Victorian Certificate of Education 2016 · 2016 FURMATH EXAM 1 4 ... Parallel boxplots would be an appropriate graphical tool to investigate the association between the monthly

FURTHER MATHEMATICSWritten examination 1

Friday 28 October 2016 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016

Victorian Certificate of Education 2016

2016FURMATHEXAM1 2

THIS PAgE IS BLANK

3 2016FURMATHEXAM1

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.Theblood pressure(low,normal,high)andtheage(under50years,50yearsorover)of110adultswererecorded.Theresultsaredisplayedinthetwo-wayfrequencytablebelow.

Blood pressureAge

Under 50 years 50 years or over

low 15 5

normal 32 24

high 11 23

Total 58 52

Question 1Thepercentageofadultsunder50yearsofagewhohavehighbloodpressureisclosesttoA. 11%B. 19%C. 26%D. 44%E. 58%

Question 2Thevariablesblood pressure(low,normal,high)andage(under50years,50yearsorover)areA. bothnominalvariables.B. bothordinalvariables.C. anominalvariableandanordinalvariablerespectively.D. anordinalvariableandanominalvariablerespectively.E. acontinuousvariableandanordinalvariablerespectively.

2016FURMATHEXAM1 4

SECTION A – continued

Question 3Thestemplotbelowdisplays30temperaturesrecordedataweatherstation.

temperature key:2|2=2.2°C

2 2 2 4 4

2 5 7 8 8 8 8 8 8 9 9 9 9

3 1 2 3 3 4 4 4

3 5 6 7 7 7 7

4 1

ThemodaltemperatureisA. 2.8°CB. 2.9°CC. 3.7°CD. 8.0°CE. 9.0°C

Use the following information to answer Questions 4 and 5.Theweightsofmaleplayersinabasketballcompetitionareapproximatelynormallydistributedwithameanof78.6kgandastandarddeviationof9.3kg.

Question 4Thereare456maleplayersinthecompetition.Theexpectednumberofmaleplayersinthecompetitionwithweightsabove60kgisclosesttoA. 3B. 11C. 23D. 433E. 445

Question 5BrettandSanjeevabothplayinthebasketballcompetition.Whentheweightsofallplayersinthecompetitionareconsidered,Bretthasastandardisedweightof z =–0.96andSanjeevahasastandardisedweightofz =–0.26Whichoneofthefollowingstatementsisnot true?A. BrettandSanjeevaarebothbelowthemeanweightforplayersinthebasketballcompetition.B. SanjeevaweighsmorethanBrett.C. IfSanjeevaincreaseshisweightby2kg,hewouldbeabovethemeanweightforplayersinthe

basketballcompetition.D. Brettweighsmorethan68kg.E. Morethan50%oftheplayersinthebasketballcompetitionweighmorethanSanjeeva.

5 2016FURMATHEXAM1


Question 6Thehistogrambelowshowsthedistributionofthenumberofbillionairespermillionpeoplefor53countries.

50

45

40

35

30

25frequency

20

15

10

5

00 2 4 6 8 10 12 14 16

number

18 20 22 24 26 28 30 32 34

Data:Gapminder

Usingthishistogram,thepercentageofthese53countrieswithlessthantwobillionairespermillionpeopleisclosesttoA. 49%B. 53%C. 89%D. 92%E. 98%

2016FURMATHEXAM1 6


Question 7Thehistogrambelowshowsthedistributionofthenumberofbillionairespermillionpeopleforthesame 53countriesasinQuestion6,butthistimeplottedonalog10scale.

30

20

frequency

10

0–4 –3 –2 –1

log10 (number)0 1 2 3

Data:Gapminder

Basedonthishistogram,thenumberofcountrieswithoneormorebillionairespermillionpeopleisA. 1B. 3C. 8D. 9E. 10

Question 8Parallelboxplotswouldbeanappropriategraphicaltooltoinvestigatetheassociationbetweenthemonthlymedianrainfall,inmillimetres,andtheA. monthlymedianwindspeed,inkilometresperhour.B. monthlymediantemperature,indegreesCelsius.C. monthoftheyear(January,February,March,etc.).D. monthlysunshinetime,inhours.E. annualrainfall,inmillimetres.

