2019 Specialist Mathematics Written examination 22019 SPECMATH EXAM 2 8 Question 14 A 4 kg mass is...

SPECIALIST MATHEMATICSWritten examination 2

Monday 11 November 2019 Reading time: 3.00 pm to 3.15 pm (15 minutes) Writing time: 3.15 pm to 5.15 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2019

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2019

STUDENT NUMBER

Letter

SECTION A – continued

2019SPECMATHEXAM2 2

Question 1

Thegraphof f x ex

x( ) =

−1doesnothavea

A. horizontalasymptote.B. verticalasymptote.C. localminimum.D. verticalaxisintercept.E. pointofinflection.

Question 2

Theasymptote(s)ofthegraphof f x xx

( ) = +−

2 12 8

hasequation(s)

A. x = 4

B. x=4andy x=

2

C. x=4andy x= +

22

D. x=8andy x=

2

E. x=8andy = 2x + 2

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 2019SPECMATHEXAM2

Question 3

Theimplieddomainofthefunctionwithrule f x x( ) sec= − +

1

4π is

A. R

B. [0,2]

C. R n n Z\ ( ) ,4 14−

∈π

D. R n n Z\ ( ) ,4 14+

∈π

E. R n n Z\ ( ) ,2 12−

∈π

Question 4Theexpressioni1! + i2! + i3! + … + i100!isequaltoA. 0B. 96C. 95+iD. 94+2iE. 98+2i

Question 5

Letz = x + yi,wherex,y ∈ R.TheraysArg( )z − =24π andArg z i− +( ) =( )5 5

6π ,wherez ∈ C,intersecton

thecomplexplaneatapoint(a,b).ThevalueofbisA. – 3B. 2 – 3C. 0D. 3E. 2 + 3


2019SPECMATHEXAM2 4

Question 6

Letz,w ∈ C,whereArg( )z =π2andArg( )w =

π4.

ThevalueofArg zw

5

4

is

A. −π2

B. π2

C. π

D. 52π

E. 72π

Question 7Thelengthofthecurvedefinedbytheparametricequationsx=3sin(t)andy=4cos(t)for0≤t≤πisgivenby

A. 9 162 20

cos ( ) sin ( )t t dt−∫π

B. 9 7 20

+∫ sin ( )t dtπ

C. 1 16 20

+∫ sin ( )t dtπ

D. 3 40

cos( ) sin( )t t dt−( )∫π

E. 3 42 20

cos ( ) sin ( )t t dt+∫π

Question 8

Withasuitablesubstitution, ( )2 1 2 11

5x x dx− +∫ canbeexpressedas

A. 12

32

12

1

5u u du+

∫

B. 232

12

3

11u u du+

∫

C. 2 232

12

1

5u u du−

∫

D. 2 232

12

3

11u u du−

∫

E. 12

232

12

3

11u u du−

∫

SECTION A – continuedTURN OVER

5 2019SPECMATHEXAM2

Question 9

y

xO−2

−8

−6

−4

−2

2

4

6

8

2 4 6 8−4−6−8

Thedifferentialequationthathasthediagramaboveasitsdirectionfieldis

A. dydx

y x= −sin( )

B. dydx

y x= −cos( )

C. dydx

x y= −sin( )

D. dydx y x

=−

1cos( )

E. dydx y x

=−

1sin( )


2019SPECMATHEXAM2 6

Question 10

h

60°

Sandfallsfromachutetoformapileintheshapeofarightcircularconewithsemi-vertexangle60°.Sandisaddedtothepileatarateof1.5m3perminute.Therateatwhichtheheighthmetresofthepileisincreasing,inmetresperminute,whentheheightofthepileis0.5m,correcttotwodecimalplaces,isA. 0.21B. 0.31C. 0.64D. 3.82E. 3.53

Question 11LetpointMhavecoordinates(a,1,–2)andletpointNhavecoordinates(–3,b,–1).

