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SPECIALIST MATHEMATICSWritten examination 2
Friday 9 June 2017 Reading time: 10.00 am to 10.15 am (15 minutes) Writing time: 10.15 am to 12.15 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.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2017
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2017
STUDENT NUMBER
Letter
SECTION A – continued
2017SPECMATHEXAM2(NHT) 2
Question 1Thenumberofasymptotesofthegraphofthefunctionwithrule f x
x xx x
( ) = − ++ −
3
27 53 4
isA. 0B. 1C. 2D. 3E. 4
Question 2Theequationx2 + y2 + 2ky+4=0,wherekisarealconstant,willrepresentacircleonlyifA. k > 2B. k < –2C. k≠±2D. k<–2ork > 2E. –2 < k < 2
Question 3Forthefunctionf:R → R,f(x)=karctan(ax – b)+c,wherek>0,c>0anda,b ∈ R,f(x)>0if
A. c k<
π2
B. c k≥
π2
C. x ba
>
D. c k+ >π2
E. c ≥ π2
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 2017SPECMATHEXAM2(NHT)
Question 4Ifsin(θ + ϕ)=aandsin(θ – ϕ)=b,thensin(θ)cos(ϕ)isequaltoA. ab
B. a b2 2+
C. ab
D. a b2 2−
E. a b+
2
Question 5GiventhatA,B,CandDarenon-zerorationalnumbers,theexpression 3 1
2 2x
x x+−( )
canberepresentedinpartialfractionformas
A. Ax
Bx
+−( )2
B. Ax
Bx
+−( )2 2
C. Ax
Bx
Cx
+−
+−( ) ( )2 2 2
D. Ax
Bx
Cx
+ +−2 2( )
E. Ax
Bxx
Cx Dx
+−
++−( ) ( )2 2 2
SECTION A – continued
2017SPECMATHEXAM2(NHT) 4
Question 6
Re(z)
Im(z)
–3 –2 –1 0 1 2 3
3
2
1
–1
–2
–3
S
TherelationthatdefinesthelineSaboveisA. z z i+ = +2 2
B. Arg( )z =34π
C. z z i− = +2 2
D. Im( )z =
+ −
Arg Arg3
4 4π π
E. z z i− = −2 2
SECTION A – continuedTURN OVER
5 2017SPECMATHEXAM2(NHT)
Question 7
sin ( )cos ( )3 2
4
3
x x dx( )∫π
π
isequivalentto
A. u u du u x4 2
12
12
−( ) =∫ where cos( )
B. − −( ) =∫ u u du u x2 4
12
12
where cos( )
C. − −( ) =∫ u u du u x2 4
4
3
π
π
where sin( )
D. u u du u x2 4
4
3
−( ) =∫π
π
where sin( )
E. − −( ) =∫ u u du u x2 4
12
12
where sin( )
SECTION A – continued
2017SPECMATHEXAM2(NHT) 6
Question 8
x
y
2
1
–1
–1 10
Thedifferentialequationthatbestrepresentsthedirectionfieldaboveis
A. dydx
x y= − 2
B. dydx
y x= −
C. dydx
y x= −2 2
D. dydx
y x= −2
E. dydx
y x= +
SECTION A – continuedTURN OVER
7 2017SPECMATHEXAM2(NHT)
Question 9ThegradientofthetangenttoacurveatanypointP(x,y)ishalfthegradientofthelinesegmentjoining PandthepointQ(–1,1).Thecoordinatesofpointsonthecurvesatisfythedifferentialequation
A. dydx
yx
=+−1
2 1( )
B. dydx
yx
=−+
2 11
( )
C. dydx
xy
=−+1
2 1( )
D. dydx
xy
=−+
2 11
( )
E. dydx
yx
=−+1
2 1( )
Question 10
Asolutiontothedifferentialequationdydx
x y x yex y=
+ − −+
cos( ) cos( )canbeobtainedfrom
A. eydy x
edx
y
xsin ( )sin ( )∫ ∫= −2
B. eydy
edx
y
xcos( )∫ ∫= 2
C. eydy x
edx
y
xcos( )cos( )∫ ∫= −2
D. eydy e x dx
yx
−−∫ ∫=sin ( )sin ( )2
E. eydy x
edx
y
xcos( )sin ( )∫ ∫= 2
Question 11
Twoparticleshavepositionsgivenby
r i j123 4= −( ) + +( )t t b and
r i j ,22 25 1= + −( )t t wheret≥0andbis
arealconstant.Theparticleswillcollideifthevalueofbis
A. 2 33
−
B. 3 1−
C. 2 33
+
D. − −2 33
E. − −3 1
SECTION A – continued
2017SPECMATHEXAM2(NHT) 8
Question 12If
u i j k= + −3 6 2 and
v i j k ,= + −2 2 thenthevectorresoluteof
u inthedirectionof
v is
A. 2073 6 2
i j k+ −( )
B. 2092 2
i j k+ −( )
C. 20493 6 2
i j k+ −( )
D. 2032 2
i j k+ −( )
E. 373 6 2
i j k+ −( )
Question 13
D
A B
C
O
LetOABCDbearightsquarepyramidwhere� � � �a b c and d= = = =OA OB OC OD, , .
