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

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.

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.

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

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( ).

7 2014MATHMETH(CAS)EXAM1

TURN OVER

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

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

9 2014MATHMETH(CAS)EXAM1

TURN OVER

b. InthemonthofApril,theprobabilityofpleasantweatherinthemorningwas58.

i. FindtheprobabilitythatonaparticularmorninginApril,SallywalkedMack. 2marks

ii. Usingyouranswerfrompartb.i.,orotherwise,findtheprobabilitythatonaparticularmorninginApril,theweatherwaspleasant,giventhatSallywalkedMackthatmorning. 2marks

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

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

2014MATHMETH(CAS)EXAM1 12

END OF QUESTION AND ANSWER BOOK

ii. Findtheminimumtotalshadedareaandthevalueofuforwhichtheareaisaminimum. 2marks

iii. Findthemaximumtotalshadedareaandthevalueofuforwhichtheareaisamaximum. 1mark

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

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