2016 Further Mathematics Written examination 2 · PDF fileFURTHER MATHEMATICS Written...
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FURTHER MATHEMATICS Written examination 2 Monday 31 October 2016 Reading time: 9.00 am to 9.15 am (15 minutes) Writing time: 9.15 am to 10.45 am (1 hour 30 minutes) QUESTION AND ANSWER BOOK Structure of book Section A – Core Number of questions Number of questions to be answered Number of marks 7 7 36 Section B – Modules Number of modules Number of modules to be answered Number of marks 4 2 24 Total 60 • Students are to write in blue or black pen. • Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers, one bound reference, one approved technology (calculator or software) and, if desired, one scientific calculator. Calculator memory DOES NOT need to be cleared. For approved computer-based CAS, full functionality may be used. • Students are NOT permitted to bring into the examination room: blank sheets of paper and/or correction fluid/tape. Materials supplied • Question and answer book of 33 pages. • Formula sheet. • Working space is provided throughout the book. Instructions • Write your student number in the space provided above on this page. • Unless otherwise indicated, the diagrams in this book are not drawn to scale. • All written responses must be in English. At the end of the examination • You may keep the 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 2016 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Victorian Certificate of Education 2016 STUDENT NUMBER Letter
Transcript of 2016 Further Mathematics Written examination 2 · PDF fileFURTHER MATHEMATICS Written...
FURTHER MATHEMATICSWritten examination 2
Monday 31 October 2016 Reading time: 9.00 am to 9.15 am (15 minutes) Writing time: 9.15 am to 10.45 am (1 hour 30 minutes)
QUESTION AND ANSWER BOOK
Structure of bookSection A – Core Number of
questionsNumber of questions
to be answeredNumber of
marks
7 7 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• Questionandanswerbookof33pages.• 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.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2016
STUDENT NUMBER
Letter
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SECTION A – Question 1 – continued
Data analysis
Question 1 (7marks)Thedotplotbelowshowsthedistributionofdailyrainfall,inmillimetres,ataweatherstationfor30daysinSeptember.
0 1 2 3 4 5 6 7 8 9daily rainfall (mm)
10 11 12 13 14 15 16 17 18 19
a. Writedownthe i. range 1mark
ii. median. 1mark
b. Circlethedatapointonthedot plot abovethatcorrespondstothethirdquartile(Q3). 1mark
(Answer on the dot plot above.)
SECTION A – Core
Instructions for Section AAnswerallquestionsinthespacesprovided.Youneednotgivenumericalanswersasdecimalsunlessinstructedtodoso.Alternativeformsmayinclude, forexample,π,surdsorfractions.In‘Recursionandfinancialmodelling’,allanswersshouldberoundedtothenearestcentunlessotherwiseinstructed.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
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c. Writedown
i. thenumberofdaysonwhichnorainfallwasrecorded 1mark
ii. thepercentageofdaysonwhichthedailyrainfallexceeded12mm. 1mark
d. UsethegridbelowtoconstructahistogramthatdisplaysthedistributionofdailyrainfallforthemonthofSeptember.Useintervalwidthsoftwowiththefirstintervalstartingat0. 2marks
222018161412frequency1086420
0 2 4 6 8 10daily rainfall (mm)
12 14 16 18
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SECTION A – Question 2 – continued
Question 2 (5marks)Theweatherstationalsorecordsdailymaximumtemperatures.
a. Thefive-numbersummaryforthedistributionofmaximumtemperaturesforthemonthofFebruaryisdisplayedinthetablebelow.
Temperature (°C)
Minimum 16
Q1 21
Median 25
Q3 31
Maximum 38
Therearenooutliersinthisdistribution.
i. Usethefive-numbersummaryabovetoconstructaboxplotonthegridbelow. 1mark
15 20 25daily temperature (°C)
30 35 40
February
ii. Whatpercentageofdayshadamaximumtemperatureof21°C,orgreater,inthisparticularFebruary? 1mark
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b. TheboxplotsbelowdisplaythedistributionofmaximumdailytemperatureforthemonthsofMayandJuly.
month
6 8 10 12temperature (°C)
14 16 18 20
July
May
i. Describetheshapesofthedistributionsofdailytemperature (includingoutliers)forJulyand forMay. 1mark
July
May
ii. DeterminethevalueoftheupperfencefortheJulyboxplot. 1mark
iii. Usingtheinformationfromtheboxplots,explainwhythemaximumdailytemperature is associatedwiththemonthoftheyear.Quotethevaluesofappropriatestatisticsinyourresponse. 1mark
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SECTION A – Question 3 – continued
Question 3 (8marks)Thedatainthetablebelowshowsasampleofactualtemperaturesandapparenttemperaturesrecordedattheweatherstation.Ascatterplotofthedataisalsoshown.Thedatawillbeusedtoinvestigatetheassociationbetweenthevariablesapparent temperature and actual temperature.
