2012 Mathematical Methods (CAS) Written examination 1 · 2012 MATHMETH(CAS) EXAM 1 10 Question 10...

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MATHEMATICAL METHODS (CAS)Written examination 1

Wednesday 7 November 2012 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,sharpeners,rulers.

• StudentsareNOTpermittedtobringintotheexaminationroom:notesofanykind,blanksheetsofpaper,whiteoutliquid/tapeoracalculatorofanytype.

Materials supplied• Questionandanswerbookof10pages,withadetachablesheetofmiscellaneousformulasinthe

centrefold.• Workingspaceisprovidedthroughoutthebook.

Instructions• Detachtheformulasheetfromthecentreofthisbookduringreadingtime.• Writeyourstudent numberinthespaceprovidedaboveonthispage.

• AllwrittenresponsesmustbeinEnglish.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2012

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2012

2012MATHMETH(CAS)EXAM1 2

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3 2012 MATHMETH(CAS) EXAM 1

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Question 1

a. If y = (x2 – 5x)4, find dydx

.

1 mark

b. If f (x) =x

xsin( ), find f ' π

2.

2 marks

InstructionsAnswer all questions in the spaces provided.In all questions where a numerical answer is required an exact value must be given unless otherwise specified.In questions where more than one mark is available, appropriate working must be shown.Unless otherwise indicated, the diagrams in this book are not drawn to scale.

2012MATHMETH(CAS)EXAM1 4

Question 2

Findananti-derivativeof 12 1 3x −( )

withrespecttox.

2marks

Question 3Theruleforfunctionhish(x)=2x3+1.Findtherulefortheinversefunctionh–1.

2marks

5 2012MATHMETH(CAS)EXAM1

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Question 4Onanygivenday,thenumberXoftelephonecallsthatDanielreceivesisarandomvariablewithprobabilitydistributiongivenby

x 0 1 2 3

Pr(X = x) 0.2 0.2 0.5 0.1

a. FindthemeanofX.

2marks

b. WhatistheprobabilitythatDanielreceivesonlyonetelephonecalloneachofthreeconsecutivedays?

1mark

c. DanielreceivestelephonecallsonbothMondayandTuesday. WhatistheprobabilitythatDanielreceivesatotaloffourcallsoverthesetwodays?

3marks

2012MATHMETH(CAS)EXAM1 6

Question 5a. Sketchthegraphoff :[0,5]→R, f (x)=–|x–3|+2.Labeltheaxesinterceptsandendpointswiththeir

coordinates.y

x

1

1 2 3 4 5 6 7

–1

–1–2–3–4–5–6–7

–2

–3

–4

–5

–6

–7

2

3

4

5

6

7

O

3marks

b. i. Findthecoordinatesoftheimageofthepoint(3,2)underareflectioninthex-axis,followedbyatranslationof5unitsinthepositivedirectionofthex-axis.

ii. Findtheequationoftheimageofthegraphoffunderareflectioninthex-axis,followedbyatranslationof5unitsinthepositivedirectionofthex-axis.

1+2=3marks

7 2012MATHMETH(CAS)EXAM1

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Question 6Thegraphsofy=cos(x)andy = asin(x),whereaisarealconstant,haveapointofintersectionat x = π

3.

a. Findthevalueofa.

2marks

b. Ifx ∈[0,2π],findthex-coordinateoftheotherpointofintersectionofthetwographs.

1mark

Question 7Solvetheequation2loge(x+2)–loge(x)=loge(2x+1),wherex>0,forx.

3marks

2012 MATHMETH(CAS) EXAM 1 8

Question 8a. The random variable X is normally distributed with mean 100 and standard deviation 4. If Pr(X < 106) = q, find Pr(94 < X < 100) in terms of q.

2 marks

b. The probability density function f of a random variable X is given by

f (x) = x x+

≤ ≤1

124

0

0

otherwise

Find the value of b such that Pr(X ≤ b) = 58

.

3 marks

9 2012 MATHMETH(CAS) EXAM 1

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Question 9a. Let f : R → R, f (x) = x sin (x). Find f '(x).

1 mark

b. Use the result of part a. to find the value of x xcos( )π

π

6

2∫ dx in the form aπ + b.

3 marks

2012MATHMETH(CAS)EXAM1 10

Question 10Let f :R→R, f (x)=e–mx + 3x,wheremisapositiverationalnumber.a. i. Find,intermsofm,thex-coordinateofthestationarypointofthegraphofy = f (x).

ii. Statethevaluesofmsuchthatthex-coordinateofthisstationarypointisapositivenumber.

2+1=3marks

b. Foraparticularvalueofm,thetangenttothegraphofy = f (x)atx=–6passesthroughtheorigin. Findthisvalueofm.

3marks

END OF QUESTION AND ANSWER BOOK

MATHEMATICAL METHODS (CAS)

Written examinations 1 and 2

FORMULA SHEET

Directions to students

Detach this formula sheet during reading time.

This formula sheet is provided for your reference.

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2012

MATHMETH (CAS) 2

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3 MATHMETH (CAS)

END OF FORMULA SHEET

Mathematical Methods (CAS)Formulas

Mensuration

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

13Ah

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 triangle: 12bc Asin

volume of a cone: 13

2π r h

Calculusddx

x nxn n( ) = −1

x dx

nx c nn n=

++ ≠ −+∫

11

11 ,

ddxe aeax ax( ) = e dx a e cax ax= +∫

1

ddx

x xelog ( )( ) = 1 1x dx x ce= +∫ 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 rule: ddxuv u dv

dxv dudx

( ) = + quotient rule: ddx

uv

v dudx

u dvdx

v

=

−

2

chain rule: dydx

dydududx

= approximation: f x h f x h f x+( ) ≈ ( ) + ′( )

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

Pr(A|B) = Pr

PrA BB∩( )( ) transition matrices: Sn = Tn × S0

mean: µ = E(X) variance: var(X) = σ 2 = E((X – µ)2) = E(X2) – 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