Victorian Certificate of Education 2018 · 2019-03-14 · 5 2018 FURMATH EXAM 1 SECTIN A –...

FURTHER MATHEMATICSWritten examination 1

Friday 2 November 2018 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• Questionbookof37pages• 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2018

Victorian Certificate of Education 2018

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TURN OVER

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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 and 2.Thedotplotbelowdisplaysthedifference in travel time betweenthemorningpeakandtheeveningpeaktraveltimesforthesamejourneyon25days.

n = 25

–5 0 5 10difference in travel time (minutes)

Question 1Thepercentageofdayswhentherewasfiveminutesdifference in travel time betweenthemorningpeakandtheeveningpeaktraveltimesisA. 0%B. 5%C. 20%D. 25%E. 28%

Question 2Themediandifference in travel timeisA. 3.0minutes.B. 3.5minutes.C. 4.0minutes.D. 4.5minutes.E. 5.0minutes.

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SECTION A – continuedTURN OVER

Use the following information to answer Questions 3–5.ThepulseratesofapopulationofYear12studentsareapproximatelynormallydistributedwithameanof69beatsperminuteandastandarddeviationof4beatsperminute.

Question 3Astudentselectedatrandomfromthispopulationhasastandardisedpulserateofz=–2.5Thisstudent’sactualpulserateisA. 59beatsperminute.B. 63beatsperminute.C. 65beatsperminute.D. 73beatsperminute.E. 79beatsperminute.

Question 4Anotherstudentselectedatrandomfromthispopulationhasastandardisedpulserateofz=–1.ThepercentageofstudentsinthispopulationwithapulserategreaterthanthisstudentisclosesttoA. 2.5%B. 5%C. 16%D. 68%E. 84%

Question 5Asampleof200studentswasselectedatrandomfromthispopulation.Thenumberofthesestudentswithapulserateoflessthan61beatsperminuteorgreaterthan 73beatsperminuteisclosesttoA. 19B. 37C. 64D. 95E. 190

Question 6Datawascollectedtoinvestigatetheassociationbetweenthefollowingtwovariables:• age(29andunder,30–59,60andover)• usespublictransport(yes,no)

Whichoneofthefollowingisappropriatetouseinthestatisticalanalysisofthisassociation?A. ascatterplotB. parallelboxplotsC. aleastsquareslineD. asegmentedbarchartE. thecorrelationcoefficient r

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Use the following information to answer Questions 7–9.Thescatterplotbelowdisplaystherestingpulserate, inbeatsperminute,andthetimespentexercising, inhoursperweek,of16students.Aleastsquareslinehasbeenfittedtothedata.

68676665646362616059585756

resting pulse rate(beats per minute)

time spent exercising (hours per week)1 2 3 4 5 6 7 8 9 10 11

Question 7Usingthisleastsquareslinetomodeltheassociationbetweenrestingpulserateandtimespentexercising,theresidualforthestudentwhospentfourhoursperweekexercisingisclosesttoA. –2.0beatsperminute.B. –1.0beatsperminute.C. –0.3beatsperminute.D. 1.0beatsperminute.E. 2.0beatsperminute.

Question 8TheequationofthisleastsquareslineisclosesttoA. restingpulserate=67.2–0.91×timespentexercisingB. restingpulserate=67.2–1.10×timespentexercisingC. restingpulserate=68.3–0.91×timespentexercisingD. restingpulserate=68.3–1.10×timespentexercising E. restingpulserate=67.2+1.10×timespentexercising

Question 9Thecoefficientofdeterminationis0.8339Thecorrelationcoefficientr isclosesttoA. –0.913B. –0.834C. –0.695D. 0.834E. 0.913

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Question 10Inastudyoftheassociationbetweenaperson’sheight,incentimetres, andbody surface area,insquaremetres,thefollowingleastsquareslinewasobtained.

body surface area=–1.1+0.019× height

Whichoneofthefollowingisaconclusionthatcanbemadefromthisleastsquaresline?A. Anincreaseof1m2inbody surface areaisassociatedwithanincreaseof0.019cmin height.B. Anincreaseof1cminheight isassociatedwithanincreaseof0.019m2inbody surface area.C. Thecorrelationcoefficientis0.019D. Aperson’sbody surface area,insquaremetres,canbedeterminedbyadding1.1cmtotheirheight.E. Aperson’sheight,incentimetres,canbedeterminedbysubtracting1.1fromtheirbody surface area,

insquaremetres.

