MATHEMATICAL METHODSWritten examination 2

Thursday 9 November 2017 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• Questionandanswerbookof22pages• 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2017

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2017

STUDENT NUMBER

Letter

2017MATHMETHEXAM2 2

SECTION A – continued

Question 1Letf :R → R,f (x)=5sin(2x)–1.TheperiodandrangeofthisfunctionarerespectivelyA. πand[−1,4]B. 2πand[−1,5]C. πand[−6,4]D. 2πand[−6,4]E. 4πand[−6,4]

Question 2Partofthegraphofacubicpolynomialfunctionfandthecoordinatesofitsstationarypointsareshownbelow.

O 5

50

–50

–6

(–3, 36)

x

y

5 400,3 27

−

f′(x)<0fortheintervalA. (0,3)

B. ( , ) ,−∞ − ∪( )5 0 3

C. ( , ) ,−∞ − ∪ ∞

3 5

3

D. −

3 5

3,

E. −

40027

36,

SECTION A – Multiple-choice questions

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

3 2017MATHMETHEXAM2

SECTION A – continuedTURN OVER

Question 3Aboxcontainsfiveredmarblesandthreeyellowmarbles.Twomarblesaredrawnatrandomfromtheboxwithoutreplacement.Theprobabilitythatthemarblesareofdifferentcoloursis

A. 58

B. 35

C. 1528

D. 1556

E. 3028

Question 4Let fandgbefunctionssuchthat f (2)=5, f (3)=4,g(2)=5,g(3)=2andg(4)=1.Thevalueof f (g(3))isA. 1B. 2C. 3D. 4E. 5

Question 5The95%confidenceintervalfortheproportionofferryticketsthatarecancelledontheintendeddeparturedayiscalculatedfromalargesampletobe(0.039,0.121).ThesampleproportionfromwhichthisintervalwasconstructedisA. 0.080B. 0.041C. 0.100D. 0.062E. 0.059

2017MATHMETHEXAM2 4

SECTION A – continued

Question 6Partofthegraphofthefunctionf isshownbelow.Thesamescalehasbeenusedonbothaxes.

x

y

Thecorrespondingpartofthegraphoftheinversefunctionf −1isbestrepresentedby

x

y

x

y

x

y

x

y

x

yA. B.

D. E.

C.

Question 7Theequation(p–1)x2 + 4x=5–p hasnorealrootswhenA. p2–6p+6<0B. p2–6p+1>0C. p2–6p–6<0D. p2–6p+1<0E. p2–6p+6>0

5 2017MATHMETHEXAM2

SECTION A – continuedTURN OVER

Question 8Ify=ab–4x+2,wherea>0,thenxisequalto

A. 14

2b ya− −( )log ( )

B. 14

2b ya− +( )log ( )

C. b ya− +

log ( )1

42

D. b ya42− −log ( )

E. 14

2b ya+ −( )log ( )

Question 9Theaveragerateofchangeofthefunctionwiththerulef (x)=x2–2xovertheinterval[1,a], wherea>1,is8.ThevalueofaisA. 9

B. 8

C. 7

D. 4

E. 1 2+

Question 10AtransformationT:R2 → R2withruleT

xy

xy

=

2 0

0 13

mapsthegraphof y x= +

3 2

4sin π

ontothegraphof

A. y=sin(x + π)

B. y x= −

sin π

2

C. y=cos(x + π)

D. y=cos(x)

E. y x= −

cos π

2

2017MATHMETHEXAM2 6

SECTION A – continued

Question 11Thefunctionf :R → R,f (x)=x3 + ax2 + bxhasalocalmaximumatx=–1andalocalminimumatx=3.ThevaluesofaandbarerespectivelyA. –2and–3B. 2and1C. 3and–9D. –3and–9E. –6and–15

Question 12Thesumofthesolutionsof sin( )2 3

2x = overtheinterval[–π,d]is–π.

ThevalueofdcouldbeA. 0

B. 6π

C. 34π

D. 76π

E. 32π

Question 13Let h R h x

x: ( , ) , ( ) .− → =

−1 1

11

Whichoneofthefollowingstatementsabouthisnottrue?A. h(x)h(–x)=–h(x2)B. h(x)+h(–x)=2h(x2)C. h(x)–h(0)=xh(x)D. h(x)–h(–x)=2xh(x2)E. (h(x))2=h(x2)

Question 14

TherandomvariableXhasthefollowingprobabilitydistribution,where0 13

< <p .

x –1 0 1

Pr(X=x) p 2p 1–3p

ThevarianceofXisA. 2p(1–3p)B. 1–4pC. (1–3p)2

D. 6p–16p2

E. p(5–9p)

7 2017MATHMETHEXAM2

SECTION A – continuedTURN OVER

Question 15ArectangleABCDhasverticesA(0,0),B(u,0),C(u, v)andD(0,v),where(u, v)liesonthegraphof y=−x3+8,asshownbelow.

y

8D C(u, v)

A B 2x

Themaximumareaoftherectangleis

A. 23

B. 6 23

C. 16D. 8

E. 3 23

Question 16ForrandomsamplesoffiveAustralians,P̂ istherandomvariablethatrepresentstheproportionwholiveinacapitalcity.

