2012 mathematical methods (cas) written examination 2€¦ · mathematical methods (cas) written...
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STUDENT NUMBER Letter
MATHEMATICAL METHODS (CAS)
Written examination 2Thursday 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 bookSection Number of
questionsNumber of questions
to be answeredNumber of
marks
12
225
225
2258
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• 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 Certificate of Education 2012
2012MATHMETH(CAS)EXAM2 2
SECTION 1–continued
Question 1Thefunctionwithrulef (x)=–3sin π x
5
hasperiod
A. 3
B. 5
C. 10
D. π5
E. π10
Question 2Forthefunctionwithrulef (x)=x3–4x,theaveragerateofchangeoff (x)withrespecttoxontheinterval [1,3]isA. 1B. 3C. 5D. 6E. 9
Question 3Therangeofthefunctionf :[–2,3)→ R, f (x)=x2 – 2x–8isA. RB. (−9,–5]C. (–5,0)D. [−9,0]E. [–9,–5)
SECTION 1
Instructions for Section 1Answerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1,anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.
3 2012MATHMETH(CAS)EXAM2
SECTION 1–continuedTURN OVER
Question 4Giventhatgisadifferentiablefunctionandkisarealnumber,thederivativeofthecompositefunction g (ekx)isA. kg′(ekx)ekx
B. kg (ekx)
C. kekx g (ekx)
D. kekx g′(ex)
E. 1k
ekxg′(ekx)
Question 5Lettheruleforafunctiongbeg (x)=loge((x–2)
2).Forthefunctiong,theA. maximaldomain=R+andrange=RB. maximaldomain=R\{2}andrange=RC. maximaldomain=R\{2}andrange=(–2,∞)D. maximaldomain=[2,∞)andrange=(0,∞)E. maximaldomain=[2,∞)andrange=[0,∞)
Question 6Asectionofthegraphoff isshownbelow.
y
x
4π
2π3
4π−
2π−
4π− O
TheruleoffcouldbeA. f (x)=tan(x)
B. f (x)=tan x −
π4
C. f (x)=tan 24
x −
π
D. f (x)=tan 22
x −
π
E. f (x)=tan12 2
x −
π
2012MATHMETH(CAS)EXAM2 4
SECTION 1–continued
Question 7Thetemperature,T°C,insideabuildingthoursaftermidnightisgivenbythefunction
f :[0,24]→ R,T(t)=22–10cosπ12
2t −( )
Theaveragetemperatureinsidethebuildingbetween2amand2pmisA. 10°CB. 12°CC. 20°CD. 22°CE. 32°C
Question 8Thefunctionf :R→R,f (x)=ax3 + bx2 + cx,whereaisanegativerealnumberandbandcarerealnumbers.Fortherealnumbersp < m < 0 < n < q,wehavef (p)=f (q)=0andf ′(m)=f ′(n)=0.Thegradientofthegraphofy=f (x)isnegativeforA. (–∞,m)∪(n,∞)B. (m,n)C. (p,0)∪(q,∞)D. (p,m)∪(0,q)E. (p,q)
Question 9Thenormaltothegraphof y b x= − 2 hasagradientof3whenx =1.Thevalueofbis
A. −109
B. 109
C. 4
D. 10
E. 11
Question 10Theaveragevalueofthefunctionf :[0,2π]→ R,f (x)=sin2(x)overtheinterval[0,a]is0.4.Thevalueofa,tothreedecimalplaces,isA. 0.850B. 1.164C. 1.298D. 1.339E. 4.046
5 2012MATHMETH(CAS)EXAM2
SECTION 1–continuedTURN OVER
Question 11Theweightsofbagsofflourarenormallydistributedwithmean252gandstandarddeviation12g.Themanufacturersaysthat40%ofbagsweighmorethanxg.ThemaximumpossiblevalueofxisclosesttoA. 249.0B. 251.5C. 253.5D. 254.5E. 255.0
Question 12Demelzaisabadmintonplayer.Ifshewinsagame,theprobabilitythatshewillwinthenextgameis0.7.Ifshelosesagame,theprobabilitythatshewilllosethenextgameis0.6.Demelzahasjustwonagame.TheprobabilitythatshewillwinexactlyoneofhernexttwogamesisA. 0.33B. 0.35C. 0.42D. 0.49E. 0.82
Question 13AandBareeventsofasamplespaceS.
