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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 4
SECTION A – continued
SECTION A – Core
Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrectforthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenot drawntoscale.
Data analysis
Use the following information to answer Questions 1–3.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
2017FURMATHEXAM1 6
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.
7 2017FURMATHEXAM1
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.
2017FURMATHEXAM1 10
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
13 2017FURMATHEXAM1
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%
15 2017FURMATHEXAM1
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.
17 2017FURMATHEXAM1
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
19 2017FURMATHEXAM1
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
2017FURMATHEXAM1 20
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.
2017FURMATHEXAM1 22
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
top related