7 2016FURMATHEXAM1


Use the following information to answer Questions 9 and 10.Thescatterplotbelowshowslifeexpectancyinyears(life expectancy)plottedagainsttheHumanDevelopmentIndex(HDI)foralargenumberofcountriesin2011.Aleastsquareslinehasbeenfittedtothedataandtheresultingresidualplotisalsoshown.

8580757065

lifeexpectancy

(years) 60555045

25 35 45 55HDI

65 75 85 95

20151050residual

–5–10–15–20

25 35 45 55HDI

65 75 85 95

Data:Gapminder

Theequationofthisleastsquareslineis

life expectancy=43.0+0.422×HDI

Thecoefficientofdeterminationisr 2=0.875

Question 9Giventheinformationabove,whichoneofthefollowingstatementsisnot true?A. Thevalueofthecorrelationcoefficientiscloseto0.94B. 12.5%ofthevariationinlifeexpectancyisnotexplainedbythevariationintheHuman

DevelopmentIndex.C. Onaverage,lifeexpectancyincreasesby43.0yearsforeach10-pointincreaseintheHuman

DevelopmentIndex.D. Ignoringanyoutliers,theassociationbetweenlifeexpectancyandtheHumanDevelopment

Indexcanbedescribedasstrong,positiveandlinear.E. UsingtheleastsquareslinetopredictthelifeexpectancyinacountrywithaHuman

DevelopmentIndexof75isanexampleofinterpolation.

Question 10In2011,lifeexpectancyinAustraliawas81.8yearsandtheHumanDevelopmentIndexwas92.9WhentheleastsquareslineisusedtopredictlifeexpectancyinAustralia,theresidualisclosesttoA. –0.6B. –0.4C. 0.4D. 11.1E. 42.6

2016FURMATHEXAM1 8


Question 11ThetablebelowgivestheHumanDevelopmentIndex(HDI)andthemeannumberofchildrenperwoman(children)for14countriesin2007.Ascatterplotofthedataisalsoshown.

HDI Children

27.3 7.6

31.3 6.1

39.5 4.9

41.6 3.9

44.0 3.8

50.8 4.3

52.3 2.7

62.5 3.0

69.1 2.4

74.6 2.1

78.9 1.9

85.6 1.8

92.0 1.9

83.4 1.6

8

7

6

5

4children

3

2

1

020 30 40 50 60

HDI70 80 90 100

Data:Gapminder

Thescatterplotisnon-linear.Alogtransformationappliedtothevariablechildrencanbeusedtolinearisethescatterplot.WithHDIastheexplanatoryvariable,theequationoftheleastsquareslinefittedtothelineariseddataisclosesttoA. log(children)=1.1–0.0095×HDIB. children=1.1–0.0095×log(HDI)C. log(children)=8.0–0.77×HDID. children=8.0–0.77×log(HDI)E. log(children)=21–10×HDI

9 2016FURMATHEXAM1


Question 12 Thereisastrongpositiveassociationbetweenacountry’sHumanDevelopmentIndexanditscarbondioxideemissions.Fromthisinformation,itcanbeconcludedthatA. increasingacountry’scarbondioxideemissionswillincreasetheHumanDevelopmentIndexofthe

country.B. decreasingacountry’scarbondioxideemissionswillincreasetheHumanDevelopmentIndexofthe

country.C. thisassociationmustbeachanceoccurrenceandcanbesafelyignored.D. countriesthathavehigherhumandevelopmentindicestendtohavehigherlevelsofcarbondioxide

emissions.E. countriesthathavehigherhumandevelopmentindicestendtohavelowerlevelsofcarbondioxide

emissions.

Question 13Considerthetimeseriesplotbelow.

30

25

20

15mean temperature

10

5

00 12 24

month number36 48

Data:CommonwealthofAustralia,BureauofMeteorology

ThepatterninthetimeseriesplotshownaboveisbestdescribedashavingA. irregularfluctuationsonly.B. anincreasingtrendwithirregularfluctuations.C. seasonalitywithirregularfluctuations.D. seasonalitywithanincreasingtrendandirregularfluctuations.E. seasonalitywithadecreasingtrendandirregularfluctuations.

