FURTHER MATHEMATICSWritten examination 1

Friday 3 November 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• Questionbookof34pages• 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 2

THIS PAgE IS BLANK

3 2017FURMATHEXAM1

TURN OVER

THIS PAgE IS BLANK

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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 1–3.Theboxplotbelowshowsthedistributionoftheforearmcircumference,incentimetres,of252people.

20 22 24 26 28circumference (cm)

30 32 34 36 38

Question 1Thepercentageofthese252peoplewithaforearmcircumferenceoflessthan30cmisclosesttoA. 15%B. 25%C. 50%D. 75%E. 100%

Question 2Thefive-numbersummaryfortheforearmcircumference ofthese252peopleisclosesttoA. 21, 27.4, 28.7, 30, 34B. 21, 27.4, 28.7, 30, 35.9C. 24.5, 27.4, 28.7, 30, 34D. 24.5, 27.4, 28.7, 30, 35.9E. 24.5, 27.4, 28.7, 30, 36

5 2017FURMATHEXAM1

SECTION A – continuedTURN OVER

Question 3Thetablebelowshowstheforearmcircumference,incentimetres,ofasampleof10peopleselectedfromthisgroupof252people.

Circumference 26.0 27.8 28.4 25.9 28.3 31.5 28.2 25.9 27.9 27.8

Themean,x,andthestandarddeviation,sx,oftheforearmcircumferenceforthissampleofpeopleare closesttoA. x =1.58 sx =27.8B. x =1.66 sx =27.8C. x =27.8 sx =1.58D. x =27.8 sx =1.66E. x =27.8 sx =2.30

Question 4Thehistogrambelowshowsthedistributionofthelog10(area),withareainsquarekilometres,of17islands.

4

3

2

1

O 1 2 3log10 (area)

4 5 6

frequency

Themedianareaoftheseislands,insquarekilometres,isbetweenA. 2and3B. 3and4C. 10and100D. 1000and10000E. 10000and100000

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SECTION A – continued

Use the following information to answer Questions 5–7.Astudywasconductedtoinvestigatetheassociationbetweenthenumber of moths caughtinamothtrap (lessthan250,250–500,morethan500)andthetrap type(sugar,scent,light).Theresultsaresummarisedinthepercentagedsegmentedbarchartbelow.

Number of moths

more than 500

250–500

less than 250

1009080706050percentage403020100

sugar scenttrap type

light

Question 5Therewere300sugartraps.Thenumberofsugartrapsthatcaughtlessthan250mothsisclosesttoA. 30B. 90C. 250D. 300E. 500

Question 6Thedatadisplayedinthepercentagedsegmentedbarchartsupportsthecontentionthatthereisanassociationbetweenthenumber of mothscaughtinamothtrapandthetrap typebecauseA. mostofthelighttrapscontainedlessthan250moths.B. 15%ofthescenttrapscontained500ormoremoths.C. thepercentageofsugartrapscontainingmorethan500mothsisgreaterthanthepercentageof

scenttrapscontaininglessthan500moths.D. 20%ofsugartrapscontainedmorethan500mothswhile50%oflighttrapscontainedlessthan

250moths.E. 20%ofsugartrapscontainedmorethan500mothswhile10%oflighttrapscontainedmorethan

500moths.

Question 7Thevariablesnumber of moths (lessthan250,250–500,morethan500)andtrap type(sugar,scent,light)areA. bothnominalvariables.B. bothordinalvariables.C. anumericalvariableandacategoricalvariablerespectively.D. anominalvariableandanordinalvariablerespectively.E. anordinalvariableandanominalvariablerespectively.

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SECTION A – continuedTURN OVER

Use the following information to answer Questions 8–10.Thescatterplotbelowshowsthewristcircumference and anklecircumference,bothincentimetres,of13people.Aleastsquareslinehasbeenfittedtothescatterplotwith ankle circumferenceastheexplanatoryvariable.