Apparent temperature (°C)
Actual temperature (°C)
24.7 28.5
24.3 27.6
24.9 27.7
23.2 26.9
24.2 26.6
22.6 25.5
21.5 24.4
20.6 23.8
19.4 22.3
18.4 22.1
17.6 20.9
18.7 21.2
18.2 20.5
a. Usethescatterplottodescribetheassociationbetweenapparent temperature and actual temperature intermsofstrength,directionandform. 1mark
b. i. Determinetheequationoftheleastsquareslinethatcanbeusedtopredicttheapparent temperaturefromtheactual temperature.
Writethevaluesoftheinterceptandslopeofthisleastsquareslineintheappropriateboxesprovidedbelow.
Roundyouranswerstotwosignificantfigures. 3marks
apparent temperature = + × actual temperature
161718192021
apparenttemperature
(°C)
2223242526
20 21 22 23 24actual temperature (°C)
25 26 27 28 29
n = 13r2 = 0.97
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ii. Interprettheinterceptoftheleastsquareslineintermsofthevariablesapparent temperature and actual temperature. 1mark
c. Thecoefficientofdeterminationfortheassociationbetweenthevariablesapparent temperature and actual temperatureis0.97
Interpretthecoefficientofdeterminationintermsofthesevariables. 1mark
d. Theresidualplotobtainedwhentheleastsquareslinewasfittedtothedataisshownbelow.
1.5
1
0.5
0residual
–0.5
–1
–1.520 21 22 23 24
actual temperature (°C)25 26 27 28 29
i. Aresidualplotcanbeusedtotestanassumptionaboutthenatureoftheassociationbetweentwonumericalvariables.
Whatisthisassumption? 1mark
ii. Doestheresidualplotabovesupportthisassumption?Explainyouranswer. 1mark
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SECTION A – Question 4 – continued
Question 4 (4marks)Thetimeseriesplotbelowshowstheminimum rainfall recordedattheweatherstationeachmonthplottedagainstthemonth number(1=January,2=February,andsoon).Rainfallisrecordedinmillimetres.Thedatawascollectedoveraperiodofoneyear.
unsmoothedfive-mediansmoothed
Key1009080706050minimum
rainfall (mm)403020100
0 1 2 3 4 5 6 7month number
8 9 10 11 12 13
a. Five-mediansmoothinghasbeenusedtosmooththetimeseriesplotabove. Thefirstfoursmoothedpointsareshownascrosses(×).
Completethefive-mediansmoothingbymarkingsmoothedvalueswithcrosses(×)onthetime series plot above. 2marks
(Answer on the time series plot above.)
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Themaximumdailyrainfalleachmonthwasalsorecordedattheweatherstation.Thetablebelowshowsthemaximum daily rainfalleachmonthforaperiodofoneyear.
Month Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.
Month number 1 2 3 4 5 6 7 8 9 10 11 12
Maximum daily rainfall (mm) 79 123 100 156 174 186 149 162 124 140 225 119
Thedatainthetablehasbeenusedtoplotmaximum daily rainfall againstmonth numberinthetimeseriesplotbelow.
unsmoothed timeseries plottwo-mean smoothedplot with centring
Key250
200
150maximum
daily rainfall(mm) 100
50
00 1 2 3 4 5 6 7
month number8 9 10 11 12 13
b. Two-meansmoothingwithcentringhasbeenusedtosmooththetimeseriesplotabove. Thesmoothedvaluesaremarkedwithcrosses(×).
Usingthedatagiveninthetable,showthatthetwo-meansmoothedrainfallcentredonOctoberis157.25mm. 2marks
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SECTION A – continued
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Recursion and financial modelling
Question 5 (5marks)Kenhasopenedasavingsaccounttosavemoneytobuyanewcaravan.Theamountofmoneyinthesavingsaccountafternyears,Vn ,canbemodelledbytherecurrencerelationshownbelow.