2016FURMATHEXAM1 10


Use the following information to answer Questions 14–16.Thetablebelowshowsthelong-termaverageofthenumberofmealsservedeachdayatarestaurant.AlsoshownisthedailyseasonalindexforMondaythroughtoFriday.

Day of the week

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Long-term average 89 93 110 132 145 190 160

Seasonal index 0.68 0.71 0.84 1.01 1.10

Question 14TheseasonalindexforWednesdayis0.84Thistellsusthat,onaverage,thenumberofmealsservedonaWednesdayisA. 16%lessthanthedailyaverage.B. 84%lessthanthedailyaverage.C. thesameasthedailyaverage.D. 16%morethanthedailyaverage.E. 84%morethanthedailyaverage.

Question 15LastTuesday,108mealswereservedintherestaurant.ThedeseasonalisednumberofmealsservedlastTuesdaywasclosesttoA. 93B. 100C. 110D. 131E. 152

Question 16TheseasonalindexforSaturdayisclosesttoA. 1.22B. 1.31C. 1.38D. 1.45E. 1.49

11 2016FURMATHEXAM1


Recursion and financial modelling

Question 17Considertherecurrencerelationbelow.

A0=2, An+1=3An+1

ThefirstfourtermsofthisrecurrencerelationareA. 0,2,7,22…B. 1,2,7,22…C. 2,5,16,49…D. 2,7,18,54…E. 2,7,22,67…

Question 18Thevalueofanannuity,Vn,afternmonthlypaymentsof$555havebeenmade,canbedeterminedusingtherecurrencerelation

V0=100000, Vn+1=1.0025Vn–555

ThevalueoftheannuityafterfivepaymentshavebeenmadeisclosesttoA. $97225B. $98158C. $98467D. $98775E. $110224

Question 19Thepurchasepriceofacarwas$26000.Usingthereducingbalancemethod,thevalueofthecarisdepreciatedby8%eachyear.Arecurrencerelationthatcanbeusedtodeterminethevalueofthecarafternyears,Cn,isA. C0=26000, Cn+1=0.92Cn

B. C0=26000, Cn+1=1.08Cn

C. C0=26000, Cn+1=Cn+8

D. C0=26000, Cn+1=Cn–8

E. C0=26000, Cn+1=0.92Cn–8

Question 20Considertherecurrencerelationbelow.

V0=10000, Vn+1=1.04Vn+500

ThisrecurrencerelationcouldbeusedtomodelA. areducingbalancedepreciationofanassetinitiallyvaluedat$10000.B. areducingbalanceloanwithperiodicrepaymentsof$500.C. aperpetuitywithperiodicpaymentsof$500fromtheannuity.D. anannuityinvestmentwithperiodicadditionsof$500madetotheinvestment.E. aninterest-onlyloanof$10000.

2016FURMATHEXAM1 12


Question 21Juanitainvests$80000inaperpetuitythatwillprovide$4000peryeartofundascholarshipatauniversity.Thegraphthatshowsthevalueofthisperpetuityoveraperiodoffiveyearsis

100000

80000

60000value ($)

40000

20000

00 1 2 3

year4 5

100000

80000

60000value ($)

40000

20000

00 1 2 3

year4 5

100000

80000

60000value ($)

40000

20000

00 1 2 3

year4 5

100000

80000

60000value ($)

40000

20000

00 1 2 3

year4 5

100000

80000

60000value ($)

40000

20000

00 1 2 3

year4 5

A. B.

C. D.

E.

13 2016FURMATHEXAM1

END OF SECTION ATURN OVER

Question 22Thefirstthreelinesofanamortisationtableforareducingbalancehomeloanareshownbelow.Theinterestrateforthishomeloanis4.8%perannumcompoundingmonthly.Theloanistoberepaidwithmonthlypaymentsof$1500.

