2017 Further Mathematics-nht Written examination 2 · PDF fileFURTHER MATHEMATICS Written examination 2 Monday 5 June 2017 Reading time: ... 2017 FURMATH EXAM 2 (NHT) 16 SECTION B

FURTHER MATHEMATICSWritten examination 2

Monday 5 June 2017 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

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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2017

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2017

STUDENT NUMBER

Letter

2017FURMATHEXAM2(NHT) 2

THIS pAgE IS BlANK

3 2017FURMATHEXAM2(NHT)

TURN OVER

THIS pAgE IS BlANK

2017FURMATHEXAM2(NHT) 4

SECTION A – Question 1 – continued

SECTION A – Core

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

Data analysis

Question 1 (7marks)A1m2solararrayislocatedataweatherstation.Thetotalamountofenergygeneratedbythesolararray,inmegajoules,isrecordedeachmonth.ThedataforthemonthofFebruaryforthelast22yearsisdisplayedinthedotplotbelow.

n = 22

17 18 19 20 21solar energy (MJ)

22 23 24 25

a. DeterminethenumberofyearsinwhichtheenergygeneratedduringFebruarywasgreaterthan23MJ. 1mark

b. Forthedatainthedotplotabove,thefirstquartileQ1=20andthethirdquartileQ3=21.8

Showthatthedatavalue17.1isanoutlier. 2marks

c. Usingthedatainthedotplot,completetheboxplotbelow. 2marks

17 18 19 20 21solar energy (MJ)

22 23 24 25

5 2017FURMATHEXAM2(NHT)

SECTION A – continuedTURN OVER

d. ThedistributionoftheamountofenergygeneratedbythesolararrayforthemonthsofApril,MayandJuneforthelast22yearsisdisplayedintheparallelboxplotsbelow.

14

13

12

11

10

energy (MJ) 9

8

7

6

5

4April May

monthJune

Theparallelboxplotssuggestthattheamountofenergygeneratedisassociatedwiththemonthoftheyear.

Explainwhy,quotingthevaluesofanappropriatestatistic. 2marks

2017FURMATHEXAM2(NHT) 6

SECTION A – Question 2 – continued

Question 2 (8marks)Twooftheweatherindicatorscollectedattheweatherstationaretemperatureandrelativehumidity.Thescatterplotbelowshowsrelative humidity(%)plottedagainsttemperature(°C)forthe29daysofFebruaryinaparticularleapyear.Themeasurementsweretakenat3pmeachday.Aleastsquareslinehasbeenfittedtothescatterplot.

100

90

80

70

60relative humidity (%)

50

40

30

2010 12 14 16 18

temperature (°C)20 22 24

a. Thecoefficientofdeterminationis0.749

i. Writedownthevalueofthecorrelationcoefficientr. Roundyouranswertotwodecimalplaces. 1mark

ii. Whatpercentageofthevariationinrelative humidity is not explainedbythevariationintemperature?

Roundyouranswertothenearestwholenumber. 1mark

7 2017FURMATHEXAM2(NHT)

SECTION A – continuedTURN OVER

b. Theequationoftheleastsquareslineis

relative humidity=136–4.38×temperature

i. Writedowntheresponsevariable. 1mark

ii. Interprettheslopeoftheleastsquareslineintermsofrelative humidity and temperature. 2marks

iii. Whenthetemperatureis11.2°C,therelative humidityis68%.

Determinetheresidualvaluewhentheleastsquareslineisusedtopredicttherelative humidityatthistemperature.

Roundyouranswertoonedecimalplace. 2marks

iv. Theresidualplotfortheleastsquareslineisshownbelow.

20

10

0residual

–10

–2010 12 14 16 18

temperature (°C)20 22 24

Doestheresidualplotsupporttheassumptionoflinearity?Brieflyexplainyouranswer. 1mark

2017FURMATHEXAM2(NHT) 8

SECTION A – continued

Question 3 (5marks)Thenumberofrainydayspermonthisalsorecordedattheweatherstation.Inthetimeseriesplotbelow,thenumber of rainy dayspermonthisplottedforJanuary(Month1)toAugust(Month8)inthesameyear.

