examination specifications and sample examination

© VCAA 2016 – Version 4 – September 2016

VCE Further Mathematics 2016–2018

Written examinations 1 and 2 – End of year

Examination specifications

Overall conditions There will be two end-of-year examinations for VCE Further Mathematics – examination 1 and examination 2.

The examinations will be sat at a time and date to be set annually by the Victorian Curriculum and Assessment Authority (VCAA). VCAA examination rules will apply. Details of these rules are published annually in the VCE and VCAL Administrative Handbook.

Both examinations will have 15 minutes reading time and 1 hour and 30 minutes writing time.

For both examinations, students are permitted to bring into the examination room an approved technology with numerical, graphical, symbolic, financial and statistical functionality, as specified in the VCAA Bulletin and the VCE Exams Navigator. One bound reference may be brought into the examination room. This may be a textbook (which may be annotated), a securely bound lecture pad, a permanently bound student-constructed set of notes without fold-outs or an exercise book. Specifications for the bound reference are published annually in the VCE Exams Navigator.

A formula sheet will be provided with both examinations.

The examinations will be marked by a panel appointed by the VCAA.

The examinations will each contribute 33 per cent to the study score.

Content The VCE Mathematics Study Design 2016–2018 (‘Further Mathematics Units 3 and 4’) is the document for the development of the examination. All outcomes in ‘Further Mathematics Units 3 and 4’ will be examined.

All content from the areas of study, and the key knowledge and skills that underpin the outcomes in Units 3 and 4, are examinable.

Examination 1 will cover both Areas of study 1 and 2. The examination is designed to assess students’ knowledge of mathematical concepts, models and techniques, and their ability to reason, interpret and apply this knowledge in a range of contexts.

Examination 2 will cover both Areas of study 1 and 2. The examination is designed to assess students’ ability to select and apply mathematical facts, concepts, models and techniques to solve extended application problems in a range of contexts.

FURMATH (SPECIFICATIONS)


Format

Examination 1

The examination will be in the form of a multiple-choice question book.

The examination will consist of two sections.

Section A will consist of 24 multiple-choice questions derived from the core component of the course. Of these 24 questions,16 will be on data analysis and 8 will be on recursion and financial modelling. All questions will be compulsory. Section A will be worth a total of 24 marks.

Section B will consist of eight multiple-choice questions on each of the four modules in Unit 4. Students must answer questions on two modules. Section B will be worth a total of 16 marks.

The total marks for the examination will be 40.

A formula sheet will be provided with the examination. The formula sheet will be the same for examinations 1 and 2.

All answers are to be recorded on the answer sheet provided for multiple-choice questions.

Examination 2

The examination will be in the form of a question and answer book.

The examination will consist of two sections.

Section A will consist of short-answer and extended-answer questions, including multi-stage questions of increasing complexity. Questions will be derived from the core component of the course. Of these, 24 marks will be allocated to data analysis and 12 marks will be allocated to recursion and financial modelling. All questions will be compulsory. Section A will be worth a total of 36 marks.

Section B will consist of short-answer and extended-answer questions, including multi-stage questions of increasing complexity. Questions will be derived from each of the four modules in Unit 4. Each module will contain questions that total 12 marks. Students must answer questions on two modules. Section B will be worth a total of 24 marks.

The total marks for the examination will be 60.

A formula sheet will be provided with the examination. The formula sheet will be the same for examinations 1 and 2.

Answers are to be recorded in the spaces provided in the question and answer book.

Approved materials and equipment The list below applies to both examinations 1 and 2:

• normal stationery requirements (pens, pencils, highlighters, erasers, sharpeners and rulers) • an approved technology with numerical, graphical, symbolic, financial and statistical

functionality • one scientific calculator • one bound reference

FURMATH (SPECIFICATIONS)


Relevant references The following publications should be referred to in relation to the VCE Further Mathematics examinations:

• VCE Mathematics Study Design 2016–2018 (‘Further Mathematics Units 3 and 4’) • VCE Further Mathematics – Advice for teachers 2016–2018 (includes assessment advice) • VCE Exams Navigator • VCAA Bulletin

Advice During the 2016–2018 accreditation period for VCE Further Mathematics, examinations will be prepared according to the examination specifications above. Each examination will conform to these specifications and will test a representative sample of the key knowledge and skills from all outcomes in Units 3 and 4.

The following sample examinations provide an indication of the types of questions teachers and students can expect until the current accreditation period is over.

Answers to multiple-choice questions are provided at the end of examination 1.

Answers to other questions are not provided.

S A M P L E

FURTHER MATHEMATICSWritten examination 1

Day Date Reading time: *.** to *.** (15 minutes) Writing time: *.** to *.** (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• Questionbookof32pages.• Formulasheet.• Answersheetformultiple-choicequestions.• Workingspaceisprovidedthroughoutthebook.

Instructions• Checkthatyourname and student numberasprintedonyouranswersheetformultiple-choice

questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

At the end of the examination• Youmaykeepthisquestionbook.

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

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016

Version4–September2016

Victorian Certificate of Education Year

FURMATHEXAM1(SAMPLE) 2 Version4–September2016

THIS PAGE IS BLANK

Version4–September2016 3 FURMATHEXAM1(SAMPLE)

SECTION A – continuedTURN OVER

SECTION A – Core

Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrectforthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

Data analysis

Question 1Thefollowingstemplotshowstheareas,insquarekilometres,of27suburbsofalargecity.

key:1|6=1.6km2

1 5 6 7 82 1 2 4 5 6 8 9 93 0 1 1 2 2 8 94 0 4 75 0 16 1 978 4

Themedianareaofthesesuburbs,insquarekilometres,isA. 3.0B. 3.1C. 3.5D. 30.1E. 30.5

Question 2ThetimespentbyshoppersatahardwarestoreonaSaturdayisapproximatelynormallydistributedwithameanof31minutesandastandarddeviationof6minutes.If2850shoppersareexpectedtovisitthestoreonaSaturday,thenumberofshopperswhoareexpectedtospendbetween25and37minutesinthestoreisclosesttoA. 16B. 68C. 460D. 1900E. 2400


SECTION A – continued

Use the following information to answer Questions 3–6.Thefollowingtableshowsthedatacollectedfromarandomsampleofsevendriversdrawnfromthepopulationofalldriverswhousedasupermarketcarparkononeday.Thevariablesinthetableare:• distance–thedistancethateachdrivertravelledtothesupermarketfromtheirhome• sex–thesexofthedriver(female,male)• number of children–thenumberofchildreninthecar• type of car–thetypeofcar(sedan,wagon,other)• postcode–thepostcodeofthedriver’shome.