Payment number

Payment Interest Principal reduction

Balance of loan

0 0 0.00 0.00 250000.00

1 1500 1000.00 500.00 249500.00

2 1500

Theamountofpaymentnumber2thatgoestowardsreducingtheprincipaloftheloanisA. $486B. $502C. $504D. $996E. $998

Question 23Sarahinvests$5000inasavingsaccountthatpaysinterestattherateof3.9%perannumcompoundingquarterly.Attheendofeachquarter,immediatelyaftertheinteresthasbeenpaid,sheadds$200toherinvestment.Aftertwoyears,thevalueofherinvestmentwillbeclosesttoA. $5805B. $6600C. $7004D. $7059E. $9285

Question 24Maiinvestsinanannuitythatearnsinterestattherateof5.2%perannumcompoundingmonthly.Monthlypaymentsarereceivedfromtheannuity.Thebalanceoftheannuitywillbe$130784.93afterfiveyears.Thebalanceoftheannuitywillbe$66992.27after10years.ThemonthlypaymentthatMaireceivesfromtheannuityisclosesttoA. $1270B. $1400C. $1500D. $2480E. $3460

2016FURMATHEXAM1 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.......................................................................................... 19

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

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

15 2016FURMATHEXAM1

SECTION B – Module 1 – continuedTURN OVER

Question 1

Thetransposeof2 7 10

13 19 8

is

A. 13 19 82 7 10

B. 10 7 28 19 13

C. 2 137 19

10 8

D. 13 219 78 10

E. 8 1019 713 2

Question 2

Thematrixproduct

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

×

LEAPS

isequalto

A. LAPSE

B. LEAPS

C. PLEAS

D. PALES

E. PEALS

Module 1 – Matrices

Beforeansweringthesequestions,youmustshadethe‘Matrices’boxontheanswersheetfor multiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

2016FURMATHEXAM1 16

SECTION B – Module 1 – continued

Question 3Thematrixequationbelowrepresentsapairofsimultaneouslinearequations.

12 93

66m

xy

=

ThesesimultaneouslinearequationshavenouniquesolutionwhenmisequaltoA. –4B. –3C. 0D. 3E. 4

Question 4Thetablebelowshowsthenumberofeachtypeofcoinsavedinamoneybox.

Coin 5cent 10cent 20cent 50cent

Number 15 32 48 24

Thematrixproductthatdisplaysthetotalnumberofcoinsandthetotalvalueofthesecoinsis

A.

5 10 20 50

15324824

[ ]

B.

15 32 48 24

1 51 101 201 50

[ ]

C.

5 10 20 50

1 151 321 481 24

[ ]

D.

15 32 48 24

5102050

[ ]

E.

5 10 20 5015 32 48 24

1111

17 2016FURMATHEXAM1


Question 5

Let1 2 3 43 4 5 6

M =

.

TheelementinrowiandcolumnjofMismi j.TheelementsofMaredeterminedbytheruleA. mi j=i+j – 1

B. mi j=2i – j+1

C. mi j=2i+j – 2

D. mi j=i+2j – 2

E. mi j=i+j+1

Question 6Familiesinacountrytownwereaskedabouttheirannualholidays.Everyyear,thesefamilieschoosebetweenstayingathome(H),travelling(T)andcamping(C).Thetransitiondiagrambelowshowsthewayfamiliesinthetownchangetheirholidaypreferencesfromyeartoyear.

30%

25%

45%

60%

30%

40%

50%

20%

H T

C

Atransitionmatrixthatprovidesthesameinformationasthetransitiondiagramis

A. fromH T C

HTCto

0 30 0 75 0 650 75 0 50 0 200 65 0 20 0 60

. . .

. . .

. . .

B. fromH T C

HTCto

0 30 0 30 0 400 45 0 50 00 25 0 20 0 60

. . .

. .

. . .

C. fromH T C

HTCto

0 30 0 30 0 400 45 0 50 0 200 25 0 20 0 60

. . .

. . .

. . .

D. fromH T C

HTCto

0 30 0 30 0 400 45 0 50 0 200 25 0 20 0 40

. . .

. . .

. . .

E. fromH T C

HTCto

0 30 0 45 0 250 30 0 50 0 200 40 0 0 60

. . .

. . .

. .