Giventhat ( ) 1Pr 0 ,243

= =P̂ then ( )Pr 0.6 ,>P̂ correcttofourdecimalplaces,is

A. 0.0453B. 0.3209C. 0.4609D. 0.5390E. 0.7901

2017MATHMETHEXAM2 8

SECTION A – continued

Question 17Thegraphofafunctionf,wheref(−x)=f(x),isshownbelow.

y

xa b c d

Thegraphhasx-interceptsat(a,0),(b,0),(c,0)and(d,0)only.Theareaboundbythecurveandthex-axisontheinterval[a,d]is

A. ( )d

af x dx∫

B. f x dx f x dx f x dxa

b

c

b

c

d( ) ( ) ( )∫ ∫ ∫− +

C. 2 f x dx f x dxa

b

b

c( ) ( )∫ ∫+

D. 2 2f x dx f x dxa

b

b

b c( ) ( )∫ ∫−

+

E. f x dx f x dx f x dxa

b

c

b

d

c( ) ( ) ( )∫ ∫ ∫+ +

Question 18LetXbeadiscreterandomvariablewithbinomialdistributionX~Bi(n, p).Themeanandthestandarddeviationofthisdistributionareequal.Giventhat0<p<1,thesmallestnumberoftrials,n,suchthatp ≤0.01isA. 37B. 49C. 98D. 99E. 101

9 2017MATHMETHEXAM2

END OF SECTION ATURN OVER

Question 19Aprobabilitydensityfunctionfisgivenby

f xx k x k

( )cos( ) ( )

=+ < < +

1 10 elsewhere

where0<k<2.ThevalueofkisA. 1

B. 3 1

2π −

C. π−1

D. 1

2π −

E. 2π

Question 20Thegraphsof : 0, , ( ) cos( ) and : 0, , ( ) 3 sin( )

2 2πf R f x x g R g x x → = → =

π areshownbelow.

ThegraphsintersectatB.

π2

π2

3,

y x= 3sin( )

y = cos(x)

B1

O

y

xA

TheratiooftheareaoftheshadedregiontotheareaoftriangleOABisA. 9:8

B. 3 1 38

− : π

C. 8 3 – 3 : 3π

D. 3 1 34

− : π

E. 1 38

: π

2017MATHMETHEXAM2 10

SECTION B – Question 1–continued

Question 1 (11marks)Let f :R → R, f (x)=x3 –5x. Partofthegraphof f isshownbelow.

x

y

10

5–5

–10

–5

O

5

a. Findthecoordinatesoftheturningpoints. 2marks

b. A(−1,f(−1))andB(1,f(1))aretwopointsonthegraphof f.

i. FindtheequationofthestraightlinethroughAandB. 2marks

ii. FindthedistanceAB. 1mark

SECTION B

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

11 2017MATHMETHEXAM2

SECTION B – continuedTURN OVER

Letg:R → R,g(x)=x3 –kx, k ∈ R+.

c. LetC(–1,g(−1))andD(1,g(1))betwopointsonthegraphofg.

i. FindthedistanceCDintermsofk. 2marks

ii. FindthevaluesofksuchthatthedistanceCDisequaltok+1. 1mark

d. Thediagrambelowshowspartofthegraphsofgandy=x.Thesegraphsintersectatthepointswiththecoordinates(0,0)and(a,a).

y y = g(x)

y = x(a, a)

(0, 0) x

i. Findthevalueofaintermsofk. 1mark

ii. Findtheareaoftheshadedregionintermsofk. 2marks

2017 MATHMETH EXAM 2 12

SECTION B – Question 2 – continued

Question 2 (12 marks)Sammy visits a giant Ferris wheel. Sammy enters a capsule on the Ferris wheel from a platform above the ground. The Ferris wheel is rotating anticlockwise. The capsule is attached to the Ferris

wheel at point P. The height of P above the ground, h, is modelled by ( ) 65 – 55cos ,15

th t π =

where t is the time in minutes after Sammy enters the capsule and h is measured in metres.

Sammy exits the capsule after one complete rotation of the Ferris wheel.

platform

a capsule

h

P

a. State the minimum and maximum heights of P above the ground. 1 mark

b. For how much time is Sammy in the capsule? 1 mark

c. Find the rate of change of h with respect to t and, hence, state the value of t at which the rate of change of h is at its maximum. 2 marks

13 2017MATHMETHEXAM2

SECTION B – Question 2–continuedTURN OVER

AstheFerriswheelrotates,astationaryboatatB, onanearbyriver,firstbecomesvisibleatpointP1.Bis500mhorizontallyfromtheverticalaxisthroughthecentreC oftheFerriswheelandangleCBO=θ,asshownbelow.

500 m

C

y

Bθ

P1

Ox

d. Findθindegrees,correcttotwodecimalplaces. 1mark

PartofthepathofPisgivenbyy x x= − + ∈ −3025 62 5 55 55, [ , ], wherexandyareinmetres.

e. Finddydx. 1mark

2017MATHMETHEXAM2 14

SECTION B – continued

AstheFerriswheelcontinuestorotate,theboatatB isnolongervisiblefromthepointP2(u,v)onwards.ThelinethroughBandP2istangenttothepathofP,whereangleOBP2=α.

