Victorian Certificate of Education 2019 · 2020. 2. 5. · 9 2019 FURMATH EXAM 1 SECTION A – continued TURN OVER Question 12 The table below shows the values of two variables x

FURTHER MATHEMATICSWritten examination 1

Friday 1 November 2019 Reading time: 2.00 pm to 2.15 pm (15 minutes) Writing time: 2.15 pm to 3.45 pm (1 hour 30 minutes)

MULTIPLE-CHOICE QUESTION BOOK

Structure of bookSection Number of

questionsNumber of questions

to be answeredNumber of modules

Number of modulesto be answered

Number of marks

A – Core 24 24 24B – Modules 32 16 4 2 16

Total 40

• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulers,oneboundreference,oneapprovedtechnology(calculatororsoftware)and,ifdesired,onescientificcalculator.CalculatormemoryDOESNOTneedtobecleared.Forapprovedcomputer-basedCAS,fullfunctionalitymaybeused.

• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.

Materials supplied• Questionbookof35pages• Formulasheet• Answersheetformultiple-choicequestions• Workingspaceisprovidedthroughoutthebook.

Instructions• Checkthatyourname and student numberasprintedonyouranswersheetformultiple-choice

questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

At the end of the examination• Youmaykeepthisquestionbookandtheformulasheet.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2019

Victorian Certificate of Education 2019

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THIS PAgE IS BLANK

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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.Thehistogrambelowshowsthedistributionofthepopulation sizeof48countriesin2018.

0 10 20 30 40 50 60 70 80population size (millions)

90 100 110 120 130 140 150

25

20

15

10

frequency

5

0

Data:Worldometers,<www.worldometers.info/>

Question 1Thenumberofthesecountrieswithapopulation sizebetween5millionand20millionpeopleisA. 11B. 17C. 23D. 34E. 35

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SECTION A – continuedTURN OVER

Question 2The shape of this histogram is best described as A. positively skewed with no outliers. B. positively skewed with outliers.C. approximately symmetric.D. negatively skewed with no outliers.E. negatively skewed with outliers.

Question 3The histogram below shows the population size for these 48 countries plotted on a log10 scale.

0 1 2 3 4 5log10 (population size)

6 7 8 9 10

25

20

15

10

frequency

5

0

Data: Worldometers, <www.worldometers.info/>

Based on this histogram, the number of countries with a population size that is less than 100 000 people isA. 1B. 5C. 7D. 8E. 48

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Use the following information to answer Questions 4 and 5.Thestemplotbelowshowsthedistributionofmathematicstest scoresforaclassof23students.

key:4|2=42 n =23

4 0 1 4 4

5 2 7 9 9 9

6 5 6 8 8 9 9

7 0 0 5 6 7 8

8 5 9

Question 4Forthisclass,therangeoftest scoresisA. 22B. 40C. 45D. 49E. 89

Question 5Forthisclass,theinterquartilerange(IQR)oftest scoresisA. 14.5B. 17.5C. 18D. 24E. 49

Question 6 Thetimetakentotravelbetweentworegionalcitiesisapproximatelynormallydistributedwithameanof 70minutesandastandarddeviationof2minutes.Thepercentageoftraveltimesthatarebetween66minutesand72minutesisclosesttoA. 2.5%B. 34%C. 68%D. 81.5%E. 95%

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Question 7 Thevolumeofacupofsoupservedbyamachineisnormallydistributedwithameanof240mLandastandarddeviationof5mL.Afast-foodstoreusedthismachinetoserve160cupsofsoup.Thenumberofthesecupsofsoupthatareexpectedtocontainlessthan230mLofsoupisclosesttoA. 3B. 4C. 8D. 26E. 156

Question 8Percyconductedasurveyofpeopleinhisworkplace.Heconstructedatwo-wayfrequencytableinvolvingtwovariables.Oneofthevariableswasattitude towards shorter working days (for,against).TheothervariablecouldhavebeenA. age (inyears).B. sex(male,female).C. height(tothenearestcentimetre).D. income(tothenearestthousanddollars).E. timespenttravellingtowork(inminutes).