20

19

18wrist

circumference(cm)

17

1621 22 23 24

ankle circumference (cm)25 26

Question 8TheequationoftheleastsquareslineisclosesttoA. ankle =10.2+0.342×wristB. wrist =10.2+0.342×ankleC. ankle =17.4+0.342×wristD. wrist =17.4+0.342×ankleE. wrist =17.4+0.731×ankle

Question 9Whentheleastsquareslineonthescatterplotisusedtopredictthewristcircumferenceofthepersonwithananklecircumferenceof24cm,theresidualwillbeclosesttoA. –0.7B. –0.4C. –0.1D. 0.4E. 0.7

Question 10Theresidualsforthisleastsquareslinehaveameanof0.02cmandastandarddeviationof0.4cm.Thevalueoftheresidualforoneofthedatapointsisfoundtobe–0.3cm.ThestandardisedvalueofthisresidualisA. –0.8B. –0.7C. –0.3D. 0.7E. 0.8

2017FURMATHEXAM1 8

SECTION A – continued

Question 11 Whichoneofthefollowingstatisticscanneverbenegative?A. themaximumvalueinadatasetB. thevalueofaPearsoncorrelationcoefficientC. thevalueofamovingmeaninasmoothedtimeseriesD. thevalueofaseasonalindexE. thevalueoftheslopeofaleastsquareslinefittedtoascatterplot

Question 12 Datacollectedoveraperiodof10yearsindicatedastrong,positiveassociationbetweenthenumberofstraycatsandthenumberofstraydogsreportedeachyear(r =0.87)inalarge,regionalcity.Apositiveassociationwasalsofoundbetweenthepopulationofthecityandboththenumberofstraycats (r=0.61)andthenumberofstraydogs(r=0.72).Duringthetimethatthedatawascollected,thepopulationofthecitygrewfrom34564to51055.Fromthisinformation,wecanconcludethatA. ifcatownerspaidmoreattentiontokeepingdogsofftheirproperty,thenumberofstraycatsreported

woulddecrease.

B. theassociationbetweenthenumberofstraycatsandstraydogsreportedcannotbecausalbecauseonlyacorrelationof+1or–1showscausalrelationships.

C. thereisnologicalexplanationfortheassociationbetweenthenumberofstraycatsandstraydogsreportedinthecitysoitmustbeachanceoccurrence.

D. becauselargerpopulationstendtohavebothalargernumberofstraycatsandstraydogs,theassociationbetweenthenumberofstraycatsandthenumberofstraydogscanbeexplainedbyacommonresponsetoathirdvariable,whichistheincreasingpopulationsizeofthecity.

E. morestraycatswerereportedbecausepeoplearenolongerascarefulaboutkeepingtheircatsproperlycontainedontheirpropertyastheywereinthepast.

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SECTION A – continuedTURN OVER

CONTINUES OVER PAgE

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SECTION A – continued

Use the following information to answer Questions 13–15.Thewindspeedatacitylocationismeasuredthroughouttheday.Thetimeseriesplotbelowshowsthedailymaximum wind speed,inkilometresperhour,overathree-weekperiod.

50

45

40

35maximumwind speed

(km/h) 30

25

20

150 7

day14 21

Question 13ThetimeseriesisbestdescribedashavingA. seasonalityonly.B. irregularfluctuationsonly.C. seasonalitywithirregularfluctuations.D. adecreasingtrendwithirregularfluctuations.E. anincreasingtrendwithirregularfluctuations.

Question 14Theseven-mediansmoothedmaximum wind speed,inkilometresperhour,forday4isclosesttoA. 22B. 26C. 27D. 30E. 32

11 2017 FURMATH EXAM 1

SECTION A – continuedTURN OVER

Question 15The table below shows the daily maximum wind speed, in kilometres per hour, for the days in week 2.

Day 8 9 10 11 12 13 14

Maximum wind speed (km/h) 22 22 19 22 43 37 33

A four-point moving mean with centring is used to smooth the time series data above. The smoothed maximum wind speed, in kilometres per hour, for day 11 is closest toA. 22B. 24C. 26D. 28E. 30

Question 16The seasonal index for the sales of cold drinks in a shop in January is 1.6To correct the January sales of cold drinks for seasonality, the actual sales should beA. reduced by 37.5%B. reduced by 40%C. reduced by 62.5%D. increased by 60%E. increased by 62.5%

2017 FURMATH EXAM 1 12

SECTION A – continued

Recursion and financial modelling

Question 17The value of a reducing balance loan, in dollars, after n months, Vn , can be modelled by the recurrence relation shown below.

