2017 Mathematical Methods-nht Written examination 2€¦ · MATHEMATICAL METHODS Written...

MATHEMATICAL METHODSWritten examination 2

Wednesday 7 June 2017 Reading time: 2.00 pm to 2.15 pm (15 minutes) Writing time: 2.15 pm to 4.15 pm (2 hours)

QUESTION AND ANSWER BOOK

Structure of bookSection Number of

questionsNumber of questions

to be answeredNumber of

marks

A 20 20 20B 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• Questionandanswerbookof21pages.• 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(NHT) 2

SECTION A – continued

Question 1Thegradientofalineperpendiculartothelinethatpassesthrough(3,0)and(0,–6)is

A. −12

B. –2

C. 12

D. 4

E. 2

Question 2

Thefunctionwithrule f x x( ) sin=

+2

41 hasperiod

A. π4

B. π2

C. π

D. 4π

E. 8π

Question 3Thesimultaneouslinearequationsmx+7y=12and7x + my = mhaveauniquesolutiononlyforA. m=7orm=–7B. m=12orm=3C. m ∈ R\{–7,7}D. m=4orm=3E. m ∈ R\{12,1}

SECTION A – Multiple-choice questions

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

3 2017MATHMETHEXAM2(NHT)

SECTION A – continuedTURN OVER

Question 4

Thegraphofthefunction f R f x x:[ , ) ( ) ,0 413∞ → =, where isreflectedinthex-axisandthentranslated

fiveunitstotherightandsixunitsverticallydown.Whichoneofthefollowingistheruleofthetransformedgraph?

A. y x= − +4 5 613( )

B. y x= − + −4 5 613( )

C. y x= − + +4 5 613( )

D. y x= − − −4 5 613( )

E. y x= − +4 5 113( )

Question 5

Whichoneofthefollowingistheinversefunctionofthefunction f R f xx

: ( , ) , ( )−∞ → =−

+3 23

1?

A. f R f xx

− −−∞ → = −−

+1 123 4

13: ( , ) , ( )

( )

B. f R f xx

− −∞ → = −−

+1 121 4

31: ( , ) , ( )

( )

C. f R f xx

− −∞ → = −−

+1 121 4

13: ( , ) , ( )

( )

D. f R f xx

− −∞ → = − +1 121 4 3: ( , ) , ( )

E. f R R f xx

− + −→ = −−

+1 12

41

3: , ( )( )

Question 6

Let f D R f x xx

: , ( ) ,→ =−−

3 52

whereD isthemaximaldomainoff .

Whichofthefollowingaretheequationsoftheasymptotesofthegraphoff ?

A. x=2and y = 53

B. x=2andy=–3

C. x=–2andy=3

D. x=–3andy = 2

E. x=2andy=3

2017MATHMETHEXAM2(NHT) 4

SECTION A – continued

Question 7Thegraphofy = kx–2willnotintersectortouchthegraphofy = x2+3xwhenA. 3 2 2 3 2 2− < < +k

B. { : } { : }k k k k< − ∪ > +3 2 2 3 2 2

C. –5<k<11

D. 3 2 2 3 2 2− ≤ ≤ +k

E. k ∈ R+

Question 8Partofthegraphofafunctionf isshownbelow.

0

(–a, –a)

(a, a)y

x

Whichoneofthefollowingistheaveragevalueofthefunctionf overtheinterval[–a,a]?A. 0

B. 34a

C. 38a

D. a2

E. a4

5 2017MATHMETHEXAM2(NHT)

SECTION A – continuedTURN OVER

Question 9

Therangeofthefunction f R f x xe: , ( , ] , ( ) log ( )−

∪ → =

120 0 2 2 is

A. −( ]2 2 2 2log ( ), log ( )e e

B. −

log , log ( )e e

14

4

C. −∞( ], log ( )2 2e

D. R e e\ log ( ), log ( )−[ )2 2 2 2

E. R

Question 10Thetangenttothegraphofy=3sin(2x)–1isparalleltothelinewithequationy=3x+1atthepoints wherexisequalto

