SPECIALIST MATHEMATICSWritten examination 2

Monday 7 November 2016 Reading time: 11.45 am to 12.00 noon (15 minutes) Writing time: 12.00 noon to 2.00 pm (2 hours)

QUESTION AND ANSWER BOOK

Structure of bookSection Number of

questionsNumber of questions

to be answeredNumber of

marks

A 20 20 20B 6 6 60

Total 80

• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulers,aprotractor,setsquares,aidsforcurvesketching,oneboundreference,oneapprovedtechnology(calculatororsoftware)and,ifdesired,onescientificcalculator.CalculatormemoryDOESNOTneedtobecleared.Forapprovedcomputer-basedCAS,fullfunctionalitymaybeused.

• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.

Materials supplied• Questionandanswerbookof23pages.• Formulasheet.• Answersheetformultiple-choicequestions.

Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice

questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.• AllwrittenresponsesmustbeinEnglish.

At the end of the examination• Placetheanswersheetformultiple-choicequestionsinsidethefrontcoverofthisbook.• 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

SECTION A – continued

2016SPECMATHEXAM2 2

Question 1Thecartesianequationoftherelationgivenbyx =3cosec2 (t)andy =4cot(t)–1is

A. y x+( )

− =1

16 91

2 2

B. yx

+( ) =+( )1

16 33

2

C. x y2 2

91

161+

+( )=

D. 4x–3y=15

E. yx

+( ) =−( )1

16 33

2

Question 2Theimplieddomainof y x a

b=

−

arccos ,whereb>0is

A. [–1,1]B. [a–b,a + b]C. [a–1,a+1]D. [a,a + bπ]E. [–b,b]

Question 3Thestraight-lineasymptote(s)ofthegraphofthefunctionwithrule f x x ax

x( ) = −3

2 ,whereaisanon-zerorealconstant,isgivenbyA. x =0only.

B. x =0andy =0only.

C. x =0andy = xonly.

D. x =0, x a= and x a= − only.

E. x =0andy = aonly.

SECTION A – Multiple-choice questions

Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrectforthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenot drawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8

SECTION A – continuedTURN OVER

3 2016SPECMATHEXAM2

Question 4Oneoftherootsofz3 + bz2 + cz=0is3–2i,wherebandcarerealnumbers.ThevaluesofbandcrespectivelyareA. 6,13B. 3,–2C. –3,2D. 2,3E. –6,13

Question 5IfArg − +( ) = −1 2

3ai π ,thentherealnumberais

A. − 3

B. −32

C. −13

D. 13

E. 3

Question 6Thepointscorrespondingtothefourcomplexnumbersgivenby

z z z z1 2 3 423

34

2 23

=

=

= −

= −cis cis cis cisπ π π π, , ,

44

aretheverticesofaparallelograminthecomplexplane.Whichoneofthefollowingstatementsisnottrue?

A. Theacuteanglebetweenthediagonalsoftheparallelogramis512

B. Thediagonalsoftheparallelogramhavelengths2and4

C. 1 2 3 4 0z z z z =

D. 1 2 3 4 0z z z z+ + + =

E. 1 2≤ ≤z forallfourofz1,z2,z3,z4

SECTION A – continued

2016SPECMATHEXAM2 4

Question 7Giventhatx=sin(t)–cos(t)and y t= ( )1

22sin ,then dy

dxintermsoftis

A. cos(t)–sin(t)

B. cos(t)+sin(t)

C. sec(t)+cosec(t)

D. sec(t)–cosec(t)

E. cos

cos sin2t

t t( )

( ) − ( )

Question 8Usingasuitablesubstitution, x e dxx

a

b3 2 4( )∫ ,wherea andbarerealconstants,canbewrittenas

A. e duu

a

b2( )∫

B. e duu

a

b2

4

4

( )∫C.

18

e duu

a

b( )∫

D. 14

24

4

e duu

a

b( )∫

E. 18 8

8

3

3

e duu

a

b( )∫

Question 9

If f x dydx

x x( ) = = −2 2 ,wherey0 = 0 = y(2),theny3usingEuler’sformulawithstepsize0.1is

A. 0.1f (2)

B. 0.6+0.1f (2.1)

C. 1.272+0.1f (2.2)

D. 2.02+0.1f (2.3)

E. 2.02+0.1f (2.2)

SECTION A – continuedTURN OVER

5 2016SPECMATHEXAM2

Question 10

–1

O

–2

2

1

–3 –2 –1 1 2 3

y

x

Thedirectionfieldforthedifferentialequation dydx

x y+ + = 0 isshownabove.