25

20

15

10

number ofrainy days

5

00 1 2 3 4

month number5 6 7 8 9

a. Describethetrendinthetimeseriesplot. 1mark

Thetrendinthetimeseriesplotistobemodelledusingaleastsquaresline.Thedatausedtoconstructthisplotisgivenbelow.

Month number 1 2 3 4 5 6 7 8

Number of rainy days 11 9 11 15 18 17 21 19

b. Usethedataabovetodeterminetheequationoftheleastsquaresline.Writethevaluesoftheinterceptandslopeintheboxesbelow.Roundyouranswerstothreesignificantfigures. 3marks

number of rainy days = + ×month number

c. Drawtheleastsquareslineonthetime series plot above. 1mark

(Answer on the time series plot above.)

9 2017FURMATHEXAM2(NHT)

SECTION A – continuedTURN OVER

CONTINUES OVER pAgE

2017FURMATHEXAM2(NHT) 10

SECTION A – Question 4 – continued

Question 4 (4marks)Thetimeseriesplotbelowshowsthetemperature(°C)recordedattheweatherstationat3pmforthe29daysofFebruaryinaparticularleapyear.

35

30

25

20temperatureat 3 pm (°C)

15

10

5

00 5 10 15

day number20 25 30

a. Writedowntherangeforthevariabletemperature. Roundyouranswertothenearestwholenumber. 1mark

b. Determinethefive-mediansmoothedtemperatureat3pmonday14. Roundyouranswertothenearestwholenumber. 1mark

11 2017FURMATHEXAM2(NHT)

SECTION A – continuedTURN OVER

c. Three-mediansmoothinghasnowbeenusedtosmooththetimeseriesplotuptoday25.

Completethethree-mediansmoothingbymarkingeachremainingsmoothedpointwithacross(×)onthetimeseriesplotbelow. 2marks

35

30

25

20temperatureat 3 pm (°C)

15

10

5

00 5 10 15

day number20 25 30

time series plotthree-median smoothed

2017FURMATHEXAM2(NHT) 12

SECTION A – continued

Recursion and financial modelling

Question 5 (5marks)Thesnookertableatacommunitycentrewaspurchasedfor$3000.Afterpurchase,thevalueofthesnookertablewasdepreciatedusingtheflatratemethodofdepreciation.Thevalueofthesnookertable,Vn ,afternyears,canbedeterminedusingtherecurrencerelationbelow.

V0=3000, Vn+1= Vn–180

a. Whatistheannualdepreciationinthevalueofthesnookertable? 1mark

b. Userecursiontoshowthatthevalueofthesnookertableaftertwoyears,V2,is$2640. 1mark

c. Afterhowmanyyearswillthevalueofthesnookertablefirstfallbelow$2000? 1mark

d. Thevalueofthesnookertablecouldalsobedepreciatedusingthereducingbalancemethodofdepreciation.

Afteroneyear,thevalueofthesnookertableis$2760. Aftertwoyears,thevalueofthesnookertableis$2539.20

i. Showthattheannualrateofdepreciationinthevalueofthesnookertableis8%. 1mark

ii. LetSnbethevalueofthesnookertableafternyears.

Writedownarecurrencerelation,intermsofSn+1 and Sn ,thatcanbeusedtodeterminethevalueofthesnookertableafternyearsusingthisreducingbalancemethod. 1mark

13 2017FURMATHEXAM2(NHT)

SECTION A – continuedTURN OVER

Question 6 (4marks)ThecommunitycentreopenedasavingsaccountwithBankP.LetPnbethebalanceofthesavingsaccountnyearsafteritwasopened.ThevalueofPncanbedeterminedusingtherecurrencerelationmodelbelow.

P0 = A, Pn+1=1.056 ×Pn

Thebalanceofthesavingsaccountoneyearafteritwasopenedwas$1584.

a. ShowthatthevalueofAis$1500. 1mark

b. Writedownthebalanceofthesavingsaccountfouryearsafteritwasopened. 1mark

c. Thebalanceofthesavingsaccountsixyearsafteritwasopenedwas$2080.05 This$2080.05wastransferredintoasavingsaccountwithBankQ. Thissavingsaccountpaysinterestattherateof5.52%perannum,compoundingmonthly. LetQnbethebalanceofthissavingsaccountnmonthsafteritwasopened. ThevalueofQncanbedeterminedfromarule.