Distance (km) Sex (F = female, M = male)

Number of children

Type of car (1 = sedan, 2 = wagon,

3 = other)

Postcode

4.2 F 2 1 8148

0.8 M 3 2 8147

3.9 F 3 2 8146

5.6 F 1 3 8245

0.9 M 1 3 8148

1.7 F 2 2 8147

2.5 M 2 2 8145

Question 3Themean, x ,andthestandarddeviation,sx,ofthevariable,distance,forthesedriversareclosesttoA. x =2.5 sx=3.3B. x =2.8 sx=1.7C. x =2.8 sx=1.8D. x =2.9 sx=1.7E. x =3.3 sx=2.5

Question 4ThenumberofdiscretenumericalvariablesinthisdatasetisA. 0B. 1C. 2D. 3E. 4



Question 5ThenumberofordinalvariablesinthisdatasetisA. 0B. 1C. 2D. 3E. 4

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

Question 7

25

20

15

10

5

0

frequency

–1.5 –1.0 –0.5 0.0 0.5 1.0 1.5log10 (oil consumption)

Thehistogramabovedisplaysthedistributionoftheannualpercapitaoil consumption(tonnes)for58countriesplottedonalog scale.Thepercentageofcountrieswithanannualpercapitaoil consumptionofmorethan10tonnesisclosesttoA. 1%B. 2%C. 27%D. 57%E. 98%



Question 8Thedotplotbelowshowsthedistributionofthetime,inminutes,that50peoplespentwaitingtogethelpfromacallcentre.

10 20 30 40 50 60time

70 80 90 100 110

n = 50

Whichoneofthefollowingboxplotsbestrepresentsthedata?

A.

time10 20 30 40 50 60 70 80 90 100 110

B.

time10 20 30 40 50 60 70 80 90 100 110

C.

time10 20 30 40 50 60 70 80 90 100 110

D.

10 20 30 40 50 60

time70 80 90 100 110

E.

10 20 30 40 50 60

time70 80 90 100 110



Question 9Theparallelboxplotsbelowsummarisethedistributionofpopulationdensity,inpeoplepersquarekilometre,fortheinnersuburbsandtheoutersuburbsofalargecity.

population density (people per square kilometre)0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

inner suburbs

outer suburbs

Whichoneofthefollowingstatementsisnot true?A. Morethan50%oftheoutersuburbshavepopulationdensitiesbelow2000peoplepersquarekilometre.B. Morethan75%oftheinnersuburbshavepopulationdensitiesbelow6000peoplepersquarekilometre.C. Populationdensitiesaremorevariableintheoutersuburbsthanintheinnersuburbs.D. Themedianpopulationdensityoftheinnersuburbsisapproximately4400peoplepersquarekilometre.E. Populationdensitiesare,onaverage,higherintheinnersuburbsthanintheoutersuburbs.

Question 10Asingleback-to-backstemplotwouldbeanappropriategraphicaltooltoinvestigatetheassociationbetweenacar’sspeed,inkilometresperhour,andtheA. driver’sage,inyears.B. car’scolour(white,red,grey,other).C. car’sfuelconsumption,inkilometresperlitre.D. averagedistancetravelled,inkilometres.E. driver’ssex(female,male).



Question 11Theequationofaleastsquaresregressionlineisusedtopredictthefuelconsumption,inkilometresperlitreoffuel,fromacar’sweight,inkilograms.Thisequationpredictsthatacarweighing900kgwilltravel10.7kmperlitreoffuel,whileacarweighing1700kgwilltravel6.7kmperlitreoffuel.TheslopeofthisleastsquaresregressionlineisclosesttoA. –200.0B. –0.005C. –0.004D. 0.005E. 200.0

Question 12Alargestudyofsecondary-schoolmalestudentsshowsthatthereisanegativeassociationbetweenthetimespentplayingsporteachweekandthetimespentplayingcomputergames.Fromthisinformation,itcanbeconcludedthatA. malestudentswhospendalotoftimeplayingcomputergamesdonotplaysport.B. encouragingmalestudentstospendlesstimeplayingsportwillincreasethetimetheyspendplaying

computergames.C. encouragingmalestudentstospendmoretimeplayingsportwillreducethetimetheyspendplaying

computergames.D. malestudentswhotendtospendmoretimeplayingsporttendtospendlesstimeplayingcomputer

games.E. malestudentswhotendtospendmoretimeplayingsporttendtospendmoretimeplayingcomputer

games.

Question 13Theseasonalindexforheatersinwinteris1.25Tocorrectforseasonality,theactualheatersalesinwintershouldbeA. reducedby20%B. increasedby20%C. reducedby25%D. increasedby25%E. reducedby75%



Use the following information to answer Questions 14 and 15.Theseasonalindicesforthefirst11monthsoftheyearforsalesinasportingequipmentstoreareshowninthetablebelow.

Month Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.

Seasonal index 1.23 0.96 1.12 1.08 0.89 0.98 0.86 0.76 0.76 0.95 1.12

Question 14TheseasonalindexforDecemberisA. 0.89B. 0.97C. 1.02D. 1.23E. 1.29

Question 15InMay,thestoresold$213956worthofsportingequipment.ThedeseasonalisedvalueofthesesaleswasclosesttoA. $165857B. $190420C. $209677D. $218322E. $240400



Question 16Thetimeseriesplotbelowshowsthenumberofdaysthatitrainedinatowneachmonthduring2011.

Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

numberof days

1214

1086420

Usingfive-mediansmoothing,thesmoothedtimeseriesplotwilllookmostlikeA.


numberof days

1214

1086420

B.


numberof days

1214

1086420

C.


numberof days

1214

1086420

D.


numberof days

1214

1086420

E.


numberof days

1214

1086420



Recursion and financial modelling

Question 17

P0=2000, Pn + 1=1.5Pn – 500

ThefirstthreetermsofasequencegeneratedbytherecurrencerelationaboveareA. 500,2500,2000…B. 2000,1500,1000…C. 2000,2500,3000…D. 2000,2500,3250…E. 2000,3000,4500…

Question 18Whichofthefollowingrecurrencerelationswillgenerateasequencewhosevaluesdecaygeometrically?A. L0=2000, Ln + 1=Ln – 100

B. L0=2000, Ln + 1=Ln + 100

C. L0=2000, Ln + 1=0.65Ln

D. L0=2000, Ln + 1=6.5Ln

E. L0=2000, Ln + 1=0.85Ln – 100

Question 19Evahas$1200thatsheplanstoinvestforoneyear.Onecompanyofferstopayherinterestattherateof6.75%perannumcompoundingdaily.TheeffectiveannualinterestrateforthisinvestmentwouldbeclosesttoA. 6.75%B. 6.92%C. 6.96%D. 6.98%E. 6.99%

Question 20Rohaninvests$15000atanannualinterestrateof9.6%compoundingmonthly.LetVnbethevalueoftheinvestmentafternmonths.ArecurrencerelationthatcanbeusedtomodelthisinvestmentisA. V0=15000,Vn + 1=0.96Vn

B. V0=15000,Vn + 1=1.008Vn

C. V0=15000,Vn + 1=1.08Vn

D. V0=15000,Vn + 1=1.0096Vn

E. V0=15000,Vn + 1=1.096Vn



Use the following information to answer Questions 21–23.Kiminvests$400000inanannuitypaying3.2%interestperannum.Theannuityisdesignedtogiveheranannualpaymentof$47372for10years.Theamortisationtableforthisannuityisshownbelow.Someoftheinformationismissing.

