MATHEMATICAL METHODSWritten examination 1

Wednesday 2 November 2016 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

8 8 40

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

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

Materials supplied• Questionandanswerbookof13pages.• 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2016

STUDENT NUMBER

Letter

2016MATHMETHEXAM1 2

THIS PAGE IS BLANK

3 2016MATHMETHEXAM1

TURN OVER

Question 1 (4marks)

a. Let y xx

=+

cos( )2 2

.

Find dydx. 2marks

b. Let f (x)=x2e5x.

Evaluate f ′(1). 2marks

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

2016MATHMETHEXAM1 4

Question 2 (3marks)

Let f R: , ,−∞

→

12

where f x x( ) = −1 2 .

a. Find f ′(x). 1mark

b. Findtheangleθfromthepositivedirectionofthex-axistothetangenttothegraphof f atx=–1,measuredintheanticlockwisedirection. 2marks

5 2016MATHMETHEXAM1

TURN OVER

Question 3 (5marks)

Let f :R\{1} →R,where f xx

( ) = +−

2 31.

a. Sketchthegraphof f .Labeltheaxisinterceptswiththeircoordinatesandlabelanyasymptoteswiththeappropriateequation. 3marks

–3 –2 –1 O 1 2 3

4

5

3

2

1

–1

–2

–3

–4

–5

y

x

b. Findtheareaenclosedbythegraphof f ,thelinesx=2andx=4,andthex-axis. 2marks

2016MATHMETHEXAM1 6

Question 4 (3marks)Apaddockcontains10taggedsheepand20untaggedsheep.Fourtimeseachday,onesheepisselectedatrandomfromthepaddock,placedinanobservationareaandstudied,andthenreturnedtothepaddock.

a. Whatistheprobabilitythatthenumberoftaggedsheepselectedonagivendayiszero? 1mark

b. Whatistheprobabilitythatatleastonetaggedsheepisselectedonagivenday? 1mark

c. Whatistheprobabilitythatnotaggedsheepareselectedoneachofsixconsecutivedays?

Expressyouranswerintheformab

c

,wherea,bandcarepositiveintegers. 1mark

7 2016MATHMETHEXAM1

TURN OVER

CONTINUES OVER PAGE

2016MATHMETHEXAM1 8

Question 5–continued

Question 5 (11marks)Let f :(0,∞)→R,where f(x)=loge(x)andg:R→R,whereg (x)=x2+1.

a. i. Findtheruleforh,where h x f g x( ) ( )= ( ). 1mark

ii. Statethedomainandrangeofh. 2marks

iii. Showthat h x h x f g x( ) ( ) ( ) .+ − = ( )( )2 2marks

iv. Findthecoordinatesofthestationarypointofhandstateitsnature. 2marks

9 2016MATHMETHEXAM1

TURN OVER

b. Letk:(–∞,0]→R,wherek (x)=loge(x2+1).

i. Findtherulefork–1. 2marks

ii. Statethedomainandrangeofk–1. 2marks

2016MATHMETHEXAM1 10

Question 6 (5marks)Let f :[–π,π]→R,where f (x)=2sin(2x)–1.

a. Calculatetheaveragerateofchangeof f between x = −π3and x = π

6. 2marks

b. Calculatetheaveragevalueof f overtheinterval − ≤ ≤π π3 6

x . 3marks

11 2016MATHMETHEXAM1

TURN OVER

Question 7 (3marks)Acompanyproducesmotorsforrefrigerators.Therearetwoassemblylines,LineAandLineB.5%ofthemotorsassembledonLineAarefaultyand8%ofthemotorsassembledonLineBarefaulty.Inonehour,40motorsareproducedfromLineAand50motorsareproducedfromLineB.Attheendofanhour,onemotorisselectedatrandomfromallthemotorsthathavebeenproducedduringthathour.

a. Whatistheprobabilitythattheselectedmotorisfaulty?Expressyouranswerintheform1b,

wherebisapositiveinteger. 2marks

b. Theselectedmotorisfoundtobefaulty.

WhatistheprobabilitythatitwasassembledonLineA?Expressyouranswerintheform1c,

wherecisapositiveinteger. 1mark

2016MATHMETHEXAM1 12

Question 8–continued

Question 8 (6marks)LetXbeacontinuousrandomvariablewithprobabilitydensityfunction

f xx x xe( ) = − ( ) < ≤

4 0 10

logelsewhere

Partofthegraphof f isshownbelow.Thegraphhasaturningpointat xe

=1 .

0 1

y

x1e

a. Showbydifferentiationthat

xk

k xk

e2 1log ( ) −( )

isanantiderivativeof xk–1loge(x),wherekisapositiverealnumber. 2marks

13 2016MATHMETHEXAM1

END OF QUESTION AND ANSWER BOOK

b. i. CalculatePr .Xe

>

12marks

ii. Hence,explainwhetherthemedianofXisgreaterthanorlessthan1e ,giventhat

e > 52

. 2marks

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 2016

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2016

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