Download pdf - 2018 Further Mathematics-nht Written examination 2 · 2018 FURMATH EXAM 2 (NHT) 6 D O N O T W R I T E I N T H I S A R E A D O N O T W R I T E I N T H I S A R E A SECTION A – Question

FURTHER MATHEMATICSWritten examination 2

Thursday 31 May 2018 Reading time: 2.00 pm to 2.15 pm (15 minutes) Writing time: 2.15 pm to 3.45 pm (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• Questionandanswerbookof34pages• Formulasheet• Workingspaceisprovidedthroughoutthebook.

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

At the end of the examination• Youmaykeeptheformulasheet.

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

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2018

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2018

STUDENT NUMBER

Letter

2018FURMATHEXAM2(NHT) 2

SECTION A – continued

do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

SECTION A – Core

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

Data analysis

Question 1 (3marks)Thedotplotandboxplotbelowdisplaythedistributionofskull length,inmillimetres,forasampleofthesamespeciesofbird.

28.5 29.0 29.5 30.0 30.5 31.0skull length (mm)

31.5 32.0 32.5 33.0 33.5 34.0

28.5 29.0 29.5 30.0 30.5 31.0skull length (mm)

31.5 32.0 32.5 33.0 33.5 34.0

a. Writedownthemodalskulllength. 1mark

b. Useinformationfromtheplotsabovetoshowwhythebirdwithaskulllength of33.5mmisnot plottedasanoutlierintheboxplot. 2marks

3 2018FURMATHEXAM2(NHT)

SECTION A – continuedTURN OVER

do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

Question 2 (3marks)Theweightofaspeciesofbirdisapproximatelynormallydistributedwithameanof71.5gandastandarddeviationof4.5g.

a. Whatisthestandardisedweight(zscore)ofabirdweighing67.9g? 1mark

b. Usethe68–95–99.7%ruletoestimate

i. theexpectedpercentageofthesebirdsthatweighlessthan67g 1mark

ii. theexpectednumberofbirdsthatweighbetween62.5gand76.0ginaflockof200ofthesebirds. 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

SECTION A – Question 3 – continued

Question 3 (3marks)Histogram1belowdisplaystheweightdistributionof143birdsofdifferentspecieslivinginasmallzoo.

Histogram 1

110

100

90

80

70

60

50

40

30

20

10

0

frequency

0 2000 4000 6000 8000weight (g)

10000 12000 14000

a. DescribetheshapeofthedistributiondisplayedinHistogram1.Notethenumberofpossibleoutliers,ifany. 1mark

b. Whatpercentageofthesebirdsweighlessthan1000g? Roundyouranswertoonedecimalplace. 1mark



do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

c. Histogram2belowdisplaystheweightdistributionofthesame143birdsplottedona log10scale.

Histogram 2

50

45

40

35

30

25

2020

frequency

15

10

5

00 1.0 2.0 3.0

log10 (weight)4.0 5.0

Howmanyofthesebirdsweighbetween10gand100g? 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


Question 4 (4marks)Asampleof96birdsaregroupedaccordingtotheirbeak size(small,medium,large).Thepercentageofbirdsineachgroupiscalculated.TheresultsaredisplayedinTable1below.

Table 1

Beak size Percentage (%)

small 25

medium 44

large 31

Total 100

a. Howmanyofthe96birdshavesmallbeaks? 1mark

b. UsethepercentagesinTable1toconstructapercentagedsegmentedbarchart. Atemplateisprovidedbelowtoassistyouincompletingthistask.Usethekeytoindicatethe

segmentofyourbarchartthatcorrespondstoeachbeaksize. 1mark

large

medium

small

Key100

90

80

70

60

50percentage

40

30

20

10

0beak size



do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

c. Inordertoinvestigateapossibleassociationbetweenbeak size and sex,thesamebirdsaregroupedbyboththeirbeak size(small,medium,large)andtheirsex(male,female).TheresultsofthisgroupingareshowninTable2.