Asolutiontothisdifferentialequationthatincludes(0,–1)couldalsoincludeA. (3,–1)B. (3.5,–2.5)C. (–1.5,–2)D. (2.5,–1)E. (2.5,1)

SECTION A – continued

2016SPECMATHEXAM2 6

Question 11Let

a i j k= + +3 2 α and

b i j k= − +4 2α ,whereαisarealconstant.

Ifthescalarresoluteof

a inthedirectionof

b is 74273

,thenαequalsA. 1B. 2C. 3D. 4E. 5

Question 12If

a i j k= − − +2 3 and

b i j k ,= − + +m 2 wheremisarealconstant,thevector

a b− willbeperpendiculartovector

b wheremequalsA. 0onlyB. 2onlyC. 0or2D. 4.5E. 0or–2

Question 13Aparticleofmass5kgissubjecttoforces12i

newtonsand9j

newtons.Ifnootherforcesactontheparticle,themagnitudeoftheparticle’sacceleration,inms–2,isA. 3B. 2 4 1 8. .

i j+C. 4.2D. 9E. 60 45

i j+

SECTION A – continuedTURN OVER

7 2016SPECMATHEXAM2

Question 14Twolightstringsoflength4mand3mconnectamasstoahorizontalbar,asshownbelow.Thestringsareattachedtothehorizontalbar5mapart.

4 m3 m

5 m

mass

horizontal bar

T2T1

GiventhetensioninthelongerstringisT1andthetensionintheshorterstringisT2,theratioofthe

tensions TT1

2is

A. 35

B. 34

C. 45

D. 54

E. 43

Question 15 AvariableforceofFnewtonsactsona3kgmasssothatitmovesinastraightline.Attimetseconds,t≥0,itsvelocityvmetrespersecondandpositionxmetresfromtheoriginaregivenbyv =3–x2.ItfollowsthatA. F=–2xB. F=–6xC. F = 2x3–6xD. F = 6x3–18xE. F = 9x–3x3

SECTION A – continued

2016SPECMATHEXAM2 8

Question 16Acricketballishitfromthegroundatanangleof30°tothehorizontalwithavelocityof20ms–1.Theballissubjectonlytogravityandairresistanceisnegligible.Giventhatthefieldislevel,thehorizontaldistancetravelledbytheball,inmetres,tothepointofimpactis

A. 10 3

g

B. 20g

C. 100 3

g

D. 200 3

g

E. 400g

Question 17Abodyofmass3kgismovingtotheleftinastraightlineat2ms–1.Itexperiencesaforceforaperiodoftime,afterwhichitisthenmovingtotherightat2ms–1.Thechangeinmomentumoftheparticle,inkgms–1,inthedirectionofthefinalmotionisA. –6B. 0C. 4D. 6E. 12

Question 18Orangesgrownonacitrusfarmhaveameanmassof204gramswithastandarddeviationof9grams.Lemonsgrownonthesamefarmhaveameanmassof76gramswithastandarddeviationof3grams.Themassesofthelemonsareindependentofthemassesoftheoranges.Themeanmassandstandarddeviation,ingrams,respectivelyofasetofthreeoftheseorangesandtwooftheselemonsareA. 764,3 29

B. 636,12

C. 764, 33

D. 636,3 10

E. 636,33

9 2016SPECMATHEXAM2

END OF SECTION ATURN OVER

Question 19Arandomsampleof100bananasfromagivenareahasameanmassof210gramsandastandarddeviation of16grams.Assumingthestandarddeviationobtainedfromthesampleisasufficientlyaccurateestimateofthepopulationstandarddeviation,anapproximate95%confidenceintervalforthemeanmassofbananasproducedinthislocalityisgivenbyA. (178.7,241.3)B. (206.9,213.1)C. (209.2,210.8)D. (205.2,214.8)E. (194,226)

Question 20Thelifetimeofacertainbrandofbatteriesisnormallydistributedwithameanlifetimeof20hoursandastandarddeviationoftwohours.Arandomsampleof25batteriesisselected.Theprobabilitythatthemeanlifetimeofthissampleof25batteriesexceeds19.3hoursisA. 0.0401B. 0.1368C. 0.6103D. 0.8632E. 0.9599

2016SPECMATHEXAM2 10

SECTION B – Question 1–continued

Question 1 (9marks)a. Findthestationarypointofthegraphof f x

x xx

x R( ) = + +∈

4 02 3

, \{ }. Expressyouranswer

incoordinateform,givingvaluescorrecttotwodecimalplaces. 1mark

b. Findthepointofinflectionofthegraphgiveninpart a.Expressyouranswerincoordinateform,givingvaluescorrecttotwodecimalplaces. 2marks