IfthecoordinatesofthemidpointofMN are −

5 3

2, , c anda,bandcarerealconstants,thenthevaluesof

a,bandcarerespectively

A. − −13 2 12

, and

B. − −2 12

3, and

C. − − −7 2 32

, and

D. − − −2 12

3, and

E. − −7 32

, 2 and

7 2019SPECMATHEXAM2

SECTION A–continuedTURN OVER

Question 12Thevectorresoluteof

i j k+ − inthedirectionofm n p

i j k+ + is2 3

i j k− + ,wherem,nandparerealconstants.Thevaluesofm,nandpcanbefoundbysolvingtheequations

A. m m n pm n p

n m n pm n p

p m n pm n

( ) , ( ) ( )+ −+ +

=+ −+ +

= −+ −+ +2 2 2 2 2 2 2 22 3 and

pp2 1=

B. m m n pm n p

n m n pm n p

p m n pm n p

( ) , ( ) ( )+ −+ +

=+ −+ +

=+ −+ +2 2 2 2 2 2 2 21 1 and 22 1= −

C. m n p m n p m n p+ − = + − = − + − = −6 9 3, and

D. m n p m m n p n m n p p+ − = + − = + − = −3 3 3, and

E. m n p m n p m n p+ − = + − = − + − =2 3 3 3 3, and

Question 13Twoforces,

F1and

F2,bothmeasuredinnewtons,actonamassof3kg,producinganaccelerationof3 2

i j ms+ − .

Giventhat

F j1 2= ,theacuteanglebetween

F1and

F2is

A. arccos 13

B. π −

arccos 1

3

C. π6

D. arccos 12 7

E. π −

arccos 1

2 7


2019SPECMATHEXAM2 8

Question 14A4kgmassisheldatrestonasmoothsurface.Itisconnectedbyalightinextensiblestringthatpassesoverasmoothpulleytoa2kgmass,whichinturnisconnectedbythesametypeofstringtoa1kgmass.Thisisshowninthediagrambelow.

4 kg

2 kg

1 kg

Whenthe4kgmassisreleased,thetensioninthestringconnectingthe1kgand2kgmassesisTnewtons.ThevalueofTis

A. 47g

B. 37g

C. g7

D. 67g

E. g

Question 15Aparticleismovingalongthex-axiswithvelocity

v i= u ,whereuisarealconstant.Attimet=0,aforceactsontheparticle,causingittoacceleratewithacceleration

a j=α ,whereαisanegativerealconstant.Whichoneofthefollowingstatementscorrectlydescribesthemotionoftheparticlefort > 0?A. Theparticleslowsdown,stopsmomentarilyandthenbeginstomoveintheoppositedirectiontoits

originalmotion.B. Theparticlecontinuestotravelalongthex-axiswithdecreasingspeed.C. Theparticletravelsparalleltothey-axis.D. Theparticlemovesalongacirculararc.E. Theparticlemovesalongaparabola.

9 2019SPECMATHEXAM2

SECTION A–continuedTURN OVER

Question 16Avariableforceactsonaparticle,causingittomoveinastraightline.Attimetseconds,wheret≥0, itsvelocityvmetrespersecondandpositionxmetresfromtheoriginaresuchthatv = exsin(x).Theaccelerationoftheparticle,inms–2,canbeexpressedas

A. e x xx2 2 12

2sin ( ) sin( )+

B. e x x xx sin sin cos( ) ( ) ( )+( )

C. e x xx sin cos( ) ( )+( )

D. 12

2 2e xx sin ( )

E. excos(x)

Question 17AparticleisheldinequilibriumbythreecoplanarforcesofmagnitudesF1,F2andF3.Theanglesbetweentheseforcesareα,βandγ,asshowninthediagrambelow.

F1

F3

F2

Ifβ = 2α,thenFF

1

2isequalto

A. 12

sin( )α

B. 2sin( )α

C. 12

cosec( )α

D. 12

cos( )α

E. 12

sec( )α

2019SPECMATHEXAM2 10

END OF SECTION A

Question 18Themassesofarandomsampleof36trackathleteshaveameanof65kg.Thestandarddeviationofthemassesofalltrackathletesisknowntobe4kg.A98%confidenceintervalforthemeanofthemassesofalltrackathletes,correcttoonedecimalplace,wouldbeclosesttoA. (51.0,79.0)B. (63.6,66.4)C. (63.3,66.7)D. (63.4,66.6)E. (64.3,65.7)