A. n n Zππ

± ∈6

,

B. −π π3 3, only

C. π π6

56

, only

D. n n Zππ

± ∈3

,

E. nπ,n ∈ Z

Question 11Abagcontainsfivebluemarblesandfourredmarbles.Asampleoffourmarblesistakenfromthebag,withoutreplacement.Theprobabilitythattheproportionofbluemarblesinthesampleisgreaterthan

12is

A. 12

B. 29

C. 514

D. 59

E. 2563

2017 MATHMETH EXAM 2 (NHT) 6

SECTION A – continued

Question 12The maximum temperature reached by the water heated in a kettle each time it is used is normally distributed with a mean of 95 °C and a standard deviation of 2 °C.When the kettle is used, the proportion of times that the maximum temperature reached by the water is greater than 98 °C is closest toA. 0.0671B. 0.0668C. 0.8669D. 0.9332E. 0.9342

Question 13The probability distribution for the discrete random variable X is defined by the following.

x –1 0 1 2

Pr(X = x) a 3b 2a b

The mean and variance of the distribution, respectively, areA. a + 2b and 3a + 4bB. a + 2b and 3a + 4b – a2 – 4ab – 4b2

C. 3a + 2b and 3a + 4b – 9a2 – 12ab – 4b2

D. 3a + 2b and 3a + 4bE. 3a + 5b and a + b

Question 14

2a

a0 2a 3a 4a 5a

y

x

The rule of the function with the graph shown above could be

A. y a xa

a=

+cos π

2

B. y a xa

a=

+2 sin π

C. y a xa

a= −

+cos π

2

D. y aax a a= −

+sin ( )π

E. y a xa

a= −

+cos π

7 2017 MATHMETH EXAM 2 (NHT)

SECTION A – continuedTURN OVER

Question 15The graph of f where f (x) = ekx, k > 0, touches the graph of its inverse function f –1 at exactly one point, as shown below.

0

y = f (x)

y = f –1(x)

y

x

The value of k must beA. e

B. e2

C. 1

D. 12e

E. 1e

Question 16

The area under the graph of y = f (x), between x = 1 and x = 4, where f (x) >12

over this interval, is eight units.The area under the graph of y = 2 f (x) – 1 over the same interval isA. 15 units.B. 13 units.C. 10 units.D. 7 units.E. 3 units.

2017MATHMETHEXAM2(NHT) 8

END OF SECTION A

Question 17Afunctionfsatisfiestherelationf(x2)=f(x)+f(x+2).Apossibleruleforf isA. f x x( ) = + 2

B. f (x)=x + 2

C. f (x)=log10(x–1)

D. f x x( ) = −( )12

12

E. f xx

( ) =−1

1

Question 18Letf (x)=xmeax,wherea andmarenon-zerorealconstants.If(x+2)isafactoroff′(x),thenwhichoneofthefollowingmustbetrue?A. m = 2B. m=–2C. m=2–aD. m = 2aE. m=–2a

Question 19ThefirstdigitofeachhousenumberinalargeaddressdatabaseisarandomvariableD.Thepossiblevalues

ofDare1,2,…,9and Pr( ) log .D dd

= = +

10 11

TheprobabilitythatDisgreaterthan8,giventhatDisgreaterthan7,is

A. 1–log10(9)

B. 1 91 8

10

10

−−log ( )log ( )

C. 1 92 9 8

10

10 10

−− −

log ( )log ( ) log ( )

D. log1089

E. log1098

Question 20Considerthefunctionf :[2,∞)→ R,f (x)=x4+2(a–4)x2–8ax+1,wherea ∈ R.Themaximalsetofvaluesofaforwhichtheinversefunctionf –1existsisA. (–9,∞)B. (–∞,1)C. [–9,1]D. [–8,∞)E. (–∞,–8]

9 2017MATHMETHEXAM2(NHT)

TURN OVER

CONTINUES OVER PAGE

2017MATHMETHEXAM2(NHT) 10

SECTION B – Question 1–continued

Question 1 (11marks)Thetemperature,T °C,inanofficeiscontrolled.Foraparticularweekday,thetemperatureattimet,wheretisthenumberofhoursaftermidnight,isgivenbythefunction

T t t t( ) sin ( ) , .= + −

≤ ≤19 6

128 0 24π

a. Whatarethemaximumandminimumtemperaturesintheoffice? 2marks

b. Whatisthetemperatureintheofficeat6.00am? 1mark

c. Mostofthepeopleworkingintheofficearriveat8.00am.