AnequationcorrectlyrelatingthesevectorsisA.
a c b d+ = +
B.
a c . d c−( ) −( ) = 0C.
a d b c+ = +
D.
a d . c b−( ) −( ) = 0E.
a b c d+ = +
Question 14Giventhatthevectors
a i j k b i j k and c i j k= + − = − + = − +, 2 2 4λ arelinearly dependent,thevalue ofλisA. –10B. –8C. 2D. 4E. 8
SECTION A – continuedTURN OVER
9 2017SPECMATHEXAM2(NHT)
Question 15 Aparticleofmass2kghasaninitialvelocityof
i j− 6 ms–1.Afterachangeofmomentumof6i j
− 2 kgms–1,theparticle’svelocity,inms–1,isA. 3i j
−
B. 2i j
−12
C. 4i j
− 7
D. 2i j
+ 5
E. 11i j
+ 2
Question 16ApersonofmassMkgcarryingabagofmassmkgisstandinginaliftthatisacceleratingdownwards at ams–2.TheforceoftheliftflooractingonthepersonhasmagnitudeA. Mg + mgB. Mg+(M + m)aC. Mg–(M + m)aD. (M + m)(g + a)E. (M + m)(g – a)
Question 17Theacceleration,ams–2,ofaparticlemovinginastraightlineisgivenbya = v2+1,wherevisthevelocityoftheparticleatanytimet.TheinitialvelocityoftheparticlewhenatoriginOis2ms–1.ThedisplacementoftheparticlefromOwhenitsvelocityis3ms–1isA. loge(2)
B. 12
103
loge
C. 12
2log ( )e
D. 12
52
loge
E. loge45
2017SPECMATHEXAM2(NHT) 10
END OF SECTION A
Question 18Xisarandomvariablewithameanof5andastandarddeviationof4,andYisarandomvariablewithameanof3andastandarddeviationof2.IfXandYareindependentrandomvariablesandZ = X – 2Y,thenZwillhavemeanμandstandarddeviationσgivenbyA. μ=–1,σ = 0
B. μ=–1,σ = 4 2
C. μ=2,σ = 8
D. μ=2,σ = 4 2
E. μ=–1,σ = 2 6
Question 19Thepetrolconsumptionofaparticularmodelofcarisnormallydistributedwithameanof 12L/100kmandastandarddeviationof2L/100km.Theprobabilitythattheaveragepetrolconsumptionof16suchcarsexceeds13L/100kmisclosesttoA. 0.0104B. 0.0193C. 0.0228D. 0.3085E. 0.3648
Question 20Themassofsuspendedmatterintheairinaparticularlocalityisnormallydistributedwithameanof μmicrogramspercubicmetreandastandarddeviationofσ=8microgramspercubicmetre.Themeanof 100randomlyselectedairsampleswasfoundtobe40microgramspercubicmetre.Basedonthis,a90%confidenceintervalforμ,correcttotwodecimalplaces,isA. (38.68,41.32)B. (26.84,53.16)C. (38.43,41.57)D. (24.32,55.68)E. (37.93,42.06)
11 2017SPECMATHEXAM2(NHT)
SECTION B – Question 1–continuedTURN OVER
Question 1 (12marks)
a. i. Useanappropriatedoubleangleformulawith t =
tan 5
12π todeduceaquadratic
equationoftheformt2 + bt + c=0,wherebandcarerealvalues. 2marks
ii. Henceshowthat tan .512
2 3π
= + 1mark
SECTION B
Instructions for Section BAnswerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8
2017SPECMATHEXAM2(NHT) 12
SECTION B – Question 1–continued
Consider f R f x x: , , ( ) arctan3 6 3 33 6
+
→ =
−
π .