V0=15000, Vn+1=1.04×Vn
a. HowmuchmoneydidKeninitiallydepositintothesavingsaccount? 1mark
b. UserecursiontowritedowncalculationsthatshowthattheamountofmoneyinKen’ssavingsaccountaftertwoyears,V2,willbe$16224. 1mark
c. Whatistheannualpercentagecompoundinterestrateforthissavingsaccount? 1mark
d. Theamountofmoneyintheaccountafternyears,Vn,canalsobedeterminedusingarule.
i. Completetherulebelowbywritingtheappropriatenumbersintheboxesprovided. 1mark
Vn = n ×
ii. HowmuchmoneywillbeinKen’ssavingsaccountafter10years? 1mark
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Question 6 (3marks)Ken’sfirstcaravanhadapurchasepriceof$38000.Aftereightyears,thevalueofthecaravanwas$16000.
a. Showthattheaveragedepreciationinthevalueofthecaravanperyearwas$2750. 1mark
b. LetCnbethevalueofthecaravannyearsafteritwaspurchased. Assumethatthevalueofthecaravanhasbeendepreciatedusingthe flat ratemethodofdepreciation.
Writedownarecurrencerelation,intermsofCn+1 and Cn,thatmodelsthevalueofthecaravan. 1mark
c. Thecaravanhastravelledanaverageof5000kmineachoftheeightyearssinceitwaspurchased. Assumethatthevalueofthecaravanhasbeendepreciatedusingtheunit costmethodofdepreciation.
Byhowmuchisthevalueofthecaravanreducedperkilometretravelled? 1mark
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END OF SECTION A
Question 7 (4marks)Kenhasborrowed$70000tobuyanewcaravan.Hewillbechargedinterestattherateof6.9%perannum,compoundingmonthly.
a. Forthefirstyear(12months),Kenwillmakemonthlyrepaymentsof$800.
i. FindtheamountthatKenwilloweonhisloanafterhehasmade12repayments. 1mark
ii. WhatisthetotalinterestthatKenwillhavepaidafter12repayments? 1mark
b. Afterthreeyears,Kenwillmakealumpsumpaymentof$Linordertoreducethebalanceofhisloan. ThislumpsumpaymentwillensurethatKen’sloanisfullyrepaidinafurtherthreeyears. Ken’srepaymentamountremainsat$800permonthandtheinterestrateremainsat6.9%perannum,
compoundingmonthly.
WhatisthevalueofKen’slumpsumpayment,$L? Roundyouranswertothenearestdollar. 2marks
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SECTION B – continuedTURN OVER
SECTION B – Modules
Instructions for Section BSelect twomodulesandanswerallquestionswithintheselectedmodules.Youneednotgivenumericalanswersasdecimalsunlessinstructedtodoso.Alternativeformsmayinclude, forexample,π,surdsorfractions.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
Contents Page
Module1–Matrices................................................................................................................................................... 14
Module2–Networksanddecisionmathematics....................................................................................................... 20
Module3–Geometryandmeasurement.................................................................................................................... 24
Module4–Graphsandrelations................................................................................................................................ 30
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SECTION B – Module 1 – continued
Module 1 – Matrices
Question 1 (3marks)Atravelcompanyarrangesflight(F),hotel(H),performance(P)andtour(T)bookings.MatrixCcontainsthenumberofeachtypeofbookingforamonth.
C
FHPT
=
85382443
a. WritedowntheorderofmatrixC. 1mark
Abookingfee,perperson,iscollectedbythetravelcompanyforeachtypeofbooking.MatrixGcontainsthebookingfees,indollars,perbooking.