SECTION B

Instructions for Section BAnswerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8

11 2016SPECMATHEXAM2

SECTION B – continuedTURN OVER

c. Sketchthegraphof f x x xx

( ) = + +4 2 3for x∈ −[ ]3 3, ontheaxesbelow,labellingthe

turningpointandthepointofinflectionwiththeircoordinates,correcttotwodecimalplaces. 3marks

y

x

–12

–8

–4

O

4

8

12

–3 –2 –1 1 2 3

Aglassistobemodelledbyrotatingthecurvethatisthepartofthegraphwhere x∈ − −[ ]3 0 5, . aboutthey-axis,toformasolidofrevolution.

d. i. Writedownadefiniteintegral,intermsofx, whichgivesthelengthofthecurvetoberotated. 1mark

ii. Findthelengthofthiscurve,correcttotwodecimalplaces. 1mark

e. ThevolumeofthesolidformedisgivenbyV a x dyc

b= ∫ 2 .

Findthevaluesofa,bandc.Donotattempttoevaluatethisintegral. 1mark

2016SPECMATHEXAM2 12

SECTION B – Question 2–continued

Question 2 (11marks)Alineinthecomplexplaneisgivenby z z i z C− = + − ∈1 2 3 , .

a. Findtheequationofthislineintheformy = mx + c. 2marks

b. Findthepointsofintersectionoftheline z z i− = + −1 2 3 withthecircle z − =1 3. 2marks

13 2016SPECMATHEXAM2

SECTION B – continuedTURN OVER

c. Sketchboththeline z z i− = + −1 2 3 andthecircle z − =1 3 ontheArganddiagrambelow. 2marks

Im(z)

Re(z)

–4

–3

–2

–1O–1–2–3–4 4321

1

2

3

4

d. Theline z z i− = + −1 2 3 cutsthecircle z − =1 3 intotwosegments.

Findtheareaofthemajorsegment. 2marks

e. Sketchtheraygivenby Arg z( ) = − 34π ontheArganddiagraminpart c. 1mark

f. Writedowntherangeofvaluesofα α, ,∈R forwhicharaywithequationArg(z)=απ intersectstheline z z i− = + −1 2 3 . 2marks

2016SPECMATHEXAM2 14

SECTION B – Question 3–continued

Question 3 (11marks)Atankinitiallyhas20kgofsaltdissolvedin100Lofwater.Purewaterflowsintothetankatarateof10L/min.Thesolutionofsaltandwater,whichiskeptuniformbystirring,flowsoutofthetankatarateof5L/min.Ifxkilogramsistheamountofsaltinthetankaftertminutes,itcanbeshownthatthe

differentialequationrelatingxandtis dxdt

xt

++

=20

0.

a. Solvethisdifferentialequationtofindxintermsoft. 3marks

Asecondtankinitiallyhas15kgofsaltdissolvedin100Lofwater.Asolutionof160

kgof

saltperlitreflowsintothetankatarateof20L/min.Thesolutionofsaltandwater,whichiskeptuniformbystirring,flowsoutofthetankatarateof10L/min.

b. Ifykilogramsistheamountofsaltinthetankaftertminutes,writedownanexpressionfortheconcentration,inkg/L, ofsaltinthesecondtankattimet. 1mark

15 2016SPECMATHEXAM2

SECTION B – Question 3–continuedTURN OVER

c. Showthatthedifferentialequationrelatingyandtisdydt

yt

++

=10

13. 2marks

d. Verifybydifferentiationandsubstitutionintotheleftsidethat yt t

t=

+ ++

2 20 9006 10( )

satisfies

thedifferential equationinpart c.Verifythatthegivensolutionforyalsosatisfiestheinitial condition. 3marks

2016SPECMATHEXAM2 16

SECTION B – continued

e. Findwhentheconcentrationofsaltinthesecondtankreaches0.095kg/L.Giveyouranswerinminutes,correcttotwodecimalplaces. 2marks

17 2016SPECMATHEXAM2

SECTION B – continuedTURN OVER

CONTINUES OVER PAGE

2016SPECMATHEXAM2 18

SECTION B – Question 4–continued

Question 4 (10marks)Twoships,AandB,areobservedfromalighthouseatoriginO.RelativetoO,theirpositionvectorsattimethoursaftermiddayaregivenby

r i j

r i jA

B

t t

t t

= −( ) + +( )= −( ) + −( )5 1 3 1

4 2 5 2

wheredisplacementsaremeasuredinkilometres.