Question 19XandYareindependentrandomvariableswhereeachhasameanof4andavarianceof9.IftherandomvariableZ = aX + bYhasameanof8andavarianceof90,possiblevaluesofaandbareA. a=1,b=1B. a=4,b = –2C. a=3,b=–1D. a=1,b=3E. a=–2,b = 4

Question 20Therandomnumberfunctionofacalculatorisdesignedtogeneraterandomnumbersthatareuniformlydistributedfrom0to1.Whenworkingproperly,acalculatorgeneratesrandomnumbersfromapopulationwhereµ=0.5andσ=0.2887Whencheckingtherandomnumberfunctionofaparticularcalculator,asampleof100randomnumberswasgeneratedandwasfoundtohaveameanofx = 0 4725.AssumingH0:µ=0.5andH1:µ<0.5,andσ=0.2887,thepvalueforaone-sidedtestisA. 0.0953B. 0.1704C. 0.4621D. 0.8296E. 0.9525

11 2019SPECMATHEXAM2

TURN OVER

CONTINUES OVER PAGE


SECTION B – Question 1–continued

Question 1 (11marks)

Acurveisdefinedparametricallybyx t y t t= + = ∈

sec( ) , tan( ) ,1 0

2 , where π .

a. Showthatthecurvecanberepresentedincartesianformbytheruley x x= −2 2 . 2marks

b. Statethedomainandrangeoftherelationgivenbyy x x= −2 2 . 2marks

c. i. Expressdydxintermsofsin(t). 2marks

ii. Statethelimitingvalueof dydxastapproachest →π

2. 1mark

SECTION B

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


SECTION B – continuedTURN OVER

d. Sketchthecurve y x x= −2 2 ontheaxesbelowforx ∈[2,4],labellingtheendpointswiththeircoordinates. 2marks

y

x−1

−4

−3

−2

−1

1

2

3

4

1 2 3−2−3−4 4O

e. Theportionofthecurvegivenbyy x x= −2 2 forx ∈[2,4]isrotatedaboutthey-axistoformasolidofrevolution.

Writedown,butdonotevaluate,adefiniteintegralintermsoftthatgivesthevolumeofthesolidformed. 2marks


SECTION B – Question 2 –continued

Question 2 (10marks)

a. i. Showthatthesolutionsof2z2 + 4z+5=0,wherez ∈ C,arez i= − ±1 62

. 1mark

ii. Plotthesolutionsof2z2 + 4z+5=0ontheArganddiagrambelow. 1mark

Im(z)

Re(z)O−1

−3

−2

−1

1

2

3

1 2 3−2−3



Let z m n+ = ,wherem,n ∈ R,representthecircleofminimumradiusthatpassesthroughthesolutionsof2z2 + 4z+5=0.

b. i. Findmandn. 2marks

ii. Findthecartesianequationofthecircle z m n+ = . 1mark

iii. SketchthecircleontheArganddiagraminpart a.ii.Interceptswiththecoordinateaxesdonotneedtobecalculatedorlabelled. 1mark

c. Findallvaluesofd,whered ∈ R,forwhichthesolutionsof2z2 + 4z + d=0satisfytherelation z m n+ ≤ . 2marks

d. Allcomplexsolutionsofaz2 + bz + c=0havenon-zerorealandimaginaryparts. Let z p q+ = representthecircleofminimumradiusinthecomplexplanethatpassesthrough

thesesolutions,wherea,b,c,p,q ∈ R.

Findpandqintermsofa,bandc. 2marks


SECTION B – Question 3 –continued

Question 3 (9marks)a. ThegrowthanddecayofaquantityPwithrespecttotimetismodelledbythedifferential

equation

dPdt

kP=

wheret≥0.

i. GiventhatP(a)=randP(b)=s,wherePisafunctionoft,showthatka b

rse=

−

1 log . 2marks

ii. Specifythecondition(s)forwhichk>0. 2marks



b. ThegrowthofanotherquantityQwithrespecttotimetismodelledbythedifferentialequation

dQdt

et Q= −

wheret≥0andQ=1whent=0.

i. Expressthisdifferentialequationintheform f Q dQ h t dt( ) ( )=∫ ∫ . 1mark

ii. Hence,showthatQ=loge(et + e–1). 2marks

iii. ShowthatthegraphofQasafunctionoftdoesnothaveapointofinflection. 2marks



Question 4 (11marks)ThebaseofapyramidistheparallelogramABCDwithverticesatpointsA(2,–1,3),B(4,–2,1),C(a,b,c)andD(4,3,–1).Theapex(top)ofthepyramidislocatedatP(4,–4,9).

