirdsfedatlocationB.In2019,thepercentageofthebirdsthatareexpectedtofeedatlocationAisA. 32%B. 42%C. 48%D. 58%E. 62%

Question 6MatrixWisa3×2matrix.MatrixQisamatrixsuchthatQ×W = W.MatrixQcouldbe

A. [1] B. 1 00 1

C. 1 11 11 1

D. 1 0 00 1 00 0 1

E. 1 1 11 1 11 1 1



Use the following information to answer Questions 7 and 8.Afarmcontainsfourwatersources,P,Q,RandS.

Question 7Cowsonthefarmarefreetomovebetweenthefourwatersources.Thechangeinthenumberofcowsateachofthesewatersourcesfromweektoweekisshowninthetransitiondiagrambelow.

0.25

0.500.60

0.50 0.50

0.250.20

0.20

0.20

0.20

0.10

0.10

0.10

0.15

0.10

0.05P Q

S R

LetCnbethestatematrixforthelocationofthecowsinweeknof2019.

Thestatematrixforthelocationofthecowsinweek23of2019isC

PQRS

23

180200240180

=

Thestatematrixforthelocationofthecowsinweek24of2019isC

PQRS

24

160222203215

=

OfthecowsexpectedtobeatQ inweek24of2019,thepercentageofthesecowsatRinweek23of2019isclosesttoA. 8%B. 9%C. 20%D. 22%E. 25%


End of Module 1 – SECTION B –continuedTURN OVER

Question 8Sheeponthefarmarealsofreetomovebetweenthefourwatersources.ThechangeinthenumberofsheepateachwatersourcefromweektoweekisshowninmatrixTbelow.

this weekP Q R S

T =

0 4 0 3 0 2 0 10 2 0 1 0 5 0 30 1 0 3 0 1 0 20 3 0 3 0

. . . .

. . . .

. . . .

. . .22 0 4.

PQRS

next week

Inthelongterm,635sheepareexpectedtobeatSeachweek.Inthelongterm,thenumberofsheepexpectedtobeatQeachweekisclosesttoA. 371B. 493C. 527D. 607E. 635



Question 1Thegraphbelowhasfiveverticesandeightedges.

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

Question 2Considerthegraphbelow.

Euler’sformulawillbeverifiedforthisgraph.Whatvaluesofe,vand f willbeusedinthisverification?A. e=5,v=5,f = 2B. e=5,v=5,f = 3C. e=6,v=5,f = 2D. e=6,v=5,f = 3E. e=6,v=6,f = 3

Module 2 – Networks and decision mathematics

Beforeansweringthesequestions,youmustshadethe‘Networksanddecisionmathematics’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.



Question 3Fourstudents,Alice,Brad,CharliandDexter,areworkingtogetheronaschoolproject.Thisprojecthasfourparts.Eachofthestudentswillcompleteonlyonepartoftheproject.Thetablebelowshowsthetimeitwouldtakeeachstudenttocompleteeachpartoftheproject,inminutes.

Part 1 Part 2 Part 3 Part 4

Alice 5 5 3 5

Brad 3 3 6 4

Charli 6 5 4 3

Dexter 4 5 6 5

Thepartsofthisprojectmustbecompletedoneaftertheother.Whichallocationofstudenttopartmustoccurforthisprojecttobecompletedintheminimumtimepossible?

A. Part 1 Part 2 Part 3 Part 4

Brad Dexter Alice Charli

B. Part 1 Part 2 Part 3 Part 4

Brad Dexter Charli Alice

C. Part 1 Part 2 Part 3 Part 4

Dexter Alice Charli Brad

D. Part 1 Part 2 Part 3 Part 4

Dexter Brad Alice Charli

E. Part 1 Part 2 Part 3 Part 4

Dexter Brad Charli Alice



Question 4Whichoneofthefollowingflowdiagramsshowsacutthathasacapacityof19?

7

55 922 6

1

5

8

4

7

10

7

55 922 6

1

5

8

4

7

10

7

55 922 6

1

5

8

4

7

10

7

55 922 6

1

5

8

4

7

10

7

55 922 6

1

5

8

4

7

10

sourcesink

A.

sourcesink

B.

sourcesink

D.

sourcesink

C.

sourcesink

E.



