2018 Mathematical Methods Written examination 2...MATHEMATICAL METHODS Written examination 2 Thursday 8 November 2018 Reading time: 3.00 pm to 3.15 pm (15 minutes) Writing time: 3.15

MATHEMATICAL METHODSWritten examination 2

Thursday 8 November 2018 Reading time: 3.00 pm to 3.15 pm (15 minutes) Writing time: 3.15 pm to 5.15 pm (2 hours)

QUESTION AND ANSWER BOOK

Structure of bookSection Number of

questionsNumber of questions

to be answeredNumber of

marks

A 20 20 20B 5 5 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• Questionandanswerbookof25pages• 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2018

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2018

STUDENT NUMBER

Letter

2018MATHMETHEXAM2 2

SECTION A – continued

Question 1

Let f :R → R, f xx( ) cos .=

+4 2

31π

TheperiodofthisfunctionisA. 1B. 2C. 3D. 4E. 5

Question 2Themaximaldomainofthefunction f isR\{1}.Apossiblerulefor f is

A. f x xx

( ) =−−

2 51

B. f x xx

( ) =+−

45

C. f x x xx

( ) =+ +

+

2

24

1

D. f x xx

( ) =−+

51

2

E. f x x( ) = −1

SECTION A – Multiple-choice questions

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

3 2018MATHMETHEXAM2

SECTION A – continuedTURN OVER

Question 3

Considerthefunction f a b R f xx

:[ , ) , ( ) ,→ = 1 whereaandbarepositiverealnumbers.

Therangeof f is

A. 1 1a b

,

B. 1 1a b

,

C. 1 1b a

,

D. 1 1b a

,

E. [a,b)

Question 4ThepointA(3,2)liesonthegraphofthefunction f.Atransformationmapsthegraphof f tothegraphofg,

where g x f x( ) ( ).= −12

1 ThesametransformationmapsthepointAtothepointP.

ThecoordinatesofthepointPareA. (2,1)B. (2,4)C. (4,1)D. (4,2)E. (4,4)

Question 5

Consider f x x px

( ) ,= +2 x≠0,p ∈ R.

Thereisastationarypointonthegraphof f whenx=–2.ThevalueofpisA. –16B. –8C. 2D. 8E. 16

Question 6Let f andg betwofunctionssuchthat f (x)= 2x andg(x + 2)= 3x + 1.Thefunction f (g(x)) isA. 6x – 5B. 6x +1C. 6x2+1D. 6x –10E. 6x + 2

2018MATHMETHEXAM2 4


Question 7Let f:R+ → R, f(x)=k log2(x),k ∈ R.Giventhat f –1(1)=8,thevalueofkisA. 0

B. 13

C. 3

D. 8

E. 12

Question 8

If ( )g x dx1

125∫ = and g x dx ( ) ,= −∫ 6

12

5then g x dx

1

5

∫ ( ) isequalto

A. –11B. –1C. 1D. 3E. 11

Question 9Atangenttothegraphofy =loge(2x)hasagradientof2.Thistangentwillcrossthey-axisatA. 0B. –0.5C. –1D. –1–loge(2)E. –2loge(2)

Question 10Thefunction f hastheproperty f(x + f (x))=f (2x)forallnon-zerorealnumbersx.Whichoneofthefollowingisapossibleruleforthefunction?A. f(x)=1–x

B. f(x)=x–1

C. f(x)=x

D. f x x( ) =2

E. f x x( ) =−12

5 2018MATHMETHEXAM2


Question 11Thegraphofy=tan(ax),wherea ∈ R+,hasaverticalasymptotex = 3πandhasexactlyonex-interceptintheregion(0,3π).Thevalueofais

A. 16

B. 13

C. 12

D. 1

E. 2

Question 12ThediscreterandomvariableXhasthefollowingprobabilitydistribution.