Completethisrulebywritingthemissingvaluesintheboxesprovidedbelow. 2marks

Qn = × n

2017 FURMATH EXAM 2 (NHT) 14

END OF SECTION A

Question 7 (3 marks)The community centre has received a donation of $5000. The donation is deposited into another savings account. This savings account pays interest compounding monthly.Immediately after the interest has been added each month, the community centre deposits a further $100 into the savings account.After five years, the community centre would like to have a total of $14 000 in the savings account.

a. What is the annual interest rate, compounding monthly, that is required to achieve this goal? Write your answer correct to two decimal places. 1 mark

b. The interest rate for this savings account is actually 6.2% per annum, compounding monthly. After 36 deposits, the community centre stopped making the additional monthly deposits of

$100.

How much money will be in the savings account five years after it was opened? 2 marks

15 2017FURMATHEXAM2(NHT)

SECTION B – continuedTURN OVER

SECTION B – Modules

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

Contents page

Module1–Matrices......................................................................................................................................16

Module2–Networksanddecisionmathematics..........................................................................................21

Module3–Geometryandmeasurement....................................................................................................... 25

Module4–Graphsandrelations................................................................................................................... 30

2017FURMATHEXAM2(NHT) 16

SECTION B – Module 1 – Question 1 – continued

Module 1 – Matrices

Question 1 (4marks)PeoplepaytoattendconcertsattheWhiteoakTheatre.Theycanchoosetheirseatsforeachconcertfromthreeclasses,A,B or C.Thetablebelowshowsthenumberofseatsavailableineachclassandthecostperseat.

Class Number of seats available

Cost per seat($)

A 100 45

B 340 35

C 160 30

a. ThecolumnmatrixNcontainsthenumberofseatsineachclass.

NABC

=

100340160

WhatistheorderofmatrixN? 1mark

b. MatrixWcontainsthecostofeachclassofseatinthetheatre.

W = 45 35 30A B C

i. DeterminethematrixproductWN. 1mark

ii. ExplainwhatthematrixproductWNrepresents. 1mark

17 2017FURMATHEXAM2(NHT)

SECTION B – Module 1 – continuedTURN OVER

c. Thenumberofseatsthatweresoldforthefirstconcertthisyearisshowninthetablebelow.

Class Number of seats sold

Cost per seat($)

A 42 45

B 179 35

C 86 30

TheinformationinthetableisusedtoconstructthematrixP, shownbelow.

P =

42 0 00 179 00 0 86

453530

MatrixPcontainsthevalueofallseatsineachclass,indollars,thatweresoldforthefirstconcertthisyear.

AmatrixproductMP isfoundwhereM = [ ]0 1 1 .

ExplainwhatthematrixproductMPrepresents. 1mark

2017FURMATHEXAM2(NHT) 18

SECTION B – Module 1 – Question 2 – continued

Question 2 (8marks)TheWhiteoakTheatreClubhas200memberswhobuyticketsforeveryconcert.Thememberscanchooseseatsfromthreedifferentclasses,A,B or C.Foreachconcert,thechoiceofseatclassforthesememberscanbedeterminedusingthetransitionmatrixT,shownbelow.

this concertA B C

T =

0 70 0 25 0 050 30 0 65 0 650 00 0 10 0 30

. . .

. . .

. . .

ABC

next concert

a. AnincompletetransitiondiagramformatrixTisshownbelow.

0.30

0.25

0.100.05

0.650.70

B classA class

C class

0.30

Completethetransition diagram above byaddingallthemissinginformation. 2marks

(Answer on the transition diagram above.)

b. ThenumberofseatsineachclasschosenbythesemembersforthefinalconcertthisyearisshowninmatrixS0below.