Table 2

Sex

Beak size Male Female

small 1 23

medium 26 16

large 27 3

Total 54 42

Doestheinformationprovidedabovesupportthecontentionthat beak sizeisassociatedwithsex?Justifyyouranswerbyquotingappropriatepercentages.Itissufficienttoconsiderone beak sizeonlywhenjustifyingyouranswer. 2marks


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


Question 5 (7marks)Thescatterplotbelowshowstheweight,ingrams,andthehead length,inmillimetres,of110birds.

85

80

75

70

65

6052 53 54 55 56

head length (mm)57 58 59 60

weight (g)

Theequationoftheleastsquareslinefittedtothisdatais

weight=–24.83+1.739×head length

a. Drawthisleastsquareslineonthescatterplot above. 1mark

(Answer on the scatterplot above.)

b. Usetheequationtopredicttheweight,ingrams,ofabirdwithahead lengthof49.0mm. Roundyouranswertoonedecimalplace. 1mark

c. Isthepredictionmadeinpart b.anexampleofinterpolationorextrapolation?Explainyouranswerbriefly. 1mark



do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

d. Whentheleastsquareslineisusedtopredicttheweight ofabirdwithahead lengthof 59.2mm,theresidualvalueis2.78

Calculatetheactualweightofthisbird. Roundyouranswertoonedecimalplace. 2marks

e. Pearson’scorrelationcoefficient,r,isequalto0.5957

Giventhisinformation,whatpercentageofthevariationintheweightofthesebirdsisnot explainedbythevariationinhead length?

Roundyouranswertoonedecimalplace. 1mark

f. Theresidualplotobtainedwhentheleastsquareslineisfittedtothedatasetisshownbelow.

8

6

4

2

0residual

–2

–4

–6

–852 53 54 55 56

head length (mm)57 58 59 60

Whatdoestheresidualplotindicateabouttheassociationbetweenhead length and weight forthesebirds? 1mark



do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

Question 6 (4marks)Thetimeseriesdatabelowshowstheworldwidegrowthinelectricalpowergeneratedbywind, inmegawatts,fortheperiod2001–2012.Thevariablethatrepresentstime,inyears,hasbeenrescaledsothat‘1’represents2001,‘2’represents2002,andsoon.Thisnewvariableiscalledyear number.Atimeseriesplotforthedataisalsoshown.

Year number Power (MW)

1 23 900

2 31100

3 39431

4 47620

5 59091

6 73957

7 93 924

8 120696

9 159052

10 197956

11 238110

12 282850

Data:GlobalWindEnergyCouncil(GWEC),GlobalStatistics, ‘GlobalCumulativeInstalledWindCapacity2001–2016’,<www.gwec.net/>

Therelationshipbetweenpower and year numberisclearlynon-linear.Alog10transformationcanbeappliedtothevariablepowertolinearisethedata.

a. Applythistransformationtothedatatodeterminetheequationoftheleastsquareslinethatcanbeusedtopredictlog10(power)fromyear number.

Writethevaluesoftheinterceptandslopeofthisleastsquareslineintheappropriateboxesprovidedbelow.

Roundyouranswerstofoursignificantfigures. 2marks

log10(power)= + × year number

b. Usetheequationinpart a.topredicttheelectricalpower,inmegawatts,expectedtobegeneratedbywindin2020.

Roundyouranswertothenearest1000MW. 2marks

300000

250000

200000

150000power(MW)

100000

50000

00 2 4 6

year number8 10 12



do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

Recursion and financial modelling

Question 7 (4marks)Roslyninvestedsomemoneyinasavingsaccountthatearnsinterestcompoundingannually.Theinterestiscalculatedandpaidattheendofeachyear.LetVnbetheamountofmoneyinRoslyn’ssavingsaccount,indollars,afternyears.TherecursivecalculationsbelowshowtheamountofmoneyinRoslyn’ssavingsaccountafteroneyearandaftertwoyears.