Question 11Freyausesthefollowingdatatogeneratethescatterplotbelow.

x 1 2 3 4 5 6 7 8 9 10

y 105 48 35 23 18 16 12 12 9 9

120

100

80

60

40

20

00 1 2 3 4 5 6 7 8 9 10

y

x

Thescatterplotshowsthatthedataisnon-linear.Tolinearisethedata,Freyaappliesareciprocaltransformationtothevariabley.Shethenfitsaleastsquareslinetothetransformeddata.Withxastheexplanatoryvariable,theequationofthisleastsquareslineisclosestto

A. 1y =–0.0039+0.012x

B. 1y =–0.025+1.1x

C. 1y =7.8–0.082x

D. y=45.3+59.7× 1x

E. y=59.7+45.3×1x

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Question 12Alog10(y)transformationwasusedtolineariseasetofnon-linearbivariatedata.Aleastsquareslinewasthenfittedtothetransformeddata.Theequationofthisleastsquareslineis

log10(y)=3.1–2.3x

Thisequationisusedtopredictthevalueofywhenx=1.1Thevalueof yisclosesttoA. –0.24B. 0.57C. 0.91D. 1.6E. 3.7

Question 13 Thestatisticalanalysisofasetofbivariatedatainvolvingvariablesxand yresultedintheinformationdisplayedinthetablebelow.

Mean x =27.8 y =33.4

Standard deviation sx=2.33 sy=3.24

Equation of the least squares line

y =–2.84+1.31x

Usingthisinformation,thevalueofthecorrelationcoefficientrforthissetofbivariatedataisclosesttoA. 0.88B. 0.89C. 0.92D. 0.94E. 0.97

Question 14Aleastsquareslineisfittedtoasetofbivariatedata.Anotherleastsquareslineisfittedwithresponseandexplanatoryvariablesreversed.Whichoneofthefollowingstatisticswillnotchangeinvalue?A. theresidualvaluesB. thepredictedvaluesC. thecorrelationcoefficientrD. theslopeoftheleastsquareslineE. theinterceptoftheleastsquaresline

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Question 15Thetablebelowshowsthemonthlyprofit,indollars,ofanewcoffeeshopforthefirstninemonthsof2018.

Month Jan. Feb. Mar. Apr. May June July Aug. Sept.

Profit ($) 2890 1978 2402 2456 4651 3456 2823 2678 2345

Usingfour-meansmoothingwithcentring,thesmoothedprofitforMayisclosesttoA. $2502B. $3294C. $3503D. $3804E. $4651

Question 16Thequarterlysalesfiguresforalargesuburbangardencentre,inmillionsofdollars,for2016and2017aredisplayedinthetablebelow.

Year Quarter 1 Quarter 2 Quarter 3 Quarter 4

2016 1.73 2.87 3.34 1.23

2017 1.03 2.45 2.05 0.78

Usingthesesalesfigures,theseasonalindexforQuarter3isclosesttoA. 1.28B. 1.30C. 1.38D. 1.46E. 1.48

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Recursion and financial modelling

Use the following information to answer Questions 17 and 18.Thevalueofanannuityinvestment,indollars,afternyears,Vn ,canbemodelledbytherecurrencerelationshownbelow.

V0=46000, Vn+1=1.0034Vn+500

Question 17Whatisthevalueoftheregularpaymentaddedtotheprincipalofthisannuityinvestment?A. $34.00B. $156.40C. $466.00D. $500.00E. $656.40

Question 18Betweenthesecondandthirdyears,theincreaseinthevalueofthisinvestmentisclosesttoA. $656B. $658C. $661D. $1315E. $1975

Question 19Danielborrows$5000,whichheintendstorepayfullyinalumpsumafteroneyear.Theannualinterestrateandcompoundingperiodforfivedifferentcompoundinterestloansaregivenbelow:• LoanI –12.6%perannum,compoundingweekly• LoanII –12.8%perannum,compoundingweekly• LoanIII–12.9%perannum,compoundingweekly• LoanIV–12.7%perannum,compoundingquarterly• LoanV –13.2%perannum,compoundingquarterly

Whenfullyrepaid,theloanthatwillcostDanieltheleastamountofmoneyisA. LoanI.B. LoanII.C. LoanIII.D. LoanIV.E. LoanV.

