SPECIALIST MATHEMATICSWritten examination 1

Thursday 19 November 2020 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

9 9 40

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

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

Materials supplied• Questionandanswerbookof10pages• 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2020

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2020

STUDENT NUMBER

Letter

2020SPECMATHEXAM1 2

THIS PAGE IS BLANK

3 2020SPECMATHEXAM1

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

Question 1 (5marks)A2kgmassisinitiallyatrestonasmoothhorizontalsurface.Themassisthenactedonbytwoconstantforcesthatcausethemasstomovehorizontally.Oneforcehasmagnitude10Nandactsinadirection60°upwardsfromthehorizontal,andtheotherforcehasmagnitude5Nandactsinadirection30°upwardsfromthehorizontal,asshowninthediagrambelow.

30° 60°

5 N 10 N

a. Findthenormalreactionforce,innewtons,thatthesurfaceexertsonthemass. 2marks

b. Findtheaccelerationofthemass,inms−2,afteritbeginstomove. 2marks

c. Findhowfarthemasstravels,inmetres,duringthefirstfoursecondsofmotion. 1mark

2020SPECMATHEXAM1 4

Question 2 (4marks)

Evaluate111

0 +−−∫xxdx .Giveyouranswerintheforma b c+ ,wherea,b,c ∈ R.

Question 3 (3marks)

Findthecuberootsof 12

12

− i.Expressyouranswersinpolarformusingprincipalvaluesoftheargument.

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Question 4 (4marks)

Solvetheinequality3 14

− >−

xx

forx,expressingyouranswerinintervalnotation.

2020SPECMATHEXAM1 6

Question 5 (4marks)Leta i j k and b i j k~ ~ ~ ~ ~ ~ ~ ~= − + = + −2 3 m ,wheremisaninteger.

Thevectorresoluteof a~ inthedirectionofb~ is − + −

1118

i j k~ ~ ~m .

a. Findthevalueofm. 3marks

b. Findthecomponentofa~ thatisperpendiculartob~ . 1mark

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Question 6 (5marks)Letf(x)=arctan(3x−6)+π.

a. Showthatf ′f xx x

' ( ) =− +

39 36 372

. 1mark

b. Hence,showthatthegraphoffhasapointofinflectionatx =2. 2marks

c. Sketchthegraphofy = f(x)ontheaxesprovidedbelow.Labelanyasymptoteswiththeirequationsandthepointofinflectionwithitscoordinates. 2marks

y

xO

32π

π

2π

2π

−

32π

−

π−

–6 –4 –2 2 4 6

2020SPECMATHEXAM1 8

Question 7 (5marks)Considerthefunctiondefinedby

f xmx n x

xx

( ),

,=

+ <

+≥

14

112

wheremandnarerealnumbers.

a. Giventhatf (x)and f ′ (x)arecontinuousoverR,showthatm=–2andn=4. 2marks

b. Findtheareaenclosedbythegraphofthefunction,thex-axisandthelinesx=0and x = 3. 3marks

9 2020SPECMATHEXAM1

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Question 8 (5marks)

Findthevolume,V,ofthesolidofrevolutionformedwhenthegraphof y x xx x

=+ +

+ +2 1

1 1

2

2( )( )isrotated

aboutthex-axisovertheinterval 0 3, .GiveyouranswerintheformV a be= +( )2π log ( ) ,wherea,b ∈ R.

2020SPECMATHEXAM1 10

Question 9 (5marks)Considerthecurvedefinedparametricallyby

x=arcsin(t)

y t te e= + + −log ( ) log ( )114

1

wheret ∈ [0,1).

a. dydt

2

canbewrittenintheform 11

11

112 2 2a t b t c t( ) ( ) ( )+

+−

+−

,wherea,bandcarereal

numbers.

Showthata=1,b=−2andc=16. 2marks

b. Findthearclength,s,ofthecurvefromt=0tot = 12.Giveyouranswerintheform

s=loge (m)+n loge(p),wherem,n,p ∈ Q. 3marks

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 2020

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2020

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

=−

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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)

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

2020 Specialist Mathematics 1InstructionsFormula sheet