SPECIALIST MATHEMATICSWritten examination 1

Friday 10 November 2017 Reading time: 9.00 am to 9.15 am (15 minutes) Writing time: 9.15 am to 10.15 am (1 hour)

QUESTION AND ANSWER BOOK

Structure of bookNumber of questions

Number of questions to be answered

Number of marks

10 10 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2017

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2017

STUDENT NUMBER

Letter

2017SPECMATHEXAM1 2

THIS PAGE IS BLANK

3 2017SPECMATHEXAM1

TURN OVER

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

Question 1 (3marks)Findtheequationofthetangenttothecurvegivenby3xy2+2y = xatthepoint(1,–1).

Question 2 (4marks)

Find 11 21

3

x xdx

+( )∫ , expressingyouranswerintheformlog ,eab

whereaandbarepositiveintegers.

2017SPECMATHEXAM1 4

Question 3 (3marks)Letz3 + az2 + 6z + a=0,z ∈ C,whereaisarealconstant.

Giventhatz = 1 – iisasolutiontotheequation,findallothersolutions.

Question 4 (3marks)Thevolumeofsoftdrinkdispensedbyamachineintobottlesvariesnormallywithameanof298mLand astandarddeviationof3mL.Thesoftdrinkissoldinpacksoffourbottles.

Findtheapproximateprobabilitythatthemeanvolumeofsoftdrinkperbottleinarandomlyselected four-bottlepackislessthan295mL.Giveyouranswercorrecttothreedecimalplaces.

5 2017SPECMATHEXAM1

TURN OVER

Question 5 (4marks)Relativetoafixedorigin,thepointsB,CandDaredefinedrespectivelybythepositionvectors

b i j k c i j k and d i j,= − + = − + = −2 2 2, a whereaisarealconstant.

GiventhatthemagnitudeofangleBCDisπ3

,finda.

2017SPECMATHEXAM1 6

Question 6 (3marks)

Let fx

x( ) .( )

=1

arcsin

Find f ′(x)andstatethelargestsetofvaluesofxforwhich f ′(x)isdefined.

Question 7 (4marks)

Thepositionvectorofaparticlemovingalongacurveattimetisgivenby

r i j( ) cos ( ) sin ( ) , .t t t t= + ≤ ≤3 3 04π

Findthelengthofthepaththattheparticletravelsalongthecurvefromt=0tot = π4

.

7 2017SPECMATHEXAM1

TURN OVER

CONTINUES OVER PAGE

2017SPECMATHEXAM1 8

Question 8 –continued

Question 8 (4marks)

Aslopefieldrepresentingthedifferentialequationdydx

xy

=−+1 2

isshownbelow.

y

x1O 2–2 –1

2

–1

–2

1

a. Sketchthesolutioncurveofthedifferentialequationcorrespondingtothecondition y(–1)=1ontheslopefieldaboveand,hence,estimatethepositivevalueofxwheny=0. Giveyouranswercorrecttoonedecimalplace. 2marks

9 2017SPECMATHEXAM1

TURN OVER

b. Solvethedifferentialequation dydx

xy

=−+1 2

withtheconditiony(–1)=1.Expressyouranswer

intheformay3 + by + cx2 + d =0,wherea,b,canddareintegers. 2marks

2017SPECMATHEXAM1 10

Question 9 (5marks)Aparticleofmass2kgwithinitialvelocity3 2

i j+ ms–1experiencesaconstantforcefor 10seconds.Theparticle’svelocityattheendofthe10-secondperiodis 43 18

i j− ms–1.

a. Findthemagnitudeoftheconstantforceinnewtons. 2marks

b. Findthedisplacementoftheparticlefromitsinitialpositionafter10seconds. 3marks

11 2017SPECMATHEXAM1

END OF QUESTION AND ANSWER BOOK

Question 10 (7marks)

a. Showthat ddx

x xa

xa

x

a xarccos arccos ,

=

−

−2 2wherea>0. 1mark

b. Statethemaximaldomainandtherangeof f x x( ) arccos .=

2

2marks

c. Findthevolumeofthesolidofrevolutiongeneratedwhentheregionboundedbythegraphofy = f(x),andthelinesx=–2andy=0,isrotatedaboutthex-axis. 4marks

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 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 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)

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 zsnx z s

n− +

,

distribution of sample mean Xmean E X( ) = µvariance var X

n( ) =σ 2

SPECMATH EXAM 4

END OF FORMULA SHEET

Calculus

ddx

x nxn n( ) = −1 x dx n x c nn n=

++ ≠ −+∫ 1 1 1

1 ,

ddxe aeax ax( ) = e dx a e c

ax ax= +∫ 1

ddx

xxe

log ( )( ) = 1 1xdx 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 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 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 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= = + +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

2017 Specialist Mathematics 1InstructionsFormula sheet