Payment number (n)

Payment made

Interest earned

Reduction in principal

Balance of annuity

0 0 0.00 0.00 400000.00

1 47372.00 12800.00 34572.00

2 47372.00 11693.70 35678.30 329749.70

3 47372.00 10551.99 36820.01 292929.69

4 47372.00 9373.75 37998.25 254931.44

5 47372.00 8157.81 215717.24

6 47372.00 6902.95 40469.05 175248.19

7 47372.00 5607.94 41764.06 133484.14

8 47372.00 90383.63

9 47372.00 2892.28 44479.72 45903.90

10 47372.00 1468.92 45903.08 0.83

Question 21ThebalanceoftheannuityafteronepaymenthasbeenmadeisA. $339828.00B. $352628.00C. $365428.00D. $387200.00E. $400000.00

Question 22Thereductionintheprincipaloftheannuityafterpaymentnumber5isA. $36820.01B. $37998.25C. $39214.19D. $40469.05E. $41764.06

Question 23Theamountofpaymentnumber8thatistheinterestearnedisclosesttoA. $3799.82B. $4074.67C. $4271.49D. $4836.57E. $5607.94


END OF SECTION ATURN OVER

Question 24Thefollowinggraphshowsthedecreasingvalueofanassetovereightyears.

O 1 2 3 4 5 6 7 8

250 000

200 000

150 000

100 000

50 000

n (years)

P (dollars)

LetPnbethevalueoftheassetafternyears,indollars.AruleforevaluatingPncouldbeA. Pn=250000×(1+0.14)

n B. Pn=250000×1.14×nC. Pn=250000×(1–0.14)×nD. Pn=250000×(0.14)

n

E. Pn=250000×(1–0.14)n


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....................................................................................................... 24

Module4–Graphsandrelations................................................................................................................... 28


SECTION B – Module 1–continuedTURN OVER

Question 1MatrixB,below,showsthenumberofphotography(P),art(A)andcooking(C)booksownedbySteven(S),Trevor(T),Ursula(U),Veronica(V)andWilliam(W).

P A C

B

STUVW

=

8 5 41 4 53 3 44 2 21 4 1

TheelementinrowiandcolumnjofmatrixBisbi j .Theelementb32isthenumberofA. artbooksownedbyTrevor.B. artbooksownedbyUrsula.C. artbooksownedbyVeronica.D. cookingbooksownedbyUrsula.E. cookingbooksownedbyTrevor.

Question 2Thetotalcostofoneice-creamandthreesoftdrinksatCatherine’sshopis$9.Thetotalcostoftwoice-creamsandfivesoftdrinksis$16.Letxbethecostofanice-creamandybethecostofasoftdrink.

Thematrixxy

isequalto

A. 1 32 5

xy

B. 1 32 5

916

C. 1 23 5

916

D. −−

5 23 1

916

E. −−

5 32 1

916

Module 1 – Matrices

Beforeansweringthesequestions,youmustshadethe‘Matrices’boxontheanswersheetfor multiple-choicequestionsandwritethenameofthemoduleintheboxprovided.


SECTION B – Module 1–continued

Question 3 Considerthefollowingfourstatements.Apermutationmatrixisalways: I asquarematrix II abinarymatrixIII adiagonalmatrixIV equaltothetransposeofitself.

Howmanyofthestatementsabovearetrue?A. 0B. 1C. 2D. 3E. 4

Question 4Fourpeople,Ash(A),Binh(B),Con(C)andDan(D),competedinatabletennistournament.Inthistournament,eachcompetitorplayedeachoftheothercompetitorsonce.Theresultsofthetournamentaresummarisedinthematrixbelow.A1inthematrixshowsthattheplayernamedinthatrowdefeatedtheplayernamedinthatcolumn.Forexample,the1inrow3showsthatCondefeatedAsh.

loserA B C D

ABCD

winner

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

Inthetournament,eachcompetitorwasgivenarankingthatwasdeterminedbycalculatingthesumoftheirone-stepandtwo-stepdominances.Thecompetitorwiththehighestsumisrankednumberone(1).Thecompetitorwiththesecond-highestsumwasrankednumbertwo(2),andsoon.Usingthismethod,therankingsofthecompetitorsinthistournamentwereA. Dan(1),Ash(2),Con(3),Binh(4).B. Dan(1),Ash(2),Binh(3),Con(4).C. Con(1),Dan(2),Ash(3),Binh(4).D. Ash(1),Dan(2),Binh(3),Con(4).E. Ash(1),Dan(2),Con(3),Binh(4).



Question 5ThematrixSn + 1isdeterminedfromthematrixSnusingtheruleSn + 1=T Sn – C,whereT,S0andCaredefinedasfollows.

T S C=

=

=

0 5 0 60 5 0 4

100250

20200

. .

. ., and

Giventhisinformation,thematrixS2equals

A. 100250

B. 148122

C. 170140

D. 180130

E. 190160

Question 6AandBaresquarematricessuchthatAB=BA=I,whereIisanidentitymatrix.Whichoneofthefollowingstatementsisnottrue?A. ABA=AB. AB2A=IC. BmustequalAD. BistheinverseofAE. bothAandBhaveinverses


End of Module 1 – SECTION B–continued

Question 7TheorderofmatrixXis3×2.TheelementinrowiandcolumnjofmatrixXisxi janditisdeterminedbytherule

xi j=i + j

ThematrixXisA. 1 2

3 45 6

B. 2 34 56 7

C. 2 3 43 4 5

D. 1 23 34 4

E. 2 33 44 5

Question 8Atransitionmatrix,T,andastatematrix,S2,aredefinedasfollows.

T =

0 5 0 0 50 5 0 5 00 0 5 0 5

. .

. .. .

S2

300200100

=

IfS2=TS1,thestatematrixS1is

A. 200250150

B. 300200100

C. 3000300

D. 4000200

E. undefined



Question 1Thegraphbelowshowstheroadsconnectingfourtowns:Kelly,Lindon,MiltonandNate.

Kelly Lindon

Nate

Milton

AbusstartsatKelly,travelsthroughNateandLindon,thenstopswhenitreachesMilton.ThemathematicaltermforthisrouteisA. aloop.B. anEuleriantrail.C. anEuleriancircuit.D. aHamiltonianpath.E. aHamiltoniancycle.

Question 2

YA

B

C

E

D

Inthedirectedgraphabove,theonlyvertexwithalabelthatcanbereachedfromvertexYisA. vertexA.B. vertexB.C. vertexC.D. vertexD.E. vertexE.

Module 2 – Networks and decision mathematics

Beforeansweringthesequestions,youmustshadethe‘Networksanddecisionmathematics’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.



Question 3Thefollowingnetworkshowsthedistances,inkilometres,alongaseriesofroadsthatconnectTownA to TownB.

1

3

5

43 6

1

4

24

6

1

3

4 4

2

2

6

5

Town A Town B

UsingDijkstra’salgorithm,orotherwise,theshortestdistance,inkilometres,fromTownAtoTownBisA. 9B. 10C. 11D. 12E. 13



Question 4

C

D EA B

FG

H

IJ

8 109 11

14

1312

16

13

17

12 8

1510

6 7

9

Whichoneofthefollowingistheminimalspanningtreefortheweightedgraphshownabove?

A.

A B

C

D E

FG

H

IJ

B.

