2016 specialist mathematics written examination 2€¦ · specialist mathematics written...
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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