2012 Mathematical Methods (CAS) Exam 2

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STUDENT NUMBER Letter

MATHEMATICAL METHODS (CAS)

Written examination 2

Thursday 8 November 2012

Reading time: 11.45 am to 12.00 noon (15 minutes)

Writing time: 12.00 noon to 2.00 pm (2 hours)

QUESTION AND ANSWER BOOK

Structure of book

Section Number of

questions

Number of questions

to be answered

Number of

marks

1

2

22

5

22

5

22

58

Total 80

• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,

sharpeners,rulers,aprotractor,set-squares,aidsforcurvesketching,oneboundreference,one

approvedCAScalculator(memoryDOESNOTneedtobecleared)and,ifdesired,onescientic

calculator.Forapprovedcomputer-basedCAS,theirfullfunctionalitymaybeused.

• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orwhite

outliquid/tape.

Materials supplied• Questionandanswerbookof24pageswithadetachablesheetofmiscellaneousformulasinthe

centrefold.

• Answersheetformultiple-choicequestions.

Instructions

• Detachtheformulasheetfromthecentreofthisbookduringreadingtime.

• Writeyourstudent numberinthespaceprovidedaboveonthispage.

• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice

questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.

• AllwrittenresponsesmustbeinEnglish.

At the end of the examination

• Placetheanswersheetformultiple-choicequestionsinsidethefrontcoverofthisbook.

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



2012MATHMETH(CAS)EXAM2 2

SECTION 1–continued

Question 1

Thefunctionwithrule f ( x)=–3sinπ x

5

hasperiod

A. 3

B. 5

C. 10

D.π

5

E.π

10

Question 2

Forthefunctionwithrule f ( x)= x3–4 x,theaveragerateofchangeof f ( x)withrespectto xontheinterval

[1,3]is

A. 1

B. 3

C. 5

D. 6

E. 9

Question 3

Therangeofthefunction f :[–2,3)→ R, f ( x)= x2 – 2 x–8is

A. R

B. (−9,–5]

C. (–5,0)

D. [−9,0]

E. [–9,–5)

SECTION 1

Instructions for Section 1

Answerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.

Choosetheresponsethatiscorrect forthequestion.

Acorrectanswerscores1,anincorrectanswerscores0.

Markswillnotbedeductedforincorrectanswers.

Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.







Question 7

Thetemperature,T °C,insideabuildingt hoursaftermidnightisgivenbythefunction

f :[0,24]→ R,T (t )=22–10cosπ

122t −( )

Theaveragetemperatureinsidethebuildingbetween2amand2pmis

A. 10°C

B. 12°C

C. 20°C

D. 22°C

E. 32°C

Question 8

Thefunction f : R→ R, f ( x)=ax3 + bx2 + cx,whereaisanegativerealnumberandbandcarerealnumbers.

Fortherealnumbers p < m < 0 < n < q,wehave f ( p)= f (q)=0and f ′ (m)= f ′ (n)=0.

Thegradientofthegraphof y= f ( x)isnegativefor A. (–∞,m)∪(n,∞)

B. (m,n)

C. ( p,0)∪(q,∞)

D. ( p,m)∪(0,q)

E. ( p,q)

Question 9

Thenormaltothegraphof y b x= −2hasagradientof3when x =1.

Thevalueofbis

A. −10

9

B.10

9

C. 4

D. 10

E. 11

Question 10

Theaveragevalueofthefunction f :[0,2π ]→ R, f ( x)=sin2( x)overtheinterval[0,a]is0.4.

Thevalueofa,tothreedecimalplaces,is

A. 0.850

B. 1.164

C. 1.298

D. 1.339

E. 4.046



5 2012MATHMETH(CAS)EXAM2


TURN OVER

Question 11

Theweightsofbagsofourarenormallydistributedwithmean252gandstandarddeviation12g.The

manufacturersaysthat40%ofbagsweighmorethan xg.

Themaximumpossiblevalueof xisclosestto

A. 249.0

B. 251.5

C. 253.5

D. 254.5

E. 255.0

Question 12

Demelzaisabadmintonplayer.Ifshewinsagame,theprobabilitythatshewillwinthenextgameis0.7.If

shelosesagame,theprobabilitythatshewilllosethenextgameis0.6.Demelzahasjustwonagame.