Use the following information to answer Questions 9 and 10.Aleastsquareslineisusedtomodeltherelationshipbetweenthemonthlyaverage temperatureandlatitude recordedatsevendifferentweatherstations.Theequationoftheleastsquareslineisfoundtobe

average temperature=42.9842–0.877447×latitude

Question 9Whenthenumbersinthisequationarecorrectlyroundedtothreesignificantfigures,theequationwillbeA. average temperature=42.984–0.877×latitudeB. average temperature=42.984–0.878×latitudeC. average temperature=43.0–0.878×latitudeD. average temperature=42.9–0.878×latitudeE. average temperature=43.0–0.877×latitude

Question 10Thecoefficientofdeterminationwascalculatedtobe0.893743Thevalueofthecorrelationcoefficient,roundedtothreedecimalplaces,isA. –0.945B. –0.898C. 0.806D. 0.898E. 0.945

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Question 11Astudywasconductedtoinvestigatetheeffectofdrinkingcoffeeonsleep.Inthisstudy,theamountofsleep,inhours,andtheamountofcoffeedrunk,incups,onagivendaywererecordedforagroupofadults.Thefollowingsummarystatisticsweregenerated.

Sleep (hours) Coffee (cups)

Mean 7.08 2.42

Standard deviation 1.12 1.56

Correlation coefficient (r) –0.770

Onaverage,foreachadditionalcupofcoffeedrunk,theamountofsleep A. decreasedby0.55hours.B. decreasedby0.77hours.C. decreasedby1.1hours.D. increasedby1.1hours.E. increasedby2.3hours.

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Question 12Thetablebelowshowsthevaluesoftwovariablesxandy.Theassociatedscatterplotisalsoshown.Theexplanatoryvariableisx.

x y

1 7.6

3 3.4

5 12.1

7 23.4

9 43.6

11 51.8

13 95.4

15 108

16 145

17 172

18 168

020406080y

100120140160180

0 2 4 6 8 10x

12 14 16 18

Thescatterplotisnon-linear.Asquaredtransformationappliedtothevariablexcanbeusedtolinearisethescatterplot.TheequationoftheleastsquareslinefittedtothelineariseddataisclosesttoA. y =–1.34+0.546xB. y =–1.34+0.546x2

C. y =3.93–0.00864x2

D. y=34.6–10.5xE. y=34.6–10.5x2

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Use the following information to answer Questions 13 and 14.Thetime,inminutes,thatLivraneachdaywasrecordedforninedays.Thesetimesareshowninthetablebelow.

Day number 1 2 3 4 5 6 7 8 9

Time (minutes) 22 40 28 51 19 60 33 37 46

Thetimeseriesplotbelowwasgeneratedfromthisdata.

60

70

50

40

30time

(minutes)

20

10

00 1 2 3 4 5

day number6 7 8 9 10

Question 13Boththree-mediansmoothingandfive-mediansmoothingarebeingconsideredforthisdata.Bothofthesemethodsresultinthesamesmoothedvalueonday numberA. 3B. 4C. 5D. 6E. 7

Question 14Aleastsquareslineistobefittedtothetimeseriesplotshownabove.Theequationofthisleastsquaresline,withday numberastheexplanatoryvariable,isclosesttoA. day number=23.8+2.29×time B. day number =28.5+1.77×timeC. time =23.8+1.77×day numberD. time=23.8+2.29×day numberE. time=28.5+1.77×day number

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Use the following information to answer Questions 15 and 16.Thetimeseriesplotbelowshowsthemonthly rainfallataweatherstation,inmillimetres,foreachmonthin2017.

300

250

200

150

100

50

0

monthlyrainfall

(mm)

Jan. Feb. Mar. Apr. May Junemonth

July Aug. Sep. Oct. Nov. Dec.

Question 15Themedianmonthly rainfallfor2017wasclosesttoA. 53mmB. 82mmC. 96mmD. 103mmE. 111mm

Question 16Ifseven-meansmoothingisusedtosmooththistimeseriesplot,thenumberofsmootheddatapointswouldbeA. 3B. 5C. 6D. 8E. 10

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Recursion and financial modelling

Question 17Considertherecurrencerelationshownbelow.