x 0 1 2 3 6

Pr(X = x)14

920

110

120

320

Let μbethemeanofX.Pr(X < μ)is

A. 12

B. 14

C. 1720

D. 45

E. 7

10

2018MATHMETHEXAM2 6


Question 13Inaparticularscoringgame,therearetwoboxesofmarblesandaplayermustrandomlyselectonemarblefromeachbox.Thefirstboxcontainsfourwhitemarblesandtworedmarbles.Thesecondboxcontainstwowhitemarblesandthreeredmarbles.Eachwhitemarblescores−2pointsandeachredmarblescores+3points.Thepointsobtainedfromthetwomarblesrandomlyselectedbyaplayerareaddedtogethertoobtainafinalscore.Whatistheprobabilitythatthefinalscorewillequal+1?

A. 23

B. 15

C. 25

D. 2

15

E. 8

15

Question 14Twoevents,AandB,areindependent,wherePr(B)=2Pr(A)andPr(A ∪ B)=0.52Pr(A)isequaltoA. 0.1B. 0.2C. 0.3D. 0.4E. 0.5

Question 15Aprobabilitydensityfunction, f,isgivenby

f xx x x

( )( )

=− ≤ ≤

112

8 0 2

0

3

elsewhere

Themedian,m,ofthisfunctionsatisfiestheequationA. – m4+16m2 – 6 = 0

B. – m4 + 4m2 – 6 = 0

C. m4–16m2 = 0

D. m4–16m2+24=0.5

E. m4–16m2 + 24 = 0

7 2018MATHMETHEXAM2


Question 16Jamieapproximatestheareabetweenthex-axisandthegraphofy=2cos(2x)+3,overtheinterval 0

2, ,π

usingthethreerectanglesshownbelow.

y

x

y = 2cos(2x) + 3

π6

π3

π2

6

5

4

3

2

1

0

Jamie’sapproximationasafractionoftheexactareais

A. 59

B. 79

C. 9

11

D. 1118

E. 73

Question 17Theturningpointoftheparabolay = x2 – 2bx+1isclosesttotheoriginwhenA. b = 0

B. b=–1orb=1

C. b b= − =12

12

or

D. b b= = −12

12

or

E. b b= = −14

14

or

2018MATHMETHEXAM2 8


Question 18Considerthefunctions f R: ,R f x x

pq

( )+→ = and g R R g x xmn: , ( ) ,+→ = wherep,q,mandnarepositive

integers,andpq and

mn arefractionsinsimplestform.

If{x:f (x)>g(x)}=(0,1)and{x:g(x)>f (x)}=(1,∞),whichofthefollowingmustbefalse?A. q>nandp = mB. m>pandq = nC. pn < qmD. f ′(c)=g′(c)forsomec ∈(0,1)E. f ′(d)=g′(d)forsomed ∈(1,∞)

Question 19Thegraphs f R R f x x: , ( ) cos→ =

π2

andg:R → R,g(x)=sin(π x)areshowninthediagrambelow.

13

53

1 3x

y

g

f

0

Anintegralexpressionthatgivesthetotalareaoftheshadedregionsis

A. sin( ) cosππx x dx−

∫ 20

3

B. 225

3

3sin cosπ

πx x dx( ) −

∫

C. ( )cos sin( ) cos sin( )ππ

ππ

x x dx x x2

220

13

−

−

−

∫ −

−

∫ ∫dx x x dx1

3

1

53

3

2cos sinπ

π

D. 22

221

53 cos sin( ) cos sin( )π

ππ

πx x dx x x

−

−

−

∫

∫ dx5

3

3

E. cos sin( ) sin( ) cosππ π

πx x dx x x2

220

13

−

+ −

∫

+

−

∫ ∫dx x x dx1

3

1

53

3

2cos sin( )π

π

9 2018MATHMETHEXAM2

END OF SECTION ATURN OVER

Question 20Thedifferentiablefunction f :R → Risaprobabilitydensityfunction.Itisknownthatthemedianoftheprobabilitydensityfunction fisatx=0and f′(0)=4.ThetransformationT :R2 → R2mapsthegraphof f tothegraphofg,whereg :R → Risaprobabilitydensityfunctionwithamedianatx=0andg′(0)=–1.ThetransformationTcouldbegivenby