2016FURMATHEXAM1 18

End of Module 1 – SECTION B–continued

Question 7Eachweek,the300studentsataprimaryschoolchooseart(A),music(M)orsport(S)asanafternoonactivity.Thetransitionmatrixbelowshowshowthestudents’choiceschangefromweektoweek.

thisweekA M S

TAMSnext w=

0 5 0 4 0 10 3 0 4 0 40 2 0 2 0 5

. . .

. . .

. . .eeek

Basedontheinformationabove,itcanbeconcludedthat,inthelongtermA. nostudentwillchoosesport.B. allstudentswillchoosetostayinthesameactivityeachweek.C. allstudentswillhavechosentochangetheiractivityatleastonce.D. morestudentswillchoosetodomusicthansport.E. thenumberofstudentschoosingtodoartandmusicwillbethesame.

Question 8Thematrixbelowshowstheresultofeachmatchbetweenfourteams,A,B,CandD,inabowlingtournament.Eachteamplayedeachotherteamonceandtherewerenodraws.

loserA B C D

winner

ABCD

0 0 1 01 0 0 10 1 0 11 0 0 0

Inthistournament,eachteamwasgivenarankingthatwasdeterminedbycalculatingthesumofitsone-stepandtwo-stepdominances.Theteamwiththehighestsumwasrankednumberone(1).Theteamwiththesecond-highestsumwasrankednumbertwo(2),andsoon.Usingthismethod,teamCwasrankednumberone(1).TeamAwouldhavebeenrankednumberone(1)ifthewinnerofonematchhadlostinstead.ThatmatchwasbetweenteamsA. AandB.B. AandD.C. BandC.D. BandD.E. CandD.

19 2016FURMATHEXAM1


Question 1Lee,Mandy,Nola,OscarandPieterareeachtobeallocatedoneparticulartaskatwork.Thebipartitegraphbelowshowswhichtask(s),1–5,eachpersonisabletocomplete.

Task 1

Task 2

Task 3

Task 4

Lee

Mandy

Nola

Oscar

Pieter Task 5

Worker Task

Eachpersoncompletesadifferenttask.Task4mustbecompletedbyA. Lee.B. Mandy.C. Nola.D. Oscar.E. Pieter.

Module 2 – Networks and decision mathematics

Beforeansweringthesequestions,youmustshadethe‘Networksanddecisionmathematics’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

2016FURMATHEXAM1 20


Question 2Thefollowingdirectedgraphshowstheflowofwater,inlitresperminute,inasystemofpipesconnectingthesourcetothesink.

4

6 8

2

10source sink

Themaximumflow,inlitresperminute,fromthesourcetothesinkisA. 10B. 14C. 18D. 20E. 22

Question 3Thefollowinggraphwithfiveverticesisacompletegraph.

Edgesareremovedsothatthegraphwillhavetheminimumnumberofedgestoremainconnected.ThenumberofedgesthatareremovedisA. 4B. 5C. 6D. 9E. 10

21 2016FURMATHEXAM1


Question 4Theminimumspanningtreeforthenetworkbelowincludestheedgewithweightlabelledk.

5 6

8

7

6

1

72

62

4

k

3

5

Thetotalweightofalledgesfortheminimumspanningtreeis33.ThevalueofkisA. 1B. 2C. 3D. 4E. 5

2016FURMATHEXAM1 22


Question 5Considertheplanargraphbelow.

Whichoneofthefollowinggraphscanberedrawnastheplanargraphabove?

A. B.

C. D.

E.

23 2016FURMATHEXAM1


Use the following information to answer Questions 6 and 7.Thedirectedgraphbelowshowsthesequenceofactivitiesrequiredtocompleteaproject.Alltimesareinhours.

D, 2

I, 4

L, 8

K, 3

J, 2

E, 5

F, 4

G, 1

H, 6

B, 3

A, 5

C, 2

start finish

Question 6ThenumberofactivitiesthathaveexactlytwoimmediatepredecessorsisA. 0B. 1C. 2D. 3E. 4

Question 7Thereisonecriticalpathforthisproject.ThreecriticalpathswouldexistifthedurationofactivityA. Iwerereducedbytwohours.B. Ewerereducedbyonehour.C. Gwereincreasedbysixhours.D. Kwereincreasedbytwohours.E. Fwereincreasedbytwohours.