Theprobabilitythatshewillwinexactlyoneofhernexttwogamesis

A. 0.33

B. 0.35C. 0.42

D. 0.49

E. 0.82

Question 13

Aand BareeventsofasamplespaceS .

Pr( A∩ B)=2

5andPr( A∩ B′ )=

3

7.

Pr( B′ | A)isequalto

A.6

35

B.15

29

C.14

35

D.29

35

E.

2

3





Question 14

Thegraphof f : R+∪{0}→ R, f ( x)= x isshownbelow.

Inordertondanapproximationtotheareaoftheregionboundedbythegraphof f ,the y-axisandtheline

y=4,Zoedrawsfourrectangles,asshown,andcalculatestheirtotalarea.

4

3

2

1

O

y x=

x

y

Zoe’sapproximationtotheareaoftheregionis

A. 14

B. 21

C. 29

D. 30

E. 64

3





TURN OVER

Question 15

If f ′ ( x)=3 x2–4,whichoneofthefollowinggraphscouldrepresentthegraphof y= f ( x)?

A. y

x

B. y

x

C. y

x

D. y

x

E.

y

x

Question 16

Thegraphofacubicfunction f hasalocalmaximumat(a,–3)andalocalminimumat(b,–8).

Thevaluesofc,suchthattheequation f ( x)+c=0hasexactlyonesolution,are

A. 3<c < 8

B. c >–3orc < –8

C. –8 < c <–3

D. c <3orc > 8

E. c < –8





Question 17

Asystemofsimultaneouslinearequationsisrepresentedbythematrixequation

m

m

x

y m

3

1 2

1

+

=

.

Thesystemofequationswillhaveno solutionwhenA. m=1

B. m=–3

C. m ∈{1,–3}

D. m ∈ R\{1}

E. m ∈{1,3}

Question 18

Thetangenttothegraphof y=loge( x)atthepoint(a,log

e(a))crossesthe x-axisatthepoint(b,0),whereb <0.

Whichofthefollowingisfalse?

A. 1 < a < e

B. Thegradientofthetangentispositive

C. a > e

D. Thegradientofthetangentis1

aE. a > 0

Question 19

Afunction f hasthefollowingtwopropertiesforallrealvaluesofθ .

f (π – θ )=– f (θ )and f (π – θ )=– f (– θ )

Apossiblerulefor f is

A. f ( x)=sin( x)

B. f ( x)=cos( x)

C. f ( x)=tan( x)

D. f ( x)=sin x

2

E. f ( x)=tan(2 x)

Question 20

Adiscreterandomvariable X hastheprobabilityfunctionPr( X =k )=(1– p)k p, wherek isanon-negative

integer.

Pr( X >1)isequalto

A. 1 – p + p2

B. 1 – p2

C. p – p2

D. 2 p – p2

E. (1– p)2




END OF SECTION 1

TURN OVER

Question 21

Thevolume,V cm3,ofwaterinacontainerisgivenbyV =1

3πh3wherehcmisthedepthofwaterinthe

containerattimet minutes.Waterisdrainingfromthecontainerataconstantrateof300cm3/min.Therate

ofdecreaseofh,incm/min,whenh=5is

A.12

π

B.4

π

C. 25π

D.60

π

E. 30π

Question 22

Thegraphofadifferentiablefunction f hasalocalmaximumat(a,b),wherea <0andb >0,andalocal

minimumat(c,d ),wherec>0andd <0.

Thegraphof y=–| f ( x–2)|has

A. alocalminimumat(a–2,– b)andalocalmaximumat(c–2,d )

B. localminimaat(a+2,– b)and(c+2,d )

C. localmaximaat(a+2,b)and(c+2,–d )

D. alocalminimumat(a–2,– b)andalocalmaximumat(a–2,–d )

E. localminimaat(c+2,–d )and(a+2, –b)




SECTION 2 – Question 1–continued

Question 1

Asolidblockintheshapeofarectangularprismhasabaseofwidth xcm.Thelengthofthebaseis

two-and-a-halftimesthewidthofthebase.

5

2

xcm

x cm

h cm

Theblockhasatotalsurfaceareaof6480sqcm.

a. Showthatiftheheightoftheblockishcm,h=6480 5

7

2− x

x.