a. Showthatthetwoshipswillnotcollide,clearlystatingyourreason. 2marks

b. Sketchandlabelthepathofeachshipontheaxesbelow.Showthedirectionofmotionofeachshipwithanarrow. 3marks

y

x

10

8

6

4

2

O–10 –8 –6 –4 –2 2 4 6 8 10–2

–4

–6

19 2016SPECMATHEXAM2

SECTION B – continuedTURN OVER

c. Findtheobtuseanglebetweenthepathsofthetwoships.Giveyouranswerindegrees,correcttoonedecimalplace. 2marks

d. i. Findthevalueoft,correcttothreedecimalplaces,whentheshipsareclosest. 2marks

ii. Findtheminimumdistancebetweenthetwoships,inkilometres,correcttotwodecimalplaces. 1mark

2016SPECMATHEXAM2 20

SECTION B – Question 5–continued

Question 5 (10marks)Amodelrocketofmass2kgislaunchedfromrestandtravelsverticallyup,withaverticalpropulsionforceof(50–10t)newtonsaftertsecondsofflight,wheret∈[ ]0 5, .Assumethattherocketissubjectonlytotheverticalpropulsionforceandgravity,andthatairresistanceisnegligible.

a. Letvms–1bethevelocityoftherockettsecondsafteritislaunched.

Writedownanequationofmotionfortherocketandshowthatdvdt

t= −765

5 . 1mark

b. Findthevelocity,inms–1,oftherocketafterfiveseconds. 2marks

c. Findtheheightoftherocketafterfiveseconds.Giveyouranswerinmetres,correcttotwodecimalplaces. 2marks

21 2016SPECMATHEXAM2

SECTION B – continuedTURN OVER

d. Afterfiveseconds,whentheverticalpropulsionforcehasstopped,therocketissubjectonlytogravity.

Findthemaximumheightreachedbytherocket.Giveyouranswerinmetres,correcttotwodecimalplaces. 2marks

e. Havingreacheditsmaximumheight,therocketfallsdirectlytotheground.

Assumingnegligibleairresistanceduringthisfinalstageofmotion,findthetimeforwhichtherocketwasinflight.Giveyouranswerinseconds,correcttoonedecimalplace. 3marks

2016SPECMATHEXAM2 22

SECTION B – Question 6–continued

Question 6 (9marks)Themeanlevelofpollutantinariverisknowntobe1.1mg/Lwithastandarddeviationof 0.16mg/L.

a. LettherandomvariableX representthemeanlevelofpollutantinthemeasurementsfromarandomsampleof25sitesalongtheriver.

WritedownthemeanandstandarddeviationofX . 2marks

Afterachemicalspill,themeanlevelofpollutantfromarandomsampleof25sitesisfoundtobe1.2mg/L.Todeterminewhetherthissampleprovidesevidencethatthemeanlevelofpollutanthasincreased, astatisticaltestiscarriedout.

b. WritedownsuitablehypothesesH0andH1totestwhetherthemeanlevelofpollutanthasincreased. 2marks

c. i. Findthepvalueforthistest,correcttofourdecimalplaces. 2marks

ii. Statewithareasonwhetherthesamplesupportsthecontentionthattherehasbeenanincreaseinthemeanlevelofpollutantafterthespill.Testatthe5%levelofsignificance. 1mark

23 2016SPECMATHEXAM2

d. Forthistest,whatisthesmallestvalueofthesamplemeanthatwouldprovideevidencethatthemeanlevelofpollutanthasincreased?Thatis,find xc suchthatPr | . .X xc> =( ) =µ 1 1 0 05.Giveyouranswercorrecttothreedecimalplaces. 1mark

e. Supposethatforalevelofsignificanceof2.5%,wefindthatxc =1 163. .Thatis, Pr . | . .X > =( ) =1 163 1 1 0 025µ

Ifthemeanlevelofpollutantintheriver,μ,isinfact1.2mg/Lafterthespill,find Pr . | . .X < =( )1 163 1 2µ Giveyouranswercorrecttothreedecimalplaces. 1mark

END OF QUESTION AND ANSWER BOOK

SPECIALIST 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 Certificate of Education 2016

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2016

SPECMATH EXAM 2

Specialist Mathematics formulas

Mensuration

area of a trapezium 12 a b h+( )

curved surface area of a cylinder 2π rh

volume of a cylinder π r2h

volume of a cone 13π r2h

volume of a pyramid 13 Ah

volume of a sphere 43π r3

area of a triangle 12 bc Asin ( )

sine ruleaA

bB

cCsin ( ) sin ( ) sin ( )