Question 5Inthegraphbelow,theverticesrepresentelectricitytransformersubstations.Thenumbersontheedgesofthegraphshowthelength,inkilometres,ofcablesthatconnectthesesubstations.

13

1114

12

11

9

8

107

811

107

Whatistheminimumlengthofcable,inkilometres,thatisnecessarytomakesurethateachsubstationremainsconnectedtothenetwork?A. 65B. 71C. 73D. 74E. 77

Question 6Fourgraphsareshownbelow.

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


End of Module 2 – SECTION B–continued

Question 7Agraphhasfivevertices,A,B,C,DandE.Theadjacencymatrixforthisgraphisshownbelow.

A B C D EABCDE

0 1 0 1 21 0 1 0 10 1 1 0 11 0 0 0 12 1 1 1 0

Whichoneofthefollowingstatementsaboutthisgraphisnottrue?A. Thegraphisconnected.B. ThegraphcontainsanEuleriantrail.C. ThegraphcontainsanEuleriancircuit.D. ThegraphcontainsaHamiltoniancycle.E. Thegraphcontainsaloopandmultipleedges.

Question 8Alandscapegardenerisplanningtheconstructionofagarden.Thisprojectrequires13activities,I to U.Thedirectedgraphbelowshowseachoftheseactivitiesrepresentedbyedges.Thenumbersontheedgesarethedurationsoftheactivities,indays.

start finish

I, 2

J, 3

M, 8

L, 4

N, 1K, 5

R, 5

Q, 7

P, 6

O, 6

U, 1

T, 9

S, 1

Thecostofreducingthecompletiontimeofanyactivityinthisprojectis$1000perday.Thelandscapegardenerhasamaximumof$3000tospendonreducingtheminimumcompletiontimeofthisproject.ThetotalcompletiontimeoftheprojectcanbereducedbythreedaysbyreducingA. activityKbyonedayandactivityQbytwodays.B. activityKbytwodaysandactivityQbyoneday.C. activityKbytwodaysandactivityMbyoneday.D. activityKbyonedayandactivityMbytwodays.E. activityKbyoneday,activityMbyonedayandactivityQbyoneday.



Question 1Apieceofcardboardisshowninthediagrambelow.

1 cm

5 cm

5 cm

Theareaofthecardboard,insquarecentimetres,isA. 4B. 5C. 21D. 25E. 29

Question 2WhichoneofthefollowinglocationsisclosesttotheGreenwichmeridian?A. 32°S,40°EB. 32°S,80°EC. 32°S,60°WD. 32°S,120°EE. 32°S,160°W

Question 3Awaterfallinanationalparkis4kmeastofacampsite.Alookouttoweris4kmsouthofthewaterfall.ThebearingofthecampsitefromthelookouttowerisA. 045°B. 090°C. 135°D. 300°E. 315°

Module 3 – Geometry and measurement

Beforeansweringthesequestions,youmustshadethe‘Geometryandmeasurement’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.



Question 4Apizzaintheshapeofacirclehasbeencutinto12equalslices.Theareaofthetopsurfaceofonesliceis9.42cm2,asshownshadedinthediagrambelow.

area = 9.42 cm2

Theperimeterofthetopsurfaceofonesliceofpizza,incentimetres,isclosesttoA. 9.14B. 15.14C. 18.00D. 21.99E. 40.84

Question 5ThecitiesofLimaandWashington,DChavethesamelongitudeof77°W.TheshortestgreatcircledistancebetweenLimaandWashington,DCis5697km.AssumethattheradiusofEarthis6400km.Limahasalatitudeof12°SandislocatedduesouthofWashington,DC.WhatisthelatitudeofWashington,DC?A. 39°NB. 51°SC. 51°ND. 63°NE. 65°S



Question 6Acakeintheshapeofthreecylindricalsectionsisshowninthediagrambelow.

8 cm

8 cm

8 cm

A

B

C

Eachsectionofthecakehasaheightof8cm,asshowninthediagram.Themiddlesectionofthecake,B,hastwicethevolumeofthetopsectionofthecake,A.Thebottomsectionofthecake,C,hastwicethevolumeofthemiddlesectionofthecake,B.Thevolumeofthetopsectionofthecake,A,is900cm3.Thediameterofthebottomsectionofthecake,C,incentimetres,isclosesttoA. 12B. 18C. 24D. 36E. 48

Question 7Twosimilarright-angledtrianglesareshowninthediagrambelow.