F H P TG = [ ]40 25 15 30
b. i. CalculatethematrixproductJ = G × C. 1mark
ii. WhatdoesmatrixJrepresent? 1mark
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SECTION B – Module 1 – continuedTURN OVER
Question 2 (2marks)Thetravelcompanyhasfiveemployees,Amara(A),Ben(B),Cheng(C),Dana(D)andElka(E).ThecompanyallowseachemployeetosendadirectmessagetoanotheremployeeonlyasshowninthecommunicationmatrixGbelow.ThematrixG2isalsoshownbelow.
receiverA B C D E
G sender
ABCDE
=
0 1 1 1 11 0 1 0 01 1 0 1 00 1 0 0 10 0 0 1 0
receiverA B C D E
G sender
ABCDE
2
2 2 1 2 11 2 1 2 11 2 2 1 21 0 1 1 00 1 0 0 1
=
The‘1’inrowE,columnDofmatrixGindicatesthatElka(sender)cansendadirectmessageto Dana(receiver).The‘0’inrowE,columnCofmatrixGindicatesthatElkacannotsendadirectmessagetoCheng.
a. TowhomcanDanasendadirectmessage? 1mark
b. ChengneedstosendamessagetoElka,butcannotdothisdirectly.
WritedownthenamesoftheemployeeswhocansendthemessagefromChengdirectlytoElka. 1mark
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SECTION B – Module 1 – Question 3 – continued
Question 3 (7marks)Thetravelcompanyisstudyingthechoicebetweenair(A),land(L),sea(S)orno(N)travelbysomeofitscustomerseachyear.MatrixT,shownbelow,containsthepercentagesofcustomerswhoareexpectedtochangetheirchoiceoftravelfromyeartoyear.
this yearA L S N
T =
0 65 0 25 0 25 0 500 15 0 60 0 20 0 150 05 0 10 0 25
. . . .
. . . .
. . . 00 200 15 0 05 0 30 0 15
.. . . .
ALSN
next year
LetSnbethematrixthatshowsthenumberofcustomerswhochooseeachtypeoftravelnyearsafter2014.MatrixS0belowshowsthenumberofcustomerswhochoseeachtypeoftravelin2014.
S
ALSN
0
5203208080
=
MatrixS1belowshowsthenumberofcustomerswhochoseeachtypeoftravelin2015.
S TSdef
ALSN
1 0
478
= =
a. WritethevaluesmissingfrommatrixS1(d,e,f)intheboxesprovidedbelow. 1mark
d = e = f =
b. Writeacalculationthatshowsthat478customerswereexpectedtochooseairtravelin2015. 1mark
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SECTION B – Module 1 – Question 3 – continuedTURN OVER
c. Considerthecustomerswhochoseseatravelin2014.
Howmanyofthesecustomerswereexpectedtochooseseatravelin2015? 1mark
d. Considerthecustomerswhowereexpectedtochooseairtravelin2015.
Whatpercentageofthesecustomershadalsochosenairtravelin2014? Roundyouranswertothenearestwholenumber. 1mark
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End of Module 1 – SECTION B – continued
In2016,thenumberofcustomersstudiedwasincreasedto1360.MatrixR2016,shownbelow,containsthenumberofthesecustomerswhochoseeachtypeoftravelin2016.
R
ALSN
2016
64646516485
=
Thecompanyintendstoincreasethenumberofcustomersinthestudyin2017andin2018.Thematrixthatcontainsthenumberofcustomerswhoareexpectedtochooseeachtypeoftravelin 2017(R2017)and2018(R2018)canbedeterminedusingthematrixequationsshownbelow.
R2017 = TR2016 + B R2018 = TR2017 + B
this yearA L S N
Twhere =
0 65 0 25 0 25 0 500 15 0 60 0 20 0 150 05 0 1
. . . .
. . . .
. . 00 0 25 0 200 15 0 05 0 30 0 15
808040. .
. . . .
=
ALSN
next year B
−−
80
ALSN
e. i. TheelementinthefourthrowofmatrixBis–80.
Explainthisnumberinthecontextofselectingcustomersforthestudiesin2017and2018. 1mark
ii. Determinethenumberofcustomerswhoareexpectedtochooseseatravelin2018. Roundyouranswertothenearestwholenumber. 2marks
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SECTION B – continuedTURN OVER
CONTINUES OVER PAGE
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SECTION B – Module 2 – continued
Module 2 – Networks and decision mathematics
Question 1 (3marks)Amapoftheroadsconnectingfivesuburbsofacity,Alooma(A),Beachton(B),Campville(C), Dovenest(D)andEasyside(E),isshownbelow.
Campville
Alooma
Beachton
Easyside
Dovenest
a. StartingatBeachton,whichtwosuburbscanbedriventousingonlyoneroad? 1mark
Agraphthatrepresentsthemapoftheroadsisshownbelow.
