MATHEMATICAL METHODS (CAS)Written examination 1
Wednesday 6 November 2013 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• Questionandanswerbookof14pages,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.
VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2013
Figures
Words
STUDENT NUMBER Letter
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2013
2013MATHMETH(CAS)EXAM1 2
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Question 1 (5marks)
a. Ify=x2loge(x),finddydx . 2marks
b. Letf (x)=ex2.
Findf ' (3). 3marks
InstructionsAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmustbeshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
2013MATHMETH(CAS)EXAM1 4
Question 2 (2marks)Findananti-derivativeof(4–2x)–5withrespecttox.
Question 3 (2marks)Thefunctionwithrule g(x)hasderivativeg′(x)=sin(2πx).
Giventhatg(1)=1π,findg(x).
5 2013MATHMETH(CAS)EXAM1
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Question 4 (2marks)
Solvetheequation sin x2
12
= − forx [2π,4π].
Question 5 (4marks)a. Solvetheequation2log3(5)–log3(2)+log3(x)=2forx. 2marks
b. Solvetheequation3–4x =96–xforx. 2marks
2013MATHMETH(CAS)EXAM1 6
Question 7–continued
Question 6 (3marks)Letg:R → R, g(x)=(a–x)2,whereaisarealconstant.
Theaveragevalueofgontheinterval[–1,1]is3112 .
Findallpossiblevaluesofa.
Question 7 (6marks)Theprobabilitydistributionofadiscreterandomvariable,X,isgivenbythetablebelow.
x 0 1 2 3 4
Pr(X=x) 0.2 0.6p2 0.1 1- p 0.1
a. Showthatp=23 orp=1. 3marks
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b. Letp=23 .
i. CalculateE(X). 2marks
ii. FindPr(X ≥E(X)). 1mark
2013MATHMETH(CAS)EXAM1 8
Question 8 (3marks)Acontinuousrandomvariable,X,hasaprobabilitydensityfunction
f xx x
( ) =
∈[ ]
π π4 4
0 2
0
cos ,if
otherwise
Giventhatddx
x x x x xsin cos sinπ π π π4 4 4 4
=
+
,findE(X).
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CONTINUES OVER PAGE
2013MATHMETH(CAS)EXAM1 10
Question 9–continued
Question 9 (6marks)Thegraphoff (x)=(x -1)2-2,x [-2,2],isshownbelow.Thegraphintersectsthex-axiswherex=a.
7
8
9
6
5
4
3
2
1
O
–1
–2
–3
–3 –2 –1 1 2 3
y
xa
a. Findthevalueofa. 1mark
b. Ontheaxesabove,sketchthegraphofg(x)=| f (x)|+1,forx [-2,2].Labeltheendpointswiththeircoordinates. 2marks
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c. Thefollowingsequenceoftransformationsisappliedtothegraphofthefunction g:[–2,2]→ R, g(x)=| f (x)|+1.
• atranslationofoneunitinthenegativedirectionofthex-axis
• atranslationofoneunitinthenegativedirectionofthey-axis
• adilationfromthex-axisoffactor13
Find i. theruleoftheimageofg afterthesequenceoftransformationshasbeenapplied 2marks
ii. thedomainoftheimageofgafterthesequenceoftransformationshasbeenapplied. 1mark
2013MATHMETH(CAS)EXAM1 12
Question 10 –continued
Question 10 (7marks)
Letf:[0,∞)→R,f(x)=2 5ex- .
Aright-angledtriangleOQPhasvertexOattheorigin,vertexQonthex-axisandvertexPonthegraphoff,asshown.ThecoordinatesofPare(x,f(x)).
y = f(x)
P(x, f(x))
y
O Qx
a. Findthearea,A,ofthetriangleOQPintermsofx. 1mark
13 2013MATHMETH(CAS)EXAM1
Question 10 –continuedTURN OVER
b. FindthemaximumareaoftriangleOQPandthevalueofxforwhichthemaximumoccurs. 3marks
2013MATHMETH(CAS)EXAM1 14
END OF QUESTION AND ANSWER BOOK
c. LetSbethepointonthegraphoffonthey-axisandletTbethepointonthegraphoffwith
they-coordinate12.
FindtheareaoftheregionboundedbythegraphoffandthelinesegmentST. 3marks
y = f(x)
S
y
xO
12
T
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 2013
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(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