2016FURMATHEXAM1 24


Question 8Fivechildren,Alan,Brianna,Chamath,DeidreandEwen,areeachtobeassignedadifferentjobbytheirteacher.Thetablebelowshowsthetime,inminutes,thateachchildwouldtaketocompleteeachofthefivejobs.

Alan Brianna Chamath Deidre Ewen

Job 1 5 8 5 8 7

Job 2 5 7 6 7 4

Job 3 9 5 7 5 9

Job 4 7 7 9 8 5

Job 5 4 4 4 4 3

Theteacherwantstoallocatethejobssoastominimisethetotaltimetakentocompletethefivejobs.Indoingso,shefindsthattwoallocationsarepossible.Ifeachchildstartstheirallocatedjobatthesametime,thenthefirstchildtofinishcouldbeeitherA. AlanorBrianna.B. BriannaorDeidre.C. ChamathorDeidre.D. ChamathorEwen.E. DeidreorEwen.

25 2016FURMATHEXAM1


Question 1Considerthediagrambelow.

10 cm

12 cm

5 cm

5 cm

Theshadedarea,insquarecentimetres,isA. 35B. 45C. 60D. 85E. 95

Question 2TriangleABCissimilartotriangleDEF.

B

CA

2.4 cm

3.6 cm

E

D F

1.8 cm

ThelengthofDF,incentimetres,isA. 0.9B. 1.2C. 1.8D. 2.7E. 3.6

Module 3 – geometry and measurement

Beforeansweringthesequestions,youmustshadethe‘Geometryandmeasurement’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

2016FURMATHEXAM1 26


Question 3

60° N

40° S

equator

AB

C20° E

Thediagramaboveshowsthepositionofthreecities,A,BandC.AccordingtothediagramA. Aislocatedonalongitudeof60°NB. Bislocatedonalongitudeof20°EC. Cislocatedonalongitudeof40°SD. AandBarelocatedonthesamelineoflongitudeE. AandCarelocatedonthesamelineoflongitude

Question 4 AlltownsinthestateofVictoriaareinthesametimezone.Mallacoota(38°S,150°E)andPortland(38°S,142°E)aretwocoastaltownsinthestateofVictoria.OnonedayinJanuary,thesunroseinMallacootaat6.03am.Assumingthat15°oflongitudeequatestoaone-hourtimedifference,thetimethatthesunwasexpectedtoriseinPortlandisA. 5.31am.B. 5.55am.C. 6.03am.D. 6.11am.E. 6.35am.

27 2016FURMATHEXAM1


Question 5Awatertankintheshapeofacylinderwithahemisphericaltopisshownbelow.

2 m2 m

5 m

Thevolumeofwaterthatthistankcanhold,incubicmetres,isclosesttoA. 80B. 88C. 96D. 105E. 121

Question 6Marcusisontheoppositesideofalargelakefromahorseanditsstable.Thestableis150mdirectlyeastofthehorse.Marcusisonabearingof170°fromthehorseandonabearingof205°fromthestable.

north

horse stable

Marcus

Thestraight-linedistance,inmetres,betweenMarcusandthehorseisclosesttoA. 45B. 61C. 95D. 192E. 237

2016FURMATHEXAM1 28


Question 7Thediagrambelowshowsarectangular-basedrightpyramid,ABCDE.Inthispyramid,AB=DC=24cm,AD=BC=10cmandAE=BE=CE=DE=28cm.

E

A B

CD

28 cm

10 cmO

24 cm

Theheight,OE,ofthepyramid,incentimetres,isclosesttoA. 10.4B. 13.0C. 24.8D. 25.3E. 30.9

Question 8Astringofsevenflagsconsistingofequilateraltrianglesintwosizesishangingattheendofaracetrack, asshowninthediagrambelow.

Theedgelengthofeachblackflagistwicetheedgelengthofeachwhiteflag.Forthisstringofsevenflags,thetotalareaoftheblackflagswouldbeA. twotimesthetotalareaofthewhiteflags.