A. Txy

xy

=

–2 0

0 12

B. Txy

xy

= −

2 0

0 12

C. Txy

xy

=

2 0

0 12

D. Txy

xy

=

−

12

0

0 2

E. Txy

xy

=

−

12

0

0 2

2018MATHMETHEXAM2 10

SECTION B – Question 1–continued

Question 1 (13marks)Considerthequartic f:R → R, f (x)=3x4 + 4x3–12x2andpartofthegraphofy = f (x)below.

y

x

y = f (x)

M

O

a. FindthecoordinatesofthepointM,atwhichtheminimumvalueofthefunction f occurs. 1mark

b. Statethevaluesofb ∈ Rforwhichthegraphofy = f (x)+bhasnox-intercepts. 1mark

SECTION B

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

11 2018MATHMETHEXAM2

SECTION B – Question 1–continuedTURN OVER

Partofthetangent,l,toy = f (x)atx = − 13isshownbelow.

y

x

y = f (x)l

O

c. Findtheequationof thetangentl. 1mark

d. Thetangentl intersectsy = f (x)atx = − 13andattwootherpoints.

Statethex-valuesofthetwootherpointsofintersection.Expressyouranswersintheforma bc± , wherea,bandcareintegers. 2marks

e. Findthetotalareaoftheregionsboundedbythetangentl andy = f (x).Expressyouranswer

intheforma bc

, wherea,bandcarepositiveintegers. 2marks


SECTION B – continued

Letp :R → R,p(x)=3x4 + 4x3+6(a–2)x2–12ax + a2,a ∈ R.

f. Statethevalueofaforwhich f (x)=p(x)forallx. 1mark

g. Findallsolutionstop′(x)=0,intermsofawhereappropriate. 1mark

h. i. Findthevaluesofaforwhichp hasonlyonestationarypoint. 1mark

ii. Findtheminimumvalueofp whena=2. 1mark

iii. Ifp hasonlyonestationarypoint,findthevaluesofaforwhichp(x)=0hasnosolutions. 2marks


SECTION B – continuedTURN OVER

CONTINUES OVER PAGE



Question 2 (10marks)Adrug,X,comesin500milligram(mg)tablets.Theamount,b,ofdrugXinthebloodstream,inmilligrams,thoursafteronetabletisconsumedisgivenbythefunction

( ) ( )b t e et t

( ) = −

− −45007

5910

a. Findthetime,inhours,ittakesfordrugXtoreachamaximumamountinthebloodstreamafteronetabletisconsumed.Expressyouranswerintheformaloge(c),wherea,c ∈ R. 2marks

Thegraphofy = b(t)isshownbelowfor0≤t≤6.

500

400

300

200

100

0 1 2 3time (hours)

4 5 6

y

amount ofdrug X (mg)

t

y = b(t)

b. FindtheaveragerateofchangeoftheamountofdrugXinthebloodstream,inmilligramsperhour,overtheinterval[2,6].Giveyouranswercorrecttoonedecimalplace. 2marks


SECTION B–continuedTURN OVER

c. FindtheaverageamountofdrugXinthebloodstream,inmilligrams,duringthefirstsixhoursafteronetabletisconsumed.Giveyouranswercorrecttothenearestmilligram. 2marks

d. Sixhoursafterone500milligramtabletofdrugXisconsumed(Tablet1),asecondidenticaltabletisconsumed(Tablet2).TheamountofdrugXinthebloodstreamfromeachtabletconsumedindependentlyisshowninthegraphbelow.

