Download pdf - 2018 Specialist Mathematics Written examination 1

SPECIALIST MATHEMATICSWritten examination 1

Tuesday 5 June 2018 Reading time: 2.00 pm to 2.15 pm (15 minutes) Writing time: 2.15 pm to 3.15 pm (1 hour)

QUESTION AND ANSWER BOOK

Structure of bookNumber of questions

Number of questions to be answered

Number of marks

9 9 40

• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpenersandrulers.

• StudentsareNOTpermittedtobringintotheexaminationroom:anytechnology(calculatorsorsoftware),notesofanykind,blanksheetsofpaperand/orcorrectionfluid/tape.

Materials supplied• Questionandanswerbookof11pages• Formulasheet• Workingspaceisprovidedthroughoutthebook.

Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.• AllwrittenresponsesmustbeinEnglish.

At the end of the examination• Youmaykeeptheformulasheet.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2018

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2018

STUDENT NUMBER

Letter

2018SPECMATHEXAM1(NHT) 2

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3 2018SPECMATHEXAM1(NHT)

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InstructionsAnswerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8

Question 1 (3marks)Alightinextensiblestringhangsoverafrictionlesspulleyconnectingmassesof3kgand7kg, asshownbelow.

3 kg

7 kg

a. Drawalloftheforcesactingonthetwomassesonthediagramabove. 1mark

b. Calculatethetensioninthestring. 2marks


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Question 2 (3marks)Let

a i j k= − +3 2 m and

b i j k ,= − +2 3 wherem ∈ R.

Findthevalue(s)ofmsuchthatthemagnitudeofthevectorresoluteof

a parallelto

b isequalto 14.

Question 3 (3marks)

Findsin(t)giventhat t =

+

arccos arctan .12

1334


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Question 4 (4marks)Throughoutthisquestion,useanintegermultipleofstandarddeviationsincalculations.Thestandarddeviationofallscoresonaparticulartestis21.0

a. Fromtheresultsofarandomsampleofnstudents,a95%confidenceintervalforthemeanscoreforallstudentswascalculatedtobe(44.7,51.7).

Calculatethemeanscoreandthesizeofthisrandomsample. 2marks

b. Determinethesizeofanotherrandomsampleforwhichtheendpointsofthe95%confidenceintervalforthepopulationmeanoftheparticulartestwouldbe1.0eithersideofthesamplemean. 2marks


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Question 5 (4marks)

Evaluate 12 52

1

2 3 1

x xdx

+ +

−

∫ .

Question 6 (4marks)Giventhaty=(x–1)e2xisasolutiontothedifferentialequationa d y

dxb dydx

y2

2 + = ,findthevaluesofaandb,whereaandbarerealconstants.


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Question 7 (4marks)

a. Find ddx

x( ) .1 212−

2marks

b. Hence,findthelengthofthecurvespecifiedby y x= −1 2 from x = 12to x = 3

2. Give

youranswerintheformkπ,k ∈ R. 2marks


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Question 8–continued

Question 8 (6marks)Acircleinthecomplexplaneisgivenbytherelation|z–1–i |=2,z ∈ C.

a. SketchthecircleontheArganddiagrambelow. 1mark

4

3

2

1

O

–1

–2

–3

–4

–4 –3 –2 –1 1 2 3 4

Im(z)

Re(z)

b. i. Writetheequationofthecircleintheform(x – a)2+(y – b)2 = candshowthatthe

gradientofatangenttothecirclecanbeexpressedas dydx

xy

=−−

11

. 2marks

ii. Findthegradientofthetangenttothecirclewherex=2inthefirstquadrantofthecomplexplane. 1mark


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c. FindtheequationsofallraysthatareperpendiculartothecircleintheformArg(z)=α. 2marks


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Question 9–continued

Question 9 (9marks)a. i. Giventhatcot(2θ)=a,showthattan2(θ)+2atan(θ)–1=0. 2marks

ii. Showthattan( ) .θ = − ± +a a2 1 1mark

iii. Hence,showthattan ,π12

2 3

= − giventhatcot( ) ,2 3θ = whereθ ∈(0,π). 1mark

b. Findthegradientofthetangenttothecurve y=tan(θ) atθ π=

12. 2marks


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END OF QUESTION AND ANSWER BOOK

c. Asolidofrevolutionisformedbyrotatingtheregionbetweenthegraphofy=tan(θ),

the horizontalaxis,andthelinesθ π=

12andθ π

=3aboutthehorizontalaxis.

Findthevolumeofthesolidofrevolution. 3marks

SPECIALIST MATHEMATICS

Written examination 1

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 2018

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2018

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