A0=3, An+1=2An+4

ThevalueofA3inthesequencegeneratedbythisrecurrencerelationisgivenbyA. 2×3+4B. 2×4+4C. 2×10+4D. 2×24+4E. 2×52+4

Question 18Thevalueofacompoundinterestinvestment,indollars,afternyears,Vn ,canbemodelledbytherecurrencerelationshownbelow.

V0=100000, Vn+1=1.01Vn

Theinterestrate,perannum,forthisinvestmentisA. 0.01%B. 0.101%C. 1%D. 1.01%E. 101%

Question 19Geoffpurchasedacomputerfor$4500.Hewilldepreciatethevalueofhiscomputerbyaflatrateof10%ofthepurchasepriceperannum.ArecurrencerelationthatGeoffcanusetodeterminethevalueofthecomputerafternyears,Vn ,isA. V0=4500, Vn+1=Vn–450B. V0=4500, Vn+1=Vn+450C. V0=4500, Vn+1=0.9VnD. V0=4500, Vn+1=1.1VnE. V0=4500, Vn+1=0.1(Vn–450)



Question 20 Consider the following amortisation table for a reducing balance loan.

Payment number Payment Interest Principal reduction Balance

0 0.00 0.00 0.00 300 000.00

1 1050.00 900.00 150.00 299 850.00

2 1050.00 899.55 150.45 299 699.55

3 1050.00 899.10 150.90 299 548.65

The annual interest rate for this loan is 3.6%.Interest is calculated immediately before each payment.For this loan, the repayments are madeA. weekly.B. fortnightly.C. monthly.D. quarterly.E. yearly.

Question 21The graph below shows the value, in dollars, of a compound interest investment after n compounding periods, Vn , for a period of four compounding periods.

1000

800

600

400

200

(0, 500.00)

1 2 3 4

(1, 580.00)(2, b)

(3, 780.45)

(4, 905.32)

Vn

nO

The coordinates of the point where n = 2 are (2, b).The value of b is A. 660.00B. 670.00C. 672.80D. 678.40E. 685.60

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END OF SECTION A

Question 22Amachineispurchasedfor$30000.Itproduces24000itemseachyear.Thevalueofthemachineisdepreciatedusingaunitcostmethodofdepreciation.Afterthreeyears,thevalueofthemachineis$18480.Aruleforthevalueofthemachineafternunitsareproduced,Vn ,isA. Vn=0.872nB. Vn=24000n–3840C. Vn=30000–24000nD. Vn=30000–0.872nE. Vn=30000–0.16n

Question 23 Josephborrowed$50000tobuyanewcar.Interestonthisloanischargedattherateof7.5%perannum,compoundingmonthly.Josephwillfullyrepaythisloanwith60monthlyrepaymentsoverfiveyears.Immediatelyafterthe59threpaymentismade,Josephstillowes$995.49Thevalueofhisfinalrepayment,tothenearestcent,willbeA. $995.49B. $998.36C. $1001.71D. $1001.90E. $1070.15

Question 24Millieinvested$20000inanaccountatherbankwithinterestcompoundingmonthly.Afteroneyear,thebalanceofMillie’saccountwas$20732.ThedifferencebetweentherateofinterestperannumusedbyherbankandtheeffectiveannualrateofinterestforMillie’sinvestmentisclosesttoA. 0.04%B. 0.06%C. 0.08%D. 0.10%E. 0.12%

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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................................................................................................................... 31

2019 FURMATH EXAM 1 16

SECTION B – Module 1 – continued

Question 1Consider the following four matrix expressions.

812

42

+

812

4 00 2

+

8 012 0

42

+

8 012 0

4 00 2

+

How many of these four matrix expressions are defined?A. 0B. 1C. 2D. 3E. 4

Question 2There are two rides called The Big Dipper and The Terror Train at a carnival.The cost, in dollars, for a child to ride on each ride is shown in the table below.

Ride Cost ($)

The Big Dipper 7

The Terror Train 8

Six children ride once only on The Big Dipper and once only on The Terror Train.The total cost of the rides, in dollars, for these six children can be determined by which one of the following calculations?