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Question 20Thegraphbelowshowsthevalue,Vn,ofanassetasitdepreciatesoveraperiodoffivemonths.

0 1 2 3 4 5

value ($)

n (months)

Whichoneofthefollowingdepreciationsituationsdoesthisgraphbestrepresent?A. flatratedepreciationwithadecreaseindepreciationrateaftertwomonthsB. flatratedepreciationwithanincreaseindepreciationrateaftertwomonthsC. unitcostdepreciationwithadecreaseinunitsusedpermonthaftertwomonthsD. reducingbalancedepreciationwithanincreaseintherateofdepreciationaftertwomonthsE. reducingbalancedepreciationwithadecreaseintherateofdepreciationaftertwomonths

Question 21Whichoneofthefollowingrecurrencerelationscouldbeusedtomodelthevalueofaperpetuityinvestment,Pn,afternmonths?A. P0=120000, Pn+1=1.0029×Pn–356B. P0=180000, Pn+1=1.0047×Pn–846C. P0=210000, Pn+1=1.0071×Pn–1534D. P0=240000, Pn+1=0.0047×Pn – 2232E. P0=250000, Pn+1=0.0085×Pn–2125

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Question 22Adamhasahomeloanwithapresentvalueof$175260.56TheinterestrateforAdam’sloanis3.72%perannum,compoundingmonthly.Hismonthlyrepaymentis$3200.Theloanistobefullyrepaidafterfiveyears.Adamknowsthattheloancannotbeexactlyrepaidwith60repaymentsof$3200.Tosolvethisproblem,Adamwillmake59repaymentsof$3200.Hewillthenadjustthevalueofthefinalrepaymentsothattheloanisfullyrepaidwiththe60threpayment.Thevalueofthe60threpaymentwillbeclosesttoA. $368.12B. $2831.88C. $3200.56D. $3557.09E. $3568.12

Question 23Fivelinesofanamortisationtableforareducingbalanceloanwithmonthlyrepaymentsareshownbelow.

Repayment number Repayment Interest Principal

reduction Balance of loan

25 $2200.00 $972.24 $1227.76 $230256.78

26 $2200.00 $967.08 $1232.92 $229023.86

27 $2200.00 $961.90 $1238.10 $227785.76

28 $2200.00 $1002.26 $1197.74 $226588.02

29 $2200.00 $996.99 $1203.01 $225385.01

Theinterestrateforthisloanchangedimmediatelybeforerepaymentnumber28.ThischangeininterestrateisbestdescribedasA. anincreaseof0.24%perannum.B. adecreaseof0.024%perannum.C. anincreaseof0.024%perannum.D. adecreaseof0.0024%perannum.E. anincreaseof0.00024%perannum.

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END OF SECTION ATURN OVER

Question 24 Mariskaplanstoretirefromwork10yearsfromnow.Herretirementgoalistohaveabalanceof$600000inanannuityinvestmentatthattime.Thepresentvalueofthisannuityinvestmentis$265298.48,onwhichsheearnsinterestattherateof 3.24%perannum,compoundingmonthly.Tomakethisinvestmentgrowfaster,Mariskawilladda$1000paymentattheendofeverymonth.Twoyearsfromnow,sheexpectstheinterestrateofthisinvestmenttofallto3.20%perannum,compoundingmonthly.ItisexpectedtoremainatthisrateuntilMariskaretires.Whentheinterestratedrops,shemustincreasehermonthlypaymentifsheistoreachherretirementgoal.ThevalueofthisnewmonthlypaymentwillbeclosesttoA. $1234B. $1250C. $1649D. $1839E. $1854

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SECTION B –continuedTURN OVER

SECTION B – Modules

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

Contents Page

Module1–Matrices...................................................................................................................................... 16

Module2–Networksanddecisionmathematics.......................................................................................... 22

Module3–Geometryandmeasurement.......................................................................................................28

Module4–Graphsandrelations................................................................................................................... 32

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SECTION B – Module 1 – continued

Question 1Whichoneofthefollowingmatriceshasadeterminantofzero?

