2012 Mathematical Methods (CAS) Written examination 2€¦ · MATHEMATICAL METHODS (CAS) Written examination 2 Thursday 8 November 2012 Reading time: ... , the derivative of the composite
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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)
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
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
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
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.
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.
f. Inordertoobtaintheemerald,TasmaniaJonesentersthetankusingavinetoclimbdownthewallofthetankassoonasthedepthoftheliquidisfirst2metres.Hemustleavethetankbeforethedepthisagaingreaterthan2metres.
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.