MATHEMATICAL METHODS (CAS) · 2 5 5 58 Total 80 • Students are permitted to bring into the...

MATHEMATICAL METHODS (CAS)Written examination 2

Thursday 6 November 2014 Reading time: 3.00 pm to 3.15 pm (15 minutes) Writing time: 3.15 pm to 5.15 pm (2 hours)

QUESTION AND ANSWER BOOK

Structure of bookSection Number of

questionsNumber of questions

to be answeredNumber of

marks

1 22 22 222 5 5 58

Total 80

• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulers,aprotractor,set-squares,aidsforcurvesketching,oneboundreference,oneapprovedCAScalculator(memoryDOESNOTneedtobecleared)and,ifdesired,onescientificcalculator.Forapprovedcomputer-basedCAS,theirfullfunctionalitymaybeused.

• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orwhiteoutliquid/tape.

Materials supplied• Questionandanswerbookof22pageswithadetachablesheetofmiscellaneousformulasinthe

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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2014

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2014

STUDENT NUMBER

Letter

2014MATHMETH(CAS)EXAM2 2

SECTION 1–continued

Question 1ThepointP(4,–3)liesonthegraphofafunction f. Thegraphof f istranslatedfourunitsverticallyupandthenreflectedinthey-axis.ThecoordinatesofthefinalimageofPareA. (–4,1)B. (–4,3)C. (0,–3)D. (4,–6)E. (–4,–1)

Question 2Thelinearfunction f D R f x x: ,→ ( ) = −4 hasrange[–2,6).ThedomainDofthefunctionisA. [–2,6)B. (–2,2]C. RD. (–2,6]E. [–6,2]

Question 3Theareaoftheregionenclosedbythegraphof y x x x= +( ) −( )2 4 andthex-axisis

A. 1283

B. 203

C. 2363

D. 1483

E. 36

SECTION 1

Instructions for Section 1Answerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1,anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.

3 2014MATHMETH(CAS)EXAM2

SECTION 1–continuedTURN OVER

Question 4Let f beafunctionwithdomainRsuchthat ′( ) = ′( ) < ≠f f x x5 0 0 5 and when .At x=5,thegraphof f hasaA. localminimum.B. localmaximum.C. gradientof5.D. gradientof–5.E. stationarypointofinflection.

Question 5TherandomvariableXhasanormaldistributionwithmean12andstandarddeviation0.5.IfZhasthestandardnormaldistribution,thentheprobabilitythatXislessthan11.5isequaltoA. Pr(Z>–1)B. Pr(Z<–0.5)C. Pr(Z>1)D. Pr(Z≥0.5)E. Pr(Z <1)

Question 6

Thefunction f D R: → withrule f x x x x( ) = − −2 9 1683 2 willhaveaninversefunctionforA. D = R

B. D=(7,∞)

C. D=(–4,8)

D. D=(–∞,0)

E. D = − ∞

12

,



Question 7

y

x

(–2, 3)

O–4

Theruleofthefunctionwhosegraphisshownaboveis

A. y = – 32

|x|+3

B. y = 23

|x +3|+2

C. y = 23

|2 + x|+3

D. y = – 32

|2 – x|+3

E. y = –32

|x+2|+3

Question 8

If f x dx( ) =∫ 61

4,then 5 2

1

4−( )∫ f x dx( ) isequalto

A. 3B. 4C. 5D. 6E. 16



Question 9Theinverseofthefunction f R R f x

x: ,+ → ( ) = +

1 4 is

A. f –1:(4,∞)→ R f xx

− ( ) =−( )

12

14

B. f –1:R+ → R f xx

− ( ) = +12

1 4

C. f –1:R+ → R f x x− ( ) = +( )1 24

D. f –1:(–4,∞)→ R f xx

− ( ) =+( )

12

14

E. f –1:(–∞,4)→ R f xx

− ( ) =−( )

12

14

Question 10Whichoneofthefollowingfunctionssatisfiesthefunctionalequation f f x x( )( ) = foreveryrealnumberx?A. f x x( ) = 2

B. f x x( ) = 2

C. f x x( ) = 2

D. f x x( ) = − 2

E. f x x( ) = −2

Question 11Abagcontainsfiveredmarblesandfourbluemarbles.Twomarblesaredrawnfromthebag,withoutreplacement,andtheresultsarerecorded.Theprobabilitythatthemarblesaredifferentcoloursis