A B

C

D E

FG

H

IJ

C.

A B

C

D E

FG

H

IJ

D.

A B

C

D E

FG

H

IJ

E.

A B

C

D E

FG

H

IJ



Use the following information to answer Questions 5 and 6.Considerthefollowingfourgraphs.

Question 5HowmanyofthefourgraphsabovehaveanEuleriancircuit?A. 0B. 1C. 2D. 3E. 4

Question 6Howmanyofthefourgraphsaboveareplanar?A. 0B. 1C. 2D. 3E. 4

Question 7Whichoneofthefollowingstatementsaboutcriticalpathsistrue?A. Therecanbeonlyonecriticalpathinaproject.B. Acriticalpathalwaysincludesatleasttwoactivities.C. Acriticalpathwillalwaysincludetheactivitythattakesthelongesttimetocomplete.D. Reducingthetimeofanyactivityonacriticalpathforaprojectwillalwaysreducetheminimum

completiontimefortheproject.E. Iftherearenootherchanges,increasingthetimeofanyactivityonacriticalpathwillalwaysincrease

thecompletiontimeofaproject.


End of Module 2 – SECTION B–continuedTURN OVER

Question 8Anetworkoftracksconnectstwocarparksinafestivalvenuetotheexit,asshowninthedirectedgraphbelow.

75 5

10

735

11

5412

4 6 6

2

8

4

9

4

5

5

5

22exit

car park

car park

Cut A Cut B Cut C Cut D Cut E

Thearrowsshowthedirectionthatcarscantravelalongeachofthetracksandthenumbersshoweachtrack’scapacityincarsperminute.Fivecutsaredrawnonthediagram.ThemaximumnumberofcarsperminutethatwillreachtheexitisgivenbythecapacityofA. Cut A.B. Cut B.C. Cut C.D. Cut D.E. Cut E.



Question 1

2.5 cm

64°

Asectorofacircleofradius2.5cmsubtendsanangleof64°atthecentreofthecircle.Theareaofthesector,insquarecentimetres,isclosesttoA. 2.8B. 3.5C. 7.0D. 88.4E. 110.5

Question 2ThecitythatisclosesttotheequatorisA. Athens,latitude38.0°NB. Belgrade,latitude44.8°NC. Kingston,latitude45.3°SD. Pretoria,latitude25.7°SE. Brisbane,latitude27.5°S

Module 3 – Geometry and measurement

Beforeansweringthesequestions,youmustshadethe‘Geometryandmeasurement’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.



Question 3Acafesellstwosizesofcupcakeswithasimilarshape.Thelargecupcakeis6cmwideatthebaseandthesmallcupcakeis4cmwideatthebase.

4 cm6 cm

Thepriceofacupcakeisproportionaltoitsvolume.Ifthelargecupcakecosts$5.40,thenthesmallcupcakewillcostA. $1.60B. $2.32C. $2.40D. $3.40E. $3.60

Question 4

10 m

4 m

5 m

12 m

Agreenhouseisbuiltintheshapeofatrapezoidalprism,asshowninthediagramabove.Thecross-sectionofthegreenhouse(shaded)isanisoscelestrapezium.Theparallelsidesofthistrapeziumare4mand10mrespectively.Thetwoequalsidesareeach5m.Thelengthofthegreenhouseis12m.Thefiveexteriorsurfacesofthegreenhouse,notincludingthebase,aremadeofglass.Thetotalareaoftheglasssurfacesofthegreenhouse,insquaremetres,isA. 196B. 212C. 224D. 344E. 672



Use the following information to answer Questions 5 and 6.

Question 5Across-countryraceisrunonatriangularcourse.ThepointsA,BandCmarkthecornersofthecourse,asshownbelow.

N

A

BC

140°

2050 m1900 m

2250 m

ThedistancefromA to Bis2050m.ThedistancefromB to Cis2250m.ThedistancefromA to Cis1900m.ThebearingofBfromAis140°.ThebearingofCfromAisclosesttoA. 032°B. 069°C. 192°D. 198°E. 209°

Question 6TheareawithinthetriangularcourseABC,insquaremetres,canbecalculatedbyevaluating

A. 3100×1200×1050×850

B. 3100× 2250× 2050×1900

C. 6200× 4300× 4150×3950

D. 12× 2050× 2250× sin(140°)

E. 12× 2050× 2250× sin(40°)


End of Module 3 – SECTION B–continuedTURN OVER

Question 7

A

C

B, latitude 40° N

6400 km equator

AssumethattheradiusofEarthis6400km.ThediagramaboveshowsasmallcircleofEarth,withcentreatA,whoselatitudeis40°N.Theradiusofthissmallcircle,inkilometres,isclosesttoA. 4114B. 4903C. 5543D. 6400E. 7390

Question 8

E H

F G

CD

A B

Arightrectangularprismwithasquarebase,ABCD,isshownabove.Thediagonaloftheprism,AH,is8cm.Theheightoftheprism,HC,is4cm.Thevolumeofthisrectangularprism,incubiccentimetres,isA. 64 B. 96 C. 128 D. 192 E. 256



Question 1Thegraphbelowshowsthealtitude,inmetres,ofaballoonoverasix-hourflight.

3500

3000

2500

2000

1500

1000

500

O

altitude(metres)

1 2 3time (hours)

4 5 6

Overthesix-hourperiod,thelengthoftime,inhours,wherethealtitudeoftheballoonwasatleast1500misA. 3B. 4C. 5D. 6E. 7

Question 2Theverticallinethatpassesthroughthepoint(3,2)hastheequationA. x + y=5B. xy=6C. 3y=2xD. y=2E. x=3

Module 4 – Graphs and relations

Beforeansweringthesequestions,youmustshadethe‘Graphsandrelations’boxontheanswersheetformultiple-choicequestionsandwritethenameofthemoduleintheboxprovided.



Question 3

Thepoint(2,20)liesonthegraphof y kx

= , asshownbelow.

100

80

60

40

20

2O 4 6 8 10

y

x

(2, 20)

ThevalueofkisA. 5B. 10C. 20D. 40E. 80



Question 4Thedistance–timegraphbelowshowsatrain’sjourneybetweentwotowns.Duringthejourney,thetrainstoppedfor30minutes.

100

80

60

40

20

09.00 am 9.30 am 10.00 am 10.30 am 11.00 am

distance (kilometres)

time

Theaveragespeedofthetrain,inkilometresperhour,forthejourneyisclosesttoA. 45B. 50C. 60D. 65E. 80

Question 5TheDomesticsCleaningCompanyprovideshouseholdcleaningservices.Fortwohoursofcleaning,thecostis$55.Forfourhoursofcleaning,thecostis$94.Theruleforthecostofcleaningservicesis

cost = a + b×hours

whereaisafixedcharge,indollars,andbisthechargeperhourofcleaning,indollarsperhour.Usingthisrule,thecostforfivehoursofcleaningisA. $19.50B. $97.50C. $99.50D. $113.50E. $121.50



Question 6Whichoneofthefollowingstatementsrelatingtothesolutionoflinearprogrammingproblemsistrue?A. Onlyonepointcanbeasolution.B. Nopointoutsidethefeasibleregioncanbeasolution.C. Tohaveasolution,thefeasibleregionmustbebounded.D. Onlythecornerpointsofafeasibleregioncanbeasolution.E. Onlythecornerpointswithintegercoordinatescanbeasolution.