A. 6 7 8[ ]×[ ] B.6

78

[ ]×

C. 6 6 7 8[ ]×[ ] D.6 6

78

[ ]×

E. 66

7 8

×[ ]

Module 1 – Matrices

Before answering these questions, you must shade the ‘Matrices’ box on the answer sheet for multiple-choice questions and write the name of the module in the box provided.

17 2019FURMATHEXAM1

SECTION B – Module 1 – continuedTURN OVER

Question 3

ConsiderthematrixP,where P =

3 2 15 4 3

.

TheelementinrowiandcolumnjofmatrixPispi j .TheelementsinmatrixParedeterminedbytheruleA. pi j=4–j B. pi j=2i+1C. pi j=i+j+1D. pi j=i+2jE. pi j=2i – j+2

Question 4Stellacompletedamultiple-choicetestthathad10questions.Eachquestionhadfivepossibleanswers,A,B,C,DandE.Forquestionnumberone,StellachosetheanswerE.Stellachoseeachofthenineremaininganswers,inorder,byfollowingthetransitionmatrix,T,below.

this questionA B C D E

T =

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

ABCDE

next question

WhatanswerdidStellachooseforquestionnumbersix?A. AB. BC. CD. DE. E



Question 5

2 2 0 22 0 2 00 2 0 22 2 2 0

268

4

−−

−−

=−

vwxy

Which one of the following systems of simultaneous linear equations best represents the matrix equation above? A. 2v + 2w – 2y = 2

2v – 2x = 6 2v – 2y = –8 2v + 2w – 2x = 4

B. 2v + 2w – 2y = 2 2v – 2x = 6 2w – 2y = –8 2v + 2w – 2x = 4

C. 2v + 2w + 2y = 2 2v – 2x = 6 2w – 2y = –8 2v + 2w – 2x = 4

D. 2v + 2w – 2y = 2 2v – 2x = 6 2w – 2y = –8 2v + 2w + 2x = 4

E. 2v + 2w – 2y = 2 2v – 2w = 6 2w – 2y = –8 2v + 2w – 2x = 4

Question 6A water park is open from 9 am until 5 pm. There are three activities, the pool (P), the slide (S) and the water jets (W), at the water park.Children have been found to change their activity at the water park each half hour, as shown in the transition matrix, T, below.

this half hourP S W

T =

0 80 0 20 0 400 05 0 60 0 100 15 0 20 0 50

. . .

. . .

. . .

PSWnext half hour

A group of children has come to the water park for the whole day.The percentage of these children who are expected to be at the slide (S) at closing time is closest toA. 14%B. 20%C. 24%D. 25%E. 62%


End of Module 1 – SECTION B – continuedTURN OVER

Question 7The communication matrix below shows the direct paths by which messages can be sent between two people in a group of six people, U to Z.

receiverU V W X Y Z

UVWXYZ

sender

0 1 1 0 1 11 0 1 0 1 01 1 0 1 0 10 1 0 0 1 10 0 1 1 0 11 1 00 1 1 0

A ‘1’ in the matrix shows that the person named in that row can send a message directly to the person named in that column. For example, the ‘1’ in row 4, column 2 shows that X can send a message directly to V.In how many ways can Y get a message to W by sending it directly to one other person? A. 0B. 1C. 2D. 3E. 4

Question 8An airline parks all of its planes at Sydney airport or Melbourne airport overnight. The transition diagram below shows the change in the location of the planes from night to night.

0.2

0.48

0.8

0.52Melbourne Sydney

There are always m planes parked at Melbourne airport.There are always s planes parked at Sydney airport.Of the planes parked at Melbourne airport on Tuesday night, 12 had been parked at Sydney airport on Monday night.How many planes does the airline have?A. 25B. 37C. 62D. 65E. 85

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Question 1

Inthegraphshownabove,thesumofthedegreesoftheverticesisA. 5B. 6C. 10D. 11E. 12

Question 2Considerthegraphbelow.