V0 = 26 000, Vn + 1 = 1.003 Vn – 400

What is the value of this loan after five months?A. $24 380.31B. $24 706.19C. $25 031.10D. $25 355.03E. $25 678.00

Question 18The first five terms of a sequence are 2, 6, 22, 86, 342 …The recurrence relation that generates this sequence could beA. P0 = 2, Pn + 1 = Pn + 4B. P0 = 2, Pn + 1 = 2 Pn + 2C. P0 = 2, Pn + 1 = 3 PnD. P0 = 2, Pn + 1 = 4 Pn – 2E. P0 = 2, Pn + 1 = 5 Pn – 4

Use the following information to answer Questions 19 and 20.Shirley would like to purchase a new home. She will establish a loan for $225 000 with interest charged at the rate of 3.6% per annum, compounding monthly.Each month, Shirley will pay only the interest charged for that month.

Question 19After three years, the amount that Shirley will owe isA. $73 362B. $170 752C. $225 000D. $239 605E. $245 865

Question 20Let Vn be the value of Shirley’s loan, in dollars, after n months.A recurrence relation that models the value of Vn isA. V0 = 225 000, Vn + 1 = 1.003 VnB. V0 = 225 000, Vn + 1 = 1.036 VnC. V0 = 225 000, Vn + 1 = 1.003 Vn – 8100D. V0 = 225 000, Vn + 1 = 1.003 Vn – 675E. V0 = 225 000, Vn + 1 = 1.036 Vn – 675

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SECTION A – continuedTURN OVER

Question 21Aprinterwaspurchasedfor$680.Afterfouryearstheprinterhasavalueof$125.Onaverage,1920pageswereprintedeveryyearduringthosefouryears.Thevalueoftheprinterwasdepreciatedusingaunitcostmethodofdepreciation.Thedepreciationinthevalueoftheprinter,perpageprinted,isclosesttoA. 3cents.B. 4cents.C. 5cents.D. 6cents.E. 7cents.

Question 22Considerthegraphbelow.

(0, 7000)

(1, 6580)

(2, 6185.20)(3, 5814.09)

(4, 5465.24)

value ($)

years

ThisgraphcouldshowthevalueofA. apianodepreciatingataflatrateof6%perannum.B. acardepreciatingwithareducingbalancerateof6%perannum.C. acompoundinterestinvestmentearninginterestattherateof6%perannum.D. aperpetuityearninginterestattherateof6%perannum.E. anannuityinvestmentwithadditionalpaymentsof6%oftheinitialinvestmentamountperannum.

2017FURMATHEXAM1 14

END OF SECTION A

Question 23Fourlinesofanamortisationtableforanannuityinvestmentareshownbelow.Theinterestrateforthisinvestmentremainsconstant,butthepaymentvaluemayvary.

Payment number

Payment Interest Principal addition

Balance of investment

17 100.00 27.40 127.40 6977.50

18 100.00 27.91 127.91 7105.41

19 100.00 28.42 128.42 7233.83

20 7500.00

Thebalanceoftheinvestmentafterpaymentnumber20is$7500.Thevalueofpaymentnumber20isclosesttoA. $29B. $100C. $135D. $237E. $295

Question 24Xavierborrowed$245000topayforahouse.Forthefirst10yearsoftheloan,theinterestratewas4.35%perannum,compoundingmonthly.Xaviermademonthlyrepaymentsof$1800.After10years,theinterestratechanged.IfXaviernowmakesmonthlyrepaymentsof$2000,hecouldrepaytheloaninafurtherfiveyears.ThenewannualinterestrateforXavier’sloanisclosesttoA. 0.35%B. 4.1%C. 4.5%D. 4.8%E. 18.7%

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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.......................................................................................... 20

Module3–Geometryandmeasurement....................................................................................................... 26

Module4–Graphsandrelations................................................................................................................... 30

2017FURMATHEXAM1 16

SECTION B – Module 1 – continued

Question 1Kaihasapart-timejob.Eachweek,heearnsmoneyandsavessomeofthismoney.Thematrixbelowshowstheamountsearned(E)andsaved(S),indollars,ineachofthreeweeks.