V0=5000V1=1.05×5000=5250V2=1.05×5250=5512.50

a. HowmuchmoneydidRoslyninitiallyinvest? 1mark

b. Howmuchinterestintotaldidsheearnbytheendofthesecondyear? 1mark

c. LetVnbetheamountofmoneyinRoslyn’ssavingsaccount,indollars,afternyears.

Writedownarecurrencerelation,intermsofV0 ,Vn+1 and Vn ,thatcanbeusedtomodeltheamountofmoney,indollars,inRoslyn’ssavingsaccount. 1mark

d. Roslynplanstousehersavingstopayforaholiday. Theholidaywillcost$6000.

WhatminimumannualpercentageinterestratewouldhavebeenrequiredforRoslyntohavesavedthis$6000aftertwoyears?

Roundyouranswertoonedecimalplace. 1mark



do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

Question 8 (5marks)RichardwilljoinRoslynontheholiday.Hewillsellhisstereosystemtohelppayforhisholiday.Thestereosystemwasoriginallypurchasedfor$8500.Richardwillsellthestereosystematadepreciatedvalue.

a. Richardcoulduseaflatratedepreciationmethod. LetSnbethevalue,indollars,ofRichard’sstereosystemnyearsafteritwaspurchased. Thevalueofthestereosystem,Sn ,canbemodelledbytherecurrencerelationbelow.

S0=8500, Sn +1 = Sn–867

i. Usingthisdepreciationmethod,whatisthevalueofthestereosystemfouryearsafteritwaspurchased? 1mark

ii. Calculatetheannualpercentageflatrateofdepreciationforthisdepreciationmethod. 1mark

b. Richardcouldalsouseareducingbalancedepreciationmethod,withanannualdepreciationrateof8%.

Usingthisdepreciationmethod,whatisthevalueofthestereosystemfouryearsafteritwaspurchased?

Roundyouranswertothenearestcent. 1mark

c. Fouryearsafteritwaspurchased,Richardsoldhisstereosystemfor$4500.

Assumingareducingbalancedepreciationmethodwasused,whatannualpercentagerateofdepreciationdidthisrepresent?

Roundyouranswertoonedecimalplace. 2marks


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

END OF SECTION ATURN OVER

Question 9 (3marks)AndrewwillalsojoinRoslynandRichardontheholiday.Andrewborrowed$10000topayfortheholidayandforotherexpenses.Interestonthisloanwillbechargedattherateof12.9%perannum,compoundingmonthly.Immediatelyaftertheinteresthasbeencalculatedandchargedeachmonth,Andrewwillmakearepayment.

a. Forthefirstyearofthisloan,Andrewwillmakeinterest-onlyrepaymentseachmonth.

Whatisthevalueofeachinterest-onlyrepayment? 1mark

b. Forthenextthreeyearsofthisloan,Andrewwillmakeequalmonthlyrepayments. Afterthesethreeyears,thebalanceofAndrew’sloanwillbe$3776.15

Whatamount,indollars,willAndrewrepayeachmonthduringthesethreeyears? 1mark

c. Andrewwillfullyrepaytheoutstandingbalanceof$3776.15withafurther12monthlyrepayments.

Thefirst11repaymentswilleachbe$330. Thetwelfthrepaymentwillhaveadifferentvaluetoensuretheloanisrepaidexactlytothe

nearestcent.

Whatisthevalueofthetwelfthrepayment? Roundyouranswertothenearestcent. 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

SECTION B – continued

SECTION B – Modules

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

Contents Page

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

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

Module3–Geometryandmeasurement....................................................................................................... 24

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


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

SECTION B – Module 1 – continued TURN OVER

Module 1 – Matrices

Question 1 (4marks)Aregionhasfourdistricts:North(N),South(S),East(E)andWest(W).Farmersfromeachdistrictattendedaconferencein2017.MatrixF2017belowshowsthenumberoffarmersfromeachofthesefourdistrictswhoattendedthe2017conference.