TheminimumnumberofextraedgesthatarerequiredsothatanEuleriancircuitispossibleinthisgraphisA. 0B. 1C. 2D. 3E. 4

Module 2 – Networks and decision mathematics

Beforeansweringthesequestions,youmustshadethe‘Networksanddecisionmathematics’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.

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Question 3Theflowofwaterthroughaseriesofpipesisshowninthenetworkbelow.Thenumbersontheedgesshowthemaximumflowthrougheachpipeinlitresperminute.

7

8

6

3

3

6

5

5

2

sinksource

Cut Q

ThecapacityofCutQ,inlitresperminute,isA. 11B. 13C. 14D. 16E. 17

Question 4Twographs,labelledGraph1andGraph2,areshownbelow.

Graph 1 Graph 2

Whichoneofthefollowingstatementsisnottrue?A. Graph1andGraph2areisomorphic.B. Graph1hasfiveedgesandGraph2hassixedges.C. BothGraph1andGraph2areconnectedgraphs.D. BothGraph1andGraph2havethreefaceseach.E. NeitherGraph1norGraph2arecompletegraphs.

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Question 5Thefollowingdiagramshowsthedistances,inmetres,alongaseriesofcablesconnectingamainservertosevenpoints,A to G,inacomputernetwork.

A B

G

ED

F

Cmain server

36

28

38

35

35

4028

15

33

32

30

Theminimumlengthofcable,inmetres,requiredtoensurethateachofthesevenpointsisconnectedtothemainserverdirectlyorviaanotherpointisA. 175B. 203C. 208D. 221E. 236

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Question 6Themapbelowshowsalltheroadconnectionsbetweenfivetowns,P, Q, R, SandT.

P

Q

R

S

T

Theroadconnectionscouldberepresentedbytheadjacencymatrix

A. P Q R S TPQRST

1 3 0 2 23 0 1 1 10 1 0 1 02 1 1 0 22 1 0 2 0

B. P Q R S TPQRST

1 2 0 2 22 0 1 1 10 1 0 1 02 1 1 0 22 1 0 2 0

C. P Q R S TPQRST

0 3 0 2 23 0 1 1 10 1 0 1 02 1 1 0 22 1 0 2 0

D. P Q R S TPQRST

0 0 2 20 1 1 1

0 1 0 1 02 1 1 22 1 0 2 0

22

1

E. P Q R S TPQRST

1 2 0 2 22 0 1 1 10 1 0 1 02 1 1 1 12 1 0 1 0

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Question 7Aprojectinvolvesnineactivities,A to I.Theimmediatepredecessor(s)ofeachactivityisshowninthetablebelow.

Activity Immediate predecessor(s)

A –

B A

C A

D B

E B, C

F D

G D

H E, F

I G, H

Adirectednetworkforthisprojectwillrequireadummyactivity.ThedummyactivitywillbedrawnfromtheendofA. activityBtothestartofactivityC.B. activity BtothestartofactivityE.C. activity DtothestartofactivityE.D. activity EtothestartofactivityH.E. activity EtothestartofactivityF.


End of Module 2 – SECTION B – continuedTURN OVER

Question 8The directed network below shows the sequence of activities, A to S, that is required to complete a manufacturing process.The time taken to complete each activity, in hours, is also shown.

start finishA, 5

B, 4

C, 7

E, 3

D, 1 F, 8

I, 7

G, 3 K, 6

H, 4 L, 5

J, 6M, 5

N, 2

O, 7

P, 8

Q, 11

R, 1

S, 9

The number of activities that have a float time of 10 hours isA. 0B. 1C. 2D. 3E. 4



Question 1The four bases of a baseball field form four corners of a square of side length 27.43 m, as shown in the diagram below.

first base

second base

third base

home base

27.43 m

A player ran from home base to first base, then to second base, then to third base and finally back to home base.The minimum distance, in metres, that the player ran isA. 27.43B. 54.86C. 82.29D. 109.72E. 164.58

Module 3 – Geometry and measurement

Before answering these questions, you must shade the ‘Geometry and measurement’ box on the answer sheet for multiple-choice questions and write the name of the module in the box provided.

