2016 Mathematical Methods Written examination 2€¦ · MATHEMATICAL METHODS Written examination 2...

MATHEMATICAL METHODSWritten examination 2

Thursday 3 November 2016 Reading time: 11.45 am to 12.00 noon (15 minutes) Writing time: 12.00 noon to 2.00 pm (2 hours)

QUESTION AND ANSWER BOOK

Structure of bookSection Number of

questionsNumber of questions

to be answeredNumber of

marks

A 20 20 20B 4 4 60

Total 80

• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulers,aprotractor,setsquares,aidsforcurvesketching,oneboundreference,oneapprovedtechnology(calculatororsoftware)and,ifdesired,onescientificcalculator.CalculatormemoryDOESNOTneedtobecleared.Forapprovedcomputer-basedCAS,fullfunctionalitymaybeused.

• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.

Materials supplied• Questionandanswerbookof26pages.• Formulasheet.• Answersheetformultiple-choicequestions.

Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice

questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.• Unlessotherwiseindicated,thediagramsinthisbookarenot drawntoscale.• AllwrittenresponsesmustbeinEnglish.

At the end of the examination• Placetheanswersheetformultiple-choicequestionsinsidethefrontcoverofthisbook.• Youmaykeeptheformulasheet.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2016

STUDENT NUMBER

Letter

2016MATHMETHEXAM2 2

SECTION A – continued

Question 1Thelinearfunction f:D → R, f(x)=5–x hasrange[–4,5).ThedomainDisA. (0,9]B. (0,1]C. [5,–4)D. [–9,0)E. [1,9)

Question 2

Letf:R → R, f x x( ) cos= −

1 2

2π .

TheperiodandrangeofthisfunctionarerespectivelyA. 4and[–2,2]B. 4and[–1,3]C. 1and[–1,3]D. 4πand[–1,3]E. 4πand[–2,2]

SECTION A – Multiple-choice questions

Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

3 2016MATHMETHEXAM2

SECTION A – continuedTURN OVER

Question 3Partofthegraphy=f(x)ofthepolynomialfunction f isshownbelow.

13

10027

,

–3

−12 1

O

(–2, –9)

y

x

f ′(x)<0for

A. x∈ − ∪ ∞

( , ) ,2 0 1

3

B. x∈ −

9 100

27,

C. x∈ −∞ − ∪ ∞

( , ) ,2 1

3

D. x∈ −

2 1

3,

E. x∈ −∞ − ∪ ∞( )( , ] ,2 1

Question 4

Theaveragerateofchangeofthefunction f withrule f x x x( ) = − +3 2 12 ,betweenx=0andx=3,is

A. 8

B. 25

C. 539

D. 253

E. 139

2016MATHMETHEXAM2 4


Question 5

Whichoneofthefollowingistheinversefunctionofg :[3,∞)→ R, g x x( ) = −2 6 ?

A. g–1:[3,∞)→ R, g x x− =+1

2 62

( )

B. g–1:[0,∞)→ R, g–1(x)=(2x −6)2

C. g–1:[0,∞)→ R, g x x− = +1

26( )

D. g–1:[0,∞)→ R, g x x− =+1

2 62

( )

E. g–1:R → R, g x x− =+1

2 62

( )

Question 6Considerthegraphofthefunctiondefinedby f :[0,2π]→ R, f(x)=sin(2x).

Thesquareofthelengthofthelinesegmentjoiningthepointsonthegraphforwhich x = π4and x = 3

4π is

A. π 2 164+

B. π + 4

C. 4

D. 3 164

2π π+

E. 1016

2π

Question 7Thenumberofpets,X,ownedbyeachstudentinalargeschoolisarandomvariablewiththefollowingdiscreteprobabilitydistribution.

x 0 1 2 3

Pr(X=x) 0.5 0.25 0.2 0.05

Iftwostudentsareselectedatrandom,theprobabilitythattheyownthesamenumberofpetsisA. 0.3B. 0.305C. 0.355D. 0.405E. 0.8

5 2016MATHMETHEXAM2


Question 8TheUVindex,y,forasummerdayinMelbourneisillustratedinthegraphbelow,wheretisthenumberofhoursafter6am.

