SPECIALIST MATHEMATICSWritten examination 1

Thursday 8 June 2017 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

11 11 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2017

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2017

STUDENT NUMBER

Letter

2017SPECMATHEXAM1(NHT) 2

THIS PAGE IS BLANK

3 2017SPECMATHEXAM1(NHT)

TURN OVER

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

Question 1 (3marks)A5kgmassonasmoothplaneinclinedat30°isheldinequilibriumbyahorizontalforceofmagnitudePnewtons,asshowninthediagrambelow.

P

30°

a. Onthediagramabove,showallotherforcesactingonthemassandlabelthem. 1mark

b. FindP. 2marks

2017SPECMATHEXAM1(NHT) 4

Question 2 (3marks)

Findagiventhat 816

622 −=

−∫ xdx

a

elog ( ),a ∈(–2,4).

Question 3 (3marks)Findthegradientofthecurvewithequation x

yx=

=sin .

1514

when Giveyouranswerintheform a b ,wherea,b ∈ Z+.

5 2017SPECMATHEXAM1(NHT)

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Question 4 (4marks)Findthevaluesofaandbgiventhatz – 1 – i isafactorofz3+(a + b)z2+(b2 – a)z–4=0,whereaandb arerealconstants.

Question 5 (4marks)Partofthegraphof y x

x=

+

−

1

1 24isshownbelow.

0.5

1

1.5

–0.5 0 0.5

y

x

Findthevolumegeneratediftheregionboundedbythegraphof y x

x=

+

−

1

1 24,thelines x x= − =

12

12

and , andthex-axisisrotatedaboutthex-axis.

2017SPECMATHEXAM1(NHT) 6

Question 6 (3marks)Findallrealsolutionsoftan(2x)=–tan(x).

7 2017SPECMATHEXAM1(NHT)

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Question 7 (5marks)

Let dydx y= −( )4 2.

a. Expressyintermsofx,wherey(0)=3. 3marks

b. Expressd ydx

2

2 intermsofy. 2marks

2017SPECMATHEXAM1(NHT) 8

Question 8 (3marks)A3kgmasshasvelocityvms–1,where v x= 2arctan whenithasadisplacementxmetresfromtheorigin,x>0.

Findthenetforce,Fnewtons,actingonthemassintermsofx.

9 2017 SPECMATH EXAM 1 (NHT)

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Question 9 (4 marks)The random variables X and Y are independent with μX = 4, var(X) = 36 and μY = 3, var(Y) = 25.

a. The random variable Z is such that Z = 2X + 3Y.

i. Find E(Z). 1 mark

ii. Find the standard deviation of Z. 1 mark

b. Researchers have reason to believe that the mean of X has decreased. They collect a random sample of 64 observations of X and find that the sample mean is X = 3 8.

i. State the null hypothesis and the alternative hypothesis that should be used to test that the mean has decreased. 1 mark

ii. Calculate the mean and standard deviation for a distribution of sample means, X , for samples of 64 observations. 1 mark

2017SPECMATHEXAM1(NHT) 10

END OF QUESTION AND ANSWER BOOK

Question 10 (4marks)Considerthevectors

a i j k and b i j k.= − − + = + +2 3 2 c

Findthevalueofc,c ∈ R,iftheanglebetween� �a and b is π

3.

Question 11 (4marks)Findthelengthofthecurvespecifiedparametricallybyx = aθ – asin(θ),y = a – acos(θ)from

θπ

θ π= =23

2 to ,wherea ∈ R+.Giveyouranswerintermsofa.

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 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