SPECIALIST MATHEMATICSWritten examination 2

Friday 20 November 2020 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 5 5 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2020

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2020

STUDENT NUMBER

Letter

SECTION A – continued

2020SPECMATHEXAM2 2

Question 1

They-interceptofthegraphofy = f(x),where f xx a x

x( ) =

−( ) +( )−( )

32

,isalsoastationarypointwhenaequals

A. −2

B. − 65

C. 0

D. 65

E. 2

Question 2

Afunctionfhastherule f x b x a a b a b( ) cos ( )= − > >

SECTION A – continuedTURN OVER

3 2020SPECMATHEXAM2

Question 3AtrainistravellingfromStationAtoStationB.ThetrainstartsfromrestatStationAandtravelswithconstantaccelerationfor30secondsuntilitreachesavelocityof10ms−1.Itthentravelsatthisvelocityfor200seconds.Thetrainthenslowsdown,withconstantacceleration,andstopsatStationBhavingtravelledfor260secondsintotal.Letvms−1bethevelocityofthetrainattimetseconds.Thevelocityvasafunctionoftisgivenby

A. v t

t

t

t

t

t

( )

,

,

( ),

=

−

≤ ≤

< ≤

< ≤

131013

260

0 30

30 230

230 260

B. v t

t

t

t

t

t

( )

,

,

( ),

=

−

≤ ≤

< ≤

< ≤

131013

230

0 30

30 230

230 260

C. v tt

t

ttt

( ),,

( ),=

−

≤ ≤< ≤< ≤

3103 230

0 3030 230230 260

D. v tt

t

ttt

( ),,

( ),=

−

≤ ≤< ≤< ≤

3103 260

0 3030 230230 260

E. v t

t

t

t

t

t

( )

,

,

( ),

=

−

≤ ≤

< ≤

< ≤

131013

230

0 30

30 200

200 230

Question 4

Let f x xx

( ) = −1 overitsimplieddomainandg(x)=cosec2 xfor 02

<

SECTION A – continued

2020SPECMATHEXAM2 4

Question 5

Giventhecomplexnumberz = a + bi,wherea ∈ R\{0}andb ∈ R, 4 2zz

z z+( )isequivalentto

A. 12

+

Im( )Re( )

zz

B. 4 Re( ) Im( )z z×[ ]

C. 4 2 2Re( ) Im( )z z[ ] + [ ]( ) D. 4 1 2+ +( )

Re( ) Im( )z z

E. 2

2×

[ ]Im( )

Re( )

zz

Question 6ForthecomplexpolynomialP(z)=z3 + az2 + bz + cwithrealcoefficientsa,bandc,P(−2)=0andP(3i)=0.Thevaluesofa,bandcarerespectivelyA. –2,9,−18B. 3,4,12C. 2,9,18D. −3,−4,12E. 2,−9,−18

Question 7

Fornon-zerorealconstantsaandb,whereb<0,theexpression12ax x b+( )inpartialfractionformwith

lineardenominators,whereA,BandCarerealconstants,is

A. Aax

Bx Cx b

+++2

B. Aax

Bx b

Cx b

++

+−

C. Ax

Bax b

Cax b

++

+−

D. Ax

Bx b

Cx b

++

+−

E. Aax

B

x b

Cx b

++( )

++2

SECTION A – continuedTURN OVER

5 2020SPECMATHEXAM2

Question 8Giventhat(x + iy)14 = a + ib,wherex,y,a,b ∈ R,(y−ix)14forallvaluesofxandyisequaltoA. −a−ibB. b−iaC. −b + iaD. −a + ibE. b + ia

SECTION A – continued

2020SPECMATHEXAM2 6

Question 9P(x,y)isapointonacurve.Thex-interceptofatangenttopointP(x,y)isequaltothey-valueatP.Whichoneofthefollowingslopefieldsbestrepresentsthiscurve?

y

x–1 1 2–2

–2

–1

1

2

A.

–2 –1 1 2

2

1

–1

–2

y

x

B.

–2 –1 1 2

2

1

–1

–2

y

x

C.

–2 –1 1 2

2

1

–1

–2

y

x

D.

–2 –1 1 2

2

1

–1

–2

y

x

E.