SABC

0

169688

=

WhatpercentageofthesememberschoseAclassseatsforthefinalconcertthisyear? 1mark

19 2017FURMATHEXAM2(NHT)

End of Module 1 – SECTION B – continuedTURN OVER

Forthefirstconcertnextyear,somememberswillchooseadifferentseatclassfromtheseatclassthattheychoseforthefinalconcertthisyear.

c. Whatpercentageofthe200membersareexpectedtochangefromBclassseatsatthefinalconcertthisyeartoAclassseatsforthefirstconcertnextyear? 1mark

Theexpectednumberofthesemembersandtheirchoiceofseatclassforthenthconcertnextyearcanbedeterminedusingtherecurrencerelation

S0

169688

=

, Sn + 1 = TSn

d. Writedownthestatematrix,S1,fortheexpectednumberofmembersandtheirchoiceofseatclassforthefirstconcertnextyear.

Writeyouranswercorrecttoonedecimalplace. 1mark

e. Inthelongterm,howmanymemberswouldbeexpectedtobuyBclassseatsforaconcert? 1mark

f. Itisexpectedthat,beginningfromthethirdconcertnextyear,theWhiteoakTheatreClubwillhavemoremembers.

Tennewmembersareexpectedateverynewconcert. Fortheirfirstconcert,newmemberswillnotbegivenaseatchoice. MatrixK2containstheexpectednumberofmembersineachclassofseatforthesecond

concertnextyear. Theexpectednumberofmembersineachclassofseatforthethirdandfourthconcertsnext

yearcanbedeterminedby

K TK BK TK B

3 2

4 3

= +

= + where T K=

=0 70 0 25 0 050 30 0 65 0 650 00 0 10 0 30

611162

2

. . .

. . .

. . .,

33

271

=

and B

DeterminethenumberofmemberswhoareexpectedtochooseAclassseatsforthefourthconcertnextyear.

Roundyouranswertothenearestwholenumber. 2marks

2017FURMATHEXAM2(NHT) 20

SECTION B – continued

THIS pAgE IS BlANK

21 2017FURMATHEXAM2(NHT)

SECTION B – Module 2 – continuedTURN OVER

Module 2 – Networks and decision mathematics

Question 1 (5marks)Simonisbuildinganewholidayhomeforhisfamily.Thedirectednetworkbelowshowsthe10activitiesrequiredforthisprojectandtheircompletiontimes,inweeks.

start finish

A, 2

B, 4

C, 2

D, 1

E, 4

G, 5

F, 3

H, 2

I, 3

J, 4

a. WritedownthetwoactivitiesthatareimmediatepredecessorsofactivityG. 1mark

b. ForactivityD,theearlieststartingtimeandthelateststartingtimearethesame.

WhatdoesthistellusaboutactivityD? 1mark

c. Determinetheminimumcompletiontime,inweeks,forthisproject. 1mark

d. Determinethelateststartingtime,inweeks,foractivityC. 1mark

e. Whichactivitycouldbedelayedforthelongesttimewithoutaffectingtheminimumcompletiontimeoftheproject? 1mark

2017FURMATHEXAM2(NHT) 22

SECTION B – Module 2 – Question 2 – continued

Question 2 (4marks)Theholidayhomehasfourrooms,A,B,C and D.Thefloorplan,belowleft,showstheseroomsandtheoutsidearea.Thereare12doors,asshownonthefloorplan.OnlyroomCandtheoutsidearelabelled.Agraphisshowntotherightofthefloorplan.Onthisgraph,verticesrepresenttheroomsandtheoutsidearea,andedgesrepresentthedoors.

outsideoutside

outside

outside

Floor plan

outside

A

BC D

Graph

C

a. Onthefloorplanabove,roomChasalreadybeenlabelled.

UsethelettersA,B and Dtolabeltheotherthreeroomsonthefloor plan above. 1mark

(Answer on the floor plan above.)

b. SimonisinroomCandhisdaughterZofiaisoutside. SimoncallsZofiatoseehiminroomC. ZofiavisitseveryotherroomonceonherwaytoroomC.

GivethemathematicaltermthatdescribesZofia’sjourney. 1mark

23 2017FURMATHEXAM2(NHT)

SECTION B – Module 2 – continuedTURN OVER

c. Simontriestofindaroutethatpassesthrougheverydooronceonlyandfinishesbackatthestartingpoint.

i. Explainwhythisisnotpossible.Refertothegraphinyouranswer. 1mark

ii. Iftwoofthedoorsarelockedandonlytheotherdoorsareconsidered,thenSimon’sroutewillbepossible.

SimonlocksthedoorbetweenroomAandroomC.