b. Sketchthegraphoffontheaxesbelow,labellingtheendpointswiththeircoordinates. 3marks
1
0.5
–0.5
–10 –5 5 10
y
xO
c. Theregionbetweenthegraphoffandthey-axisisrotatedaboutthey-axistoformasolidofrevolution.
i. Writedownadefiniteintegralintermsofythatgivesthevolumeofthesolidformed. 2marks
ii. Findthevolumeofthesolid,correcttothenearestinteger. 1mark
13 2017SPECMATHEXAM2(NHT)
SECTION B – continuedTURN OVER
d. Afishpondthathasashapeapproximatelylikethatofthesolidofrevolutioninpart c.isbeingfilledwithwater.Whenthedepthishmetres,thevolume,Vm3,ofwaterinthepondisgivenby
V h h= +
− −tan π6
33
Ifwaterisflowingintothepondatarateof0.03m3perminute,findtherateatwhichthedepthisincreasingwhenthedepthis0.6m.Giveyouranswerinmetresperminute,correcttothreedecimalplaces. 3marks
2017SPECMATHEXAM2(NHT) 14
SECTION B – Question 2–continued
Question 2 (11marks)Onerootofaquadraticequationwithrealcoefficientsis 3 + i.
a. i. Writedowntheotherrootofthequadraticequation. 1mark
ii. Hencedeterminethequadraticequation,writingitintheformz2 + bz + c=0. 2marks
b. Plotandlabeltherootsof z z z3 22 3 4 0− + = ontheArganddiagrambelow. 3marks
Im(z)
Re(z)
3
2
1
–1
–2
–3
O–3 –2 –1 1 2 3
15 2017SPECMATHEXAM2(NHT)
SECTION B – continuedTURN OVER
c. Findtheequationofthelinethatistheperpendicularbisectorofthelinesegmentjoiningtheoriginandthepoint 3 + i.Expressyouranswerintheformy = mx + c. 2marks
d. Thethreerootsplottedinpart b.lieonacircle.
Findtheequationofthiscircle,expressingitintheform z − =α β,whereα,β ∈ R. 3marks
2017SPECMATHEXAM2(NHT) 16
SECTION B – Question 3–continued
Question 3 (12marks)BacteriaarespreadingoveraPetridishataratemodelledbythedifferentialequation
dPdt
P P= −( )21 ,0<P<1
wherePistheproportionofthedishcoveredafterthours.
a. i. Express21P P−( ) inpartialfractionform. 1mark
ii. Henceshowbyintegrationthat t c PPe
−=
−
2 1
log , wherecisaconstantofintegration. 2marks
iii. IfhalfofthePetridishiscoveredbythebacteriaatt=0,expressPintermsoft. 2marks
17 2017SPECMATHEXAM2(NHT)
SECTION B – continuedTURN OVER
Afteronehour,atoxinisaddedtothePetridish,whichharmsthebacteriaandreducestheirrateofgrowth.Thedifferentialequationthatmodelstherateofgrowthisnow
dPdt
P P P= −( ) −21
20 for t≥1
b. FindthelimitingvalueofP,whichisthemaximumpossibleproportionofthePetridishthatcannowbecoveredbythebacteria.Giveyouranswercorrecttothreedecimalplaces. 2marks
c. Thetotaltime,Thours,measuredfromtimet=0,neededforthebacteriatocover80%ofthePetridishisgivenby
T P P P dP s
q
r
= −( ) −
+
1
21
20
whereq,rands ∈ R.