E Sweekweekweek

123

300 100270 90240 80

HowmuchdidKaisaveinweek2?A. $80B. $90C. $100D. $170E. $270

Question 2Thematrixbelowshowshowfivepeople,Alan(A),Bevan(B),Charlie(C),Drew(D)andEsther(E),cancommunicatewitheachother.

receiverA B C D E

sender

ABCDE

0 1 0 1 01 0 0 0 00 0 0 1 11 0 1 0 00 0 1 0 0

A‘1’inthematrixshowsthatthepersonnamedinthatrowcansendamessagedirectlytothepersonnamedinthatcolumn.Forexample,the‘1’inrow3andcolumn4showsthatCharliecansendamessagedirectlytoDrew.EstherwantstosendamessagetoBevan.Whichoneofthefollowingshowstheorderofpeoplethroughwhichthemessageissent?A. Esther–BevanB. Esther–Charlie–BevanC. Esther–Charlie–Alan–BevanD. Esther–Charlie–Drew–BevanE. Esther–Charlie–Drew–Alan–Bevan

Module 1 – Matrices

Beforeansweringthesequestions,youmustshadethe‘Matrices’boxontheanswersheetfor multiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

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SECTION B – Module 1 – continuedTURN OVER

Question 3Whichoneofthefollowingmatrixequationshasauniquesolution?

6 –64 4

6036−

=

xy

1 11 1

210

=

xy

7 05 0

1415

=

xy

4 –26 –3

3624

=

xy

A. B.

C. D.

E.

8 –44 2

1218

=

xy

Question 4Apermutationmatrix,P,canbeusedtochange

FEARS

SAFER

into .

MatrixPis

A. 0 0 1 0 10 0 1 1 01 1 0 0 00 1 0 0 11 0 0 1 0

B. 0 0 0 1 00 0 1 0 00 1 0 0 00 0 0 0 11 0 0 0 0

C. 0 0 0 0 10 0 1 0 01 0 0 0 00 1 0 0 00 0 0 1 0

D. 1 0 0 0 10 1 1 0 01 0 1 0 00 1 0 1 00 0 0 1 1

E. 0 0 0 0 10 0 1 0 00 1 0 0 01 0 0 0 00 0 0 1 0

2017FURMATHEXAM1 18

SECTION B – Module 1 – continued

Question 5Fourteams,A,B,CandD,competedinaround-robincompetitionwhereeachteamplayedeachoftheotherteamsonce.Therewerenodraws.Theresultsareshowninthematrixbelow.

loserA B C D

winner

ABCD

f

gh

0 0 11 0 0 01 0 10 1 0

A‘1’inthematrixshowsthattheteamnamedinthatrowdefeatedtheteamnamedinthatcolumn.Forexample,the‘1’inrow2showsthatteamBdefeatedteamA.Inthismatrix,thevaluesof f,gandh areA. f=0, g=1, h = 0B. f=0, g=1, h = 1C. f=1, g=0, h = 0D. f=1, g=1, h = 0E. f=1, g=1, h = 1

Question 6Thetablebelowshowsinformationabouttwomatrices,AandB.

Matrix Order Rule

A 3×3 ai j = 2i+j

B 3×3 bi j = i – j

TheelementinrowiandcolumnjofmatrixAisai j.TheelementinrowiandcolumnjofmatrixBisbi j.ThesumA+BisA. 5 7 9

8 10 1211 13 15

B. 5 8 117 10 139 12 15

C. 3 6 93 6 93 6 9

D. 3 3 36 6 69 9 9

E. 3 6 36 3 93 9 3

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End of Module 1 – SECTION B–continuedTURN OVER

Question 7Atafishfarm:• youngfish(Y)mayeventuallygrowintojuveniles(J)ortheymaydie(D)• juveniles(J)mayeventuallygrowintoadults(A)ortheymaydie(D)• adults(A)eventuallydie(D).Theinitialstateofthispopulation, F0,isshownbelow.

