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

7 2015MATHMETH(CAS)EXAM1

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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

9 2015MATHMETH(CAS)EXAM1

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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

2015MATHMETH(CAS)EXAM1 10

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

2015MATHMETH(CAS)EXAM1 12

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

15 2015MATHMETH(CAS)EXAM1

Question 10–continuedTURN OVER

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

2015MATHMETH(CAS)EXAM1 16

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