= =

cosine rule c2 = a2 + b2 – 2ab cos (C )

Circular functions

cos2 (x) + sin2 (x) = 1

1 + tan2 (x) = sec2 (x) cot2 (x) + 1 = cosec2 (x)

sin (x + y) = sin (x) cos (y) + cos (x) sin (y) sin (x – y) = sin (x) cos (y) – cos (x) sin (y)

cos (x + y) = cos (x) cos (y) – sin (x) sin (y) cos (x – y) = cos (x) cos (y) + sin (x) sin (y)

tan ( ) tan ( ) tan ( )tan ( ) tan ( )

x y x yx y

+ =+

−1tan ( ) tan ( ) tan ( )

tan ( ) tan ( )x y x y

x y− =

−+1

cos (2x) = cos2 (x) – sin2 (x) = 2 cos2 (x) – 1 = 1 – 2 sin2 (x)

sin (2x) = 2 sin (x) cos (x) tan ( ) tan ( )tan ( )

2 21 2x x

x=

−

3 SPECMATH EXAM

TURN OVER

Circular functions – continued

Function sin–1 or arcsin cos–1 or arccos tan–1 or arctan

Domain [–1, 1] [–1, 1] R

Range −

π π2 2, [0, �] −

π π2 2,

Algebra (complex numbers)

z x iy r i r= + = +( ) =cos( ) sin ( ) ( )θ θ θcis

z x y r= + =2 2 –π < Arg(z) ≤ π

z1z2 = r1r2 cis (θ1 + θ2)zz

rr

1

2

1

21 2= −( )cis θ θ

zn = rn cis (nθ) (de Moivre’s theorem)

Probability and statistics

for random variables X and YE(aX + b) = aE(X) + bE(aX + bY ) = aE(X ) + bE(Y )var(aX + b) = a2var(X )

for independent random variables X and Y var(aX + bY ) = a2var(X ) + b2var(Y )

approximate confidence interval for μ x z snx z s

n− +

,

distribution of sample mean Xmean E X( ) = µvariance var X

n( ) = σ2

SPECMATH EXAM 4

END OF FORMULA SHEET

Calculus

ddx

x nxn n( ) = −1 x dxn

x c nn n=+

+ ≠ −+∫ 11

11 ,

ddxe aeax ax( ) = e dx

ae cax ax= +∫ 1

ddx

xxelog ( )( ) = 1 1

xdx x ce= +∫ log

ddx

ax a axsin ( ) cos( )( ) = sin ( ) cos( )ax dxa

ax c= − +∫ 1

ddx

ax a axcos( ) sin ( )( ) = − cos( ) sin ( )ax dxa

ax c= +∫ 1

ddx

ax a axtan ( ) sec ( )( ) = 2 sec ( ) tan ( )2 1ax dxa

ax c= +∫ddx

xx

sin−( ) =−

12

1

1( ) 1 0

2 21

a xdx x

a c a−

=

+ >−∫ sin ,

ddx

xx

cos−( ) = −

−

12

1

1( ) −

−=

+ >−∫ 1 0

2 21

a xdx x

a c acos ,

ddx

xx

tan−( ) =+

12

11

( ) aa x

dx xa c2 2

1

+=

+

−∫ tan

( )( )

( ) ,ax b dxa n

ax b c nn n+ =+

+ + ≠ −+∫ 11

11

( ) logax b dxa

ax b ce+ = + +−∫ 1 1

product rule ddxuv u dv

dxv dudx

( ) = +

quotient rule ddx

uv

v dudx

u dvdx

v

=

−

2

chain rule dydx

dydududx

=

Euler’s method If dydx

f x= ( ), x0 = a and y0 = b, then xn + 1 = xn + h and yn + 1 = yn + h f (xn)

acceleration a d xdt

dvdt

v dvdx

ddx

v= = = =

2

221

2

arc length 1 2 2 2

1

2

1

2

+ ′( ) ′( ) + ′( )∫ ∫f x dx x t y t dtx

x

t

t( ) ( ) ( )or

Vectors in two and three dimensions

r = i + j + kx y z

r = + + =x y z r2 2 2

� � � � �ir r i j k= = + +ddt

dxdt

dydt

dzdt

r r1 2. cos( )= = + +r r x x y y z z1 2 1 2 1 2 1 2θ

Mechanics

momentum

p v= m

equation of motion

R a= m