Pr(A ∩ B)= 25andPr(A ∩ B′ )= 3
7.
Pr(B′ |A)isequalto
A. 635
B. 1529
C. 1435
D. 2935
E. 23
2012MATHMETH(CAS)EXAM2 6
SECTION 1–continued
Question 14
Thegraphof f :R+∪{0} → R, f (x)= x isshownbelow.Inordertofindanapproximationtotheareaoftheregionboundedbythegraphoff,they-axisandtheliney=4,Zoedrawsfourrectangles,asshown,andcalculatestheirtotalarea.
4
3
2
1
O
y x=
x
y
Zoe’sapproximationtotheareaoftheregionisA. 14
B. 21
C. 29
D. 30
E. 643
7 2012MATHMETH(CAS)EXAM2
SECTION 1–continuedTURN OVER
Question 15If f ′(x)=3x2–4,whichoneofthefollowinggraphscouldrepresentthegraphofy=f (x)?
A. y
x
B. y
x
C. y
x
D. y
x
E. y
x
Question 16Thegraphofacubicfunctionfhasalocalmaximumat(a,–3)andalocalminimumat(b,–8).Thevaluesofc,suchthattheequationf (x)+c=0hasexactlyonesolution,areA. 3<c < 8B. c >–3orc < –8C. –8 < c <–3D. c <3orc > 8E. c < –8
2012MATHMETH(CAS)EXAM2 8
SECTION 1–continued
Question 17Asystemofsimultaneouslinearequationsisrepresentedbythematrixequation
mm
xy m
31 2
1+
=
.
Thesystemofequationswillhaveno solutionwhenA. m=1B. m=–3C. m ∈{1,–3}D. m ∈ R\{1}E. m ∈{1,3}
Question 18Thetangenttothegraphofy=loge(x)atthepoint(a,loge(a))crossesthex-axisatthepoint(b,0),whereb <0.Whichofthefollowingisfalse?A. 1 < a < e
B. Thegradientofthetangentispositive
C. a > e
D. Thegradientofthetangentis 1a
E. a > 0
Question 19Afunction f hasthefollowingtwopropertiesforallrealvaluesofθ .
f (π – θ )=–f (θ )and f (π – θ )=–f (–θ )
Apossiblerulefor f isA. f (x)=sin(x)
B. f (x)=cos(x)
C. f (x)=tan(x)
D. f (x)=sin x2
E. f (x)=tan(2x)
Question 20AdiscreterandomvariableXhastheprobabilityfunctionPr(X=k)=(1–p)kp, wherekisanon-negativeinteger.Pr(X >1)isequaltoA. 1 – p + p2
B. 1 – p2
C. p – p2
D. 2p – p2
E. (1–p)2
9 2012MATHMETH(CAS)EXAM2
END OF SECTION 1TURN OVER
Question 21
Thevolume,Vcm3,ofwaterinacontainerisgivenbyV=13πh3wherehcmisthedepthofwaterinthe
containerattimetminutes.Waterisdrainingfromthecontainerataconstantrateof300cm3/min.Therateofdecreaseofh,incm/min,whenh=5is
A. 12π
B. 4π
C. 25π
D. 60π
E. 30π
Question 22Thegraphofadifferentiablefunctionfhasalocalmaximumat(a,b),wherea <0andb >0,andalocalminimumat(c,d ),wherec>0andd <0.Thegraphofy=–|f (x–2)|hasA. 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)
2012MATHMETH(CAS)EXAM2 10
SECTION 2 – Question 1–continued
Question 1Asolidblockintheshapeofarectangularprismhasabaseofwidthxcm.Thelengthofthebaseis two-and-a-halftimesthewidthofthebase.