27 2019FURMATHEXAM1


Question 2TownBislocatedonabearingof060° fromTownA.Thediagramthatcouldillustratethisis

north

Town B

Town A

north

Town A

Town B

north

Town A

Town B

north

Town B

Town A

north

Town A

Town B

A.

C.

B.

D.

E.

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Question 3Anicecreamdessertisintheshapeofahemisphere.Thedesserthasaradiusof5cm.

5 cm

Thetopandthebaseofthedessertarecoveredinchocolate.Thetotalsurfacearea,insquarecentimetres,thatiscoveredinchocolateisclosesttoA. 52B. 157C. 236D. 314E. 942

Question 4TriangleM,shownbelow,hassidelengthsof3cm,4cmand5cm.

3 cm 4 cm

5 cm

Fourothertriangleshavethefollowingsidelengths:• TriangleNhassidelengthsof3cm,6cmand8cm.• TriangleOhassidelengthsof4cm,8cmand12cm.• TrianglePhassidelengthsof6cm,8cmand10cm.• TriangleQhassidelengthsof9cm,12cmand15cm.

ThetrianglesthataresimilartotriangleM areA. triangleNandtriangleO.B. triangleN,triangleOandtriangleP.C. triangleOandtriangleP.D. triangleOandtriangleQ.E. trianglePandtriangleQ.

29 2019FURMATHEXAM1


Question 5IntriangleABC,thepointDliesonthelineAC, asshowninthediagrambelow.

A D C

B

ThelengthofthelineBCisequaltothelengthofthelineBD. Whichoneofthefollowingstatementsistrue?A. TheangleBADisequaltotheangleADB.B. TheangleADBisequaltotheangleBDC.C. ThesumoftheangleACBandtheangleADBisequalto180°.D. ThesumoftheangleABDandtheangleCBDisequalto180°.E. ThesumoftheangleABCandtheangleACBisequaltotheangleADB.

Question 6 AllcitiesinChinaareinthesametimezone.Wuhan(31°N,114°E)andChengdu(31°N,104°E)arebothcitiesinChina.OnonedayinJanuary,thesunsetinWuhanat5.50pm.Assumingthat15°oflongitudeequatestoaone-hourtimedifference,thetimethesunsetinChengduonthesamedayisA. 4.20pmB. 5.10pmC. 5.50pmD. 6.30pmE. 7.20pm

Question 7Acanofdogfoodisintheshapeofacylinder.Thecanhasacircumferenceof18.85cmandavolumeof 311cm3.Theheightofthecan,incentimetres,isclosesttoA. 2.8B. 3.0C. 6.0D. 11.0E. 16.5

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End of Module 3 – SECTION B–continued

Question 8Fouridenticalcirclesofradiusraredrawninsideasquare,asshowninthediagrambelow.Theregionenclosedbythecircleshasbeenshadedinthediagram.

r

TheshadedareacanbefoundusingA. 4r2 – 2π rB. 4r2 – π r2

C. 4r – π r2

D. 2r2 – π r2

E. 2r – 2π r

31 2019FURMATHEXAM1


Question 1Thegraphbelowshowsthewaterlevelinacreek,inmetres,overa12-hourperiod.

2.5

2.0

1.5

1.0

0.5

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

water level (m)

time

Comparedtothewaterlevelat8am,thewaterlevelat12noonwasclosesttoA. 1.0mhigher.B. 1.4mhigher.C. 2.4mhigher.D. 0.9mlower.E. 1.3mlower.

Question 2Thecost,$C,ofusingKkilowatthoursofelectricitycanbecalculatedusingtheequationbelow.

C=52.00+0.15×K

Fromthisequation,itcanbeconcludedthatthereisA. nofixedchargeandtheelectricityusedischargedat$0.15perkilowatthour.B. nofixedchargeandtheelectricityusedischargedat$52.00perkilowatthour.C. afixedchargeof$0.15andtheelectricityusedischargedat$52.00perkilowatthour.D. afixedchargeof$52.00andtheelectricityusedischargedat$0.15perkilowatthour.E. afixedchargeof$52.00andtheelectricityusedischargedat$15.00perkilowatthour.