F

NSEW

2017

36202816

=

a. WhatistheorderofmatrixF2017? 1mark

b. HowmanyofthesefarmerscamefromeithertheNorthorSouthdistrict? 1mark

Thetablebelowshowsthecostperfarmer,foreachdistrict,toattendthe2017conference.

District Cost per farmer ($)

North 25

South 20

East 45

West 35

c. WritedownamatrixthatcouldbemultipliedbymatrixF2017togivethetotalcostforallfarmerswhoattendedthe2017conference. 1mark

d. Thenumberoffarmerswhoattendedthe2018conferenceincreasedby25%foreachdistrictfromthepreviousyear.

Completetheproductbelowwithascalarsothattheproductgivesthenumberoffarmersfromeachdistrictwhoattendedthe2018conference. 1mark

F2018 = × F2017


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

SECTION B – Module 1 – continued

Question 2 (2marks)Fivefarmers,A,B,C,D and E,attendedthe2018conference.Pairsofthesefarmershadpreviouslyattendedoneormoreconferencestogether.ThenumberofconferencespreviouslyattendedtogetherisshowninmatrixMbelow.Forexample,the‘1’inthebottomrowshowsthatD and Ehadattendedoneearlierconferencetogether.

A B C D E

M

ABCDE

=

01110

10213

12012

11101

03210

a. Whichtwofarmershadnotpreviouslyattendedaconferencetogether? 1mark

b. WhatdothenumbersincolumnDindicate? 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


Question 3 (4marks)Threefarmers,A,B and C,eachplacedordersforthreetypesoffertilisersfortheircornfields.ThetypesoffertilisersareKalm(K),Nitro(N)andPhate(P).Thematrixbelowshowstheamountoffertiliser,intonnes,orderedbyFarmerAandFarmerB.

PNKFarmer AFarmer B

22

45

21

a. FarmerAandFarmerBeachpaidatotalof$16000forfertiliser.

WhatconclusioncanbedrawnaboutthepricesofNitro(N)andPhate(P)? 1mark

Let xbethepricepertonneofKalm(K) ybethepricepertonneofNitro(N) zbethepricepertonneofPhate(P).

Thetotalcostofthesetwoorderscanbesummarisedbythematrixequation

22

45

21

1600016000

×

=

xyz

b. Explainwhythisequationcannotbesolvedusingthematrixinversemethod. 1mark

c. Thematrixequationbelowshowsthefertiliserordersforallthreefarmers.

2 4 22 5 11 1 1

16000160006500

×

=

xyz

i. Completethematrixequationbelowbyfillinginthemissingelements. 1mark

xyz

=

−−−

__..

__ __0 51 5

01

11

16000160006500

ii. Determinethecost,indollars,ofonetonneofPhate(P). 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

End of Module 1 – SECTION B – continued

Question 4 (2marks)Areasoffarmlandintheregionareallocatedtogrowingbarley(B),corn(C)andwheat(W).Thisallocationoffarmlandistobechangedeachyear,beginningin2019.Thetablebelowshowstheareasoffarmland,inhectares,allocatedtoeachcropin2018(n=0)and2019 (n=1).

Year 2018 2019

n 0 1

barley 2000 2100

corn 1000 1900

wheat 3000 2000

Theplannedannualchangetotheareaallocatedtoeachcropcanbemodelledby

this yearB C W

H R H Q Rn n+ = + =1 0 1 0 8 0 20 2

where 0.7 0.1 0.1

0.1 0. . .. ..7

BCW

next year

Hnrepresentsthestatematrixthatshowstheareaallocatedtoeachcropnyearsafter2018.Qisamatrixthatcontainsadditionalfixedchangestotheareathatisallocatedtoeachcropeachyear.

