2018 further mathematics-nht written examination 2 · 2018 furmath exam 2 (nht) 6 d o n o t w r i t...
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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
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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
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SECTION A – continuedTURN OVER
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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
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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
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SECTION A – continuedTURN OVER
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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
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SECTION A – Question 4 – continued
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
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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
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SECTION A – Question 5 – continued
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
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SECTION A – continuedTURN OVER
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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
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SECTION A – continued
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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
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SECTION A – continuedTURN OVER
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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
2018FURMATHEXAM2(NHT) 12
SECTION A – continued
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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
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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$330. Thetwelfthrepaymentwillhaveadifferentvaluetoensuretheloanisrepaidexactlytothe
nearestcent.
Whatisthevalueofthetwelfthrepayment? Roundyouranswertothenearestcent. 1mark
2018FURMATHEXAM2(NHT) 14
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SECTION B – continued
SECTION B – Modules
Instructions for Section BSelect twomodulesandanswerallquestionswithintheselectedmodules.Youneednotgivenumericalanswersasdecimalsunlessinstructedtodoso.Alternativeformsmayinclude,forexample,π,surdsorfractions.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
Contents Page
Module1–Matrices......................................................................................................................................15
Module2–Networksanddecisionmathematics..........................................................................................19
Module3–Geometryandmeasurement....................................................................................................... 24
Module4–Graphsandrelations................................................................................................................... 29
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SECTION B – Module 1 – continued TURN OVER
Module 1 – Matrices
Question 1 (4marks)Aregionhasfourdistricts:North(N),South(S),East(E)andWest(W).Farmersfromeachdistrictattendedaconferencein2017.MatrixF2017belowshowsthenumberoffarmersfromeachofthesefourdistrictswhoattendedthe2017conference.
F
NSEW
2017
36202816
=
a. WhatistheorderofmatrixF2017? 1mark
b. HowmanyofthesefarmerscamefromeithertheNorthorSouthdistrict? 1mark
Thetablebelowshowsthecostperfarmer,foreachdistrict,toattendthe2017conference.
District Cost per farmer ($)
North 25
South 20
East 45
West 35
c. WritedownamatrixthatcouldbemultipliedbymatrixF2017togivethetotalcostforallfarmerswhoattendedthe2017conference. 1mark
d. Thenumberoffarmerswhoattendedthe2018conferenceincreasedby25%foreachdistrictfromthepreviousyear.
Completetheproductbelowwithascalarsothattheproductgivesthenumberoffarmersfromeachdistrictwhoattendedthe2018conference. 1mark
F2018 = × F2017
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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
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SECTION B – Module 1 – continued TURN OVER
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
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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
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SECTION B – Module 2 – continued TURN OVER
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
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SECTION B – Module 2 – continued
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
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SECTION B – Module 2 – continued TURN OVER
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
(Answer on the graph above.)
c. Determinethemaximumflowofwater,inlitresperminute,fromthesourcetothesink. 1mark
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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
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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
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SECTION B – Module 3 – Question 1 – continued
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
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SECTION B – Module 3 – continued TURN OVER
Acakecookedinthecircularbakingtiniscutinto10piecesofequalsize,asshowninthediagrambelow.
θ8 cm
12 cm
Theangleθisalsoshownonthediagram.
c. Showthattheangleθisequalto36°. 1mark
d. Whatisthevolume,incubiccentimetres,ofonepieceofcake? Roundyouranswertoonedecimalplace. 1mark
2018FURMATHEXAM2(NHT) 26
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SECTION B – Module 3 – Question 2 – continued
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)
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SECTION B – Module 3 – continued TURN OVER
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
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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
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SECTION B – Module 4 – continued TURN OVER
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
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SECTION B – Module 4 – continued
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
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SECTION B – Module 4 – continued TURN OVER
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
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SECTION B – Module 4 – Question 4 – continued
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
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SECTION B – Module 4 – Question 4 – continuedTURN OVER
a. OnSaturday,100singlecheeseburgersweresold.
Whatisthemaximumnumberoftriplecheeseburgersthatcouldhavebeensoldonthesameday? 1mark
Theprofitforonesinglecheeseburgeris$1.50andtheprofitforonetriplecheeseburgeris$3.00
b. Howmanysinglecheeseburgersandhowmanytriplecheeseburgersmusttherestaurantsellinadayinordertomaximiseprofit? 1mark
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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
(Answer on the graph above.)
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