Module 4 – graphs and relations

Beforeansweringthesequestions,youmustshadethe‘Graphsandrelations’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.



Question 3A shop sells two types of discs: compact discs (CDs) and digital video discs (DVDs).CDs are sold for $7.00 each and DVDs are sold for $13.00 each.Bonnie bought a total of 16 discs for $178.00How many DVDs did Bonnie buy?A. 3B. 5C. 7D. 9E. 11

Question 4The graph below shows a straight line that crosses the x-axis at (–10, 0), passes through the point (–6, 8) and crosses the y-axis at (0, q).

(0, q)

(–6, 8)

(–10, 0) O

y

x

What is the value of q?A. 14B. 16C. 18D. 20E. 22

Question 5Giovanni makes and sells toy train sets.Each toy train set costs $40.00 to make and sells for $65.00Giovanni also has a fixed cost for making a batch of toy train sets.Last Sunday, Giovanni sold a batch of 35 toy train sets for a profit of $250.Giovanni’s fixed cost for this batch of train sets isA. $250B. $625C. $875D. $1400E. $2275



Question 6A recipe for a fruit drink lists both pineapple juice and mango juice.Let x be the number of millilitres of pineapple juice required to make the fruit drink.Let y be the number of millilitres of mango juice required to make the fruit drink.For every 200 mL of mango juice that is used, at least 300 mL of pineapple juice must be used.The inequality representing this situation is

A. x y≤

23

B. x y≤

32

C. y x≤

23

D. y x≤

32

E. y x≤

25

2019FURMATHEXAM1 34


Question 7Thefeasibleregionforalinearprogrammingproblemisshadedinthediagrambelow.

V (9, 9)

W (11, 3)

R (4, 12)

S (1, 10)

T (2, 7)

U (7, 3)

y

x

Theequationoftheobjectivefunctionforthisproblemisoftheform

P=mx+ny, wherem>0andn>0

ThedottedlinepassingthroughthepointsV and W inthediagramhasthesameslopeastheobjectivefunctionforthisproblem.Theminimum valueoftheobjectivefunctioncanbedeterminedbycalculatingitsvalueatA. pointS only.B. pointT only.C. pointU only.D. anypointalongthelinesegmentRU.E. anypointalongthelinesegmentST.

35 2019FURMATHEXAM1

END OF MULTIPLE-CHOICE QUESTION BOOK

Question 8JennyandAlan’shouseis900mfromasupermarket.JennyisatthehouseandAlanisatthesupermarket.At12noonJennyleavesthehouseandwalkstowardsthesupermarket.Atthesametime,Alanleavesthesupermarketandwalkstowardsthehouse.Jenny’splannedwalkismodelledbytheequation

jt

tt

ttt

= ++

< ≤< ≤< ≤

120100 4065 250

0 22 66 10

wherejisJenny’sdistance,inmetres,fromthehouseaftertminutes.Alan’splannedwalkismodelledbytheequation

a=–80t+900 t>0

whereaisAlan’sdistance,inmetres,fromthehouseaftertminutes.WhentheymeetA. Jennywillhavewalked359m,tothenearestmetre.B. Alanwillhavewalked360m,tothenearestmetre.C. Alanwillhavewalked382m,tothenearestmetre.D. Alanwillhavewalked518m,tothenearestmetre.E. Jennywillhavewalked541m,tothenearestmetre.

FURTHER MATHEMATICS

Written examination 1

FORMULA SHEET

Instructions

This formula sheet is provided for your reference.A multiple-choice question book is provided with this formula sheet.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2019

Victorian Certificate of Education 2019

FURMATH EXAM 2

Further Mathematics formulas

Core – Data analysis

standardised score z x xsx

=−

lower and upper fence in a boxplot lower Q1 – 1.5 × IQR upper Q3 + 1.5 × IQR

least squares line of best fit y = a + bx, where b 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

Documents

Victorian Certificate of Education 2019 · 2020. 2. 5. · 9 2019 FURMATH EXAM 1 SECTION A – continued TURN OVER Question 12 The table below shows the values of two variables x