Question 7Theconstraintsofalinearprogrammingproblemaregivenbythefollowingsetofinequalities.

x + y≤8 2x + 4y≤23 x≥0 y≥0

Thegraphbelowshowsthelinesthatrepresenttheboundariesoftheseinequalities.

9

8

7

6

5

4

3

2

1

O 1 2 3 4 5 6 7 8 9 10 11 12x

y

Thecoordinatesofthepointsthatdefinetheboundariesofthefeasibleregionforthislinearprogrammingproblemare(0,0),(0,5.75),(4.5,3.5)and(8,0).WhenmaximisingtheobjectivefunctionZ=2x + 3yfortheseconstraints,thesolutionisfoundatthepoint(4.5,3.5).Ifonlyinteger solutionsarepermittedforthisproblem,thesolutionwilloccuratA. (1,5)B. (3,4)C. (4,4)D. (5,3)E. (6,2)


END OF MULTIPLE-CHOICE QUESTION BOOK

Question 8XavierandYvetteshareajob.YvettemustworkatleasttwiceasmanyhoursasXavier.Theymustworkatleast40hourseachweek,intotal.Xaviermustworkatleast10hourseachweek.Yvettecanonlyworkforamaximumof30hourseachweek.LetxrepresentthenumberofhoursthatXavierworkseachweek.LetyrepresentthenumberofhoursthatYvetteworkseachweek.Inwhichoneofthefollowinggraphsdoestheshadedareashowthefeasibleregiondefinedbytheseconditions?

y

x

605040302010

O 10 20 30 40 50 60

y

x

605040302010

O 10 20 30 40 50 60

y

x

605040302010

O 10 20 30 40 50 60

y

x

605040302010

O 10 20 30 40 50 60

y

x

605040302010

O 10 20 30 40 50 60

A. B.

C. D.

E.

FURMATH EXAM 1 (SAMPLE – ANSWERS)


Answers to multiple-choice questions

Section A – Core

Data analysis

Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Answer B D C B A B B A C E B D A E E B


Question 17 18 19 20 21 22 23 24

Answer D C D B C C C E

Section B – Modules


Question 1 2 3 4 5 6 7 8

Answer B E C E B C E D


Question 1 2 3 4 5 6 7 8

Answer D D B A B E E D


Question 1 2 3 4 5 6 7 8

Answer B D A C E A B B


Question 1 2 3 4 5 6 7 8

Answer B E D A D B D C

S A M P L E

FURTHER MATHEMATICSWritten examination 2

Day Date Reading time: *.** to *.** (15 minutes) Writing time: *.** to *.** (1 hour 30 minutes)

QUESTION AND ANSWER BOOK

Structure of bookSection A – Core Number of

questionsNumber of questions

to be answeredNumber of

marks

9 9 36Section B – Modules Number of

modulesNumber of modules

to be answeredNumber of

marks

4 2 24 Total 60

• Studentsaretowriteinblueorblackpen.• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,

sharpeners,rulers,oneboundreference,oneapprovedtechnology(calculatororsoftware)and,ifdesired,onescientificcalculator.CalculatormemoryDOESNOTneedtobecleared.Forapprovedcomputer-basedCAS,fullfunctionalitymaybeused.

• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.

Materials supplied• Questionandanswerbookof30pages.• Formulasheet.• Workingspaceisprovidedthroughoutthebook.

Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.• AllwrittenresponsesmustbeinEnglish.

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

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016

Version4–September2016

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education Year

STUDENT NUMBER

Letter


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SECTION A – Question 1 – continued

Data analysis

Question 1 (3marks)Thesegmentedbarchartbelowshowstheagedistributionofpeopleinthreecountries,Australia,IndiaandJapan,fortheyear2010.

65 years and over

15–64 years

0–14 years

100

90

80

70

60

50

40

30

20

10

0

percentage

Australia Indiacountry

Japan

Source:AustralianBureauofStatistics,3201.0–Population by Age and Sex, Australian States and Territories,June2010

SECTION A – Core

Instructions for Section AAnswerallquestionsinthespacesprovided.Writeusingblueorblackpen.Youneednotgivenumericalanswersasdecimalsunlessinstructedtodoso.Alternativeformsmayinclude, forexample,π,surdsorfractions.In‘Recursionandfinancialmodelling’,allanswersshouldberoundedtothenearestcentunlessotherwiseinstructed.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.



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a. WritedownthepercentageofpeopleinAustraliawhowereaged0–14yearsin2010. 1mark

b. In2010,thepopulationofJapanwas128000000.

HowmanypeopleinJapanwereaged65yearsandoverin2010? 1mark

c. Fromthegraphonpage2,itappearsthatthereisnoassociationbetweenthepercentageofpeopleinthe15–64agegroupandthecountryinwhichtheylive.

Explainwhy,quotingappropriatepercentagestosupportyourexplanation. 1mark



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Question 2 (3marks)Thedevelopmentindexforacountryisawholenumberbetween0and100.Thedotplotbelowdisplaysthevaluesofthedevelopmentindicesfor28countries.

n = 28

70 71 72 73 74 75development index

76 77 78 79

a. Usingtheinformationinthedotplot,determineeachofthefollowing. 1mark

Themode Therange

b. Writedownanappropriatecalculationanduseittoexplainwhythecountrywithadevelopmentindexof70isanoutlierforthisgroupofcountries. 2marks


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SECTION A – Question 3 – continuedTURN OVER

Question 3 (6marks)Thescatterplotbelowshowsthepopulation and area(insquarekilometres)ofasampleofinnersuburbsofalargecity.

30000

25000

20000

15000population

10000

5000

00 1 2 3 4

area (km2)5 6 7 8 9

Theequationoftheleastsquaresregressionlineforthedatainthescatterplotis

population=5330+2680×area

a. Writedowntheresponsevariable. 1mark

b. Drawtheleastsquaresregressionlineonthescatterplot above. 1mark

(Answer on the scatterplot above.)

c. Interprettheslopeofthisleastsquaresregressionlineintermsofthevariablesarea and population. 2marks



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d. Wistonisaninnersuburb.Ithasanareaof4km2andapopulationof6690. Thecorrelationcoefficient,r,isequalto0.668

i. CalculatetheresidualwhentheleastsquaresregressionlineisusedtopredictthepopulationofWistonfromitsarea. 1mark

ii. Whatpercentageofthevariationinthepopulationofthesuburbsisexplainedbythevariationinarea?

Roundyouranswertoonedecimalplace. 1mark



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Question 4 (3marks)Thescatterplotandtablebelowshowthepopulation,inthousands,andthearea,insquarekilometres,forasampleof21outersuburbsofthesamecity.