O

2

4

6

8

10

2 4 6 8 10 12 14

y

t

Thegraphismostlikelytobethegraphof

A. y t= +

5 5

7cos π

B. y t= −

5 5

7cos π

C. y t= +

5 5

14cos π

D. y t= −

5 5

14cos π

E. y t= +

5 5

14sin π

Question 9

Giventhat d xedx

kx ekx

kx( )= +( )1 ,then kxxe dx isequalto

A. xekx

ckx

++

1

B. kxk

e ckx+

+

1

C. 1 kxe dxk ∫

D. 1kxe e dx ckx kx−( ) +∫

E. 12kxe e ckx kx−( ) +

2016MATHMETHEXAM2 6


Question 10Forthecurvey=x2–5,thetangenttothecurvewillbeparalleltothelineconnectingthepositivex-interceptandthey-interceptwhenxisequalto

A. 5

B. 5

C. –5

D. 52

E. 15

Question 11Thefunction f hastheproperty f (x)–f (y)=(y–x)f (xy)forallnon-zerorealnumbersxandy.Whichoneofthefollowingisapossibleruleforthefunction?

A. f (x)=x2

B. f (x)=x2 + x4

C. f (x)=xloge(x)

D. f xx

( ) = 1

E. f xx

( ) = 12

Question 12Thegraphofafunction f isobtainedfromthegraphofthefunctiongwithrule g x x( ) = −2 5 bya

reflectioninthex-axisfollowedbyadilationfromthey-axisbyafactorof12.

Whichoneofthefollowingistheruleforthefunctionf ?

A. f x x( ) = −5 4

B. f x x( ) = − − 5

C. f x x( ) = + 5

D. f x x( ) = − −4 5

E. f x x( ) = − −4 10

7 2016MATHMETHEXAM2


Question 13Considerthegraphsofthefunctions f andgshownbelow.

y = f (x)

y = g (x)

a b c d

y

x

Theareaoftheshadedregioncouldberepresentedby

A. f x g x dxa

d( ) ( )−( )∫

B. f x g x dxd

( ) ( )−( )∫0C. f x g x dx f x g x dx

b

b

c( ) ( ) ( ) ( )−( ) + −( )∫ ∫0

D. f x dx f x g x dx f x dxa

a

c

b

d( ) ( ) ( ) ( )

0∫ ∫ ∫+ −( ) +

E. f x dx g x dxd

a

c( ) ( )

0∫ ∫−

2016MATHMETHEXAM2 8


Question 14Arectangleisformedbyusingpartofthecoordinateaxesandapoint(u,v),whereu>0ontheparabolay=4–x2.

(u, v)

y

x

5

4

3

2

1

54321O

Whichoneofthefollowingisthemaximumareaoftherectangle?A. 4

B. 2 33

C. 8 3 43−

D. 83

E. 16 39

Question 15Aboxcontainssixredmarblesandfourbluemarbles.Twomarblesaredrawnfromthebox,withoutreplacement.Theprobabilitythattheyarethesamecolouris

A. 12

B. 2845

C. 715

D. 35

E. 13

9 2016MATHMETHEXAM2


Question 16Therandomvariable,X,hasanormaldistributionwithmean12andstandarddeviation0.25Iftherandomvariable,Z,hasthestandardnormaldistribution,thentheprobabilitythatXisgreaterthan12.5isequaltoA. Pr(Z<–4)B. Pr(Z<–1.5)C. Pr(Z<1)D. Pr(Z≥1.5)E. Pr(Z >2)

Question 17Insideacontainerthereareonemillioncolouredbuildingblocks.Itisknownthat20%oftheblocksarered.Asampleof16blocksistakenfromthecontainer.Forsamplesof16blocks,P̂ istherandomvariableofthedistributionofsampleproportionsofredblocks.(Donotuseanormalapproximation.)