B. fourtimesthetotalareaofthewhiteflags.

C. 43timesthetotalareaofthewhiteflags.

D. 163timesthetotalareaofthewhiteflags.

E. 169timesthetotalareaofthewhiteflags.

29 2016FURMATHEXAM1


Question 1Thegraphbelowshowsthetemperature,indegreesCelsius,between6amand6pmonagivenday.

25

20

15

10

5

6 am 8 am 10 am 12 noon 2 pm 4 pm 6 pm

temperature (°C)

time

From2pmto6pm,thetemperaturedecreasedbyA. 4°CB. 11°CC. 12°CD. 15°CE. 23°C

Question 2Aphonecompanychargesafixed,monthlylinerentalfeeof$28and$0.25percall.Letnbethenumberofcallsthataremadeinamonth.LetCbethemonthlyphonebill,indollars.Theequationfortherelationshipbetweenthemonthlyphonebill,indollars,andthenumberofcallsisA. C=28+0.25nB. C=28n+0.25C. C=n+28.25D. C=28(n+0.25)E. C=0.25(n+28)

Module 4 – graphs and relations

Beforeansweringthesequestions,youmustshadethe‘Graphsandrelations’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

2016FURMATHEXAM1 30


Question 3Thegraphbelowshowsastraightlinethatpassesthroughthepoints(6,6)and(8,9).

(8, 9)

(6, 6)

y

x

Thecoordinatesofthepointwherethelinecrossesthex-axisareA. (–3,0)B. (1,0)C. (1.5,0)D. (2,0)E. (4,0)

Question 4Thepoint(3,–2)satisfiestheinequalityA. y≥0

B. y x≥23

C. –2 < x < 3

D. y<5–x

E. y > x–5

31 2016FURMATHEXAM1


Question 5Thefeasibleregionforalinearprogrammingproblemisshadedinthediagrambelow.

y

x

A B

C

DO

Theequationoftheobjectivefunctionforthisproblemisoftheform

P=ax+by, wherea>0andb > 0

Thedottedlineinthediagramhasthesameslopeastheobjectivefunctionforthisproblem.ThemaximumvalueoftheobjectivefunctioncanbedeterminedbycalculatingitsvalueatA. pointA.B. pointB.C. pointC.D. pointD.E. anypointalonglinesegmentBC.

2016FURMATHEXAM1 32


Question 6Thepoint(2,12)liesonthegraphof y=kxn,asshownbelow.

1412108642

O 0.5 1 1.5 2

(2, 12)

x

y

Anothergraphthatrepresentsthisrelationshipbetweenyandxcouldbe

1412108642

O 0.5 1 1.5 2x

y

1412108642

O 0.5 1 1.5 2x2

y

1412108642

O 0.5 1 1.5 2x2

y

1412108642

O 0.5 1 1.5 2x3

y

1412108642

O 0.5 1 1.5 2x3

y

(1, 6) (1, 6)

(1, 6)(2, 3)

(2, 3)

A. B.

C. D.

E.

33 2016FURMATHEXAM1

END OF MULTIPLE-CHOICE QUESTION BOOK

Question 7Simongrowscucumbersandzucchinis.Letxbethenumberofcucumbersthataregrown.Letybethenumberofzucchinisthataregrown.Foreverytwocucumbersthataregrown,Simongrowsatleastthreezucchinis.Aninequalitythatrepresentsthissituationis

A. y x≥ +

23

B. y x≥ +

32

C. y x≤ +

32

D. y x≥

23

E. y x≥

32

Question 8Meganwalksfromherhousetoashopthatis800maway.Theequationfortherelationshipbetweenthedistance,inmetres,thatMeganisfromherhousetminutesafterleavingis

distance =≤ ≤< ≤< ≤

100t tt a

kt a t

0 6600 6

10

IfMeganreachestheshop10minutesafterleavingherhouse,thevalueofaisA. 7.0B. 7.5C. 8.0D. 8.5E. 9.0

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 2016

Victorian Certificate of Education 2016

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

Documents

Victorian Certificate of Education 2016 · 2016 FURMATH EXAM 1 4 ... Parallel boxplots would be an appropriate graphical tool to investigate the association between the monthly