F

YJAD

0

50000100007000

0

=

Everymonth,fishareeithersoldorboughtsothatthenumberofyoung,juvenileandadultfishinthefarmremainsconstant.Thepopulationoffishinthefishfarmafternmonths,Fn,canbedeterminedbytherecurrencerule

F Fn+ =

1

0 65 0 0 00 25 0 75 0 0

0 0 20 0 95 00 10 0 05 0 05 1

.

. .. .

. . .

nn B+

whereBisacolumnmatrixthatshowsthenumberofyoung,juvenileandadultfishboughtorsoldeachmonthandthenumberofdeadfishthatareremoved.Eachmonth,thefishfarmwillA. sell1650adultfish.B. buy1750adultfish.C. sell17500youngfish.D. buy50000youngfish.E. buy10000juvenilefish.

Question 8Considerthematrixrecurrencerelationbelow.

S S TS TVW

X Y Zn n0 1

401520

0 3 0 20 2 0 2=

= =

+,. .. .where

MatrixTisaregulartransitionmatrix.

Giventheaboveandthat S1

291333

=

,whichoneofthefollowingexpressionsisnottrue?

A. W > ZB. Y > XC. V > YD. V+W+Z = 1E. X+Y+Z > 1

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SECTION B – Module 2 – continued

Question 1Whichoneofthefollowinggraphscontainsaloop?

A. B.

C. D.

E.

Module 2 – Networks and decision mathematics

Beforeansweringthesequestions,youmustshadethe‘Networksanddecisionmathematics’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

21 2017FURMATHEXAM1

SECTION B – Module 2 – continuedTURN OVER

Question 2Twographs,labelledGraph1andGraph2,areshownbelow.

Graph 1 Graph 2

ThesumofthedegreesoftheverticesofGraph1isA. twolessthanthesumofthedegreesoftheverticesofGraph2.B. onelessthanthesumofthedegreesoftheverticesofGraph2.C. equaltothesumofthedegreesoftheverticesofGraph2.D. onemorethanthesumofthedegreesoftheverticesofGraph2.E. twomorethanthesumofthedegreesoftheverticesofGraph2.

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SECTION B – Module 2 – continued

Question 3Considerthefollowinggraph.

Z

XY

W

Theadjacencymatrixforthisgraph,withsomeelementsmissing,isshownbelow.

W X Y ZWXYZ

10

01

_ _ __ _ __ _ __ _ _

Thisadjacencymatrixcontains16elementswhencomplete.Ofthe12missingelementsA. eightare‘1’andfourare‘2’.B. fourare‘1’andeightare‘2’.C. sixare‘1’andsixare‘2’.D. twoare‘0’,sixare‘1’andfourare‘2’.E. fourare‘0’,fourare‘1’andfourare‘2’.

23 2017FURMATHEXAM1

SECTION B – Module 2 – continuedTURN OVER

Use the following information to answer Questions 4 and 5.Thedirectedgraphbelowshowsthesequenceofactivitiesrequiredtocompleteaproject.Thetimetocompleteeachactivity,inhours,isalsoshown.

A, 10

B, 4 E, 4 M, 4

O, 1

N, 3

H, 3

K, 1

J, 5G, 3

F, 6C, 4

I, 2

D, 5L, 6

start finish

Question 4Theearlieststartingtime,inhours,foractivityNisA. 3B. 10C. 11D. 12E. 13

Question 5Tocompletetheprojectinminimumtime,someactivitiescannotbedelayed.ThenumberofactivitiesthatcannotbedelayedisA. 2B. 3C. 4D. 5E. 6

2017FURMATHEXAM1 24

SECTION B – Module 2 – continued

Question 6

AnEuleriantrailforthegraphabovewillbepossibleifonlyoneedgeisremoved.Inhowmanydifferentwayscouldthisbedone?A. 1B. 2C. 3D. 4E. 5

Question 7 Agraphwithsixverticeshasnoloopsormultipleedges.Whichoneofthefollowingstatementsaboutthisgraphisnot true?A. Ifthegraphisatreeithasfiveedges.B. Ifthegraphiscompleteithas15edges.C. Ifthegraphhaseightedgesitmayhaveanisolatedvertex.D. Ifthegraphisbipartiteitwillhaveaminimumofnineedges.E. Ifthegraphhasacycleitwillhaveaminimumofthreeedges.

