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MATHEMATICAL METHODS (CAS)Written examination 1
Wednesday 5 November 2014 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• Questionandanswerbookof12pages,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.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2014
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2014
STUDENT NUMBER
Letter
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2014MATHMETH(CAS)EXAM1 2
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3 2014MATHMETH(CAS)EXAM1
TURN OVER
Question 1 (5marks)
a. If y x x= ( )2 sin ,find dydx. 2marks
b. If f x x( ) = +2 3 ,find ′( )f 1 . 3marks
InstructionsAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmustbeshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
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2014MATHMETH(CAS)EXAM1 4
Question 2 (2marks)
Let 22 14
5
xdx be−= ( )∫ log .
Findthevalueofb.
Question 3 (2marks)
Solve 2 2 3 0cos , .x x x( ) = − ≤ ≤for where π
Question 4 (2marks)
Solvetheequation 2 83 3 2x x− −= forx.
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5 2014MATHMETH(CAS)EXAM1
Question 5–continuedTURN OVER
Question 5 (7marks)
Considerthefunction f R f x x x: , , .−[ ]→ ( ) = −1 3 3 2 3
a. Findthecoordinatesofthestationarypointsofthefunction. 2marks
b. Ontheaxesbelow,sketchthegraphof f. Labelanyendpointswiththeircoordinates. 2marks
1
–1–2–3–4–5 54321
2
3
4
5
–5
–4
–3
–2
–1
O
y
x
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2014MATHMETH(CAS)EXAM1 6
c. Findtheareaenclosedbythegraphofthefunctionandthehorizontallinegivenbyy=4. 3marks
Question 6 (2marks)
Solve log loge ex x( ) − = ( )3 forx,wherex>0.
Question 7 (3marks)
If ′( ) = ( ) − ( )f x x x2 2cos sin and f π2
12
= ,find f x( ).
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7 2014MATHMETH(CAS)EXAM1
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Question 8 (4marks)Acontinuousrandomvariable,X,hasaprobabilitydensityfunctiongivenby
f xe x
x
x
( ) = ≥
<
−15
0
0 0
5
ThemedianofXism.
a. Determinethevalueofm. 2marks
b. Thevalueofmisanumbergreaterthan1.
Find Pr .X X m< ≤( )1 2marks
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2014MATHMETH(CAS)EXAM1 8
Question 9–continued
Question 9 (6marks)Sallyaimstowalkherdog,Mack,mostmornings.Iftheweatherispleasant,theprobabilitythat
shewillwalkMackis34,andiftheweatherisunpleasant,theprobabilitythatshewillwalkMack
is13.
Assumethatpleasantweatheronanymorningisindependentofpleasantweatheronanyothermorning.
a. Inaparticularweek,theweatherwaspleasantonMondaymorningandunpleasantonTuesdaymorning.
FindtheprobabilitythatSallywalkedMackonatleastoneofthesetwomornings. 2marks
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9 2014MATHMETH(CAS)EXAM1
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b. InthemonthofApril,theprobabilityofpleasantweatherinthemorningwas58.
i. FindtheprobabilitythatonaparticularmorninginApril,SallywalkedMack. 2marks
ii. Usingyouranswerfrompartb.i.,orotherwise,findtheprobabilitythatonaparticularmorninginApril,theweatherwaspleasant,giventhatSallywalkedMackthatmorning. 2marks
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2014MATHMETH(CAS)EXAM1 10
Question 10–continued
Question 10 (7marks)AlineintersectsthecoordinateaxesatthepointsU andVwithcoordinates(u,0)and(0,v),
respectively,whereuandvarepositiverealnumbersand52≤u≤6.
a. Whenu=6,thelineisatangenttothegraphof y ax bx= +2 atthepointQwithcoordinates(2,4),asshown.
O
V (0, v)
Q (2, 4)
U (u, 0)
y
x
Ifaandbarenon-zerorealnumbers,findthevaluesofaandb. 3marks
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11 2014MATHMETH(CAS)EXAM1
Question 10–continuedTURN OVER
b. TherectangleOPQRhasavertexatQontheline.ThecoordinatesofQare(2,4),asshown.
O
R
P
V (0, v)
Q (2, 4)
U (u, 0)
y
x
i. Findanexpressionforvintermsofu. 1mark
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2014MATHMETH(CAS)EXAM1 12
END OF QUESTION AND ANSWER BOOK
ii. Findtheminimumtotalshadedareaandthevalueofuforwhichtheareaisaminimum. 2marks
iii. Findthemaximumtotalshadedareaandthevalueofuforwhichtheareaisamaximum. 1mark
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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 2014
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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=
++ ≠ −+∫ 1
111 ,
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