500 m

α

y

B

O

P2(u, v)

C

x

f. FindthegradientofthelinesegmentP2Bintermsofuand,hence,findthecoordinatesof P2,correcttotwodecimalplaces. 3marks

g. Findαindegrees,correcttotwodecimalplaces. 1mark

h. Henceorotherwise,findthelengthoftime,tothenearestminute,duringwhichtheboatat B isvisible. 2marks

15 2017MATHMETHEXAM2

SECTION B – continuedTURN OVER

CONTINUES OVER PAGE

2017MATHMETHEXAM2 16

SECTION B – Question 3–continued

Question 3 (19marks)ThetimeJenniferspendsonherhomeworkeachdayvaries,butshedoessomehomework everyday.ThecontinuousrandomvariableT,whichmodelsthetime,t,inminutes,thatJenniferspends eachdayonherhomework,hasaprobabilitydensityfunctionf,where

f t

t t

t t( )

( )

( )=

− ≤ <

− ≤ ≤

1625

20 20 45

1625

70 45 70

0 elsewhere

a. Sketchthegraphoffontheaxesprovidedbelow. 3marks

150

125

200 40 60 80 100

y

t

b. FindPr(25≤T≤55). 2marks

c. FindPr(T≤25|T≤55). 2marks

17 2017MATHMETHEXAM2

SECTION B – Question 3–continuedTURN OVER

d. FindasuchthatPr(T≥a)=0.7,correcttofourdecimalplaces. 2marks

e. TheprobabilitythatJenniferspendsmorethan50minutesonherhomeworkonanygivenday

is 825

. Assumethattheamountoftimespentonherhomeworkonanydayisindependentof

thetimespentonherhomeworkonanyotherday.

i. FindtheprobabilitythatJenniferspendsmorethan50minutesonherhomeworkonmorethanthreeofsevenrandomlychosendays,correcttofourdecimalplaces. 2marks

ii. FindtheprobabilitythatJenniferspendsmorethan50minutesonherhomeworkonatleasttwoofsevenrandomlychosendays,giventhatshespendsmorethan50minutesonherhomeworkonatleastoneofthosedays,correcttofourdecimalplaces. 2marks

2017MATHMETHEXAM2 18

SECTION B – continued

LetpbetheprobabilitythatonanygivendayJenniferspendsmorethandminutesonherhomework.Letqbetheprobabilitythatontwoorthreedaysoutofsevenrandomlychosendaysshespendsmorethandminutesonherhomework.

f. Expressqasapolynomialintermsofp. 2marks

g. i. Findthemaximumvalueofq,correcttofourdecimalplaces,andthevalueofpforwhichthismaximumoccurs,correcttofourdecimalplaces. 2marks

ii. Findthevalueofd forwhichthemaximumfoundinpart g.i.occurs,correcttothenearestminute. 2marks

19 2017MATHMETHEXAM2

SECTION B – continuedTURN OVER

CONTINUES OVER PAGE

2017MATHMETHEXAM2 20

SECTION B – Question 4–continued

Question 4 (18marks)Letf :R → R :f (x)=2x +1–2.Partofthegraphoff isshownbelow.

4

3

2

1

O–1

–2–3

–4

–4 –3 –2 –1 1 2 3 4

y

x

a. ThetransformationT R R Txy

xy

cd

: ,2 2→

=

+

mapsthegraphofy=2

xontothegraphoff.

Statethevaluesofcandd. 2marks

b. Findtheruleanddomainforf −1,theinversefunctionoff. 2marks

c. Findtheareaboundedbythegraphsoffandf −1. 3marks

21 2017MATHMETHEXAM2

SECTION B – Question 4–continuedTURN OVER

d. Partofthegraphsoffandf −1areshownbelow.

O

f

f –1

y

x–2

–2

Findthegradientof fandthegradientof f −1 at x=0. 2marks

Thefunctionsofgk ,wherek ∈ R+,aredefinedwithdomainRsuchthatgk(x)=2ekx–2.

e. Findthevalueofksuchthatgk(x)=f(x). 1mark

f. Findtherulefortheinversefunctionsgk–1ofgk ,wherek ∈ R+. 1mark

2017MATHMETHEXAM2 22

END OF QUESTION AND ANSWER BOOK

g. i. Describethetransformationthatmapsthegraphofg1ontothegraphofgk. 1mark

ii. Describethetransformationthatmapsthegraphofg1–1ontothegraphofgk

–1. 1mark

h. ThelinesL1andL2arethetangentsattheorigintothegraphsofgkandgk–1respectively.

Findthevalue(s)ofkforwhichtheanglebetweenL1andL2is30°. 2marks

i. Letpbethevalueofkforwhichgk(x)=gk−1(x)hasonlyonesolution.

i. Findp. 2marks

ii. LetA(k)betheareaboundedbythegraphsofgkandgk–1forallk>p.

StatethesmallestvalueofbsuchthatA(k)<b. 1mark

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 2017

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2017

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