MATHEMATICAL METHODS (CAS) · Mathematical Methods (CAS) Formulas Mensuration area of a trapezium:...
Transcript of MATHEMATICAL METHODS (CAS) · Mathematical Methods (CAS) Formulas Mensuration area of a trapezium:...
MATHEMATICAL METHODS (CAS)Written examination 1
Wednesday 4 November 2015 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,correctionfluid/tapeoracalculatorofanytype.
Materials supplied• Questionandanswerbookof16pages,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.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2015
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2015
STUDENT NUMBER
Letter
2015MATHMETH(CAS)EXAM1 2
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3 2015MATHMETH(CAS)EXAM1
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Question 1 (4marks)a. Lety=(5x+1)7.
Find dydx. 1mark
b. Let f x xxe( ) log ( ) .= 2
i. Findf ′ (x). 2marks
ii. Evaluatef ′ (1). 1mark
InstructionsAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmustbeshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
2015MATHMETH(CAS)EXAM1 4
Question 2 (3marks)
Let ′ = −f xx
( ) ,1 3 wherex≠0.
Giventhatf (e)=–2,find f (x).
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Question 3 (2marks)
Evaluate 11
4
xdx
∫ .
2015MATHMETH(CAS)EXAM1 6
Question 4–continued
Question 4 (6marks)Considerthefunction f R f x x x: , , ( ) .−[ ]→ = + −( )3 2 1
23 43 2
a. Findthecoordinatesofthestationarypointsofthefunction. 2marks
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Theruleforfcanalsobeexpressedas f x x x( ) .= −( ) +( )12
1 2 2
b. Ontheaxesbelow,sketchthegraphoff, clearlyindicatingaxisinterceptsandturningpoints. Labeltheendpointswiththeircoordinates. 2marks
y
x–8 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 8
–8–7–6–5–4–3–2–1
1
O
2345678
c. Findtheaveragevalueoffovertheinterval0≤x≤2. 2marks
2015 MATHMETH (CAS) EXAM 1 8
Question 5 (3 marks)On any given day, the depth of water in a river is modelled by the function
h t t t( ) sin ,= +
≤ ≤14 8
120 24π
where h is the depth of water, in metres, and t is the time, in hours, after 6 am.
a. Find the minimum depth of the water in the river. 1 mark
b. Find the values of t for which h(t) = 10. 2 marks
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Question 6 (3marks)LettherandomvariableXbenormallydistributedwithmean2.5andstandarddeviation0.3LetZbethestandardnormalrandomvariable,suchthatZ~N(0,1).
a. FindbsuchthatPr(X>3.1)=Pr(Z<b). 1mark
b. Usingthefactthat,correcttotwodecimalplaces,Pr(Z<–1)=0.16,findPr(X<2.8|X>2.5). Writetheanswercorrecttotwodecimalplaces. 2marks
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Question 7 (5marks)a. Solvelog2(6–x)–log2(4–x)=2forx,wherex<4. 2marks
b. Solve3et=5+8e–tfort. 3marks
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Question 8 (3marks)
ForeventsA and B fromasamplespace,Pr( )A B =34 andPr( ) .B =
13
a. CalculatePr(A ∩B). 1mark
b. CalculatePr(A′ ∩B),whereA′denotesthecomplementofA. 1mark
c. IfeventsAandB areindependent,calculatePr(A∪B). 1mark
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Question 9–continued
Question 9 (4marks)AneggmarketingcompanybuysitseggsfromfarmAandfarmB.Letpbetheproportionofeggsthatthecompanybuysfromfarm A.Therestofthecompany’seggscomefromfarmB.Eachday,theeggsfrombothfarmsaretakentothecompany’swarehouse.
Assumethat35 ofalleggsfromfarmAhavewhiteeggshellsand
15 ofalleggsfromfarmBhave
whiteeggshells.
a. Aneggisselectedatrandomfromthesetofalleggsatthewarehouse.
Find,intermsofp,theprobabilitythattheegghasawhiteeggshell. 1mark
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b. Anothereggisselectedatrandomfromthesetofalleggsatthewarehouse.
i. Giventhattheegghasawhiteeggshell,find,intermsofp,theprobabilitythatitcamefromfarmB. 2marks
ii. IftheprobabilitythatthiseggcamefromfarmB is0.3,findthevalueofp. 1mark
2015MATHMETH(CAS)EXAM1 14
Question 10–continued
Question 10(7marks)Thediagrambelowshowsapoint,T,onacircle.Thecirclehasradius2andcentreatthepointC
withcoordinates(2,0).TheangleECTisθ,where 02
< ≤θπ .
x
Y
B (2, b)
T
D (4, d)
y
E (4, 0)C (2, 0)O Xθ
ThediagramalsoshowsthetangenttothecircleatT.ThistangentisperpendiculartoCTandintersectsthex-axisatpointXandthey-axisatpointY.
a. FindthecoordinatesofTintermsofθ. 1mark
b. FindthegradientofthetangenttothecircleatTintermsofθ. 1mark
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c. TheequationofthetangenttothecircleatTcanbeexpressedas
cos(θ)x+sin(θ)y=2+2cos(θ)
i. PointB,withcoordinates(2,b),isonthelinesegmentXY.
Findbintermsofθ. 1mark
ii. Point D,withcoordinates(4,d),isonthelinesegmentXY.
Finddintermsofθ. 1mark
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END OF QUESTION AND ANSWER BOOK
d. ConsiderthetrapeziumCEDBwithparallelsidesoflengthbandd.
FindthevalueofθforwhichtheareaofthetrapeziumCEDB isaminimum.Alsofindtheminimumvalueofthearea. 3marks
MATHEMATICAL METHODS (CAS)
Written examinations 1 and 2
FORMULA SHEET
Instructions
Detach this formula sheet during reading time.
This formula sheet is provided for your reference.
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2015
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: πr 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