3Pr16

P ≥ ̂ isclosestto

A. 0.6482B. 0.8593C. 0.7543D. 0.6542E. 0.3211

Question 18Thecontinuousrandomvariable,X,hasaprobabilitydensityfunctiongivenby

f xx x

( )cos

=

≤ ≤

14 2

3 5

0

elsewhere

π π

ThevalueofasuchthatPr( )X a< =+3 2

4is

A. 196π

B. 143π

C. 103π

D. 296π

E. 173π

2016MATHMETHEXAM2 10

END OF SECTION A

Question 19ConsiderthediscreteprobabilitydistributionwithrandomvariableXshowninthetablebelow.

x –1 0 b 2b 4

Pr(X=x) a b b 2b 0.2

ThesmallestandlargestpossiblevaluesofE(X)arerespectivelyA. –0.8and1B. –0.8and1.6C. 0and2.4D. 0.2125and1E. 0and1

Question 20ConsiderthetransformationT,definedas

T R R Txy

xy

: ,2 2 1 00 3

05

→

=

−

+

ThetransformationTmapsthegraphofy=f(x)ontothegraphofy=g(x).

If f x dx( ) =∫ 50

3,then g x dx( )

−∫ 3

0isequalto

A. 0B. 15C. 20D. 25E. 30

11 2016MATHMETHEXAM2

SECTION B – Question 1–continuedTURN OVER

Question 1 (11marks)

Let f :[0,8π]→ R, f xx( ) cos=

+2

2π .

a. Findtheperiodandrangeof f. 2marks

b. Statetheruleforthederivativefunction f ′. 1mark

c. Findtheequationofthetangenttothegraphof f at x=π. 1mark

SECTION B

Instructions for Section BAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.


SECTION B – continued

d. Findtheequationsofthetangentstothegraphof f :[0,8π]→ R, f x x( ) cos=

+2

2π that

haveagradientof1. 2marks

e. Theruleof f ′canbeobtainedfromtheruleof f underatransformationT,suchthat

T R R Txy a

xy b

: ,2 2 1 00

→

=

+

−

π

Findthevalueofaandthevalueofb. 3marks

f. Findthevaluesofx,0≤x≤8π,suchthat f (x)=2f ′(x)+π. 2marks




Considerthefunction f (x)=– 13(x+2)(x–1)2.

a. i. Giventhatg′(x)=f (x)andg (0)=1,showthat g xx x x( ) .= − + − +

4 2

12 223

1 1mark

ii. Findthevaluesofxforwhichthegraphofy=g (x)hasastationarypoint. 1mark


SECTION B – Question 2–continued

Thediagrambelowshowspartofthegraphofy=g (x),thetangenttothegraphatx=2andastraightlinedrawnperpendiculartothetangenttothegraphatx=2.Theequationofthetangentat

thepointAwithcoordinates(2,g (2))is yx

= −3 43.

Thetangentcutsthey-axisatB.Thelineperpendiculartothetangentcutsthey-axisatC.

B

A

C

O

y

x

b. i. FindthecoordinatesofB. 1mark

ii. FindtheequationofthelinethatpassesthroughAandCand,hence,findthecoordinatesofC. 2marks

iii. FindtheareaoftriangleABC. 2marks


SECTION B – continuedTURN OVER

c. ThetangentatDisparalleltothetangentatA.ItintersectsthelinepassingthroughAandC at E.