CompleteH2,thestatematrixfor2020.

H2 =

B

C

W


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


Module 2 – Networks and decision mathematics

Question 1 (2marks)Afarmer’spropertyisdividedintofourareaslabelled1to4onthediagrambelow.Theboldlinesrepresenttheboundaryfencesbetweentwoareas.

1

2

3

4

Inthegraphbelow,thefourareasofthepropertyarerepresentedasvertices.Theedgesofthegraphrepresenttheboundaryfencesbetweenareas.

1

2

3

4

Oneoftheedgesismissingfromthisgraph.

a. Onthegraph above,drawinthemissingedge. 1mark

(Answer on the graph above.)

b. Withthisedgedrawnin,whatisthesumofthedegreesoftheverticesofthegraph? 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


Question 2 (3marks)Area1ofthepropertycontainseightlargebushesthatarelabelledA to H,asshownonthegraphbelow.Thefarmer’sdogenjoysrunningaroundthisarea,stoppingateachbushontheway.Thenumbersontheedgesjoiningtheverticesgivetheshortestdistance,inmetres,betweenbushes.

A

G

F

B

C

D

E

H

30

35

40

15

60

10

5045

25

a. ExplainwhythedogcouldnotfollowanEuleriancircuitthroughthisnetwork. 1mark

b. IfthedogfollowstheshortestHamiltonianpath,nameabushatwhichthedogcouldstartandabushatwhichthedogcouldfinish. 1mark

Start Finish

c. Thesumofalldistancesshownonthegraphis310m. ThedogstartsandfinishesatbushFandrunsalongeveryedgeinthenetwork.

Whatistheshortestdistance,inmetres,thatthedogcouldhaverun? 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


Question 3 (3marks)Allareasofthepropertyrequireaconstantsupplyofwater.Thefollowingdirectedgraphrepresentsthecapacity,inlitresperminute,ofaseriesofwaterpipesonthepropertyconnectingthesourcetothesink.

10

15

10

10

10

40

20

25

25

20

sink

source

Cut 1

Whenconsideringthepossibleflowthroughthisnetwork,differentcutscanbemade.Cut1islabelledonthegraphabove.

a. WhatisthecapacityofCut1inlitresperminute? 1mark

b. Onthegraph above,drawthecut(Cut2)thathasacapacityof70litresperminute.LabelyouranswerclearlyasCut2. 1mark


c. Determinethemaximumflowofwater,inlitresperminute,fromthesourcetothesink. 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

SECTION B – Module 2 – Question 4 – continued

Question 4 (4marks)Abarnwillbebuiltontheproperty.Thisbuildingprojectwillinvolve11activities,A to K.Thedirectednetworkbelowshowstheseactivitiesandtheirdurationindays.ThedurationofactivityI isunknownatthestartoftheproject.LetthedurationofactivityIbepdays.

D, 4

H, 10

K, 4J, 5

I, p

G, 6C, 2

A, 3

B, 5

F, 5

E, 1

start

finish

a. Determinetheearlieststartingtime,indays,foractivityI. 1mark

b. Determinethevalueofp,indays,thatwouldcreatemorethanonecriticalpath. 1mark

c. Ifthevalueofpissixdays,whatwillbethefloattime,indays,ofactivityH? 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

End of Module 2 – SECTION B – continuedTURN OVER

d. Whenasecondbarnisbuiltlater,activityIwillnotbeneeded. A dummyactivityisrequired,asshownonthereviseddirectednetworkbelow.

D, 4

H, 10

K, 4J, 5

dummy

G, 6C, 2

A, 3

B, 5

F, 5

E, 1

start

finish

Explainwhatthisdummyactivityindicatesonthereviseddirectednetwork. 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


Module 3 – Geometry and measurement

Question 1 (5marks)Shannonisabaker.Oneofherbakingtinshasarectangularbaseoflength28cmandwidth20cm.Theheightofthisbakingtinis5cm,asshowninthediagrambelow.