A. 0 11 0

B. 1 00 1

C. 1 2–3 6

D. 3 62 4

E. 4 00 –2

Question 2

Thematrixproduct 4 2 04

128

[ ]×

isequalto

A. 144[ ]

B. 16240

C. 4 1 2 01

128

×[ ]×

D. 2 2 1 0264

×[ ]×

E. 4 2 1 0264

×[ ]×

Module 1 – Matrices

Beforeansweringthesequestions,youmustshadethe‘Matrices’boxontheanswersheetfor multiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

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SECTION B – Module 1 – continuedTURN OVER

Question 3Fivepeople,India(I),Jackson(J),Krishna(K),Leanne(L)andMustafa(M),competedinatabletennistournament.Eachcompetitorplayedeveryothercompetitoronceonly.Eachmatchresultedinawinnerandaloser.Thematrixbelowshowsthetournamentresults.

loserI J K L M

winner

IJKLM

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

A‘1’inthematrixshowsthatthecompetitornamedinthatrowdefeatedthecompetitornamedinthatcolumn.Forexample,the‘1’inthefourthrowshowsthatLeannedefeatedJackson.A‘0’inthematrixshowsthatthecompetitornamedinthatrowlosttothecompetitornamedinthatcolumn.Thereisanerrorinthematrix.Thewinnerofoneofthematcheshasbeenincorrectlyrecordedasa‘0’.ThismatchwasbetweenA. IndiaandMustafa.B. IndiaandKrishna.C. KrishnaandLeanne.D. LeanneandMustafa.E. JacksonandMustafa.

Question 4MatrixPisa4×4permutationmatrix.MatrixWisanothermatrixsuchthatthematrixproductP ×Wisdefined.ThismatrixproductresultsintheentirefirstandthirdrowsofmatrixWbeingswapped.ThepermutationmatrixPis

A. 0 0 0 10 1 0 00 0 1 00 0 0 1

B. 0 0 1 00 1 0 01 0 0 00 0 0 1

C. 1 0 0 00 0 0 01 0 0 00 0 0 0

D. 1 0 0 00 0 0 00 0 1 00 0 0 0

E. 1 0 0 00 1 0 01 0 0 00 0 0 1

2018FURMATHEXAM1 18


Question 5Liamcycles,runs,swimsandwalksforexerciseseveraltimesamonth.Eachtimehecycles,Liamcoversadistanceofckilometres.Eachtimeheruns,Liamcoversadistanceofrkilometres.Eachtimeheswims,Liamcoversadistanceofskilometres.Eachtimehewalks,Liamcoversadistanceofwkilometres.ThenumberoftimesthatLiamcycled,ran,swamandwalkedeachmonthoverafour-monthperiod,andthetotaldistancethatLiamtravelledineachofthosemonths,areshowninthetablebelow.

Number of times in a month

Cycle Run Swim Walk Total distance for a month (km)

Month 1 5 7 6 8 160

Month 2 8 6 9 7 172

Month 3 7 8 7 6 165

Month 4 8 8 5 5 162

ThematrixthatcontainsthedistanceeachtimeLiamcycled,ran,swamandwalked,

crsw

, is

A. 5675

B. 8619

C. 8679

D. 8898

E. 4290493146234291

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Question 6Atransitionmatrix,V,isshownbelow.

this monthL T F M

V =

0 6 0 6 0 2 0 00 1 0 2 0 0 0 10 3 0 0 0 8 0 40 0 0 2

. . . .

. . . .

. . . .

. . 00 0 0 5. .

LTFM

next month

ThetransitiondiagrambelowhasbeenconstructedfromthetransitionmatrixV.Thelabellinginthetransitiondiagramisnotyetcomplete.