Whatisthetemperatureintheofficewhentheyarrive? 1mark

SECTION B

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

11 2017MATHMETHEXAM2(NHT)

SECTION B – continuedTURN OVER

d. Forhowmanyhoursofthedayisthetemperaturegreaterthanorequalto19°C? 2marks

e. Whatistheaveragerateofchangeofthetemperatureintheofficebetween8.00amandnoon? 2marks

f. i. FindT ′(t). 1mark

ii. Atwhattimeofthedayisthetemperatureintheofficedecreasingmostrapidly? 2marks

2017MATHMETHEXAM2(NHT) 12

SECTION B – Question 2–continued

Question 2 (13marks)Letf :R → R,wheref(x)=(x–2)2(x–5).

a. Findf ′(x). 1mark

b. Forwhatvaluesofxisf ′(x)<0? 1mark

c. i. Findthegradientofthelinesegmentjoiningthepointsonthegraphofy = f (x) wherex=1andx=5. 1mark

ii. Showthatthemidpointofthelinesegmentinpart c.i.alsoliesonthegraphofy = f (x). 2marks

13 2017MATHMETHEXAM2(NHT)

SECTION B – Question 2–continuedTURN OVER

iii. Findthevaluesofxforwhichthegradientofthetangenttothegraphofy = f (x)isequaltothegradientofthelinesegmentjoiningthepointsonthegraphwherex=1andx=5. 2marks

Letg :R → R,whereg (x)=(x–2)2(x–a),wherea ∈ R.

d. ThecoordinatesofthestationarypointsofgareP(2,0)andQ(p(a+1),q(a–2)3),where pandqarerationalnumbers.

Findthevaluesofpandq. 2marks

2017MATHMETHEXAM2(NHT) 14

SECTION B – continued

e. Showthatthegradientofthetangenttothegraphofy = g (x)atthepoint(a,0)ispositivefora ∈ R\{2}. 1mark

f. i. Findthecoordinatesofanotherpointwherethetangenttothegraphofy = g (x)isparalleltothetangentatx = a. 2marks

ii. Hence,findthedistancebetweenthispointandpointQ whena>2. 1mark

15 2017 MATHMETH EXAM 2 (NHT)

SECTION B – Question 3 – continuedTURN OVER

Question 3 (18 marks)A company supplies schools with whiteboard pens.The total length of time for which a whiteboard pen can be used for writing before it stops working is called its use-time.There are two types of whiteboard pens: Grade A and Grade B.The use-time of Grade A whiteboard pens is normally distributed with a mean of 11 hours and a standard deviation of 15 minutes.

a. Find the probability that a Grade A whiteboard pen will have a use-time that is greater than 10.5 hours, correct to three decimal places. 1 mark

The use-time of Grade B whiteboard pens is described by the probability density function

f xx e x

x

( )( )( )

=− − ≤ ≤

576

12 x 1

0

0 126

otherwise

where x is the use-time in hours.

b. Determine the expected use-time of a Grade B whiteboard pen. Give your answer in hours, correct to two decimal places. 2 marks

c. Determine the standard deviation of the use-time of a Grade B whiteboard pen. Give your answer in hours, correct to two decimal places. 2 marks

2017MATHMETHEXAM2(NHT) 16

SECTION B – Question 3–continued

d. FindtheprobabilitythatarandomlychosenGradeBwhiteboardpenwillhaveause-timethatisgreaterthan10.5hours,correcttofourdecimalplaces. 2marks