A. 2081

B. 518

C. 49

D. 4081

E. 59



Question 12

ThetransformationT R R: 2 2→ withrule

Txy

xy

=

−

+ −

1 00 2

12

mapsthelinewithequation x y− =2 3 ontothelinewithequationA. x + y = 0B. x+4y = 0C. –x – y=4D. x+4y=–6E. x – 2y = 1

Question 13Thedomainofthefunctionh,where h x xa( ) = ( )cos log ( ) andaisarealnumbergreaterthan1,ischosensothat h isaone-to-onefunction.Whichoneofthefollowingcouldbethedomain?

A. a a−

π π2 2,

B. (0,p)

C. 1 2, aπ

D. a a−

π π2 2,

E. a a−

π π2 2,

Question 14IfXisarandomvariablesuchthatPr Pr ,X a X b>( ) = >( ) =5 8 and then Pr X X< <( )5 8 is

A. ab

B. a bb

−−1

C. 11−−ba

D. abb1−

E. ab−−

11



Question 15Zoehasarectangularpieceofcardboardthatis8cmlongand6cmwide.Zoecutssquaresofsidelengthxcentimetresfromeachofthecornersofthecardboard,asshowninthediagrambelow.

8 cm

6 cm

x cm

Zoeturnsupthesidestoformanopenbox.

ThevalueofxforwhichthevolumeoftheboxisamaximumisclosesttoA. 0.8B. 1.1C. 1.6D. 2.0E. 3.6

Question 16ThecontinuousrandomvariableX,withprobabilitydensityfunctionp(x),hasmean2andvariance5.

Thevalueof x p x dx2 ( )−∞

∞

∫ isA. 1B. 7C. 9D. 21E. 29

Question 17Thesimultaneouslinearequations ax–3y=5 and 3x – ay = 8 – a haveno solution forA. a=3B. a=–3C. botha=3anda=–3D. a ∈ R\{3}E. a ∈ R\[–3,3]



Question 18Thegraphof y = kx–4 intersectsthegraphof y = x2 + 2x attwodistinctpointsforA. k=6B. k>6ork < –2C. –2≤k≤6D. 6 2 3 6 2 3− ≤ ≤ +kE. k = –2

Question 19

JakeandAnitaarecalculatingtheareabetweenthegraphof y x= andthey-axisbetweeny=0andy=4.Jakeusesapartitioning,showninthediagrambelow,whileAnitausesadefiniteintegraltofindtheexactarea.

y

x

4

3

2

1

O

y x=

ThedifferencebetweentheresultsobtainedbyJakeandAnitaisA. 0

B. 223

C. 263

D. 14

E. 35



Question 20Thegraphofafunction,h,isshownbelow.

10

8

6

4

2

(6, 10)

(1, 4) (11, 4)

2 4 6 8 10 12

y = h(x)

y

x

TheaveragevalueofhisA. 4B. 5C. 6D. 7E. 10

Question 21ThetrapeziumABCDisshownbelow.ThesidesAB,BCandDAareofequallength,p.ThesizeoftheacuteangleBCDisxradians.

D C

A B

p p

p

x

Theareaofthetrapeziumisamaximumwhenthevalueofxis

A. 12

B. 6

C. 4

D. 3

E. 512


END OF SECTION 1

Question 22JohnandRebeccaareplayingdarts.Theresultofeachoftheirthrowsisindependentoftheresultofany

otherthrow.TheprobabilitythatJohnhitsthebullseyewithasinglethrowis14 .Theprobabilitythat

Rebeccahitsthebullseyewithasinglethrowis12 .JohnhasfourthrowsandRebeccahastwothrows.

TheratiooftheprobabilityofRebeccahittingthebullseyeatleastoncetotheprobabilityofJohnhittingthebullseyeatleastonceisA. 1:1B. 32:27C. 64:85D. 2:1E. 192:175


SECTION 2 –continuedTURN OVER

Question 1 (7marks)Thepopulationofwombatsinaparticularlocationvariesaccordingtotherule

n t t( ) = +

1200 400

3cos π ,wherenisthenumberofwombatsandtisthenumberofmonthsafter

1March2013.

a. Findtheperiodandamplitudeofthefunctionn. 2marks

b. Findthemaximumandminimumpopulationsofwombatsinthislocation. 2marks

c. Findn(10). 1mark

d. Overthe12monthsfrom1March2013,findthefractionoftimewhenthepopulationofwombatsinthislocationwaslessthann(10). 2marks

SECTION 2

Instructions for Section 2Answerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.