52x cm
x cm
h cm
Theblockhasatotalsurfaceareaof6480sqcm.
a. Showthatiftheheightoftheblockishcm,h= 6480 57
2− xx
.
2marks
SECTION 2
Instructions for Section 2Answerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequiredanexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
11 2012MATHMETH(CAS)EXAM2
SECTION 2–continuedTURN OVER
b. Thevolume,Vcm3,oftheblockisgivenbyV(x)= 5 6480 514
2x x( ).−
GiventhatV(x)>0andx >0,findthepossiblevaluesofx.
2marks
c. Find dVdx
,expressingyouranswerintheform dVdx
=ax2 + b,whereaandbarerealnumbers.
3marks
d. Findtheexactvaluesofxandhiftheblockistohavemaximumvolume.
2marks
2012MATHMETH(CAS)EXAM2 12
SECTION 2 – Question 2–continued
Question 2
Letf :R\{2} → R, f (x)= 12 4x −
+3.
a. Sketchthegraphofy=f (x)onthesetofaxesbelow.Labeltheaxesinterceptswiththeircoordinatesandlabeleachoftheasymptoteswithitsequation.
x
y
O
3marks
b. i. Findf ′(x).
ii. Statetherangeoff ′.
iii. Usingtheresultof part ii. explainwhyfhasnostationarypoints.
1+1+1=3marks
13 2012MATHMETH(CAS)EXAM2
SECTION 2 – Question 2–continuedTURN OVER
c. If(p,q)isanypointonthegraphofy=f (x),showthattheequationofthetangenttoy=f (x)atthispointcanbewrittenas(2p–4)2(y–3)=–2x+4p–4.
2marks
2012MATHMETH(CAS)EXAM2 14
SECTION 2 – Question 2–continued
d. Findthecoordinatesofthepointsonthegraphofy=f (x)suchthatthetangentstothegraphatthese
pointsintersectat −
1, 7
2.
4marks
15 2012 MATHMETH(CAS) EXAM 2
SECTION 2 – continuedTURN 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) = 1x
has rule Txy
a xy
cd
= +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 3Steve, 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. WhatistheprobabilitythatStevewillanswerthefirstthreequestionsofthissetcorrectly?
ii. Find,tofourdecimalplaces,theprobabilitythatStevewillansweratleast10ofthefirst20 questions of this set correctly.
iii. Use the fact that the variance of X is 7516
to show that the value of n is 25.
1 + 2 + 1 = 4 marks
17 2012MATHMETH(CAS)EXAM2
SECTION 2 – Question 3–continuedTURN OVER
IfKaterinaanswersaquestioncorrectly,theprobabilitythatshewillanswerthenextquestioncorrectly is 3
4 . Ifsheanswersaquestionincorrectly,theprobabilitythatshewillanswerthenextquestion incorrectlyis 2
3 .
Inaparticularset,KaterinaanswersQuestion1incorrectly.b. i. CalculatetheprobabilitythatKaterinawillanswerQuestions3,4and5correctly.
ii. FindtheprobabilitythatKaterinawillanswerQuestion25correctly.Giveyouranswercorrecttofourdecimalplaces.
3+2=5marks
2012MATHMETH(CAS)EXAM2 18
SECTION 2–continued
c. TheprobabilitythatJesswillansweranyquestioncorrectly,independentlyofheranswertoanyotherquestion,isp (p > 0). LettherandomvariableYbethenumberofquestionsthatJessanswerscorrectlyinanysetof25.
IfPr(Y >23)=6Pr(Y =25),showthatthevalueofp is 56
.
2marks
d. Fromthesesetsof25questionsbeingcompletedbymanystudents,ithasbeenfoundthatthetime,inminutes,thatanystudenttakestoanswereachsetof25questionsisanotherrandomvariable,W,whichisnormally distributedwithmeana andstandarddeviationb.
Itturnsoutthat,forJess,Pr(Y≥18)=Pr(W≥20)andalsoPr(Y≥22)=Pr(W≥25). Calculatethevaluesofaandb,correcttothreedecimalplaces.