0.4

0.2 0.1

x

Theproportionforoneofthetransitionsislabelledx.ThevalueofxisA. 0.2B. 0.5C. 0.6D. 0.7E. 0.8

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Question 7A study of the antelope population in a wildlife park has shown that antelope regularly move between three locations, east (E), north (N) and west (W).Let An be the state matrix that shows the population of antelope in each location n months after the study began.The expected population of antelope in each location can be determined by the matrix recurrence rule

An + 1 = T An – D

where

this monthE N W

TENW

=

0 4 0 2 0 20 3 0 6 0 30 3 0 2 0 5

. . .

. . .

. . .next month

and

DENW

=

505050

The state matrix, A3 , below shows the population of antelope three months after the study began.

AENW

3

161628002134

=

The number of antelope in the west (W) location two months after the study began, as found in the state matrix A2 , is closest toA. 2060B. 2130C. 2200D. 2240E. 2270

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End of Module 1 – SECTION B –continuedTURN OVER

Question 8Apubliclibraryorganised500ofitsmembersintofivecategoriesaccordingtothenumberofbookseachmemberborrowseachmonth.Thesecategoriesare

J = nobooksborrowedpermonthK = onebookborrowedpermonthL = twobooksborrowedpermonthM = threebooksborrowedpermonthN = fourormorebooksborrowedpermonth

Thetransitionmatrix,T,belowshowshowthenumberofbooksborrowedpermonthbythemembersisexpectedtochangefrommonthtomonth.

this monthJ K L M N

T =

0 1 0 2 0 2 0 00 5 0 2 0 3 0 1 00 3 0 3 0 4 0 1 0 20 1

. . .

. . . .

. . . . .

. 00 2 0 1 0 6 0 30 0 1 0 0 2 0 5

. . . .

. . .

JKLMN

next month

Inthelongterm,whichcategoryisexpectedtohaveapproximately96memberseachmonth?A. JB. KC. LD. ME. N

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Question 1Considerthegraphwithfiveisolatedverticesshownbelow.

Toformatree,theminimumnumberofedgesthatmustbeaddedtothegraphisA. 1B. 4C. 5D. 6E. 10

Question 2Nikodrivesfromhishometouniversity.Thenetworkbelowshowsthedistances,inkilometres,alongaseriesofstreetsconnectingNiko’shometotheuniversity.TheverticesA,B,C,DandErepresenttheintersectionofthesestreets.

2

23 3

1

22

5

home

university

4B

2A

CD

E

TheshortestpathforNikofromhishometotheuniversitycouldbefoundusingA. aminimumcut.B. Prim’salgorithm.C. Dijkstra’salgorithm.D. criticalpathanalysis.E. theHungarianalgorithm.

Module 2 – Networks and decision mathematics

Beforeansweringthesequestions,youmustshadethe‘Networksanddecisionmathematics’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

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Question 3Aplanargraphhasfivefaces.ThisgraphcouldhaveA. eightverticesandeightedges.B. sixverticesandeightedges. C. eightverticesandfiveedges.D. eightverticesandsixedges.E. fiveverticesandeightedges.

Question 4Considerthegraphbelow.

Q

SRP

T

Whichoneofthefollowingisnotapathforthisgraph?A. PRQTSB. PQRTSC. PRTSQD. PTQSRE. PTRQS

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Question 5Thedirectednetworkbelowshowsthesequenceof11activitiesthatareneededtocompleteaproject.Thetime,inweeks,thatittakestocompleteeachactivityisalsoshown.

A, 3

D, 7

H, 2E, 2

I, 5

J, 2F, 6

B, 4

G, 1

C, 4

K, 3start finish

Howmanyoftheseactivitiescouldbedelayedwithoutaffectingtheminimumcompletiontimeoftheproject?A. 3B. 4C. 5D. 6E. 7

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Question 6 Whichoneofthefollowinggraphsisnotaplanargraph?

A. B.

C.

E.

D.

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Question 7Aprojectrequiresnineactivities(A–I)tobecompleted.Theduration,inhours,andtheimmediatepredecessor(s)ofeachactivityareshowninthetablebelow.