3 m

6 m

12 m

θ

θ

Thetotalperimeterofthetwotriangles,inmetres,isA. 21B. 30C. 36D. 42E. 45


End of Module 3 – SECTION B–continued

Question 8Ayoungtreeisprotectedbyatreeguardintheshapeofasquare-basedpyramid.Theheightofthetreeguardis54cm,asshowninthediagrambelow.

54 cm

Thetopsectionofthetreeguardisremovedalongthedottedlinetoallowthetreetogrow.Removingthistopsectiondecreasestheheightofthetreeguardto45cm,asshowninthediagrambelow.

45 cm

TheratioofthevolumeofthetreeguardthatisremovedtothevolumeofthetreeguardthatremainsisA. 1:6B. 1:35C. 1:36D. 1:215E. 1:216



Question 1Thegraphbelowshowsthetemperature,indegreesCelsius,over24hours.

temperature (°C)

time (hours)

40

30

20

10

10 15 2050

Thedifferencebetweenthehighestandlowesttemperaturesonthisday,indegreesCelsius,isclosesttoA. 16B. 20C. 24D. 28E. 32

Question 2Twostraightlineshavetheequations3x – 2y =3and−2x+5y =9.Theselineshaveonepointofintersection.AnotherlinethatalsopassesthroughthispointofintersectionhastheequationA. y = –x B. y = xC. y = –2xD. y = 2xE. y = 3x

Module 4 – Graphs and relations

Beforeansweringthesequestions,youmustshadethe‘Graphsandrelations’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.



Question 3Thegraphbelowshowsarelationshipbetweenxandy.

y

x2

109876

4321

5

5 6 7 8 94321 10O

Therulethatrepresentsthisrelationshipbetweenxandyis

A. y x=23

B. y x=32

C. y x=23

2

D. y x=32

2

E. y x=32

Question 4Afarmhasxcowsandysheep.Onthisfarmtherearealwaysatleasttwiceasmanysheepascows.Therelationshipbetweenthenumberofcowsandthenumberofsheeponthisfarmcanberepresentedbytheinequality

A. x y≤2

B. y x≤2

C. 2x≥y

D. 2y≥x

E. xy≥2

31 2019 FURMATH EXAM 1 (NHT)


Question 5The revenue, R, in dollars, that a company receives from selling n caps is given by the equation

Rn nn c n

=≤

+ >

25 100020 1000

The graph of this revenue equation consists of two straight lines that intersect at the point where n = 1000.What is the value of c in this revenue equation?A. 0B. 1000C. 2000D. 4000E. 5000

Question 6In January 2018, an online shop had 260 customer accounts.In June 2018, the shop had 500 customer accounts.The graph below shows the number of customer accounts, a, with the online shop n months after January 2018, for a period of 10 months.

a

n

400

500

600

700

800

300

200

100

05 6 7 8 94321 100

The growth in the number of customer accounts that this graph shows is expected to continue beyond these 10 months, following the same trend.How many customer accounts can the online shop expect to have at the end of December 2019 (n = 23)?A. 1364B. 1376C. 1388D. 1400E. 1412


END OF MULTIPLE-CHOICE QUESTION BOOK

Question 7Theshadedareainthegraphbelowrepresentsthefeasibleregionforalinearprogrammingproblem.

B

A

C

D

10

5

5 10

y

x

ThemaximumvalueoftheobjectivefunctionZ = –2x – 2yoccursatA. pointConly.B. anypointalonglinesegmentBC.C. anypointalonglinesegmentAD.D. anypointalonglinesegmentAB.E. anypointalonglinesegmentDC.

Question 8JamiesoldbottlesofhomemadelemonadetohisneighboursonSaturday.Therevenue,indollars,hemadefromsellingnbottlesoflemonadeisgivenby

revenue =3.5n

Thecost,indollars,ofmakingnbottlesoflemonadeisgivenby

cost =60+n

TheprofitmadebyJamieonSaturdaycouldhavebeenA. $162B. $165C. $168D. $173E. $177

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 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 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

victorian certificate of education 2019 · 2019 furmath exam 1 (nht) 6 section a continued question...

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