2marks

SECTION 2

Instructions for Section 2

Answerallquestionsinthespacesprovided.

Inallquestionswhereanumericalanswerisrequiredanexactvaluemustbegivenunlessotherwise

specied.

Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.

Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.





TURN OVER

b. Thevolume,V cm3,oftheblockisgivenbyV ( x)=5 6480 5

14

2 x x( ).

−

GiventhatV ( x)>0and x >0,ndthepossiblevaluesof x.

2marks

c. FinddV

dx,expressingyouranswerintheform

dV

dx=ax2 + b,whereaandbarerealnumbers.

3marks

d. Findtheexactvaluesof xandhiftheblockistohavemaximumvolume.

2marks







TURN OVER

c. If( p,q)isanypointonthegraphof y= f ( x),showthattheequationofthetangentto y= f ( x)atthis

pointcanbewrittenas(2 p–4)2( y–3)=–2 x+4 p–4.

2marks





d. Findthecoordinatesofthepointsonthegraphof y= f ( x)suchthatthetangentstothegraphatthese

pointsintersectat −

1,

7

2.

4marks



15 2012 MATHMETH(CAS) EXAM 2

SECTION 2 – continued

TURN OVER

e. A transformation T : R2 → R2 that maps the graph of f to the graph of the function

g : R\{0}→ R, g ( x) =1

xhas rule T

x

y

a x

y

c

d = +

0

0 1, where a, c and d are non-zero real numbers.

Find the values of a, c and d .

2 marks



2012 MATHMETH(CAS) EXAM 2 16

SECTION 2 – Question 3 – continued

Question 3

Steve, Katerina and Jess are three students who have agreed to take part in a psychology experiment. Each

student is to answer several sets of multiple-choice questions. Each set has the same number of questions,

n, where n is a number greater than 20. For each question there are four possible options (A, B, C or D), of

which only one is correct.

a. Steve decides to guess the answer to every question, so that for each question he chooses A, B, C or D

at random.Let the random variable X be the number of questions that Steve answers correctly in a particular set.

i. WhatistheprobabilitythatStevewillanswertherstthreequestionsofthissetcorrectly?

ii. Find,tofourdecimalplaces,theprobabilitythatStevewillansweratleast10oftherst20 questions of this set correctly.

iii. Use the fact that the variance of X is75

16 to show that the value of n is 25.

1 + 2 + 1 = 4 marks





TURN OVER

IfKaterinaanswersaquestioncorrectly,theprobabilitythatshewillanswerthenextquestioncorrectly

is3

4. Ifsheanswersaquestionincorrectly,theprobabilitythatshewillanswerthenextquestion

incorrectlyis2

3.

Inaparticularset,KaterinaanswersQuestion1incorrectly.

b. i. CalculatetheprobabilitythatKaterinawillanswerQuestions3,4and5correctly.

ii. FindtheprobabilitythatKaterinawillanswerQuestion25correctly.Giveyouranswercorrectto

fourdecimalplaces.

3+2=5marks





c. TheprobabilitythatJesswillansweranyquestioncorrectly,independentlyofheranswertoanyother

question,is p ( p > 0). LettherandomvariableY bethenumberofquestionsthatJessanswerscorrectly

inanysetof25.

IfPr(Y >23)=6Pr(Y =25),showthatthevalueof p is5

6.

2marks

d. Fromthesesetsof25questionsbeingcompletedbymanystudents,ithasbeenfoundthatthetime,inminutes,thatanystudenttakestoanswereachsetof25questionsisanotherrandomvariable,W ,which

isnormally distributedwithmeana andstandarddeviationb.

Itturnsoutthat,forJess,Pr(Y ≥18)=Pr(W ≥20)andalsoPr(Y ≥22)=Pr(W ≥25).

Calculatethevaluesofaandb,correcttothreedecimalplaces.

4marks





TURN OVER

Question 4

TasmaniaJonesisinthejungle,searchingfortheQuetzalotltribe’svaluableemeraldthathasbeenstolenand

hiddenbyaneighbouringtribe.Tasmaniahasheardthattheemeraldhasbeenhiddeninatankshapedlikean

invertedcone,withaheightof10metresandadiameterof4metres(asshownbelow).