Writedownthetworoomsthatarejoinedbytheotherdoorthatmustbelocked. 1mark

2017FURMATHEXAM2(NHT) 24

End of Module 2 – SECTION B – continued

Question 3 (3marks)Simonbuilthisholidayhomeonanestate.Theestatehasone-waystreetsbetweentheentranceandtheexit.Therearerestrictionsonthenumberoftrucksthatareallowedtotravelalongeachstreetperday.Onthedirectedgraphbelow,theverticesrepresenttheintersectionsoftheone-waystreets.Thenumberoneachedgeisthemaximumnumberoftrucksthatareallowedtotravelalongthatstreetperday.

entrance

exit

Cut A

9

9

9

9

11

10

10

7

7

7

6

5

4

6

8

8

8

8

8

83

Whenconsideringthepossibleflowoftrucksthroughthisnetwork,manydifferentcutscanbemade.

a. DeterminethecapacityofCutA,shownabove. 1mark

b. Findthemaximumnumberoftrucksthatcouldtravelfromtheentrancetotheexitperday. 1mark

c. Acompanywouldliketosendonegroupoftrucksfromtheentrancetotheexit. Alltrucksinthisgroupmustfolloweachotherandtravelalongthesameroute. Thetrucksinthisgroupwillbetheonlytruckstousethesestreetsonthatday.

Whatisthemaximumnumberoftrucksthatcouldbeinthisgroup? 1mark

25 2017FURMATHEXAM2(NHT)

SECTION B – Module 3 – continuedTURN OVER

Module 3 – geometry and measurement

Question 1 (3marks)Adairyfarmissituatedonalargeblockofland.Theshadedareainthediagrambelowrepresentstheblockofland.

2.2 km 4 km

2.3 km

0.8 km

6.2 km

3.1 kmd

a. Showthatthelengthdis3.2km,roundedtoonedecimalplace. 1mark

b. Usingd =3.2, calculatetheperimeter,inkilometres,ofthisblockofland. 1mark

c. Calculatetheareaofthisblockofland. Roundyouranswertothenearestsquarekilometre. 1mark

2017FURMATHEXAM2(NHT) 26

SECTION B – Module 3 – continued

Question 2 (2marks)Thedairyfarmhasafarmhouse,amilkingshedandamanufacturingbuilding.Thefarmhouseislocateddueeastofthemilkingshed.Themanufacturingbuildingislocatedduesouthofthefarmhouse.Themanufacturingbuildingis160mfromthemilkingshed,asshownbelow.

60°

160 m

milking shed

manufacturing building

N farmhouse

a. Howfareastofthemilkingshedisthemanufacturingbuildinglocated? 1mark

b. Astoragefacilityislocated900meastand400mnorthofthemanufacturingbuilding,asshownbelow.

storage facility

N

manufacturing building

400 m

900 m

Whatisthebearingofthestoragefacilityfromthemanufacturingbuilding? Roundyouranswertothenearestdegree. 1mark

27 2017FURMATHEXAM2(NHT)

SECTION B – Module 3 – continuedTURN OVER

Question 3 (3marks)ThedairyfarmislocatedinthetownofMilkdale(34°S,141°E).

a. ThecowsaremilkedearlyinthemorninginMilkdale. Oneday,thesunrisesinanothertown,Creamville(36°S,147°E),at6.42am. AssumeMilkdaleandCreamvilleareinthesametimezone.

AtwhattimewillthesunriseinMilkdaleonthisday? 2marks

b. AssumethattheradiusofEarthis6400km.

DeterminetheshortestdistancefromMilkdaletotheSouthPole. Roundyouranswertothenearestkilometre. 1mark

2017FURMATHEXAM2(NHT) 28

End of Module 3 – SECTION B – continued

Question 4 (4marks)Milkismadeintocheeseinthemanufacturingbuilding.Therearetwosizesofcheese,eachmadeintheshapeofacylinderandofequalheight.Asmallcylinderofcheesehasaradiusof55mmandalargecylinderofcheesehasaradiusof75mm.

radius = 55 mm radius = 75 mm

small cylinder of cheese large cylinder of cheese

a. Thepriceofacylinderofcheeseisproportionaltoitsvolume. Thepriceofasmallcylinderofcheeseis$12.10

Whatisthepriceofalargecylinderofcheese? 2marks

OA

Bh

75 mm

b. Alargecylinderofcheeseiscutintofiveequalpiecesandonepieceisremoved,asshownabove.