25 2017FURMATHEXAM1

End of Module 2 – SECTION B–continuedTURN OVER

Question 8 Theflowofoilthroughaseriesofpipelines,inlitresperminute,isshowninthenetworkbelow.

11

7

7

2

3

3

15

10

10

1012

35

6x

x

x 5 9

4

98

source

sink

Cut A

Cut B Cut CCut D Cut E

Theweightingsofthreeoftheedgesarelabelledx.FivecutslabelledA–Eareshownonthenetwork.Themaximumflowofoilfromthesourcetothesink,inlitresperminute,isgivenbythecapacityofA. CutAifx = 1B. CutBifx = 2C. CutCifx = 2D. CutDifx = 3E. CutEifx = 3

2017FURMATHEXAM1 26

SECTION B – Module 3 – continued

Question 1Awheelhasfivespokesequallyspacedaroundacentralhub,asshowninthediagrambelow.

θ°

Theangleθbetweentwospokesislabelledonthediagram.Whatistheangleθ?A. 5°B. 36°C. 60°D. 72°E. 90°

Question 2Aright-angledtriangle,XYZ,hassidelengthsXY =38.5cmandYZ =24.0cm,asshowninthediagrambelow.

X

Y

Z

38.5 cm

24.0 cm

ThelengthofXZ, incentimetres,isclosesttoA. 24.8B. 30.1C. 38.8D. 45.4E. 62.5

Module 3 – geometry and measurement

Beforeansweringthesequestions,youmustshadethe‘Geometryandmeasurement’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

27 2017FURMATHEXAM1

SECTION B – Module 3 – continuedTURN OVER

Question 3Thelocationsoffourcitiesaregivenbelow.

Adelaide(35°S,139°E) BuenosAires(35°S,58°W)Melilla(35°N,3°W) Heraklion(35°N,25°E)

Inwhichorder,fromfirsttolast,willthesunriseinthesecitiesonNewYear’sDay2018?A. Adelaide,Heraklion,BuenosAires,MelillaB. Adelaide,Heraklion,Melilla,BuenosAiresC. Heraklion,Adelaide,Melilla,BuenosAiresD. Melilla,Adelaide,Heraklion,BuenosAiresE. Melilla,Adelaide,BuenosAires,Heraklion

Question 4Agrainstoragesilointheshapeofacylinderwithaconicaltopisshowninthediagrambelow.

1.8 m

16 m

10 m

Thevolumeofthissilo,incubicmetres,isclosesttoA. 550B. 1304C. 1327D. 1398E. 2560

2017FURMATHEXAM1 28

SECTION B – Module 3 – continued

Question 5Asegmentisformedbyanangleof75°inacircleofradius112mm.Thissegmentisshownshadedinthediagrambelow.

O112 mm75°

Whichoneofthefollowingcalculationswillgivetheareaoftheshadedsegment?

A. π × ×

112 75

3602

B. π × ×

112 285

3602

C. π × ×

− × × °112 75

36012

112 752 2 sin( )

D. π × ×

− × × °112 285

36012

112 752 2 sin( )

E. π × ×

− × × °112 75

18012

112 752 2 sin( )

Question 6Ahemisphericalbowlofradius10cmisshowninthediagrambelow.

10 cm

2 cm

Thebowlcontainswaterwithamaximumdepthof2cm.Theradiusofthesurfaceofthewater,incentimetres,isA. 2B. 6C. 8D. 9E. 10

29 2017FURMATHEXAM1

End of Module 3 – SECTION B–continuedTURN OVER

Question 7AtriangleABChas:• oneside,AB,oflength4cm• oneside,BC ,oflength7cm• oneangle,∠ACB,of26°.

Whichoneofthefollowingangles,correcttothenearestdegree,couldnotbeanotherangleintriangleABC?A. 24°B. 50°C. 104°D. 130°E. 144°

Question 8Threecirclesofradius50mmareplacedsothattheyjusttoucheachother.Theregionenclosedbythecirclesisshadedinthediagrambelow.