Findthevaluesofq,rands,givingthevalueofqcorrecttotwodecimalplaces. 2marks
d. GiventhatP=0.75whent =3,useEuler’smethodwithastepsizeof0.5toestimatethevalueofPwhent=3.5.Giveyouranswercorrecttothreedecimalplaces. 3marks
2017SPECMATHEXAM2(NHT) 18
SECTION B – Question 4–continued
Question 4 (8marks)Acricketerhitsaballattimet=0secondsfromanoriginOatgroundlevelacrossalevelplayingfield.Thepositionvector
r( )t ,fromO,oftheballaftertsecondsisgivenby
r i j( ) .t t t t= + −( )15 15 3 4 9 2 ,where
i isaunitvectorintheforwarddirection,
j isaunitvector
verticallyupanddisplacementcomponentsaremeasuredinmetres.
a. Findtheinitialvelocityoftheballandtheinitialangle,indegrees,ofitstrajectorytothehorizontal. 2marks
b. Findthemaximumheightreachedbytheball,givingyouranswerinmetres,correcttotwodecimalplaces. 2marks
c. Findthetimeofflightoftheball.Giveyouranswerinseconds,correcttothreedecimalplaces. 1mark
19 2017SPECMATHEXAM2(NHT)
SECTION B – continuedTURN OVER
d. Findtherangeoftheballinmetres,correcttoonedecimalplace. 1mark
e. Afielder,morethan40mfromO,catchestheballataheightof2mabovetheground.
HowfarhorizontallyfromOisthefielderwhentheballiscaught?Giveyouranswer inmetres,correcttoonedecimalplace. 2marks
2017SPECMATHEXAM2(NHT) 20
SECTION B – Question 5–continued
Question 5 (10marks)A5kgmassisinitiallyheldatrestonasmoothplanethatisinclinedat30°tothehorizontal.Themassisconnectedbyalightinextensiblestringpassingoverasmoothpulleytoa3kgmass,whichinturnisconnectedtoa2kgmass.The5kgmassisreleasedfromrestandallowedtoaccelerateuptheplane.Takeaccelerationtobepositiveinthedirectionsindicated.
positiveacceleration
5 kg
T1
T1
3 kg
2 kg
T230°
positive acceleration
a. Writedownanequationofmotion,inthedirectionofmotion,foreachmass. 3marks
b. Showthattheaccelerationofthe5kgmassisg4 ms–2. 1mark
21 2017 SPECMATH EXAM 2 (NHT)
SECTION B – continuedTURN OVER
c. Find the tensions T1 and T2 in the string in terms of g. 2 marks
d. Find the momentum of the 5 kg mass, in kg ms–1, after it has moved 2 m up the plane, giving your answer in terms of g. 2 marks
e. A resistance force R acting parallel to the inclined plane is added to hold the system in equilibrium, as shown in the diagram below.
5 kg
T1
T1
3 kg
2 kg
T230°
R
Find the magnitude of R in terms of g. 2 marks
2017SPECMATHEXAM2(NHT) 22
SECTION B – Question 6–continued
Question 6 (7marks)Abankclaimsthattheamountitlendsforhousingisnormallydistributedwithameanof$400000andastandarddeviationof$30000.Aconsumerorganisationbelievesthattheaverageloanamountishigherthanthebankclaims.Tocheckthis,theconsumerorganisationexaminesarandomsampleof25loansandfindsthesamplemeantobe$412000.
a. Writedownthetwohypothesesthatwouldbeusedtoundertakeaone-sidedtest. 1mark
b. Writedownanexpressionforthepvalueforthistestandevaluateittofourdecimalplaces. 2marks
c. Statewithareasonwhetherthebank’sclaimshouldberejectedatthe5%levelofsignificance. 1mark
d. Whatisthelargestvalueofthesamplemeanthatcouldbeobservedbeforethebank’s claimwasrejectedatthe5%levelofsignificance?Giveyouranswercorrecttothenearest 10dollars. 1mark
23 2017SPECMATHEXAM2(NHT)
e. Iftheaverageloanmadebythebankisactually$415000andnot$400000asoriginallyclaimed,whatistheprobabilitythatarandomselectionof25loanshasasamplemeanthatisatmost$410000?Giveyouranswercorrecttothreedecimalplaces. 2marks
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 2017
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2017
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