O

O

O

O

O

SECTION A – continuedTURN OVER

7 2020SPECMATHEXAM2

Question 10Atankinitiallycontains300gramsofsaltthatisdissolvedin50Lofwater.Asolutioncontaining15gramsofsaltperlitreofwaterispouredintothetankatarateof2Lperminuteandthemixtureinthetankiskeptwellstirred.Atthesametime,5Lofthemixtureflowsoutofthetankperminute.Adifferentialequationrepresentingthemass,mgrams,ofsaltinthetankattimetminutes,foranon-zerovolumeofmixture,is

A. dmdt

= 0

B. dmdt

mt

= −−

550 5

C. dmdt

m= −30

10

D. dmdt

mt

= −−

30 550 3

E. dmdt

mt

= −−

30 550 5

Question 11

Withasuitablesubstitutionsec

sec tan

2

24

33 1

xx x

dx( )( ) − ( ) +∫π

π

canbeexpressedas

A. 11

121

13

u udu

−−

−

∫

B. 13 3

131

3

u udu

−( )−

∫

C. 12

111

3

u udu

−−

−

∫

D. 1

11

21

3

u udu

−−

−

∫

E. 13 1

13 2

4

3u u

du−( )

−+( )

∫π

π

SECTION A – continued

2020SPECMATHEXAM2 8

Question 12

Ifdydx

e x= cos( )andy0 = ewhenx0=0,then,usingEuler’sformulawithstepsize0.1,y3isequalto

A. e e+ +( )0 1 1 0 1. cos( . )

B. e e e+ + +( )0 1 1 0 1 0 2. cos( . ) cos( . )

C. e e e e+ + +( )0 1 0 1 0 2. cos( . ) cos( . )

D. e e e e+ + +( )0 1 0 1 0 2 0 3. cos( . ) cos( . ) cos( . )

E. e e e e e+ + + +( )0 1 0 1 0 2 0 3. cos( . ) cos( . ) cos( . )

Question 13Thevectorsa i j k , b i j k and c i k~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~= + − = + + = +2 3 2λ willbelinearly dependentwhenthevalueofλisA. 1B. 2C. 3D. 4E. 5

Question 14 ThemagnitudeofthecomponentoftheforceF i j 8k~ ~ ~ ~= + −6 1 thatactsinthedirectiond i j k~ ~ ~ ~= − −2 3 6 is

A. 9219

B. 927

C. 124

7

D. 9211

E. 187

SECTION A – continuedTURN OVER

9 2020SPECMATHEXAM2

Question 15Twoforces,F i j and F i jA B~ ~ ~ ~ ~ ~= − = +4 2 2 5 ,actonaparticleofmass3kg.Theparticleisinitiallyatrestat

position i j~ ~+ .Allforcecomponentsaremeasuredinnewtonsanddisplacementsaremeasuredinmetres.

Thecartesianequationofthepathoftheparticleis

A. y x=2

B. y x= −2

12

C. yx

=+( )

+1

21

2

D. yx

=−( )

+1

21

2

E. y x= +2

12

Question 16Let a i j k and b i j k~ ~ ~ ~ ~ ~ ~ ~= + + = − +2 2 2 4 4 ,wheretheacuteanglebetweenthesevectorsisθ.

Thevalueofsin(2θ)is

A. 19

B. 4 59

C. 4 581

D. 8 581

E. 2 4625

Question 17

Thevelocity,vms–1,ofaparticleattimet ≥0secondsandatpositionx ≥ 1mfromtheoriginis vx

=1 .

Theaccelerationoftheparticle,inms–2,whenx=2is

A. −14

B. −18

C. 18

D. 12

E. 14

2020SPECMATHEXAM2 10

END OF SECTION A

Question 18Aparticleofmassmkilogramshangsfromastringthatisattachedtoafixedpoint.TheparticleisactedonbyahorizontalforceofmagnitudeFnewtons.Thesystemisinequilibriumwhenthestringmakesanangleαtothehorizontal,asshowninthediagrambelow.ThetensioninthestringhasmagnitudeTnewtons.

m F

Thevalueoftan(α)is

A. mgT

B. T

mg

C. TF

D. F

mg

E. mgF

Question 19Acricketballofmass0.02kg,movingwithvelocity2 10 1i j ms~ ~−

− ,ishitandafterimpacttravelswithvelocity2 7 1i j ms~ ~−

− .Themagnitudeofthechangeinmomentumofthecricketball,inkgms−1,isclosesttoA. 0.04B. 0.06C. 0.10D. 0.24E. 0.34

Question 20Anobjectofmass2kgissuspendedfromaspringbalancethatisinsidealifttravellingdownwards.Ifthereadingonthespringbalanceis30N,theaccelerationoftheliftisA. 5.2ms−2upwards.B. 5.2ms−2downwards.C. 9.8ms−2downwards.D. 10.4ms−2upwards.E. 10.4ms−2downwards.

