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
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