2012 mathematical methods (cas) exam 2
Post on 14-Apr-2018
223 Views
Preview:
TRANSCRIPT
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 1/27
Figures
Words
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 2/27
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.
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 3/27
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 4/27
2012MATHMETH(CAS)EXAM2 4
SECTION 1–continued
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 5/27
5 2012MATHMETH(CAS)EXAM2
SECTION 1–continued
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 6/27
2012MATHMETH(CAS)EXAM2 6
SECTION 1–continued
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 7/27
7 2012MATHMETH(CAS)EXAM2
SECTION 1–continued
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 8/27
2012MATHMETH(CAS)EXAM2 8
SECTION 1–continued
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 9/27
9 2012MATHMETH(CAS)EXAM2
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)
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 10/27
2012MATHMETH(CAS)EXAM2 10
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.
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 11/27
11 2012MATHMETH(CAS)EXAM2
SECTION 2–continued
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 12/27
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 13/27
13 2012MATHMETH(CAS)EXAM2
SECTION 2 – Question 2–continued
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 14/27
2012MATHMETH(CAS)EXAM2 14
SECTION 2 – Question 2–continued
d. Findthecoordinatesofthepointsonthegraphof y= f ( x)suchthatthetangentstothegraphatthese
pointsintersectat −
1,
7
2.
4marks
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 15/27
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 16/27
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 17/27
17 2012MATHMETH(CAS)EXAM2
SECTION 2 – Question 3–continued
TURN OVER
IfKaterinaanswersaquestioncorrectly,theprobabilitythatshewillanswerthenextquestioncorrectly
is3
4. Ifsheanswersaquestionincorrectly,theprobabilitythatshewillanswerthenextquestion
incorrectlyis2
3.
Inaparticularset,KaterinaanswersQuestion1incorrectly.
b. i. CalculatetheprobabilitythatKaterinawillanswerQuestions3,4and5correctly.
ii. FindtheprobabilitythatKaterinawillanswerQuestion25correctly.Giveyouranswercorrectto
fourdecimalplaces.
3+2=5marks
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 18/27
2012MATHMETH(CAS)EXAM2 18
SECTION 2–continued
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 19/27
19 2012MATHMETH(CAS)EXAM2
SECTION 2 – Question 4–continued
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 20/27
2012MATHMETH(CAS)EXAM2 20
SECTION 2 – Question 4–continued
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 21/27
21 2012MATHMETH(CAS)EXAM2
SECTION 2–continued
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 22/27
2012MATHMETH(CAS)EXAM2 22
SECTION 2 – Question 5–continued
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.
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 23/27
23 2012MATHMETH(CAS)EXAM2
SECTION 2 – Question 5–continued
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 24/27
2012MATHMETH(CAS)EXAM2 24
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 25/27
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
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 26/27
MATHMETH (CAS) 2
This page is blank
7/27/2019 2012 Mathematical Methods (CAS) Exam 2
http://slidepdf.com/reader/full/2012-mathematical-methods-cas-exam-2 27/27
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
top related