35

30

25

20population(thousands)

15

10

5

00 20 40 60 80

area (km2)100 120 140

Area (km2) Population (thousands)

1.6 5.2 4.4 14.3 4.6 7.5 5.6 11.0 6.3 17.1 7.0 19.4 7.3 15.5 8.0 11.3 8.8 17.1 11.1 19.7 13.0 17.9 18.5 18.7 21.3 24.6 24.2 15.2 27.0 13.6 62.1 26.1 66.5 16.4101.4 26.2119.2 16.5130.7 18.9135.4 31.3

Intheoutersuburbs,therelationshipbetweenpopulation and areaisnon-linear.A logtransformationcanbeappliedtothevariableareatolinearisethescatterplot.

a. Applythelogtransformationtothedataanddeterminetheequationoftheleastsquaresregressionlinethatallowsthepopulationofanoutersuburbtobepredictedfromthelogarithmofitsarea.

Writetheslopeandinterceptofthisleastsquaresregressionlineintheboxesprovidedbelow. Roundyouranswerstotwosignificantfigures. 2marks

population = + log(area)

b. Usetheequationoftheleastsquaresregressionlineinpart a.topredictthepopulationofanoutersuburbwithanareaof90km2.

Roundyouranswertothenearestonethousandpeople. 1mark



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Question 5 (4marks)Thereisanegativeassociationbetweenthevariablespopulation density,inpeoplepersquarekilometre,and area,insquarekilometres,of38innersuburbsofthesamecity.Forthisassociation,r2=0.141

a. Writedownthevalueofthecorrelationcoefficientforthisassociationbetweenthevariablespopulation density and area.

Roundyouranswertothreedecimalplaces. 1mark

b. Themeanandstandarddeviationofthevariablespopulation density and areaforthese 38innersuburbsareshowninthetablebelow.

Population density (people per km2)

Area (km2)

Mean 4370 3.4

Standard deviation 1560 1.6

Oneofthesesuburbshasapopulationdensityof3082peoplepersquarekilometre.

i. Determinethestandardz-scoreofthissuburb’spopulationdensity. Roundyouranswertoonedecimalplace. 1mark

ii. Interpretthez-scoreofthissuburb’spopulationdensitywithreferencetothemeanpopulationdensity. 1mark

iii. Assumetheareasoftheseinnersuburbsareapproximatelynormallydistributed. Howmanyofthese38suburbsareexpectedtohaveanareathatistwostandarddeviationsor

moreabovethemean? Roundyouranswertothenearestwholenumber. 1mark


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SECTION A – Question 6 – continuedTURN OVER

Question 6 (5marks)Table1showstheAustraliangrossdomesticproduct(GDP)perperson,indollars,atfiveyearlyintervals(year)fortheperiod1980to2005.

Table 1

Year 1980 1985 1990 1995 2000 2005

GDP 20 900 22 300 25000 26 400 30 900 33800

1975 1980 1985 1990 1995 2000 2005 2010year

36 000

34 000

32 000

30 000

28 000

26 000

24 000

22 000

20 000

GDP (dollars)

a. Completethetime series plot abovebyplottingtheGDPfortheyears2000and2005. 1mark

(Answer on the time series plot above.)

b. Brieflydescribethegeneraltrendinthedata. 1mark



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c. InTable2,thevariableyearhasbeenrescaledusing1980=0,1985=5,andsoon.Thenewvariableis time.

Table 2

Year 1980 1985 1990 1995 2000 2005

Time 0 5 10 15 20 25

GDP 20 900 22 300 25000 26 400 30 900 33800

i. Usethevariablestime and GDPtowritedowntheequationoftheleastsquaresregressionlinethatcanbeusedtopredictGDPfromtime.Taketimeastheexplanatoryvariable. 2marks

ii. Theleastsquaresregressionlineinpart c.i.abovehasbeenusedtopredicttheGDPin2010.

Explainwhythispredictionisunreliable. 1mark



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Question 7 (4marks)Hugoisaprofessionalbikerider.Thevalueofhisbikewillbedepreciatedovertimeusingtheflatratemethodofdepreciation.ThevalueofHugo’sbike,indollars,afternyears,Vn,canbemodelledusingtherecurrencerelationbelow.

V0=8400, Vn+1 = Vn–1200

a. Usingtherecurrencerelation,writedowncalculationstoshowthatthevalueofHugo’sbikeaftertwoyearsis$6000. 1mark

Hugowillsellhisbikewhenitsvaluereducesto$3600.

b. AfterhowmanyyearswillHugosellhisbike? 1mark

TheunitcostmethodcanalsobeusedtodepreciatethevalueofHugo’sbike.Aruleforthevalueofthebike,indollars,aftertravellingnkilometresis

Vn=8400–0.25n

c. Whatisthedepreciationofthebikeperkilometre? 1mark

Aftertwoyears,thevalueofthebikewhendepreciatedbytheunitcostmethodwillbethesameasthevalueofthebikewhendepreciatedbytheflatratemethod.

d. Howmanykilometreshasthebiketravelledaftertwoyears? 1mark



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Question 8 (5marks)Hugowon$5000inaroadrace.Hedepositedthismoneyintoasavingsaccount.ThevalueofHugo’ssavingsafternmonths,Sn,canbemodelledbytherecurrencerelationbelow.

S0=5000, Sn+1=1.004Sn

a. Whatistheannualinterestrate(compoundingmonthly)forHugo’ssavingsaccount? 1mark

b. WhatwouldbethevalueofHugo’ssavingsafter12months? 1mark

Usingadifferentinvestmentstrategy,Hugocoulddeposit$3000intoanaccountearningcompoundinterestattherateof4.2%perannum,compoundingmonthly,andmakeadditionalpaymentsof$200aftereverymonth.LetTnbethevalueofHugo’sinvestmentafternmonthsusingthisstrategy.Themonthlyinterestrateforthisaccountis0.35%.

c. i. Writedownarecurrencerelation,intermsofTn+1 and Tn,thatmodelsthevalueofHugo’sinvestmentusingthisstrategy. 1mark

ii. WhatisthetotalinterestHugowouldhaveearnedaftersixmonths? 2marks


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END OF SECTION ATURN OVER

Question 9 (3marks)Hugoneedstobuyanewbike.Heborrowed$7500topayforthebikeandwillbechargedinterestattherateof5.76%perannum,compoundingmonthly.Hugowillfullyrepaythisloanwithrepaymentsof$430eachmonth.

a. Howmanyrepaymentsarerequiredtofullyrepaythisloan? Roundyouranswertothenearestwholenumber. 1mark

Afterthefifthrepayment,Hugoincreasedhismonthlyrepaymentsothattheloanwasfullyrepaidwithafurthersevenrepayments(thatis,12repaymentsintotal).

b. i. WhatistheminimumvalueofHugo’snewmonthlyrepayment? 1mark

ii. Whatisthevalueofthefinalrepaymentrequiredtoensuretheloanisfullyrepaidafter12repayments? 1mark


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SECTION B – continued

SECTION B – Modules

Instructions for Section BSelect twomodulesandanswerallquestionswithintheselectedmodules.Writeusingblueorblackpen.Youneednotgivenumericalanswersasdecimalsunlessinstructedtodoso.Alternativeformsmayinclude, forexample,π,surdsorfractions.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

Contents Page

Module1–Matrices................................................................................................................................................... 15

Module2–Networksanddecisionmathematics....................................................................................................... 19

Module3–Geometryandmeasurement.................................................................................................................... 23

Module4–Graphsandrelations................................................................................................................................ 28


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SECTION B – Module 1 – continuedTURN OVER


Question 1 (2marks)Fivetrout-breedingponds,P,Q,R,X and V,areconnectedbypipes,asshowninthediagrambelow.

