SPECIALIST MATHEMATICSWritten examination 1

Friday 8 November 2019 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• Questionandanswerbookof12pages• 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2019

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2019

STUDENT NUMBER

Letter

2019SPECMATHEXAM1 2

THIS PAGE IS BLANK

3 2019SPECMATHEXAM1

TURN OVER

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

Question 1 (4marks)

Solvethedifferentialequationdydx

yee

x

x=+

21

2

2 giventhaty(0)=π.

Question 2 (3marks)

Findallvaluesofxforwhich| | x x− = +4

27.

2019SPECMATHEXAM1 4

Question 3 (3marks)Amachineproduceschocolateintheformofacontinuouscylinderofradius0.5cm.Smallercylindricalpiecesarecutparalleltoitsend,asshowninthediagrambelow.Thelengthsofthepiecesvarywithameanof3cmandastandarddeviationof0.1cm.

a. Findtheexpectedvolumeofapieceofchocolateincm3. 1mark

b. Findthevarianceofthevolumeofapieceofchocolateincm6. 1mark

c. Findtheexpectedsurfaceareaofapieceofchocolateincm2. 1mark

5 2019SPECMATHEXAM1

TURN OVER

Question 4 (3marks)ThepositionvectorsoftwoparticlesAandBattimetsecondsaftertheyhavestartedmovingaregivenby

r i j and r iA Bt t a t t t t t( ) ( ) arccos~ ~ ~= −( ) + +

= −( ) + 2 31

3 2

j

~respectively,whereaisarealconstant

and0≤t≤2.

Findthevalueofaiftheparticlescollideaftertheyhavestartedmoving.

2019SPECMATHEXAM1 6

Question 5 (6marks)Thegraphof f (x)=cos2(x)+cos(x)+1overthedomain0≤x≤2πisshownbelow.

2 23

252 3

−2O

y

x

4

3

2

1

–1

a. i. Find f ′(x). 1mark

ii. Hence,findthecoordinatesoftheturningpointsofthegraphintheinterval(0,2π). 2marks

b. Sketchthegraphofyf x

=1( )

onthesetofaxesabove.Clearlylabeltheturningpointsand

endpointsofthisgraphwiththeircoordinates. 3marks

7 2019SPECMATHEXAM1

TURN OVER

Question 6 (3marks)Findthevalueofdforwhichthevectors

a i j k b i j k= − + = − + −2 3 4 2 4 8, and

c i j k= − + +6 2 d are linearly dependent.

2019SPECMATHEXAM1 8

Question 7 (5marks)

a. Showthat3 3 2 36

− = −

i cis π . 1mark

b. Find 3 33

−( )i ,expressingyouranswerintheformx+iy,wherex,y ∈ R. 2marks

c. Findtheintegervaluesofnforwhich 3 3−( )i nisreal. 1mark

d. Findtheintegervaluesofnforwhich 3 3−( ) =i ain

,whereaisarealnumber. 1mark

9 2019SPECMATHEXAM1

Question 8 (4marks)

Findthevolumeofthesolidofrevolutionformedwhenthegraphofy = 1 21 2++

xxisrotatedaboutthex-axis

overtheinterval[0,1].

TURN OVER

2019SPECMATHEXAM1 10

Question 9–continued

Question 9 (4marks)a. Alightinextensiblestringisconnectedateachendtoahorizontalceiling.Amassof

mkilogramshangsinequilibriumfromasmoothringonthestring,asshowninthediagrambelow.Thestringmakesanangleαwiththeceiling.

α

m

α

Expressthetension,Tnewtons,inthestringintermsofm,gandα. 1mark

11 2019SPECMATHEXAM1

TURN OVER

b. Adifferentlightinextensiblestringisconnectedateachendtoahorizontalceiling.Amassof mkilogramshangsfromasmoothringonthestring.AhorizontalforceofFnewtonsisappliedtothering.Thetensioninthestringhasaconstantmagnitudeandthesystemisinequilibrium.Atoneendthestringmakesanangleβ withtheceilingandattheotherendthestringmakesanangle2β withtheceiling,asshowninthediagrambelow.

m

2

F

ShowthatF = mg 1−

cos( )sin ( )

ββ

. 3marks

2019SPECMATHEXAM1 12

Question 10 (5marks)

Finddydxatthepoint

π π6 3

,

forthecurvedefinedbytherelationsin( ) cos( )x y xy2 2 3 2

+ =π

.

Giveyouranswerintheformππ−

( )a b

a b + ,wherea,b ∈ Z+.

END OF QUESTION AND ANSWER BOOK

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 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 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