D

A

CE

O

y

x

i. FindthecoordinatesofD. 2marks

ii. FindthelengthofAE. 3marks

2016 MATHMETH EXAM 2 16

SECTION B – Question 3 – continued

Question 3 (16 marks)A school has a class set of 22 new laptops kept in a recharging trolley. Provided each laptop is correctly plugged into the trolley after use, its battery recharges.On a particular day, a class of 22 students uses the laptops. All laptop batteries are fully charged at the start of the lesson. Each student uses and returns exactly one laptop. The probability that a student does not correctly plug their laptop into the trolley at the end of the lesson is 10%. The correctness of any student’s plugging-in is independent of any other student’s correctness.

a. Determine the probability that at least one of the laptops is not correctly plugged into the trolley at the end of the lesson. Give your answer correct to four decimal places. 2 marks

b. A teacher observes that at least one of the returned laptops is not correctly plugged into the trolley.

Giventhis,findtheprobabilitythatfewerthanfivelaptopsarenot correctly plugged in. Give your answer correct to four decimal places. 2 marks



Thetimeforwhichalaptopwillworkwithoutrecharging(thebatterylife)isnormallydistributed,withameanofthreehoursand10minutesandstandarddeviationofsixminutes.Supposethatthelaptopsremainoutoftherechargingtrolleyforthreehours.

c. Foranyonelaptop,findtheprobabilitythatitwillstopworkingbytheendofthesethreehours.Giveyouranswercorrecttofourdecimalplaces. 2marks

Asupplieroflaptopsdecidestotakeasampleof100newlaptopsfromanumberofdifferentschools.Forsamplesofsize100fromthepopulationoflaptopswithameanbatterylifeofthreehoursand10minutesandstandarddeviationofsixminutes, P̂ istherandomvariableofthedistributionofsampleproportionsoflaptopswithabatterylifeoflessthanthreehours.

d. FindtheprobabilitythatPr(P̂ ≥0.06|P̂ ≥0.05).Giveyouranswercorrecttothreedecimalplaces.Donotuseanormalapproximation. 3marks

Itisknownthatwhenlaptopshavebeenusedregularlyinaschoolforsixmonths,theirbatterylifeisstillnormallydistributedbutthemeanbatterylifedropstothreehours.Itisalsoknownthatonly12%ofsuchlaptopsworkformorethanthreehoursand10minutes.

e. Findthestandarddeviationforthenormaldistributionthatappliestothebatterylifeoflaptopsthathavebeenusedregularlyinaschoolforsixmonths,correcttofourdecimalplaces. 2marks

2016 MATHMETH EXAM 2 18

SECTION B – Question 3 – continued

The laptop supplier collects a sample of 100 laptops that have been used for six months from a number of different schools and tests their battery life. The laptop supplier wishes to estimate the proportion of such laptops with a battery life of less than three hours.

f. Suppose the supplier tests the battery life of the laptops one at a time.

Findtheprobabilitythatthefirstlaptopfoundtohaveabatterylifeoflessthanthreehoursisthe third one. 1 mark

Thelaptopsupplierfindsthat,inaparticularsampleof100laptops,sixofthemhaveabatterylifeof less than three hours.

g. Determinethe95%confidenceintervalforthesupplier’sestimateoftheproportionofinterest.Give values correct to two decimal places. 1 mark


SECTION B – continuedTURN OVER

h. Thesupplieralsoprovideslaptopstobusinesses.Theprobabilitydensityfunctionforbatterylife,x(inminutes),ofalaptopaftersixmonthsofuseinabusinessis

f xx e

x

x

( )( )

=−

≤ ≤

−

210400

0

0 210

21020

elsewhere

i. Findthemeanbatterylife,inminutes,ofalaptopwithsixmonthsofbusinessuse,correcttotwodecimalplaces. 1mark

ii. Findthemedianbatterylife,inminutes,ofalaptopwithsixmonthsofbusinessuse,correcttotwodecimalplaces. 2marks




a. Express 2 12

xx++

intheform a bx

++ 2

,whereaandbarenon-zerointegers. 2marks

b. Let f :R\{–2}→ R, f x xx

( ) = ++

2 12.

i. Findtheruleanddomainof f –1,theinversefunctionof f. 2marks

ii. Partofthegraphsof f andy=xareshowninthediagrambelow.

y = x

y = f (x)

2

1

–1

–2

1O 2–2 –1

y

x

Findtheareaoftheshadedregion. 1mark



iii. Partofthegraphsof f and f –1areshowninthediagrambelow.

y = f –1 (x)

y = f (x)

2

1

–1

–2

1 2–2 –1

y

xO

Findtheareaoftheshadedregion. 1mark



c. Partofthegraphof f isshowninthediagrambelow.