a. Findthevaluesofa,bandc. 2marks

b. FindthecosineoftheanglebetweenthevectorsAB→

andAD→

. 2marks

c. Findtheareaofthebaseofthepyramid. 2marks



d. Showthat6 2 5

i j k+ + isperpendiculartobothAB→

andAD→

,andhencefindaunitvectorthatisperpendiculartothebaseofthepyramid. 3marks

e. Findthevolumeofthepyramid. 2marks

2019 SPECMATH EXAM 2 20

SECTION B – Question 5 – continued

Question 5 (10 marks)A mass of m1 kilograms is initially held at rest near the bottom of a smooth plane inclined at θ degrees to the horizontal. It is connected to a mass of m2 kilograms by a light inextensible string parallel to the plane, which passes over a smooth pulley at the end of the plane. The mass m2 is 2 m above the horizontal floor.The situation is shown in the diagram below.

m1

m2

θ°2 m

a. After the mass m1 is released, the following forces, measured in newtons, act on the system:• weight forces W1 and W2• the normal reaction force N• the tension in the string T

On the diagram above, show and clearly label the forces acting on each of the masses. 1 mark

b. If the system remains in equilibrium after the mass m1 is released, show that sin( )θ =mm

2

1. 1 mark



c. Afterthemassm1isreleased,themassm2fallstothefloor.

i. Forwhatvaluesofθwillthisoccur?Expressyouranswerasaninequalityintermsofm1 andm2. 1mark

ii. Findthemagnitudeoftheacceleration,inms–2,ofthesystemafterthemassm1 isreleasedandbeforethemassm2hitsthefloor.Expressyouranswerintermsof m1,m2andθ. 2marks

d. Afterthemassm1isreleased,itmovesuptheplane.

Findthemaximumdistance,inmetres,thatthemassm1willmoveuptheplaneif

m m1 22 14

= = and sin( )θ . 5marks



Question 6 (9marks)Acompanyproducespacketsofnoodles.Itisknownfrompastexperiencethatthemassofapacketofnoodlesproducedbyoneofthecompany’smachinesisnormallydistributedwithameanof 375gramsandastandarddeviationof15grams.Tochecktheoperationofthemachineaftersomerepairs,thecompany’squalitycontrolemployeesselecttwoindependentrandomsamplesof50packetsandcalculatethemeanmassofthe 50packetsforeachrandomsample.

a. Assumethatthemachineisworkingproperly.Findtheprobabilitythatatleastonerandomsamplewillhaveameanmassbetween370gramsand375grams.Giveyouranswercorrecttothreedecimalplaces. 2marks

b. Assumethatthemachineisworkingproperly.Findtheprobabilitythatthemeansofthetworandomsamplesdifferbylessthan2grams.Giveyouranswercorrecttothreedecimalplaces. 3marks

Totestwhetherthemachineisworkingproperlyaftertherepairsandisstillproducingpacketswithameanmassof375grams,thetworandomsamplesarecombinedandthemeanmassofthe100packetsisfoundtobe372grams.Assumethatthestandarddeviationofthemassofthepacketsproducedisstill15grams.Atwo-tailedtestatthe5%levelofsignificanceistobecarriedout.

c. WritedownsuitablehypothesesH0andH1forthistest. 1mark

d. Findthepvalueforthetest,correcttothreedecimalplaces. 1mark


e. Doesthemeanmassofthesampleof100packetssuggestthatthemachineisworkingproperlyatthe5%levelofsignificanceforatwo-tailedtest?Justifyyouranswer. 1mark

f. Whatisthesmallestvalueofthemeanmassofthesampleof100packetsforH0tobenot rejected?Giveyouranswercorrecttoonedecimalplace. 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 2019

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2019

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 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 sn

x z sn

− +

,

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 ,

ddx

e aeax ax( ) = e dxa

e 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 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 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

2019 Specialist Mathematics Written examination 22019 SPECMATH EXAM 2 8 Question 14 A 4 kg mass is...

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