A
B
C
DE
OneoftheedgesthatconnectstovertexEismissingfromthegraph.
b. i. Addthemissingedgetothegraph above. 1mark
(Answer on the graph above.)
ii. ExplainwhattheloopatDrepresentsintermsofadriverwhoisdepartingfromDovenest. 1mark
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SECTION B – Module 2 – continuedTURN OVER
Question 2 (3marks)ThesuburbofAloomahasaskateboardparkwithsevenramps.TherampsareshownasverticesT,U,V,W,X,Y and Zonthegraphbelow.
V
WU
YT
Z X
rough
rough
ThetracksbetweenrampsU and VandbetweenrampsW and Xarerough,asshownonthegraphabove.
a. NathanbeginsskatingatrampWandfollowsanEuleriantrail.
AtwhichrampdoesNathanfinish? 1mark
b. ZoebeginsskatingatrampXandfollowsaHamiltonianpath. Thepathshechoosesdoesnotincludethetworoughtracks.
WritedownapaththatZoecouldtakefromstarttofinish. 1mark
c. Birracanskateoveranyofthetracks,includingtheroughtracks. HebeginsskatingatrampXandwillcompleteaHamiltoniancycle.
Inhowmanywayscouldhedothis? 1mark
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SECTION B – Module 2 – Question 3 – continued
Question 3 (6marks)AnewskateboardparkistobebuiltinBeachton.Thisprojectinvolves13activities,A to M.Thedirectednetworkbelowshowstheseactivitiesandtheircompletiontimesindays.
D, 5
K, 5A, 4 E, 4 I, 2
L, 4F, 7B, 3
G, 4 J, 6 M, 3C, 1
H, 9
start finish
a. DeterminetheearlieststarttimeforactivityM. 1mark
b. Theminimumcompletiontimefortheskateboardparkis15days.
Writedownthecriticalpathforthisproject. 1mark
c. Whichactivityhasafloattimeoftwodays? 1mark
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End of Module 2 – SECTION B – continuedTURN OVER
d. ThecompletiontimesforactivitiesE,F,G,I and Jcaneachbereducedbyoneday. Thecostofreducingthecompletiontimebyonedayfortheseactivitiesisshowninthetablebelow.
Activity Cost ($)
E 3000
F 1000
G 5000
I 2000
J 4000
Whatistheminimumcosttocompletetheprojectintheshortesttimepossible? 1mark
e. Theskateboardparkprojectonpage22willberepeatedatCampville,butwiththeadditionofoneextraactivity.
Thenewactivity,N,willtakesixdaystocompleteandhasafloattimeofoneday. ActivityNwillfinishatthesametimeastheproject.
i. AddactivityNtothenetworkbelow. 1mark
D, 5
K, 5A, 4 E, 4 I, 2
L, 4F, 7B, 3
G, 4 J, 6 M, 3C, 1
H, 9
start finish
ii. WhatisthelateststarttimeforactivityN? 1mark
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SECTION B – Module 3 – continued
Module 3 – Geometry and measurement
Question 1 (2marks)Agolfballissphericalinshapeandhasaradiusof21.4mm,asshowninthediagrambelow.
r = 21.4 mm
Assumethatthesurfaceofthegolfballissmooth.
a. Whatisthesurfaceareaofthegolfballshown? Roundyouranswertothenearestsquaremillimetre. 1mark
b. Golfballsaresoldinarectangularboxthatcontainsfiveidenticalgolfballs,asshowninthediagrambelow.
r = 21.4 mm
Whatistheminimumlength,inmillimetres,ofthebox? 1mark
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SECTION B – Module 3 – continuedTURN OVER
Question 2 (2marks)SalenapractisesgolfatadrivingrangebyhittinggolfballsfrompointT.ThefirstballthatSalenahitstravelsdirectlynorth,landingatpointA.ThesecondballthatSalenahitstravels50monabearingof030°,landingatpointB.Thediagrambelowshowsthepositionsofthetwoballsaftertheyhavelanded.
A
T
Bnorth
50 m
30°
a. Howfarapart,inmetres,arethetwogolfballs? 1mark
b. Afenceispositionedattheendofthedrivingrange. Thefenceis16.8mhighandis200mfromthepointT.