P Q

V

R X

ThematrixWisusedtorepresenttheinformationinthisdiagram.

P Q R X V

W

PQRXV

=

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

InmatrixW:• the1inrow2,column1,forexample,indicatesthatpondPisdirectlyconnectedbyapipetopondQ• the0inrow5,column1,forexample,indicatesthatpondPisnotdirectlyconnectedbyapipeto

pondV.

a. Intermsofthebreedingpondsdescribed,whatdoesthesumoftheelementsinrow3ofmatrixW represent? 1mark

ThematrixW 2isshownbelow.

P Q R X V

W

PQRXV

2

2 0 0 2 10 3 2 1 10 2 2 0 12 1 0 3 11 1 1 1 2

=

b. MatrixW 2hasa2inrow2(Q),column3(R).

ExplainwhatthisnumbertellsusaboutthepipeconnectionsbetweenQ and R. 1mark


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SECTION B – Module 1 – Question 2 – continued

Question 2 (10marks)10000trouteggs,1000babytroutand800adulttroutareplacedinapondtoestablishatroutpopulation.Inestablishingthispopulation:• eggs(E)maydie(D)ortheymayliveandeventuallybecomebabytrout(B)• babytrout(B)maydie(D)ortheymayliveandeventuallybecomeadulttrout(A)• adulttrout(A)maydie(D)ortheymayliveforaperiodoftimebutwilleventuallydie.

Fromyeartoyear,thissituationcanberepresentedbythetransitionmatrixT,where

this yearE B A D

T

EB

=

0 0 0 00 4 0 0 00 0 25 0 5 00 6 0 75 0 5 1

.. .

. . .AAD

next year

a. UsetheinformationinthetransitionmatrixT to

i. determinethenumberofeggsinthispopulationthatdieinthefirstyear 1mark

ii. completethetransitiondiagrambelow,showingtherelevantpercentages. 2marks

40%

60%

B

EA

D


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SECTION B – Module 1 – Question 2 – continuedTURN OVER

Theinitialstatematrixforthistroutpopulation,S0,canbewrittenas

S

EBAD

0

1000010008000

=

LetSnrepresentthestatematrixdescribingthetroutpopulationafternyears.

b. UsingtheruleSn+1 = T Sn,determine

i. S1 1mark

ii. thenumberofadulttroutpredictedtobeinthepopulationafterfouryears. Roundyouranswertothenearestwholenumberoftrout. 1mark

c. ThetransitionmatrixTpredictsthat,inthelongterm,alloftheeggs,babytroutandadulttroutwilldie.

i. Howmanyyearswillittakeforalloftheadulttrouttodie(thatis,whenthenumberofadulttroutinthepopulationisfirstpredictedtobelessthanone)? 1mark

ii. Whatisthelargestnumberofadulttroutthatispredictedtobeinthepondinanyoneyear? 1mark

d. Determinethenumberofeggs,babytroutandadulttroutthat,ifaddedtoorremovedfromthepondattheendofeachyear,willensurethatthenumberofeggs,babytroutandadulttroutinthepopulationremainsconstantfromyeartoyear. 2marks


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End of Module 1 – SECTION B – continued

TheruleSn+1 = T Snthatwasusedtodescribethedevelopmentofthetroutinthisponddoesnottakeintoaccountneweggsaddedtothepopulationwhentheadulttroutbegintobreed.Totakebreedingintoaccount,assumethateveryyear50%oftheadulttrouteachlay500eggs.Thematrixdescribingthepopulationafternyears,Sn,isnowgivenbythenewrule

Sn+1 = T Sn+500M Sn

where

T M=

=

0 0 0 00 40 0 0 00 0 25 0 50 00 60 0 75 0 50 1 0

0 0 0 5.

. .. . . .

,

. 00 00 0 0 00 0 0 00 0 0 0

1000010008000

0

=

and S

e. UsethisnewruletodetermineS2. 1mark


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Question 1 (6marks)Waterwillbepumpedfromadamtoeightlocationsonafarm.Thepumpandtheeightlocations(includingthehouse)areshownasverticesinthenetworkdiagrambelow.Thenumbersontheedgesjoiningtheverticesgivetheshortestdistances,inmetres,betweenlocations.

pump

7040

50house40

8080

9070

40

80

5080

90

6070

120

70

dam

a. i. Determinetheshortestdistancebetweenthehouseandthepump. 1mark

ii. Howmanyverticesonthenetworkdiagramhaveanodddegree? 1mark

iii. Thetotallengthofalledgesinthenetworkis1180m. Ajourneystartsandfinishesatthehouseandtravelsalongeveryedgeinthenetwork.

Determinetheshortestdistancetravelled. 1mark

iv. AHamiltonianpath,beginningatthehouse,isdeterminedforthisnetwork.

Howmanyedgesdoesthispathinvolve? 1mark


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SECTION B – Module 2 – continued

Thetotallengthofpipethatsupplieswaterfromthepumptotheeightlocationsonthefarmisaminimum.Thisminimumlengthofpipeislaidalongsomeoftheedgesinthenetwork.

b. i. Onthediagrambelow,drawtheminimumlengthofpipethatisneededtosupplywatertoalllocationsonthefarm. 1mark

pump

7040

50house40

8080

9070

40

80

5080

90

6070

120

70

dam

ii. Whatisthemathematicaltermthatisusedtodescribethisminimumlengthofpipeinpart b.i.? 1mark


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Question 2 (6marks)Aprojectwillbeundertakenonthefarm.Thisprojectinvolvesthe13activitiesshowninthetablebelow.Theduration,inhours,andpredecessor(s)ofeachactivityarealsoincludedinthetable.

Activity Duration (hours)

Predecessor(s)

A 5 –

B 7 –

C 4 –

D 2 C

E 3 C

F 15 A

G 4 B,D,H

H 8 E

I 9 F,G

J 9 B,D,H

K 3 J

L 11 J

M 8 I, K

ActivityGismissingfromthenetworkdiagramforthisproject,whichisshownbelow.

start finish

F, 15 I, 9

M, 8K, 3

L, 11

B, 7

A, 5

D, 2

H, 8

J, 9

C, 4

E, 3

a. Completethenetwork diagram abovebyinsertingactivityG. 1mark

(Answer on the network diagram above.)

b. DeterminetheearlieststartingtimeofactivityH. 1mark


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End of Module 2 – SECTION B – continued

c. GiventhatactivityGisnotonthecriticalpath

i. writedowntheactivitiesthatareonthecriticalpathintheorderthattheyarecompleted 1mark

ii. findthelateststartingtimeforactivityD. 1mark

d. Considerthefollowingstatement: ‘Ifjustoneoftheactivitiesinthisprojectiscrashedbyonehour,thentheminimumtimetocomplete

theentireprojectwillbereducedbyonehour.’

Explainthecircumstancesunderwhichthisstatementwillbetrueforthisproject. 1mark

e. AssumeactivityFiscrashedbytwohours.