Activity Duration (hours) Immediate predecessor(s)

A 4 –

B 3 A

C 7 A

D 2 A

E 5 B

F 2 C

G 4 E, F

H 5 D

I 3 G, H

Theminimumcompletiontimeforthisproject,inhours,isA. 14B. 19C. 20D. 24E. 35

27 2018 FURMATH EXAM 1

End of Module 2 – SECTION B – continuedTURN OVER

Question 8Annie, Buddhi, Chuck and Dorothy work in a factory.Today each worker will complete one of four tasks, 1, 2, 3 and 4.The usual completion times for Annie, Chuck and Dorothy are shown in the table below.

Task 1 Task 2 Task 3 Task 4

Annie 7 3 8 2

Buddhi k k 3 k

Chuck 5 6 9 2

Dorothy 4 8 5 3

Buddhi takes 3 minutes for Task 3.He takes k minutes for each other task.Today the factory supervisor allocates the tasks as follows:• Task1toDorothy• Task2toAnnie• Task3toBuddhi• Task4toChuck

This allocation will achieve the minimum total completion time if the value of k is at leastA. 0B. 1C. 2D. 3E. 4

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Question 1Henryfliesakiteattachedtoalongstring,asshowninthediagrambelow.

string15 m

8 m

ThehorizontaldistanceofthekitetoHenry’shandis8m.TheverticaldistanceofthekiteaboveHenry’shandis15m.Thelengthofthestring,inmetres,isA. 13B. 17C. 23D. 161E. 289

Module 3 – geometry and measurement

Beforeansweringthesequestions,youmustshadethe‘Geometryandmeasurement’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

29 2018FURMATHEXAM1


Question 2AtrianglePQRisshowninthediagrambelow.

Q

P R

18 cm

26 cm30°

Thelengthoftheside QRis 18cm.The lengthoftheside PRis 26cm.TheangleQRPis30°.TheareaoftrianglePQR,insquarecentimetres,isclosesttoA. 117B. 162C. 171D. 234E. 468

Question 3ThecityofKarachiinPakistanhaslatitude25°Nandlongitude67°E.AssumethattheradiusofEarthis6400km.TheshortestdistancealongthesurfaceofEarthbetweenKarachiandtheNorthPole,inkilometres,canbefoundbyevaluatingwhichoneofthefollowingproducts?

A. 23

1806400× ×π

B. 25180

6400× ×π

C. 65

1806400× ×π

D. 67

1806400× ×π

E. 23

3606400× ×π

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Question 4Thecourseforayachtraceistriangularinshapeandismarkedbythreebuoys,T,U andV.

U

V

30°

north

T 69°

47°

StartingfrombuoyT,theyachtssailonabearingof030°tobuoyU.FrombuoyUtheyachtssailtobuoyV andthentobuoyT.TheangleUTV is69°andtheangleUVT is47°.ThebearingofbuoyU frombuoyVisA. 034°B. 047°C. 133°D. 279°E. 326°

Question 5OnthefirstdayofJune2019inthecityofStPetersburg,Russia,(60°N,30°E)thesunisexpectedtorise at3.48am.Onthissameday,thesunisexpectedtoriseinHelsinki,Finland,(60°N,25°E)approximatelyA. atthesametime.B. 20minutesearlier.C. 20minuteslater.D. onehourearlier.E. onehourlater.

Question 6Aaliyahisbushwalking.Shewalks5.4kmfromastartingpointonabearingof045°untilshereachesahut.Fromthishut,shewalks2.8kmonabearingof300°untilshereachesariver.Fromtheriver,sheturnsandwalksbackdirectlytothestartingpoint.Thetotaldistancethatshewalks,inkilometres,isclosesttoA. 8.2B. 13.2C. 13.6D. 14.1E. 14.9

31 2018 FURMATH EXAM 1

End of Module 3 – SECTION B – continuedTURN OVER

Question 7A windscreen wiper blade can clean a large area of windscreen glass, as shown by the shaded area in the diagram below.

110°

30 cm

9 cm

The windscreen wiper blade is 30 cm long and it is attached to a 9 cm long arm.The arm and blade move back and forth in a circular arc with an angle of 110° at the centre.The area cleaned by this blade, in square centimetres, is closest toA. 786B. 1382C. 2573D. 2765E. 4524

Question 8A cone with a radius of 2.5 cm is shown in the diagram below.