SECTION 2 – Question 2–continued

Question 2 (13marks)On1January2010,TasmaniaJoneswaswalkingthroughanice-coveredregionofGreenlandwhenhefoundalargeicecylinderthatwasmadeathousandyearsagobytheVikings.Astatuewasinsidetheicecylinder.Thestatuewas1mtallanditsbasewasatthecentreofthebaseofthecylinder.

1 m

h metres

d metres

Thecylinderhadaheightofhmetresandadiameterofdmetres.TasmaniaJonesfoundthatthevolumeofthecylinderwas216m3.Atthattime,1January2010,thecylinderhadnotchangedinathousandyears.ItwasexactlyasitwaswhentheVikingsmadeit.

a. Writeanexpressionforh intermsofd. 2marks


SECTION 2 – Question 2–continuedTURN OVER

b. Showthatthesurfaceareaofthecylinderexcludingthebase,Ssquaremetres,isgivenbythe

rule S dd

= +π 2

4864 . 1mark

TasmaniafoundthattheVikingsmadethecylindersothatS isaminimum.

c. FindthevalueofdforwhichSisaminimumandfindthisminimumvalueofS. 2marks

d. FindthevalueofhwhenSisaminimum. 1mark



On1January2010,TasmaniabelievedthatduetorecenttemperaturechangesinGreenland,theiceofthecylinderhadjuststartedmelting.Therefore,hedecidedtoreturnon1Januaryeachyeartomeasuretheicecylinder.Heobservesthatthevolumeoftheicecylinderdecreasesbyaconstantrateof10m3peryear.Assumethatthecylindricalshapeisretainedandd = 2h atthebeginningandasthecylindermelts.

e. WritedownanexpressionforVintermsofh. 1mark

f. Find dhdt

intermsofh. 3marks

g. Findtherateatwhichtheheightofthecylinderwillbedecreasingwhenthetopofthestatueisjustexposed. 1mark



h. Findtheyearinwhichthetopofthestatuewilljustbeexposed.(Assumethatthemeltingstartedon1January2010.) 2marks



Question 3 (11marks)Inacontrolledexperiment,Juantooksomemedicineat8pm.Theconcentrationofmedicineinhisbloodwasthenmeasuredatregularintervals.TheconcentrationofmedicineinJuan’sblood

ismodelledbythefunction c t tet

( ) = −52

32 ,t ≥0,wherecistheconcentrationofmedicineinhis

blood,inmilligramsperlitre,thoursafter8pm.Partofthegraphofthefunctioncisshownbelow.

c

0.5

O t

a. WhatwasthemaximumvalueoftheconcentrationofmedicineinJuan’sblood,inmilligramsperlitre,correcttotwodecimalplaces? 1mark

b. i. Findthevalueoft,inhours,correcttotwodecimalplaces,whentheconcentrationofmedicineinJuan’sbloodfirstreached0.5milligramsperlitre. 1mark

ii. FindthelengthoftimethattheconcentrationofmedicineinJuan’sbloodwasabove0.5milligramsperlitre.Expresstheanswerinhours,correcttotwodecimalplaces. 2marks



c. i. Whatwasthevalueoftheaveragerateofchangeoftheconcentrationofmedicinein

Juan’sbloodovertheinterval 23

3,

?Expresstheanswerinmilligramsperlitre

perhour,correcttotwodecimalplaces. 2marks

ii. Attimest1andt2 ,theinstantaneousrateofchangeoftheconcentrationofmedicinein

Juan’sbloodwasequaltotheaveragerateofchangeovertheinterval 23

3,

.