Theemeraldisonashelf.Thetankhasapoisonousliquidinit.

r m

4 m

tap

shelf

10 m

h m

2 m

a. Ifthedepthoftheliquidinthetankishmetres

i. ndtheradius,r metres,ofthesurfaceoftheliquidintermsofh

ii. showthatthevolumeoftheliquidinthetankisπ h

3

75m3.

1+1=2marks





Thetankhasatapatitsbasethatallowstheliquidtorunoutofit.Thetankisinitiallyfull.Whenthetapis

turnedon,theliquidowsoutofthetankatsucharatethatthedepth,hmetres,oftheliquidinthetankis

givenby

h t t = + −101

160012003( ) ,

wheret minutesisthelengthoftimeafterthetapisturnedonuntilthetankisempty.

b. Showthatthetankisemptywhent =20.

1mark

c. Whent =5minutes,nd

i. thedepthoftheliquidinthetank

ii. therateatwhichthevolumeoftheliquidisdecreasing,correcttoonedecimalplace.

1+3=4marks





TURN OVER

d. Theshelfonwhichtheemeraldisplacedis2metresabovethevertexofthecone.

Fromthemomenttheliquidstartstoowfromthetank,ndhowlong,inminutes,ittakesuntilh=2.

(Giveyouranswercorrecttoonedecimalplace.)

2marks

e. Assoonasthetankisempty,thetapturnsitselfoffandpoisonousliquidstartstoowintothetankata

rateof0.2m3/minute.

Howlong,inminutes,afterthetankisrstemptywilltheliquidonceagainreachadepthof2metres?

2marks

f. Inordertoobtaintheemerald,TasmaniaJonesentersthetankusingavinetoclimbdownthewallof

thetankassoonasthedepthoftheliquidisrst2metres.Hemustleavethetankbeforethedepthis

againgreaterthan2metres.

Findthelengthoftime,inminutes,correcttoonedecimalplace,thatTasmaniaJoneshasfromthetime

heentersthetanktothetimeheleavesthetank.

1mark





Question 5

TheshadedregioninthediagrambelowistheplanofaminesitefortheBlackPossumminingcompany.

Alldistancesareinkilometres.

Twooftheboundariesoftheminesiteareintheshapeofthegraphsofthefunctions

f : R → R, f ( x)=e x and g : R+ → R, g ( x)=loge( x).

y

x –3 –2 –1 1 2 3

–3

–2

–1

O

1

2

3

x = 1

y = f ( x) y = g ( x)

y = –2

a. i. Evaluate f x dx( ) .−∫

2

0

ii. Hence,orotherwise,ndtheareaoftheregionboundedbythegraphof g ,the xand yaxes,and

theline y=–2.





TURN OVER

iii. Findthetotalareaoftheshadedregion.

1+1+1=3marks

b. Theminingengineer,Victoria,decidesthatabettersiteforthemineistheregionboundedbythegraph

of g andthatofanewfunctionk :(–∞,a)→ R,k ( x)=–loge(a – x),whereaisapositiverealnumber.

i. Find,intermsofa,the x-coordinatesofthepointsofintersectionofthegraphsof g andk .

ii. Hence,ndthesetofvaluesofa,forwhichthegraphsof g andk havetwodistinctpointsof

intersection.

2+1=3marks




c. Forthenewminesite,thegraphsof g andk intersectattwodistinctpoints, Aand B.Itisproposedto

startminingoperationsalongthelinesegment AB,whichjoinsthetwopointsofintersection.

Victoriadecidesthatthegraphofk willbesuchthatthe x-coordinateofthemidpointof ABis 2 .

Findthevalueofainthiscase.

2marks

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

This page is blank



3 MATHMETH (CAS)

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: 2π rh volume of a sphere:4

3

3π r

volume of a cylinder: π r 2h area of a triangle:1

2bc Asin

volume of a cone:1

3

2π r 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

ae c

ax ax= +∫ 1

d

dx x

xelog ( )( ) =

1

1

xdx x ce= +∫ log

d

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

aax 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( A ∪ B) = Pr( A) + Pr( B) – Pr( A ∩ B)

Pr( A| B) =Pr

Pr

A B

B

∩( )

( )transition matrices: S

n= T n × S

0

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 d x( ) σ µ 2 2= −

−∞

∞

∫ ( ) ( ) x f x dx

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2012 Mathematical Methods (CAS) Exam 2