Theareaoftheshadedregion,insquaremillimetres,isclosesttoA. 403B. 436C. 1309D. 2844E. 4330

2017FURMATHEXAM1 30

SECTION B – Module 4 – continued

Question 1Theequationofthelinethatpassesthroughthepoints(0,4)and(2,4)isA. x = 4B. y = 4C. y = 4xD. y = 4x+2E. y = 2x+4

Question 2Thegraphbelowshowsthevolumeofwaterinawatertankbetween7amand5pmononeday.

volume (litres)

7 am 8 am 9 am 10 am 11 am 12 noon 1 pm 2 pm 3 pm 4 pm 5 pmtime

1000

800

600

400

200

Whichoneofthefollowingstatementsistrue?A. Thevolumeofwaterinthetankdecreasesbetween8amand11am.B. Thevolumeofwaterinthetankincreasesatthegreatestratebetween4pmand5pm.C. Thevolumeofwaterinthetankisconstantbetween12noonand2pm.D. Thetankisfilledwithwaterataconstantrateof100Lperhour.E. Morewaterentersthetankduringthefirstfivehoursthanduringthelastfivehours.

Module 4 – graphs and relations

Beforeansweringthesequestions,youmustshadethe‘Graphsandrelations’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

31 2017FURMATHEXAM1

SECTION B – Module 4 – continuedTURN OVER

Question 3Thepoint(5,100)liesonthegraphofy = kx2,asshownbelow.

400

300

200

100

1O 2 3 4 5 6 7 8 9 10x

y

(5, 100)

Thevalueofkis

A. 14

B. 4

C. 5

D. 20

E. 40

2017FURMATHEXAM1 32

SECTION B – Module 4 – continued

Question 4Theannualfeeformembershipofacarclub,indollars,basedonyears of membershipoftheclubisshowninthestepgraphbelow.

fee ($)

years ofmembership

600

500

400

300

200

100

5 10 15 20 25 30 35 40 45 50O

IntheMartinfamily:• Hayleyhasbeenamemberoftheclubforfouryears• Johnhasbeenamemberoftheclubfor20years• Sharonhasbeenamemberoftheclubfor25years.

WhatisthetotalfeeformembershipofthecarclubfortheMartinfamily?A. $200B. $600C. $720D. $900E. $940

33 2017FURMATHEXAM1

SECTION B – Module 4 – continuedTURN OVER

Question 5Achildcarecentrerequiresatleastoneteacheremployedforevery15childrenenrolled.Letxbethenumberofteachersemployed.Lety bethenumberofchildrenenrolled.Whichoneofthefollowingistheinequalitythatrepresentsthissituation?A. y x≥15

B. y x≤

15

C. yx

≤15

D. y x≤15

E. y x≥

15

Question 6Theticketofficeatacircussellsadultticketsandchildtickets:• ThePaynefamilyboughttwoadultticketsandthreechildticketsfor$69.50• TheTranfamilyboughtoneadultticketandfivechildticketsfor$78.50• TheSaundersfamilyboughtthreeadultticketsandfourchildtickets.

WhatisthetotalamountspentbytheSaundersfamily?A. $83.40B. $87.50C. $98.00D. $101.50E. $112.00

Question 7Connormakes200meatpiestosellathislocalmarket.Thecost,$C,ofproducingn piescanbedeterminedfromtherulebelow.

C=0.8n+250

Connorsellsthefirst150piesatfullpriceandsellstheremaining50piesathalf-price.Tobreakeven,thefullpriceofeachpiemustbeclosesttoA. $1.85B. $2.05C. $2.35D. $2.50E. $2.75

2017FURMATHEXAM1 34

END OF MULTIPLE-CHOICE QUESTION BOOK

Question 8Theshadedareainthegraphbelowshowsthefeasibleregionforalinearprogrammingproblem.

B

D

E

C

A

2019181716151413121110987654321

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

y

xO

Theobjectivefunctionisgivenby

Z = mx + ny

Whichoneofthefollowingstatementsisnottrue?A. Whenm =4andn =1,theminimumvalueofZisatpointA.B. Whenm =1andn =6,themaximumvalueofZisatpointB.C. Whenm =2andn =5,theminimumvalueofZisatpointC.D. Whenm =2andn =6,themaximumvalueofZisatpointD.E. Whenm =12andn =1,themaximumvalueofZisatpointE.

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