500

400

300amount of

drug X (mg)

200

100

0 1 2 3 4 5 6time (hours)

7 8 9 10 11 12

y

t

Tablet 1 Tablet 2

i. Onthegraphabove,sketchthetotalamountofdrugXinthebloodstreamduringthefirst12hoursafterTablet1isconsumed. 2marks

ii. FindthemaximumamountofdrugXinthebloodstreaminthefirst12hoursandthetimeatwhichthismaximumoccurs.Giveyouranswerscorrecttotwodecimalplaces. 2marks



Question 3 (11marks)Ahorizontalbridgepositioned5mabovelevelgroundis110minlength.Thebridgealsotouchesthetopofthreearches.Eacharchbeginsandendsatgroundlevel.Thearchesare5mapartatthebase,asshowninthediagrambelow.Letxbethehorizontaldistance,inmetres,fromtheleftsideofthebridgeandletybetheheight,inmetres,abovegroundlevel.

y

x

6

5

4

3

2

1

0 10 20 30 40 50 60 70 80 90 100 110

Arch 1 Arch 2 Arch 3

bridge

Arch1canbemodelledbythefunction h R h x x1 15 35 5 5

30: [ , ] , ( ) sin ( ) .→ =

−

π

Arch2canbemodelledbythefunction h R h x x2 240 5 40

30: [ , ] , ( ) sin ( ) . 70 → =

−

π

Arch3canbemodelledbythefunction h a R h x x a3 3 5

30: [ , ] , ( ) sin ( ) . 105 → =

−

π

a. Statethevalueofa,wherea ∈ R. 1mark

b. Describethetransformationthatmapsthegraphofy = h2(x)toy = h3(x). 1mark



Theareaabovegroundlevelbetweenthearchesandthebridgeisfilledwithstone.Thestoneisrepresentedbytheshadedregionsshowninthediagrambelow.

y

x

6

5

4

3

2

1

0 10 20 30 40 50 60 70 80 90 100 110


bridge

c. Findthetotalareaoftheshadedregions,correcttothenearestsquaremetre. 3marks


SECTION B – Question 3 –continued

A secondbridgehasaheightof5mabovethegroundatitsleft-mostpointandisinclinedata

constantangleofelevationofπ90

radians,asshowninthediagrambelow.Thesecondbridge

alsohasthreearchesbelowit,whichareidenticaltothearchesbelowthefirstbridge,andspansahorizontaldistanceof110m.

Letxbethehorizontaldistance,inmetres,fromtheleftsideofthesecondbridgeandletybetheheight,inmetres,abovegroundlevel.

�90


Q

y

10987654321

0 10 20 30 40 50 60 70 80 90 100 110x

P

secondbridge

d. Statethegradientofthesecondbridge,correcttothreedecimalplaces. 1mark

PisapointonArch5.ThetangenttoArch5atpointP hasthesamegradientasthesecondbridge.

e. FindthecoordinatesofP,correcttotwodecimalplaces. 2marks



f. AsupportingrodconnectsapointQ onthesecondbridgetopointPonArch5.Therodfollowsastraightlineandrunsperpendiculartothesecondbridge,asshowninthediagramonpage18.

FindthedistancePQ,inmetres,correcttotwodecimalplaces. 3marks



Question 4 (16marks)Doctorsarestudyingtherestingheartrateofadultsintwoneighbouringtowns:MathslandandStatsville.Restingheartrateismeasuredinbeatsperminute(bpm).TherestingheartrateofadultsinMathslandisknowntobenormallydistributedwithameanof 68bpmandastandarddeviationof8bpm.

a. FindtheprobabilitythatarandomlyselectedMathslandadulthasarestingheartratebetween60bpmand90bpm.Giveyouranswercorrecttothreedecimalplaces. 1mark

Thedoctorsconsiderapersontohaveaslowheartrateiftheperson’srestingheartrateislessthan60bpm.TheprobabilitythatarandomlychosenMathslandadulthasaslowheartrateis0.1587Itisknownthat29%ofMathslandadultsplaysportregularly.Itisalsoknownthat9%ofMathslandadultsplaysportregularlyandhaveaslowheartrate.LetSbetheeventthatarandomlyselectedMathslandadultplayssportregularlyandletHbetheeventthatarandomlyselectedMathslandadulthasaslowheartrate.

b. i. FindPr(H|S),correcttothreedecimalplaces. 1mark

ii. AretheeventsHandSindependent?Justifyyouranswer. 1mark



c. i. Findtheprobabilitythatarandomsampleof16Mathslandadultswillcontainexactlyonepersonwithaslowheartrate.Giveyouranswercorrecttothreedecimalplaces. 2marks

ii. Forrandomsamplesof16Mathslandadults,P̂ istherandomvariablethatrepresentstheproportionofpeoplewhohaveaslowheartrate.