Whatwillbetheminimumcompletiontimefortheproject? 1mark


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Question 1 (3marks)Oneofthefieldeventsatathleticscompetitionsisthediscus.Thefieldmarkingsforthediscuseventconsistofacircularthrowingring,foullinesandtheboundarylineofthefield,asshowninthediagrambelow.Theshadedareaonthediagramisthelandingregionforadiscusthrow.

boundary line

landing region

foul line foul line

throwing ring

A B

65 m 65 m

θ

ThefoullinesmeettheboundarylineatpointsA and B,65mfromthecentreofthethrowingring.Theangleθis34.92°.

a. WhatisthelengthoftheboundarylinefrompointAtopointB? Writeyouranswerinmetres,roundedtotwodecimalplaces. 1mark

b. Calculatetheareaofthelandingregion. Roundyouranswertothenearestsquaremetre. 2marks


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Question 2 (5marks)DaniellivesinMildura(34°S,142°E).HewillflytoSydney(34°S,151°E)andthenflyontoRome(42°N,12°E)tocompeteinthediscuseventataninternationalathleticscompetition.Inthisquestion,assumethattheradiusofEarthis6400km.

a. FindtheshortestgreatcircledistancetotheSouthPolefromMildura(34°S,142°E). Roundyouranswertothenearestkilometre. 1mark

b. TheflightfromMildura(34°S,142°E)toSydney(34°S,151°E)travelsalongasmallcircle.

i. Findtheradiusofthissmallcircle. Roundyouranswertotwodecimalplaces. 1mark

ii. FindthedistancetheplanetravelsbetweenMildura(34°S,142°E)andSydney(34°S,151°E). Roundyouranswertothenearestkilometre. 1mark


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c. HowlongafterthesunrisesinSydney(34°S,151°E)willthesunriseinRome(42°N,12°E)? Roundyouranswertothenearestminute. 1mark

d. Daniel’sflighttoRomeleavesSydneyairportonSunday,6Marchat10.20am,localtime.TheflightarrivesinRomeonMonday,7Marchat2.30am.AssumethetimedifferencebetweenSydneyandRomeis10hours.

HowlongdoestheflighttaketotravelfromSydneytoRome? Roundyouranswertothenearestminute. 1mark


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SECTION B – Module 3 – continued

Question 3 (2marks)Danielwillcompeteintheintermediatedivisionofthediscuscompetition.Competitorsintheintermediatedivisionuseasmallerdiscusthantheoneusedintheseniordivision,butofasimilarshape.Thetotalsurfaceareaofeachdiscusisgivenbelow.

Intermediate discus

total surface area 500 cm2

Senior discus

total surface area 720 cm2

Bywhatvaluecanthevolumeoftheintermediatediscusbemultipliedtogivethevolumeoftheseniordiscus?


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End of Module 3 – SECTION B – continuedTURN OVER

Question 4 (2marks)Danielhasqualifiedforthefinalsofthediscuscompetition.Onhisfirstthrow,DanielthrewthediscustopointA,adistanceof53.32monabearingof057°.Onhissecondthrowfromthesamepoint,Danielthrewthediscusadistanceof57.51m.ThesecondthrowlandedatpointB,onabearingof125°,measuredfrompointA.

Determinethedistance,inmetres,betweenpointsA and B.Roundyouranswertoonedecimalplace.


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Question 1 (8marks)FastgrowandBoosteraretwotomatofertilisersthatcontainthenutrientsnitrogenandphosphorus.TheamountofnitrogenandphosphorusineachkilogramofFastgrowandBoosterisshowninthetablebelow.

1 kg of Booster 1 kg of Fastgrow

Nitrogen 0.05kg 0.05kg

Phosphorus 0.02kg 0.06kg

a. Howmanykilogramsofphosphorusarein2kgofBooster? 1mark

b. If100kgofBoosterand400kgofFastgrowaremixed,howmanykilogramsofnitrogenwouldbeinthemixture? 1mark

Arthurisafarmerwhogrowstomatoes.HemixesquantitiesofBoosterandFastgrowtomakehisownfertiliser.LetxbethenumberofkilogramsofBoosterinArthur’sfertiliser.LetybethenumberofkilogramsofFastgrowinArthur’sfertiliser.Inequalities1to4representthenitrogenandphosphorusrequirementsofArthur’stomatofield.

Inequality1 x≥0Inequality2 y≥0Inequality3(nitrogen) 0.05x+0.05y≥200Inequality4(phosphorus) 0.02x+0.06y≥120

Arthur’stomatofieldalsorequiresatleast180kgofthenutrientpotassium.EachkilogramofBoostercontains0.06kgofpotassium.EachkilogramofFastgrowcontains0.04kgofpotassium.

c. Inequality5representsthepotassiumrequirementsofArthur’stomatofield.

WritedownInequality5intermsofx and y. 1mark

Inequality5(potassium)


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ThelinesthatrepresenttheboundariesofInequalities3,4and5areshowninthegraphbelow.

1000O 2000 3000 4000 5000 6000

1000

2000

3000

4000

5000

6000

y

x

A

d. i. Usingthegraphabove,writedowntheequationoflineA. 1mark

ii. Onthegraph above,shadetheregionthatsatisfiesInequalities1to5. 1mark

(Answer on the graph above.)

ArthurwouldliketousetheleastamountofhisownfertilisertomeetthenutrientrequirementsofhistomatofieldandstillsatisfyInequalities1to5.Hewill,therefore,minimisethetotalweightoffertiliser,W,whereW = x+y.Thesliding-linemethodistobeusedtodeterminetheweightofBoosterandFastgrowfertilisershewilluse.

e. i. WritedownthegradientoftheobjectivefunctionW = x+y. 1mark

ii. Onthegraph above,drawthelinethatpassesthroughthepoint(0,1000)usingthegradientfrom part e.i. 1mark


iii. Hence,showonthegraph abovethepoint(s)wherethesolutionforminimumweightoccurs. 1mark



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Question 2 (4marks)Ashopownerbought100kgofArthur’stomatoestosellinhershop.Sheboughtthetomatoesfor$3.50perkilogram.Theshopownerwillofferadiscounttohercustomersbasedonthenumberofkilogramsoftomatoestheybuyinonebag.Therevenue,indollars,thattheshopownerreceivesfromsellingthetomatoesisgivenbythepiecewisedefinedrelationbelow

revenue =< ≤

+ − < ≤+ − < <

5 4 0 210 8 4 2 2 10

2 10 10 100

.. ( )( )

n nn n

a n n

wherenisthenumberofkilogramsoftomatoesthatacustomerbuysinonebag.

a. Whatistherevenuethattheshopownerreceivesfromselling8kgoftomatoesinonebag? 1mark

b. Arevenueof$46.80isreceivedfromselling12kgoftomatoesinonebag.

Showthatahasthevalue42.8intherevenueequationabove. 1mark

c. Findthemaximumnumberofkilogramsoftomatoesthatacustomercanbuyinonebag,sothattheshopownernevermakesaloss. 2marks

END OF QUESTION AND ANSWER BOOK

examination specifications and sample examination

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