2012 mathematical methods (cas) written examination 1 · 2012 mathmeth(cas) exam 1 10 question 10...
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MATHEMATICAL METHODS (CAS)Written examination 1
Wednesday 7 November 2012 Reading time: 9.00 am to 9.15 am (15 minutes) Writing time: 9.15 am to 10.15 am (1 hour)
QUESTION AND ANSWER BOOK
Structure of bookNumber of questions
Number of questions to be answered
Number of marks
10 10 40
• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulers.
• StudentsareNOTpermittedtobringintotheexaminationroom:notesofanykind,blanksheetsofpaper,whiteoutliquid/tapeoracalculatorofanytype.
Materials supplied• Questionandanswerbookof10pages,withadetachablesheetofmiscellaneousformulasinthe
centrefold.• Workingspaceisprovidedthroughoutthebook.
Instructions• Detachtheformulasheetfromthecentreofthisbookduringreadingtime.• Writeyourstudent numberinthespaceprovidedaboveonthispage.
• AllwrittenresponsesmustbeinEnglish.
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)EXAM1 2
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3 2012 MATHMETH(CAS) EXAM 1
TURN OVER
Question 1
a. If y = (x2 – 5x)4, find dydx
.
1 mark
b. If f (x) =x
xsin( ), find f ' π
2.
2 marks
InstructionsAnswer all questions in the spaces provided.In all questions where a numerical answer is required an exact value must be given unless otherwise specified.In questions where more than one mark is available, appropriate working must be shown.Unless otherwise indicated, the diagrams in this book are not drawn to scale.
2012MATHMETH(CAS)EXAM1 4
Question 2
Findananti-derivativeof 12 1 3x −( )
withrespecttox.
2marks
Question 3Theruleforfunctionhish(x)=2x3+1.Findtherulefortheinversefunctionh–1.
2marks
5 2012MATHMETH(CAS)EXAM1
TURN OVER
Question 4Onanygivenday,thenumberXoftelephonecallsthatDanielreceivesisarandomvariablewithprobabilitydistributiongivenby
x 0 1 2 3
Pr(X = x) 0.2 0.2 0.5 0.1
a. FindthemeanofX.
2marks
b. WhatistheprobabilitythatDanielreceivesonlyonetelephonecalloneachofthreeconsecutivedays?
1mark
c. DanielreceivestelephonecallsonbothMondayandTuesday. WhatistheprobabilitythatDanielreceivesatotaloffourcallsoverthesetwodays?
3marks
2012MATHMETH(CAS)EXAM1 6
Question 5a. Sketchthegraphoff :[0,5]→R, f (x)=–|x–3|+2.Labeltheaxesinterceptsandendpointswiththeir
coordinates.y
x
1
1 2 3 4 5 6 7
–1
–1–2–3–4–5–6–7
–2
–3
–4
–5
–6
–7
2
3
4
5
6
7
O
3marks
b. i. Findthecoordinatesoftheimageofthepoint(3,2)underareflectioninthex-axis,followedbyatranslationof5unitsinthepositivedirectionofthex-axis.
ii. Findtheequationoftheimageofthegraphoffunderareflectioninthex-axis,followedbyatranslationof5unitsinthepositivedirectionofthex-axis.
1+2=3marks
7 2012MATHMETH(CAS)EXAM1
TURN OVER
Question 6Thegraphsofy=cos(x)andy = asin(x),whereaisarealconstant,haveapointofintersectionat x = π
3.
a. Findthevalueofa.
2marks
b. Ifx ∈[0,2π],findthex-coordinateoftheotherpointofintersectionofthetwographs.
1mark
Question 7Solvetheequation2loge(x+2)–loge(x)=loge(2x+1),wherex>0,forx.
3marks
2012 MATHMETH(CAS) EXAM 1 8
Question 8a. The random variable X is normally distributed with mean 100 and standard deviation 4. If Pr(X < 106) = q, find Pr(94 < X < 100) in terms of q.
2 marks
b. The probability density function f of a random variable X is given by
f (x) = x x+
≤ ≤1
124
0
0
otherwise
Find the value of b such that Pr(X ≤ b) = 58
.
3 marks
9 2012 MATHMETH(CAS) EXAM 1
TURN OVER
Question 9a. Let f : R → R, f (x) = x sin (x). Find f '(x).
1 mark
b. Use the result of part a. to find the value of x xcos( )π
π
6
2∫ dx in the form aπ + b.
3 marks
2012MATHMETH(CAS)EXAM1 10
Question 10Let f :R→R, f (x)=e–mx + 3x,wheremisapositiverationalnumber.a. i. Find,intermsofm,thex-coordinateofthestationarypointofthegraphofy = f (x).
ii. Statethevaluesofmsuchthatthex-coordinateofthisstationarypointisapositivenumber.
2+1=3marks
b. Foraparticularvalueofm,thetangenttothegraphofy = f (x)atx=–6passesthroughtheorigin. Findthisvalueofm.
3marks
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
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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