11 2020SPECMATHEXAM2

TURN OVER

CONTINUES OVER PAGE

2020SPECMATHEXAM2 12

SECTION B – Question 1–continued

Question 1 (12marks)Aparticlemovesinthex-yplanesuchthatitspositionintermsofxandymetresattsecondsisgivenbytheparametricequations

x=2sin(2t)

y=3cos(t)

wheret≥0.

a. Findthedistance,inmetres,oftheparticlefromtheoriginwhent = π6. 2marks

b. i. Expressdydxintermsoftand,hence,findtheequationofthetangenttothepathofthe

particleatt = πseconds. 3marks

SECTION B

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

13 2020SPECMATHEXAM2

SECTION B – continuedTURN OVER

ii. Findthevelocity,v~ ,inms–1,oftheparticlewhent = π. 2marks

iii. Findthemagnitudeoftheacceleration,inms–2,whent = π. 2marks

c. Findthetime,inseconds,whentheparticlefirstpassesthroughtheorigin. 1mark

d. Expressthedistance,dmetres,travelledbytheparticlefromt = 0 to t = π6asadefinite

integralandfindthisdistancecorrecttothreedecimalplaces. 2marks

2020SPECMATHEXAM2 14

SECTION B – Question 2 –continued

Question 2 (11marks)Twocomplexnumbers,uandv,aredefinedasu=−2−iandv=−4−3i.

a. Expresstherelation|z − u |=|z − v |inthecartesianformy = mx + c,wherem,c ∈ R. 3marks

b. Plotthepointsthatrepresentuandvandtherelation|z − u |=|z − v |ontheArganddiagrambelow. 2marks

–5

5

5–5 O

Im(z)

Re(z)

c. Stateageometricalinterpretationofthegraphof|z − u |=|z − v |inrelationtothepointsthatrepresentuandv. 1mark

15 2020SPECMATHEXAM2

SECTION B – continuedTURN OVER

d. i. SketchtheraygivenbyArg( )z u− = π4ontheArganddiagraminpart b. 1mark

ii. WritedownthefunctionthatdescribestherayArg( )z u− = π4,givingtherulein

cartesianform. 1mark

e. Thepointsrepresentinguandvand–5ilieonthecirclegivenby|z – zc|=r,wherezcisthecentreofthecircleandristheradius.

Findzc intheforma + ib,wherea,b ∈ R,andfindtheradiusr. 3marks

2020SPECMATHEXAM2 16

SECTION B – Question 3 –continued

Question 3 (10marks)Letf(x)=x2e−x.

a. Findanexpressionfor f ′ (x)andstatethecoordinatesofthestationarypointsof f (x). 2marks

b. Statetheequation(s)ofanyasymptotesof f (x). 1mark

c. Sketchthegraphofy = f (x)ontheaxesprovidedbelow,labellingthelocalmaximumstationarypointandallpointsofinflectionwiththeircoordinates,correcttotwodecimalplaces. 3marks

–3

–2

–1

1

2

3

–5 –4 –3 –2 –1 O 1 2 3 4 5

y

x

17 2020SPECMATHEXAM2

SECTION B – continuedTURN OVER

Letg(x)=xne−x,wheren ∈ Z.

d. Writedownanexpressionforg″(x). 1mark

e. i. Findthenon-zerovaluesofxforwhichg″(x)=0. 1mark

ii. Completethefollowingtablebystatingthevalue(s)ofnforwhichthegraphofg(x)hasthegivennumberofpointsofinflection. 2marks

Number of points of inflection

Value(s) of n (where n ∈ Z )

0

1

2

3

2020SPECMATHEXAM2 18

SECTION B – Question 4 –continued

Question 4 (14marks)Apilotisperformingatanairshow.Thepositionofheraeroplaneattimetrelativetoafixedorigin

Oisgivenby r iA~ ~( ) sin costt t

= −

+ −

450 1506

400 2006

π π

j , where i~ ~isaunitvectorina

horizontaldirection, j~isaunitvectorverticallyup,displacementcomponentsaremeasuredin

metresandtimetismeasuredinsecondswheret≥0.