4marks
19 2012MATHMETH(CAS)EXAM2
SECTION 2 – Question 4–continuedTURN OVER
Question 4TasmaniaJonesisinthejungle,searchingfortheQuetzalotltribe’svaluableemeraldthathasbeenstolenandhiddenbyaneighbouringtribe.Tasmaniahasheardthattheemeraldhasbeenhiddeninatankshapedlikeaninvertedcone,withaheightof10metresandadiameterof4metres(asshownbelow).Theemeraldisonashelf.Thetankhasapoisonousliquidinit.
r m
4 m
tap
shelf
10 m
h m
2 m
a. Ifthedepthoftheliquidinthetankishmetres i. findtheradius,rmetres,ofthesurfaceoftheliquidintermsofh
ii. showthatthevolumeoftheliquidinthetankisπh3
75m3.
1+1=2marks
2012MATHMETH(CAS)EXAM2 20
SECTION 2 – Question 4–continued
Thetankhasatapatitsbasethatallowstheliquidtorunoutofit.Thetankisinitiallyfull.Whenthetapisturnedon,theliquidflowsoutofthetankatsucharatethatthedepth,hmetres,oftheliquidinthetankisgivenby
h t t= + −10 11600
12003( ),
wheret minutesisthelengthoftimeafterthetapisturnedonuntilthetankisempty.b. Showthatthetankisemptywhent=20.
1mark
c. Whent =5minutes,find i. thedepthoftheliquidinthetank
ii. therateatwhichthevolumeoftheliquidisdecreasing,correcttoonedecimalplace.
1+3=4marks
21 2012MATHMETH(CAS)EXAM2
SECTION 2–continuedTURN OVER
d. Theshelfonwhichtheemeraldisplacedis2metresabovethevertexofthecone. Fromthemomenttheliquidstartstoflowfromthetank,findhowlong,inminutes,ittakesuntilh=2.
(Giveyouranswercorrecttoonedecimalplace.)
2marks
e. Assoonasthetankisempty,thetapturnsitselfoffandpoisonousliquidstartstoflowintothetankatarateof0.2m3/minute.
Howlong,inminutes,afterthetankisfirstemptywilltheliquidonceagainreachadepthof2metres?
2marks
f. Inordertoobtaintheemerald,TasmaniaJonesentersthetankusingavinetoclimbdownthewallofthetankassoonasthedepthoftheliquidisfirst2metres.Hemustleavethetankbeforethedepthisagaingreaterthan2metres.
Findthelengthoftime,inminutes,correcttoonedecimalplace,thatTasmaniaJoneshasfromthetimeheentersthetanktothetimeheleavesthetank.
1mark
2012MATHMETH(CAS)EXAM2 22
SECTION 2 – Question 5–continued
Question 5TheshadedregioninthediagrambelowistheplanofaminesitefortheBlackPossumminingcompany. Alldistancesareinkilometres.Twooftheboundariesoftheminesiteareintheshapeofthegraphsofthefunctions
f :R → R,f (x)=ex 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,findtheareaoftheregionboundedbythegraphofg,thexandyaxes,andtheliney=–2.
23 2012MATHMETH(CAS)EXAM2
SECTION 2 – Question 5–continuedTURN OVER
iii. Findthetotalareaoftheshadedregion.
1+1+1=3marks
b. Theminingengineer,Victoria,decidesthatabettersiteforthemineistheregionboundedbythegraphofgandthatofanewfunctionk:(–∞,a)→ R,k(x)=–loge(a – x),whereaisapositiverealnumber.
i. Find,intermsofa,thex-coordinatesofthepointsofintersectionofthegraphsofgandk.
ii. Hence,findthesetofvaluesofa,forwhichthegraphsofgandkhavetwodistinctpointsofintersection.
2+1=3marks
2012MATHMETH(CAS)EXAM2 24
c. Forthenewminesite,thegraphsofgandkintersectattwodistinctpoints,AandB.ItisproposedtostartminingoperationsalongthelinesegmentAB,whichjoinsthetwopointsofintersection.
Victoriadecidesthatthegraphofkwillbesuchthatthex-coordinateofthemidpointofABis 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)
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