TheareaofsectorOAB(shaded)is3534.3mm2. Thetotalsurfaceareaofthispieceis12200mm2.

Whatistheheight,h,ofthispiece? Roundyouranswertothenearestmillimetre. 2marks

29 2017FURMATHEXAM2(NHT)

SECTION B – continuedTURN OVER

CONTINUES OVER pAgE

2017 FURMATH EXAM 2 (NHT) 30

SECTION B – Module 4 – Question 1 – continued

Module 4 – Graphs and relations

Question 1 (4 marks)Anita often drives to a farmers’ market.The graph below shows the relationship between the average speed for the journey (in km/h) and the time (in hours) it takes her to reach the farmers’ market.

average speed (km/h)

time (hours)

120

100

80

60

40

20

O 1 2 3

a. One day, it took Anita two hours to reach the farmers’ market.

What was her average speed in kilometres per hour? 1 mark

b. In March, Anita travelled to the farmers’ market at an average speed of 80 km/h. In April, she travelled to the farmers’ market at an average speed of 40 km/h due to

roadworks.

How much longer did it take Anita to reach the farmers’ market in April compared to the time she took in March? 1 mark

31 2017FURMATHEXAM2(NHT)

SECTION B – Module 4 – continuedTURN OVER

c. Theequationfortherelationshipbetweenaverage speed and timehastheform

average speed ktime

� � �=

i. Findthevalueofk. 1mark

ii. Ontheaxesprovidedbelow,drawagraphoftherelationshipbetweenaverage speed

and 1time

. 1mark

120

100

80

60

40

20

1 21

time

average speed (km/h)

2017FURMATHEXAM2(NHT) 32

SECTION B – Module 4 – Question 3 – continued

Question 2 (4marks)Anitasellsbottlesoftomatojuiceatthefarmers’market.Therevenue,indollars,thatshemakesfromsellingnbottlesoftomatojuiceisgivenby

revenue =6.5n

Thecost,indollars,ofmakingnbottlesoftomatojuiceisgivenby

cost=2.5n + 60

a. Whatisthesellingpriceofeachbottleoftomatojuice? 1mark

b. HowmanybottlesoftomatojuicewillAnitaneedtoselltobreakeven? 1mark

c. Anitawouldliketomakeaprofitof$300fromthesaleof75bottlesoftomatojuice.

Forthistooccur,whatwouldthesellingpriceofeachbottleoftomatojuicehavetobe? 2marks

Question 3 (4marks)Anitaproducestwonewflavoursofjuice,BreakfastBlastandMorningShine.LetxbethenumberofbottlesofBreakfastBlast producedeachweek.Lety bethenumberofbottlesofMorningShineproducedeachweek.EachbottleofBreakfastBlastcontainsthejuiceofthreeapplesandoneorange.EachbottleofMorningShinecontainsthejuiceoftwoapplesandtwooranges.TheconstraintsontheproductionofjuiceeachweekaregivenbyInequalities1to4.

Inequality1 x≥40Inequality2 y≥30Inequality3(apples) 3x + 2y≤250Inequality4(oranges) x + 2y≤138

a. Whatisthemaximumnumberoforangesavailabletoproducethetwoflavoursofjuiceeachweek? 1mark

33 2017FURMATHEXAM2(NHT)

END OF QUESTION AND ANSWER BOOK

b. ThegraphbelowshowsthelinesthatrepresenttheboundariesofInequalities1to4.

Onthegraph,shadetheregionthatcontainsthepointsthatsatisfytheseinequalities. 1mark

20

40

60

80

100

120

20O 40 60 80 100 120 140x

y

c. Anitamakesaprofitof$4.80fromeverybottleofBreakfastBlastthatsheproducesand $3.20fromeverybottleofMorningShinethatsheproduces.

WhatisthesmallesttotalnumberofbottlesofthetwojuicesthatAnitacanproducetomakethemaximumprofit? 2marks

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 2017

Victorian Certificate of Education 2017

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