MATHEMATICAL METHODSWritten examination 1

Wednesday 8 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

9 9 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2017

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2017

STUDENT NUMBER

Letter

2017MATHMETHEXAM1 2

THIS PAGE IS BLANK

3 2017MATHMETHEXAM1

TURN OVER

Question 1 (4marks)

a. Letf:(–2,∞)→R,f(x)= xx + 2

.

Differentiatefwithrespecttox. 2marks

b. Letg(x)=(2–x3)3.

Evaluateg′ (1). 2marks

InstructionsAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegiven,unlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmustbeshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

2017 MATHMETH EXAM 1 4

Question 2 (4 marks)Let y = x loge(3x).

a. Find dydx

. 2 marks

b. Hence, calculate (log ( ) ) .e x dx1

23 1∫ + Express your answer in the form loge(a), where a is a

positive integer. 2 marks

5 2017MATHMETHEXAM1

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Question 3 (4marks)Letf:[–3,0]→R,f (x)=(x+2)2(x–1).

a. Showthat(x+2)2(x–1)=x3+3x2–4. 1mark

b. Sketchthegraphoffontheaxesbelow.Labeltheaxisinterceptsandanystationarypointswiththeircoordinates. 3marks

6

4

2

O–1–2–3–4 1 2

–2

–4

–6

x

y

2017MATHMETHEXAM1 6

Question 4 (2marks)Inalargepopulationoffish,theproportionofangelfishis

14.

LetP̂ betherandomvariablethatrepresentsthesampleproportionofangelfishforsamplesofsizendrawnfromthepopulation.

FindthesmallestintegervalueofnsuchthatthestandarddeviationofP̂ islessthanorequalto1

100.

7 2017MATHMETHEXAM1

TURN OVER

Question 5 (4marks)ForJactologontoacomputersuccessfully,Jacmusttypethecorrectpassword.Unfortunately,Jachasforgottenthepassword.IfJactypesthewrongpassword,Jaccanmakeanotherattempt.Theprobabilityofsuccessonanyattemptis2

5.Assumethattheresultofeachattemptisindependent

oftheresultofanyotherattempt.Amaximumofthreeattemptscanbemade.

a. WhatistheprobabilitythatJacdoesnotlogontothecomputersuccessfully? 1mark

b. CalculatetheprobabilitythatJaclogsontothecomputersuccessfully.Expressyouranswerintheforma

b,whereaandbarepositiveintegers. 1mark

c. CalculatetheprobabilitythatJaclogsontothecomputersuccessfullyonthesecondoronthethirdattempt.Expressyouranswerintheform c

d,wherecanddarepositiveintegers. 2marks

2017MATHMETHEXAM1 8

Question 6 (3marks)

Let tan( ) sin( ) cos( ) sin( ) cos( ) .θ θ θ θ θ− −( ) +( ) =1 3 3 0( )a. Stateallpossiblevaluesoftan(θ ). 1mark

b. Hence,findallpossiblesolutionsfor tan( ) sin ( ) cos ( ) ,θ θ θ− −( ) =1 3 02 2( ) where0≤θ≤π. 2marks

9 2017MATHMETHEXAM1

TURN OVER

Question 7 (5marks)

Let f R f x x: [ , ) , ( ) .0 1∞ → = +

a. Statetherangeoff. 1mark

b. Letg:(–∞,c]→R,g(x)=x2+4x+3,wherec<0.

i. Findthelargestpossiblevalueofcsuchthattherangeofgisasubsetofthedomainoff. 2marks

ii. Forthevalueofcfoundinpart b.i.,statetherangeoff(g(x)). 1mark

c. Leth:R→R,h(x)=x2+3.

Statetherangeof f(h(x)). 1mark

2017MATHMETHEXAM1 10

Question 8 (5marks)

ForeventsAandBfromasamplespace,Pr and PrA B B A| | .( ) = ( ) =15

14Let Pr ( ) .A B p∩ =

a. FindPr(A)intermsofp. 1mark

b. FindPr ( )′∩ ′A B intermsofp. 2marks

c. GiventhatPr ( ) ,A B∪ ≤15statethelargestpossibleintervalforp. 2marks

11 2017MATHMETHEXAM1

Question 9 –continuedTURN OVER

Question 9 (9marks)Thegraphof f R f x xx: , , ( )(0 1 1[ ]→ = – isshownbelow.

y x= )1 – x

y

x0 1

a. Calculatetheareabetweenthegraphoffandthex-axis. 2marks

b. Forxintheinterval(0,1),showthatthegradientofthetangenttothegraphoffis1 32− x

x. 1mark

2017MATHMETHEXAM1 12

END OF QUESTION AND ANSWER BOOK

Theedgesoftheright-angled triangleABCarethelinesegmentsACandBC,whicharetangenttothegraphoff,andthelinesegmentAB,whichispartofthehorizontalaxis,asshownbelow.LetθbetheanglethatACmakeswiththepositivedirectionofthehorizontalaxis,where45°≤θ<90°.

y x= )1 – x

y

x0A B

C

θ

c. FindtheequationofthelinethroughBandCintheformy=mx+c,forθ=45°. 2marks

d. FindthecoordinatesofCwhenθ=45°. 4marks

MATHEMATICAL METHODS

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

MATHMETH EXAM 2

Mathematical Methods formulas

Mensuration

area of a trapezium 12a b h+( ) volume of a pyramid 1

3Ah

curved surface area of a cylinder 2π  rh volume of a sphere

43

3π r

volume of a cylinder π r 2h area of a triangle12bc Asin ( )

volume of a cone13

2π r h

Calculus

ddx

x nxn n( ) = −1 x dxn

x c nn n=+

+ ≠ −+∫ 11

11 ,

ddx

ax b an ax bn n( )+( ) = +( ) −1 ( )( )

( ) ,ax b dxa n

ax b c nn n+ =+

+ + ≠ −+∫ 11

11

ddxe aeax ax( ) = e dx a e cax ax= +∫ 1

ddx

x xelog ( )( ) = 11 0x dx x c xe= + >∫ log ( ) ,

ddx

ax a axsin ( ) cos( )( ) = sin ( ) cos( )ax dx a ax c= − +∫ 1

ddx

ax a axcos( )( ) −= sin ( ) cos( ) sin ( )ax dx a ax c= +∫ 1

ddx

ax aax

a axtan ( )( )

( ) ==cos

sec ( )22

product ruleddxuv u dv

dxv dudx

( ) = + quotient ruleddx

uv

v dudx

u dvdx

v

=

−

2

chain ruledydx

dydududx

=

3 MATHMETH EXAM

END OF FORMULA SHEET

Probability

Pr(A) = 1 – Pr(A′) Pr(A ∪ B) = Pr(A) + Pr(B) – Pr(A ∩ B)

Pr(A|B) = Pr

PrA BB∩( )( )

mean µ = E(X) variance var(X) = σ 2 = E((X – µ)2) = E(X 2) – µ2

Probability distribution Mean Variance

discrete Pr(X = x) = p(x) µ = ∑ x p(x) σ 2 = ∑ (x – µ)2 p(x)

continuous Pr( ) ( )a X b f x dxa

b< < = ∫ µ =

−∞

∞

∫ x f x dx( ) σ µ2 2= −−∞

∞

∫ ( ) ( )x f x dx

Sample proportions

P Xn

=̂ mean E(P̂ ) = p

standard deviation

sd P p pn

(ˆ ) ( )=

−1 approximate confidence interval

,p zp p

np z

p pn

−−( )

+−( )

1 1ˆ ˆ ˆˆˆ ˆ