T200 m
16.8 mfence
WhatistheangleofelevationfromTtothetopofthefence? Roundyouranswertothenearestdegree. 1mark
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SECTION B – Module 3 – continued
Question 3 (2marks)AgolftournamentisplayedinStAndrews,Scotland,atlocation56°N,3°W.
a. AssumethattheradiusofEarthis6400km.
FindtheshortestgreatcircledistancetotheequatorfromStAndrews. Roundyouranswertothenearestkilometre. 1mark
b. ThetournamentbeginsonThursdayat6.32aminStAndrews,Scotland. ManypeopleinMelbournewillwatchthetournamentliveontelevision. AssumethatthetimedifferencebetweenMelbourne(38°S,145°E)andStAndrews(56°N,3°W)is
10hours.
OnwhatdayandatwhattimewillthetournamentbegininMelbourne? 1mark
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SECTION B – Module 3 – continuedTURN OVER
Question 4 (3marks)Duringagameofgolf,Salenahitsaballtwice,fromP to QandthenfromQ to R.Thepathoftheballaftereachhitisshowninthediagrambelow.
P
Q
R130°
50° 54°80 m 100 m
north
AfterSalena’sfirsthit,theballtravelled80monabearingof130°frompointPtopointQ.AfterSalena’ssecondhit,theballtravelled100monabearingof054°frompointQtopointR.
a. AnotherballishitandtravelsdirectlyfromP to R.
Usethecosineruletofindthedistancetravelledbythisball. Roundyouranswertothenearestmetre. 2marks
b. WhatisthebearingofRfromP? Roundyouranswertothenearestdegree. 1mark
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SECTION B – Module 3 – Question 5 – continued
Question 5 (3marks)Agolfcoursehasasprinklersystemthatwatersthegrassintheshapeofasector,asshowninthediagrambelow.
100°S
d
AsprinklerispositionedatpointSandcanturnthroughanangleof100°.Theshadedareaonthediagramshowstheareaofgrassthatiswateredbythesprinkler.
a. If147.5m2ofgrassiswatered,whatisthemaximumdistance,dmetres,thatthewaterreaches fromS?
Roundyouranswertothenearestmetre. 1mark
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End of Module 3 – SECTION B – continuedTURN OVER
b. Anothersprinklercanwateralargerareaofgrass. Thissprinklerwillwaterasectionofgrassasshowninthediagrambelow.
M
N
L
100°
area watered by sprinkler
4.5 m12 m
Thesectionofgrassthatiswateredis4.5mwideatallpoints. Watercanreachamaximumof12mfromthesprinkleratL.
Whatistheareaofgrassthatthissprinklerwillwater? Roundyouranswertothenearestsquaremetre. 2marks
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SECTION B – Module 4 – continued
Module 4 – Graphs and relations
Question 1 (4marks)Mariaisahockeyplayer.Sheispaidabonusthatdependsonthenumberofgoalsthatshescoresinaseason.ThegraphbelowshowsthevalueofMaria’sbonusagainstthenumberofgoalsthatshescoresinaseason.
4000
3000
2000
1000
O 5 10 15 20 25 30
bonus ($)
number of goals
a. WhatisthevalueofMaria’sbonusifshescoressevengoalsinaseason? 1mark
b. WhatistheleastnumberofgoalsthatMariamustscoreinaseasontoreceiveabonusof$2500? 1mark
Anotherplayer,Bianca,ispaidabonusof$125foreverygoalthatshescoresinaseason.
c. WhatisthevalueofBianca’sbonusifshescoreseightgoalsinaseason? 1mark
d. Attheendoftheseason,bothplayershavescoredthesamenumberofgoalsandreceivethesamebonusamount.
HowmanygoalsdidMariaandBiancaeachscoreintheseason? 1mark
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SECTION B – Module 4 – continuedTURN OVER
Question 2 (3marks)Thebonusmoneyisprovidedbyacompanythatmanufacturesandsellshockeyballs.Thecost,indollars,ofmanufacturingacertainnumber of ballscanbefoundusingtheequation
cost=1200+1.5×number of balls
a. Howmanyballswouldbemanufacturedifthecostis$1650? 1mark
b. Onthegridbelow,sketchthegraphoftherelationshipbetweenthemanufacturingcostandthe number of ballsmanufactured. 1mark
3000
2500
2000
1500
1000
500
O 100 200 300 400 500 600
cost ($)
number of balls
c. Thecompanywillbreakevenonthesaleofhockeyballswhenitmanufacturesandsells200hockeyballs.