a. Findthemaximumspeedoftheaeroplane.Giveyouranswerinms–1. 3marks

b. i. UserA~ ( )t toshowthatthecartesianequationofthepathoftheaeroplaneisgivenby

( ) ( )x y− + − =45022500

40040000

12 2

. 2marks

19 2020 SPECMATH EXAM 2

SECTION B – Question 4 – continuedTURN OVER

ii. Sketch the path of the aeroplane on the axes provided below. Label the position of the aeroplane when t = 0, using coordinates, and use an arrow to show the direction of motion of the aeroplane. 3 marks

y

x2000 400 600 800 1000 1200 1400

800

600

400

200

A friend of the pilot launches an experimental jet-powered drone to take photographs of the air show. The position of the drone at time t relative to the fixed origin is given by

r i jD~ ~ ~( ) ( )t t t t= + − +( )30 402 , where t is in seconds and 0 ≤ t ≤ 40, i~ is a unit vector in the same horizontal direction, j

~ is a unit vector vertically up, and displacement components are measured in

metres.

c. Sketch the path of the drone on the axes provided in part b.ii. Using coordinates, label the points where the path of the drone crosses the path of the aeroplane, correct to the nearest metre. 3 marks

2020SPECMATHEXAM2 20

SECTION B–continued

d. Determinewhetherthedronewillmakecontactwiththeaeroplane.Givereasonsforyouranswer. 3marks

21 2020SPECMATHEXAM2

SECTION B – continuedTURN OVER

CONTINUES OVER PAGE

2020SPECMATHEXAM2 22

SECTION B – Question 5–continued

Question 5 (13marks)Twoobjects,eachofmassmkilograms,areconnectedbyalightinextensiblestringthatpassesoverasmoothpulley,asshownbelow.TheobjectontheplatformisinitiallyatpointAand,whenitisreleased,itmovestowardspointC.ThedistancefrompointAtopointCis10m.Theplatformhasaroughsurfaceand,whenitmovesalongtheplatform,theobjectexperiencesahorizontalforceopposingthemotionofmagnitudeF1newtonsinthesectionABandahorizontalforceopposingthemotionofmagnitudeF2newtonswhenitmovesinthesectionBC.

m

m

A CB

10 m

a. Onthediagramabove,markallforcesthatactoneachobjectoncetheobjectontheplatformhasbeenreleasedandthesystemisinmotion. 2marks

TheforceF1isgivenbyF1 = kmg,k ∈ R+.

b. i. Showthatanexpressionfortheacceleration,inms–2,oftheobjectontheplatform,in

termsofk,asitmovesfrompointAtopointBisgivenbyg k1

2−( ) . 2marks

ii. Thesystemwillonlybeinmotionforcertainvaluesofk.

Findthesevaluesofk. 1mark

23 2020SPECMATHEXAM2

PointBismidwaybetweenpointsAandC.

c. Find,intermsofk,thetimetaken,inseconds,fortheobjectontheplatformtoreachpointB. 2marks

d. Express,intermsofk,thespeedvB ,inms–1,oftheobjectontheplatformwhenitreaches

pointB. 2marks

e. WhentheobjectontheplatformisatpointB,thestringbreaks.Thevelocityoftheobjectat pointBisvB=2.5ms

–1.TheforcethatopposesmotionfrompointBtopointCis F2=0.075mg+0.4mv

2,wherevisthevelocityoftheobjectwhenitisadistanceofxmetresfrompointB.TheobjectontheplatformcomestorestbeforepointC.

Findtheobject’sdistancefrompointCwhenitcomestorest.Giveyouranswerinmetres,correcttotwodecimalplaces. 4marks

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 2020

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2020

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 rulea

Ab

Bc

Csin ( ) 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 2

x xx

=−

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)

SPECMATH EXAM 4

END OF FORMULA SHEET

Calculus

ddx

x nxn n( ) = −1 x dx n x c nn n=

++ ≠ −+∫ 1 1 1

1 ,

ddx

e aeax ax( ) = e dx a e cax ax= +∫ 1

ddx

xxe

log ( )( ) = 1 1x

dx x ce= +∫ logddx

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 xa c a−=

+ >

−∫ sin ,ddx

xx

cos−( ) = −−

12

1

1( ) −

−=

+ >

−∫ 1 02 2 1a x dxxa c acos ,

ddx

xx

tan−( ) =+

12

11

( ) aa x

dx xa c2 21

+=

+

−∫ tan( )

( )( ) ,ax b dx

a nax b c nn n+ =

++ + ≠ −+∫ 1 1 1

1

( ) logax b dxa

ax b ce+ = + +−∫ 1 1

product rule ddx

uv u dvdx

v dudx

( ) = +

quotient rule ddx

uv

v dudx

u dvdx

v

=

−

2

chain rule dydx

dydu

dudx

=

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 12 2 2

1

2

1

2

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

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= = + +d

dtdxdt

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

2020 Specialist Mathematics 2Section A – Multiple-choice questionsSection BFormula sheet