5 cm 20 cm

28 cm

a. Whatisthevolumeofthistin,incubiccentimetres? 1mark

Anotherbakingtinhasacircularbasewitharadiusof12cm.Theheightofthisbakingtinis8cm,asshowninthediagrambelow.

12 cm

8 cm

b. Shannonneedstocovertheinsideofboththebaseandsideofthistinwithbakingpaper.

Whatistheareaofbakingpaperrequired,insquarecentimetres? Roundyouranswertoonedecimalplace. 2marks


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


Acakecookedinthecircularbakingtiniscutinto10piecesofequalsize,asshowninthediagrambelow.

θ8 cm

12 cm

Theangleθisalsoshownonthediagram.

c. Showthattheangleθisequalto36°. 1mark

d. Whatisthevolume,incubiccentimetres,ofonepieceofcake? Roundyouranswertoonedecimalplace. 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


Question 2 (4marks)ShannonplanstotraveltoParis,Beijing,BrasiliaandVancouvertotrythelocalcakespecialties:• Paris(49°N,2°E)inFrance• Beijing(40°N,116°E)inChina• Brasilia(16°S,48°W)inBrazil• Vancouver(49°N,123°W)inCanada

ThediagrambelowshowsthepositionofParisatlatitude49°Nandlongitude2°E.Thethreeothercitiesareindicatedonthediagramas1,2and3.

Paris2

equator

Greenwich meridian

N

S

1

3

a. Completethetablebelowbymatchingthecitywiththecorrespondingcitynumber (1,2and3)giveninthediagramabove. 1mark

City City number

Beijing(40°N,116°E)

Brasilia(16°S,48°W)

Vancouver(49°N,123°W)


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


b. ShannontravelledfromSydneytoParisonWednesday,30May.SheleftSydneyat 10.50am.

TheflighttoParistook22hoursand25minutes. ThetimedifferencebetweenSydney(34°S,151°E)andParis(49°N,2°E)iseighthours.

OnwhatdayandatwhattimewillShannonarriveinParis? 1mark

c. OnthedaythatShannonarrivesinParis,thesunwillriseat5.54am. Assumethat15°oflongitudeequatestoaone-hourtimedifference.

HowlongafterthesunrisesinParis(49°N,2°E)willthesunriseinVancouver (49°N,123°W)?

Writeyouranswerinhoursandminutes. 1mark

d. ShannontravelstotheFrenchcitiesofLyon(46°N,5°E)andMarseille(43°N,5°E). AssumethattheradiusofEarthis6400km.

FindtheshortestgreatcircledistancebetweenLyonandMarseille. Roundyouranswertothenearestkilometre. 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

End of Module 3 – SECTION B – continued

Question 3 (3marks)Afterreturningfromhertravels,Shannondecidestodesignaninterestingpackageforsomeofhersmallercircularcakes.Shedesignsatriangularboxwithsidelengthsof16cm,asshowninthediagrambelow.

16 cm 16 cm

16 cm

w cm

a. Showthatthevalueofwonthediagramis13.9,roundedtoonedecimalplace. 1mark

b. Onecircularcakeisplacedinthetriangularbox.

16 cm 16 cm

16 cm

Whatisthediameter,incentimetres,ofthelargestcakethatwillfitinthetriangularbox? Roundyouranswertoonedecimalplace. 2marks


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


Module 4 – Graphs and relations

Question 1 (2marks)Ahamburgerrestaurantrecordedthenumberofseatedcustomerseachhourfrom11amto10pm.Thegraphbelowshowsthenumberofcustomersseatedeachhourononeparticularday.

40

30

20

10

0 time

number ofseated customers

11 am 12 noon 1 pm 2 pm 3 pm 4 pm 5 pm 6 pm 7 pm 8 pm 9 pm 10 pm

a. Howmanycustomerswereseatedat4pm? 1mark

b. Howmanytimesdidtherestaurantrecordhaving30ormoreseatedcustomers? 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


Question 2 (4marks)Therestaurantmakesandsellsbaconburgers.Theprofit,P,indollars,obtainedfrommakingandsellingnbaconburgersisshownbythelineinthegraphbelow.