Findthesellingpriceofonehockeyball. 1mark
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SECTION B – Module 4 – Question 3 – continued
Question 3 (5marks)Thecompanyalsoproducestwotypesofhockeystick,the‘Flick’andthe‘Jink’.LetxbethenumberofFlickhockeysticksthatareproducedeachmonth.LetybethenumberofJinkhockeysticksthatareproducedeachmonth.Eachmonth,upto500hockeysticksintotalcanbeproduced.Theinequalitiesbelowrepresentconstraintsonthenumberofeachhockeystickthatcanbeproducedeachmonth.
Constraint1 x≥0 Constraint2 y≥0Constraint 3 x + y≤500 Constraint4 y≤2x
a. InterpretConstraint4intermsofthenumberofFlickhockeysticksandthenumberofJinkhockeysticksproducedeachmonth. 1mark
Thereisanotherconstraint,Constraint5,onthenumberofeachhockeystickthatcanbeproducedeachmonth.Constraint5isboundedbyLineA,shownonthegraphbelow.
600
500
400
300
200
100
O 100 200 300 400 500 600
y
x
Line A
y = 2x
x + y = 500
Theshadedregionofthegraphcontainsthepointsthatsatisfyconstraints1to5.
b. WritedowntheinequalitythatrepresentsConstraint5. 1mark
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Theprofit,P,thatthecompanymakesfromthesaleofthehockeysticksisgivenby
P = 62x + 86y
c. Findthemaximumprofitthatthecompanycanmakefromthesaleofthehockeysticks. 1mark
d. ThecompanywantstochangethesellingpriceoftheFlickandJinkhockeysticksinordertoincreaseitsmaximumprofitto$42000.
AlloftheconstraintsonthenumbersofFlickandJinkhockeysticksthatcanbeproducedeachmonthremainthesame.
Theprofit,Q,thatismadefromthesaleofhockeysticksisnowgivenby
Q = mx + ny
TheprofitmadeontheFlickhockeysticksismdollarsperhockeystick. TheprofitmadeontheJinkhockeysticksisndollarsperhockeystick. Themaximumprofitof$42000ismadebyselling400Flickhockeysticksand100Jinkhockey
sticks.
Whatarethevaluesofm and n? 2marks
END OF QUESTION AND ANSWER BOOK
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 2016
Victorian Certificate of Education 2016
FURMATH EXAM 2
Further Mathematics formulas
Core – Data analysis
standardised score z x xsx
=−
lower and upper fence in a boxplot lower Q1 – 1.5 × IQR upper Q3 + 1.5 × IQR
least squares line of best fit y = a + bx, where b rssy
x= and a y bx= −
residual value residual value = actual value – predicted value
seasonal index seasonal index = actual figuredeseasonalised figure
Core – Recursion and financial modelling
first-order linear recurrence relation u0 = a, un + 1 = bun + c
effective rate of interest for a compound interest loan or investment
r rneffective
n= +
−
×1
1001 100%
Module 1 – Matrices
determinant of a 2 × 2 matrix A a bc d=
, det A
acbd ad bc= = −
inverse of a 2 × 2 matrix AAd bc a
− =−
−
1 1det
, where det A ≠ 0
recurrence relation S0 = initial state, Sn + 1 = T Sn + B
Module 2 – Networks and decision mathematics
Euler’s formula v + f = e + 2
3 FURMATH EXAM
END OF FORMULA SHEET
Module 3 – Geometry and measurement
area of a triangle A bc=12
sin ( )θ
Heron’s formula A s s a s b s c= − − −( )( )( ), where s a b c= + +12
( )
sine ruleaA
bB
cCsin ( ) sin ( ) sin ( )
= =
cosine rule a2 = b2 + c2 – 2bc cos (A)
circumference of a circle 2π r
length of an arc r × × °π
θ180
area of a circle π r2
area of a sector πθr2
360×
°
volume of a sphere43π r 3
surface area of a sphere 4π r2
volume of a cone13π r 2h
volume of a prism area of base × height
volume of a pyramid13
× area of base × height
Module 4 – Graphs and relations
gradient (slope) of a straight line m y y
x x=
−−
2 1
2 1
equation of a straight line y = mx + c
