2012 mathematical methods (cas) exam 1

7/27/2019 2012 Mathematical Methods (CAS) Exam 1

1/13

Figures

Words

STUDENT NUMBER Letter

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 book

Number of

questions

Number of questions

to be answered

Number of

marks

10 10 40

Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,

sharpeners,rulers.

StudentsareNOTpermittedtobringintotheexaminationroom:notesofanykind,blanksheetsof

paper,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 Certicate of Education

2012


2/13

2012MATHMETH(CAS)EXAM1 2

This page is blank


3/13

3 2012 MATHMETH(CAS) EXAM 1

TURN OVER

Question 1

a. Ify = (x2 5x)4, fnddy

dx.

1 mark

b. Iff(x) =x

xsin( ), fndf '

2.

2 marks

Instructions

Answerall questions in the spaces provided.

In all questions where a numerical answer is required an exact value must be given unless otherwise

specifed.

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.


4/13


Question 2

Findananti-derivativeof1

2 13

x ( )withrespecttox.

2marks

Question 3

Theruleforfunctionhish(x)=2x3+1.Findtherulefortheinversefunctionh1.

2marks


5/13

5 2012MATHMETH(CAS)EXAM1

TURN OVER

Question 4

Onanygivenday,thenumberXoftelephonecallsthatDanielreceivesisarandomvariablewithprobability

distributiongivenby

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


6/13


Question 5

a. Sketchthegraphoff:[0,5]R,f(x)=|x3|+2.Labeltheaxesinterceptsandendpointswiththeir

coordinates.

y

x

1

1 2 3 4 5 6 7

1

1234567

2

3

4

5

6

7

2

3

4

5

6

7

O

3marks

b. i. Findthecoordinatesoftheimageofthepoint(3,2)underareectioninthex-axis,followedbya

translationof5unitsinthepositivedirectionofthex-axis.

ii. Findtheequationoftheimageofthegraphoffunderareectioninthex-axis,followedbya

translationof5unitsinthepositivedirectionofthex-axis.

1+2=3marks


7/13

7 2012MATHMETH(CAS)EXAM1

TURN OVER

Question 6

Thegraphsofy=cos(x)andy = asin(x),whereaisarealconstant,haveapointofintersectionatx =

3.

a. Findthevalueofa.

2marks

b. Ifx[0,2],ndthex-coordinateoftheotherpointofintersectionofthetwographs.

1mark

Question 7

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

3marks


8/13

2012 MATHMETH(CAS) EXAM 1 8

Question 8

a. The random variableXis normally distributed with mean 100 and standard deviation 4.

If Pr(X< 106) = q, fnd Pr(94


9/13

9 2012 MATHMETH(CAS) EXAM 1

TURN OVER

Question 9

a. Let f:R R, f(x) =x sin (x).

Findf '(x).

1 mark

b. Use the result ofparta. to nd the value of x xcos( )

6

2 dx in the form a+ b.

3 marks


10/13


Question 10

Letf:RR,f(x)=emx + 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


11/13

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


12/13

MATHMETH (CAS) 2

This page is blank


13/13

3 MATHMETH (CAS)

END OF FORMULA SHEET

Mathematical Methods (CAS)

Formulas

Mensuration

area of a trapezium:1

2 a b h+( ) volume of a pyramid:

1

3 Ah

curved surface area of a cylinder: 2rh volume of a sphere:4

3

3r

volume of a cylinder: r2h area of a triangle:1

2bc Asin

volume of a cone:1

3

2r h

Calculus

d

dx

x nxn n

( )=

1

x dx

n

x c nn n=

+

+ +

1

1

11

,

d

dxe ae

ax ax( ) = e dx a e cax ax= +

1

d

dxx

xelog ( )( ) =

1

1

xdx x ce= + log

d

dxax a axsin( ) cos( )( ) = sin( ) cos( )ax dx a ax c= +

1

d

dxax a axcos( )( ) = sin( )

cos( ) sin( )ax dx

aax c= +

1

d

dxax

a

axa axtan( )

( )

( ) ==

cos

sec ( )2

2

product rule:d

dxuv u

dv

dxv

du

dx( ) = + quotient rule:

d

dx

u

v

vdu

dxudv

dx

v

=

2

chain rule:dy

dx

dy

du

du

dx= approximation: f x h f x h f x+( ) ( ) + ( )

Probability

Pr(A) = 1 Pr(A) Pr(AB) = Pr(A) + Pr(B) Pr(AB)

Pr(A|B) =Pr

Pr

A B

B

( )

( )transition matrices: S

n= Tn S0

mean: = E(X) variance: var(X) = 2 = E((X)2) = E(X2) 2

Probability distribution Mean Variance

discrete Pr(X=x) =p(x) = xp(x) 2 = (x )2p(x)

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

b( ) =

x f x d x( ) 2 2=

( ) ( )x f x dx

2012 mathematical methods (cas) exam 1

Documents