Aworkeratthecompanyfindstwoboxesofwhiteboardpensthatarenotlabelled,butknowsthatoneboxcontainsonlyGradeAwhiteboardpensandtheotherboxcontainsonlyGradeBwhiteboardpens.Theworkerdecidestorandomlyselectawhiteboardpenfromoneoftheboxes.Iftheselectedwhiteboardpenhasause-timethatisgreaterthan10.5hours,thentheboxthatitcamefromwillbelabelledGradeAandtheotherboxwillbelabelledGradeB.Otherwise,theboxthatitcamefromwillbelabelledGradeBandtheotherboxwillbelabelledGradeA.

e. Findtheprobability,correcttothreedecimalplaces,thattheworkerlabelstheboxesincorrectly. 2marks

f. Findtheprobability,correcttothreedecimalplaces,thatthewhiteboardpenselectedwasGradeB,giventhattheboxeshavebeenlabelledincorrectly. 2marks

17 2017MATHMETHEXAM2(NHT)

SECTION B – continuedTURN OVER

Asawhiteboardpenages,itstipmaydrytothepointwherethewhiteboardpenbecomes defective(unusable).Thecompanyhasstockthatistwoyearsoldand,atthatage,itisknownthat5%ofGradeAwhiteboardpenswillbedefective.

g. AschoolpurchasesaboxofGradeAwhiteboardpensthatistwoyearsoldandaclassof26studentsisthefirsttousethem.

Ifeverystudentreceivesawhiteboardpenfromthisbox,findtheprobability,correcttofourdecimalplaces,thatatleastonestudentwillreceiveadefectivewhiteboardpen. 2marks

h. LetP̂ AbetherandomvariableofthedistributionofsampleproportionsofdefectiveGradeAwhiteboardpensinboxesof100.Theboxescomefromthestockthatistwoyearsold.

Find(P̂ A>0.04|P̂ A<0.08).Giveyouranswercorrecttofourdecimalplaces.Donotuseanormalapproximation. 3marks

i. Aboxof100GradeAwhiteboardpensthatistwoyearsoldisselectedanditisfoundthatsixofthewhiteboardpensaredefective.

Determinea90%confidenceintervalforthepopulationproportionfromthissample,correcttotwodecimalplaces. 2marks

2017MATHMETHEXAM2(NHT) 18

SECTION B – Question 4–continued

Question 4 (18marks)

Let f R f x x x:[ , ] , ( ) .0 16 4→ = −

a. Findthevalueofxatwhichfhasamaximumandstatethemaximumvalue. 2marks

b. Sketchthegraphofy = f (x)ontheaxesbelow. 1mark

0

2

4

6

2 4 6 8 10 12 14 16

y

x

19 2017MATHMETHEXAM2(NHT)

SECTION B – Question 4–continuedTURN OVER

c. i. FindtheareaAoftheregionenclosedbythegraphoffandthex-axis. 1mark

ii. LetXbethepoint(16,0).TrianglescanbeformedwithverticesO,XandC(c,f (c)), whereCisapointonthegraphofy = f (x).

FindthevalueofcforwhichtriangleOCXhasmaximumareaandfindthismaximumarea. 2marks

2017MATHMETHEXAM2(NHT) 20

SECTION B – Question 4–continued

Letaandbbetwovaluesinthedomainoffsuchthatf (b)=f (a),andassumeb > a.

d. Showthatb a= −( )42. 1mark

e. Arectangleisformedwithvertices(a,0),(b,0),(b,f (b))and(a,f (a)).

i. Findtheareaofthisrectangleintermsofa. 2marks

ii. Findthevaluesofaandbforwhichtherectanglehasmaximumarea. 3marks

iii. Findthevalueofthemaximumareaintheform m np

,wherem,nandparepositiveintegers. 2marks

21 2017MATHMETHEXAM2(NHT)

END OF QUESTION AND ANSWER BOOK

f. AtrapeziumODBXisformedwithverticesatO,X(16,0),B(b,f (b))andD(a,f (a)),asshowninthediagrambelow,whereDBisparalleltoOX.

O X(16, 0)

D(a, f (a)) B(b, f (b))

y

x

i. Findthevalueofa forwhichthetrapeziumhasmaximumareaATandfindthismaximumarea. 3marks

ii. FindAAT ,whereAistheareaoftheregionenclosedbythegraphoffandthex-axis. 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 ( )( ) = 11 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