450

400

350

300

250

200

150

100

50n

–50

O

–100

–150

–200

–250

20

$P

40 60 80 100 120 140 160 180 200 220

a. Determinetheprofitobtainedfrommakingandselling110baconburgers. 1mark

b. Howmanybaconburgersmustbemadeandsoldtobreakeven? 1mark

c. Theprofitobtainedfromselling100baconburgersis$160. Thecost,C,indollars,ofmakingnbaconburgersisgivenbytheequationC=1.5n+240.

Calculatethesellingpriceofeachbaconburger. 2marks


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


Question 3 (2marks)Therestaurantalsosellsmealpacksforlargegroups.Therestaurantcharges$10perpackforthefirst80packs.Foreverypackbeyondthefirst80packs,thepricereducesto$8perpack.Therevenue,R,indollars,receivedfromsellingnmealpackscanbedeterminedasfollows.

Rn n

n c n=

< ≤+ >

10 0 808 80

Arevenueof$960isreceivedfromselling100mealpacks.

a. Showthatc=160. 1mark

b. Whatrevenuewilltherestaurantreceivefromsellingasingleorderfor130mealpacks? 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea


Question 4 (4marks)Therestaurantmakesandsellstwotypesofcheeseburgers:asinglecheeseburgerandatriplecheeseburger.Letxbethenumberofsinglecheeseburgersmadeandsoldinoneday.Letybethenumberoftriplecheeseburgersmadeandsoldinoneday.Eachsinglecheeseburgercontainsonebun,onemeatpattyandonecheeseslice.Eachtriplecheeseburgercontainsonebun,threemeatpattiesandtwocheeseslices.TheconstraintsontheproductionofcheeseburgerseachdayaregivenbyInequalities1to5.

Inequality1 x≥0Inequality2 y≥0Inequality3(buns) x+y≤250Inequality4(meatpatties) x+3y≤450Inequality5(cheeseslices) x+2y≤350

ThegraphbelowshowsthelinesthatrepresenttheboundariesofInequalities1to5.Thefeasibleregionhasbeenshaded.

300

250

200

150

100

50

O 50 100 150 200 250 300 350 400 450

y

x


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

SECTION B – Module 4 – Question 4 – continuedTURN OVER

a. OnSaturday,100singlecheeseburgersweresold.

Whatisthemaximumnumberoftriplecheeseburgersthatcouldhavebeensoldonthesameday? 1mark

Theprofitforonesinglecheeseburgeris$1.50andtheprofitforonetriplecheeseburgeris$3.00

b. Howmanysinglecheeseburgersandhowmanytriplecheeseburgersmusttherestaurantsellinadayinordertomaximiseprofit? 1mark


do

no

t w

rit

e i

n t

his

ar

ea

do

no

t w

rit

e i

n t

his

ar

ea

END OF QUESTION AND ANSWER BOOK

OnSunday,30cheesesliceswerefoundtobemouldyandcouldnotbeused.ThischangedInequality5to

x+2y≤320

ThegraphbelowshowsthelinesthatrepresenttheboundariesofInequalities1to4.

300

250

200

150

100

50

O 50 100 150 200 250 300 350 400 450

y

x

c. Sketchthelinex+2y=320onthegraph above. 1mark


d. ThemaximumprofitpossibleonthisSundaywas$480.

Calculatetheminimumtotalnumberofcheeseburgersthatneedtobesoldtomakethisprofit. 1mark

FURTHER MATHEMATICS

Written examination 2

FORMULA SHEET

Instructions

This formula sheet is provided for your reference.A question and answer 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 2018

Victorian Certificate of Education 2018

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