FindtheprobabilitythatP̂ isgreaterthan10%,correcttothreedecimalplaces. 2marks

iii. ForrandomsamplesofnMathslandadults,P̂nistherandomvariablethatrepresentstheproportionofpeoplewhohaveaslowheartrate.

Findtheleastvalueofnforwhich Pr .nn >

>

1 0 99P̂ 2marks



ThedoctorstookalargerandomsampleofadultsfromthepopulationofStatsvilleandcalculatedanapproximate95%confidenceintervalfortheproportionofStatsvilleadultswhohaveaslowheartrate.Theconfidenceintervaltheyobtainedwas(0.102,0.145).

d. i. Determinethesampleproportionusedinthecalculationofthisconfidenceinterval. 1mark

ii. ExplainwhythisconfidenceintervalsuggeststhattheproportionofadultswithaslowheartrateinStatsvillecouldbedifferentfromtheproportioninMathsland. 1mark

EveryyearatMathslandSecondaryCollege,studentshiketothetopofahillthatrisesbehindtheschool.ThetimetakenbyarandomlyselectedstudenttoreachthetopofthehillhastheprobabilitydensityfunctionMwiththerule

M tte t

t( ) =

≥

<

350 50 00 0

2 t−

50

3

wheretisgiveninminutes.

e. Findtheexpectedtime,inminutes,forarandomlyselectedstudentfromMathslandSecondaryCollegetoreachthetopofthehill.Giveyouranswercorrecttoonedecimalplace. 2marks

Studentswhotakelessthan15minutestogettothetopofthehillarecategorisedas‘elite’.

f. FindtheprobabilitythatarandomlyselectedstudentfromMathslandSecondaryCollegeiscategorisedaselite.Giveyouranswercorrecttofourdecimalplaces. 1mark



g. TheYear12studentsatMathslandSecondaryCollegemakeup 17ofthetotalnumberof

studentsattheschool.OftheYear12studentsatMathslandSecondaryCollege,5%arecategorisedaselite.

Findtheprobabilitythatarandomlyselectednon-Year12studentatMathslandSecondaryCollegeiscategorisedaselite.Giveyouranswercorrecttofourdecimalplaces. 2marks



Question 5 (10marks)Considerfunctionsoftheform

f R R f x x a xa

: , ( ) ( )→ =

−

814

2

4

and

h R R h x xa

: , ( )→ = 92 2

whereaisapositiverealnumber.

a. Findthecoordinatesofthelocalmaximumof f intermsofa. 2marks

b. Findthex-valuesofallofthepointsofintersectionbetweenthegraphsof f andh,intermsof awhereappropriate. 1mark

c. Determinethetotalareaoftheregionsboundedbythegraphsofy = f(x)andy = h(x). 2marks


Considerthefunctiong a R g x x a xa

: , , ( ) ( ) ,0 23

814

2

4

→ =− whereaisapositiverealnumber.

d. Evaluate23

23

a g a×

. 1mark

e. Findtheareaboundedbythegraphofg–1,thex-axisandtheline x g a=

23

. 2marks

f. Findthevalueofa forwhichthegraphsofgandg–1havethesameendpoints. 1mark

g. Findtheareaenclosedbythegraphsofgandg–1whentheyhavethesameendpoints. 1mark

END OF QUESTION AND ANSWER BOOK

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 2018

© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2018

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

Documents

2018 Mathematical Methods Written examination 2...MATHEMATICAL METHODS Written examination 2 Thursday 8 November 2018 Reading time: 3.00 pm to 3.15 pm (15 minutes) Writing time: 3.15