Findthevaluesoft1andt2 ,inhours,correcttotwodecimalplaces. 2marks

Aliciatookpartinasimilarcontrolledexperiment.However,sheusedadifferentmedicine.Theconcentrationofthisdifferentmedicinewasmodelledbythefunction n t Ate kt( ) = − ,t ≥0, whereAandk ∈ R+.

d. IfthemaximumconcentrationofmedicineinAlicia’sbloodwas0.74milligramsperlitreatt=0.5hours,findthevalueofA,correcttothenearestinteger. 3marks



Question 4 (14marks)Patriciaisagardenerandsheownsagardennursery.Shegrowsandsellsbasilplantsandcorianderplants.Theheights,incentimetres,ofthebasilplantsthatPatriciaissellingaredistributednormallywithameanof14cmandastandarddeviationof4cm.Thereare2000basilplantsinthenursery.

a. Patriciaclassifiesthetallest10percentofherbasilplantsassuper.

Whatistheminimumheightofasuperbasilplant,correcttothenearestmillimetre? 1mark

Patriciadecidesthatsomeofherbasilplantsarenotgrowingquicklyenough,sosheplanstomovethemtoaspecialgreenhouse.Shewillmovethebasilplantsthatarelessthan9cminheight.

b. HowmanybasilplantswillPatriciamovetothegreenhouse,correcttothenearestwholenumber? 2marks



Theheightsofthecorianderplants,xcentimetres,followtheprobabilitydensityfunction h x( ),where

h xx x

( ) =

< <

π π100 50

0 50

0

sin

otherwise

c. Statethemeanheightofthecorianderplants. 1mark

Patriciathinksthatthesmallest15percentofhercorianderplantsshouldbegivenanewtypeofplantfood.

d. Findthemaximumheight,correcttothenearestmillimetre,ofacorianderplantifitistobegiventhenewtypeofplantfood. 2marks

Patriciaalsogrowsandsellstomatoplantsthatsheclassifiesaseithertallorregular.Shefindsthat20percentofhertomatoplantsaretall.Acustomer,Jack,selectsntomatoplantsatrandom.

e. LetqbetheprobabilitythatatleastoneofJack’sntomatoplantsistall.

Findtheminimumvalueofnsothatq isgreaterthan0.95. 2marks



Inanothersectionofthenursery,acraftsmanmakesplantpots.Thepotsareclassifiedassmoothorrough.Thecraftsmanfinisheseachpotbeforestartingonthenext.Overaperiodoftime,itisfoundthatifoneplantpotissmooth,theprobabilitythatthenextoneissmoothis0.7,whileifoneplantpotisrough,theprobabilitythatthenextoneisroughisp,where0<p <1.Thevalueofpstaysfixedforaweekatatime,butcanvaryfromweektoweek.Thefirstpotmadeeachweekisalwaysasmoothpot.

f. i. Find,intermsofp,theprobabilitythatthethirdpotmadeinagivenweekissmooth. 2marks

ii. Inoneparticularweek,theprobabilitythatthethirdpotmadeissmoothis0.61.

Calculatethevalueofpinthisweek. 2marks

g. If,inanotherweek,p=0.8,findtheprobabilitythatthefifthpotmadethatweekissmooth. 2marks



Question 5 (13marks)

Let and f R R f x x x x g R R g x x x: , : , .→ ( ) = −( ) −( ) +( ) → ( ) = −3 1 3 82 4

a. Expressx4 – 8x intheform x x a x b c−( ) + +( )( )2 . 2marks

b. Describethetranslationthatmapsthegraphof y f x= ( ) ontothegraphof y g x= ( ) . 1mark

c. Findthevaluesofdsuchthatthegraphof y f x d= +( ) has i. onepositivex-axisintercept 1mark

ii. twopositivex-axisintercepts. 1mark

d. Findthevalueofn forwhichtheequation g x n( ) = hasonesolution. 1mark


END OF QUESTION AND ANSWER BOOK

e. Atthepoint u g u, ( ) ,( ) thegradientof y g x= ( ) ismandatthepoint v g v, ( ) ,( ) thegradientis–m,wheremisapositiverealnumber.

i. Findthevalueof u3 + v3. 2marks

ii. Finduandvif u + v=1. 1mark

f. i. Findtheequationofthetangenttothegraphof y g x= ( ) atthepoint p g p, ( )( ) . 1mark

ii. Findtheequationsofthetangentstothegraphof y g x= ( ) thatpassthroughthepointwithcoordinates 3

212, −

. 3marks

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 2014

MATHMETH (CAS) 2

THIS PAGE IS BLANK

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=

++ ≠ −+∫ 1

111 ,

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(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

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