2.5 cm

x

The slant edge, x, of this cone is also shown.The volume of this cone is 36 cm3.The surface area of this cone, including the base, can be found using the rule surface area = πr(r + x).The total surface area of this cone, including the base, in square centimetres, is closest toA. 20B. 42C. 63D. 67E. 90

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Question 1Thegraphbelowshowsalineintersectingthex-axisat(4,0)andthey-axisat(0,2).

y

x

(0, 2)

(4, 0)O

ThegradientofthislineisA. –4

B. –2

C. −12

D. 12

E. 4

Module 4 – graphs and relations

Beforeansweringthesequestions,youmustshadethe‘Graphsandrelations’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

33 2018FURMATHEXAM1


Question 2Stevenisaweddingphotographer.Hechargeshisclientsafixedfeeof$500,plus$250perhourofphotography.Theequationthatrepresentsthetotalamount,$C,Stevencharges,forthoursofphotographyisA. C=250tB. C=500tC. C=750tD. C=500+250tE. C=250+500t

Question 3Thethreeinequalitiesbelowwereusedtoconstructthefeasibleregionforalinearprogrammingproblem.

x < 3 y < 6x+y > 6

ApointthatlieswithinthisfeasibleregionisA. (0,6)B. (1,5)C. (2,4)D. (2,5)E. (3,6)

2018FURMATHEXAM1 34


Question 4Ateamoffourstudentscompetesina4×100mrelayrace.Eachstudentintheraceruns100m.Theorderinwhicheachstudentrunsintheraceisshowninthetablebelow.

Order Name

first Joanne

second Elle

third Sam

fourth Kristen

Thefollowingline-segmentgraphrepresentstheraceforJoanne,SamandKristen,wheredisthedistance,inmetres,fromthestartingpointandtistherecordedtime,inseconds.Elle’slinesegmentismissing.

(81, 400)

(61, 300)

(45, 200)

(20, 100)

distance from starting point (m)

time (s)

400

300

200

100

O 10 20 30 40 50 60 70 80 90

TheequationrepresentingElle’slinesegmentisA. d = 4tB. d = 4t+20C. d=5tD. d=5t–5E. d=6.25t–81.25

35 2018FURMATHEXAM1


Question 5Thegraphbelowshowsarelationshipbetweenyand 1

2x.

(1, 2)

12x

y

O

Thegraphthatshowsthesamerelationshipbetweenyandxis

y

(1, 2)

x

y

(2, 1)

x

y

(2, 4)

x

y

(4, 1)

x

y

(4, 2)

x

A. B.

C. D.

E.

O O

O

O

O

2018 FURMATH EXAM 1 36


Question 6Amy makes and sells quilts.The fixed cost to produce the quilts is $800.Each quilt costs an additional $35 to make.Amy made and sold a batch of 80 quilts for a profit of $1200.The selling price of each quilt wasA. $15.00B. $22.50C. $30.00D. $45.00E. $60.00

Question 7In the diagram below, the shaded region (with boundaries included) represents the feasible region for a linear programming problem.

M

N

10

5x

y

O

The objective function, Z, has minimum values at both point M and point N.Which one of the following could be the objective function?A. Z = x – 2yB. Z = x + 2yC. Z = 2x + yD. Z = –2x + yE. Z = 2x + 2y

37 2018FURMATHEXAM1

END OF MULTIPLE-CHOICE QUESTION BOOK

Question 8Aride-sharecompanyhasafeethatincludesafixedcostandacostthatdependsonboththetimespenttravelling,inminutes,andthedistancetravelled,inkilometres.Thefixedcostofarideis$2.55Judy’sridecost$16.75andtookeightminutes.Thedistancetravelledwas10km.Pat’sridecost$30.35andtook20minutes.Thedistancetravelledwas18km.Roy’sridetook10minutes.Thedistancetravelledwas15km.ThecostofRoy’sridewasA. $17.00B. $19.55C. $20.50D. $23.05E. $25.60

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 2018

Victorian Certificate of Education 2018

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

Victorian Certificate of Education 2018 · 2019-03-14 · 5 2018 FURMATH EXAM 1 SECTIN A –...

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