P(c, d)y = f (x)

12

1

–1

–2

2

x

y

1 2–2 –1−

12

O

ThepointP(c,d)isonthegraphof f.

Findtheexactvaluesofcanddsuchthatthedistanceofthispointtotheoriginisaminimum,andfindthisminimumdistance. 3marks



Letg :(–k,∞)→ R, g x kxx k

( ) = ++

1,wherek>1.

d. Showthatx1<x2impliesthatg (x1)<g (x2),wherex1 ∈(–k,∞)andx2 ∈(–k,∞). 2marks



e. i. LetXbethepointofintersectionofthegraphsofy=g (x)andy=–x.

FindthecoordinatesofXintermsofk. 2marks

ii. FindthevalueofkforwhichthecoordinatesofXare −

12

12

, . 2marks



iii. LetZ(–1,–1),Y(1,1)andX betheverticesofthetriangleXYZ.Lets(k)bethesquareoftheareaoftriangleXYZ.

y

x

Y

X

1

–1

1–1 O

Z

y = g (x)

y = x

Findthevaluesofksuchthats(k)≥1. 2marks


END OF QUESTION AND ANSWER BOOK

f. Thegraphofgandtheliney=xenclosearegionoftheplane.Theregionisshownshadedinthediagrambelow.

y

x

y = x

y = g (x)

1–1

1

–1O

LetA(k)betheruleofthefunctionAthatgivestheareaofthisenclosedregion.ThedomainofAis(1,∞).

i. GivetheruleforA(k). 2marks

ii. Showthat0<A(k)<2forallk>1. 2marks

MATHEMATICAL METHODS

Written examination 2

FORMULA SHEET

Instructions

This formula sheet is provided for your reference.A question and answer book is provided with this formula sheet.

Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.

Victorian Certificate of Education 2016

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2016

MATHMETH EXAM 2

Mathematical Methods formulas

Mensuration

area of a trapezium 12a b h+( ) volume of a pyramid 1

3Ah

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 triangle12bc Asin ( )

volume of a cone13

2π r h

Calculus

ddx

x nxn n( ) = −1 x dxn

x c nn n=+

+ ≠ −+∫ 11

11 ,

ddx

ax b an ax bn n( )+( ) = +( ) −1 ( )( )

( ) ,ax b dxa n

ax b c nn n+ =+

+ + ≠ −+∫ 11

11

ddxe aeax ax( ) = e dx a e cax ax= +∫ 1

ddx

x xelog ( )( ) = 1 1 0x dx x c xe= + >∫ 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 ruleddxuv u dv

dxv dudx

( ) = + quotient ruleddx

uv

v dudx

u dvdx

v

=

−

2

chain ruledydx

dydududx

=

3 MATHMETH EXAM

END OF FORMULA SHEET

Probability

Pr(A) = 1 – Pr(A′) Pr(A ∪ B) = Pr(A) + Pr(B) – Pr(A ∩ B)

Pr(A|B) = Pr

PrA BB∩( )( )

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

Sample proportions

P Xn

=̂ mean E(P̂ ) = p

standard deviation

sd P p pn

(ˆ ) ( )=

−1 approximate confidence interval

,p zp p

np z

p pn

−−( )

+−( )

1 1ˆ ˆ ˆˆˆ ˆ

2016 Mathematical Methods Written examination 2€¦ · MATHEMATICAL METHODS Written examination 2...

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