2017 Mathematical Methods-nht Written examination 15 2017 MATHMETH EXAM 1 (NHT) TURN OVER Question 3 (3 marks) a. State the smallest positive value of k such that x = 3 4 π is a solution

MATHEMATICAL METHODSWritten examination 1

Tuesday 6 June 2017 Reading time: 2.00 pm to 2.15 pm (15 minutes) Writing time: 2.15 pm to 3.15 pm (1 hour)

QUESTION AND ANSWER BOOK

Structure of bookNumber of questions

Number of questions to be answered

Number of marks

8 8 40

• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpenersandrulers.

• StudentsareNOTpermittedtobringintotheexaminationroom:anytechnology(calculatorsorsoftware),notesofanykind,blanksheetsofpaperand/orcorrectionfluid/tape.

Materials supplied• Questionandanswerbookof13pages.• Formulasheet.• Workingspaceisprovidedthroughoutthebook.

Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.• AllwrittenresponsesmustbeinEnglish.

At the end of the examination• 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

2017MATHMETHEXAM1(NHT) 2

THIS PAGE IS BLANK

3 2017MATHMETHEXAM1(NHT)

TURN OVER

Question 1 (4marks)

a. Let y e xx=

2

2cos .

Find dydx

. 2marks

b. Let f :(0,π)→R,where f (x)=loge(sin(x)).

Evaluate ′

f

π3

. 2marks

InstructionsAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegiven,unlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmustbeshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

2017MATHMETHEXAM1(NHT) 4

Question 2 (5marks)a. Findanantiderivativeofcos(1–x)withrespecttox. 1mark

b. Evaluate 3 42 21

2x

xdx+

∫ . 2marks

c. Find f (x)giventhat f(4)=25and ′ = − + >−f x x x x( ) ,38

10 1 0212 . 2marks

5 2017MATHMETHEXAM1(NHT)

TURN OVER

Question 3 (3marks)

a. Statethesmallestpositivevalueofksuchthat x = 34π isasolutionoftan(x)=cos(kx). 1mark

b. Solve2sin2(x)+3sin(x)–2=0,where0≤x≤2π. 2marks

2017MATHMETHEXAM1(NHT) 6

Question 4 –continued

Question 4 (5marks)

Let f : ,−

π π2 2

→R,where f(x)=tan(2x)+1.

a. Sketchthegraphof f ontheaxesbelow.Labelanyasymptoteswiththeappropriateequation,andlabeltheendpointsandtheaxisinterceptswiththeircoordinates. 4marks

0

2

–2

–4

4

y

x

2π

−4π

−4π

2π

–1

1

–3

3

7 2017MATHMETHEXAM1(NHT)

TURN OVER

b. Usefeaturesofthegraphinpart a.tofindtheaveragevalueof f between x = −π8and

x = π8. 1mark

2017MATHMETHEXAM1(NHT) 8

Question 5 (6marks)Recordsofthearrivaltimesoftrainsatabusystationhavebeenkeptforalongperiod.TherandomvariableXrepresentsthenumberofminutesafterthescheduledtimethatatrainarrivesatthisstation,thatis,thelatenessofthetrain.Assumethatthelatenessofonetrainarrivingatthisstationisindependentofthelatenessofanyothertrain.ThedistributionofXisgiveninthetablebelow.

x –1 0 1 2

Pr(X=x) 0.1 0.4 0.3 p

a. Findthevalueofp. 1mark

b. FindE(X ). 1mark

c. Findvar(X ). 2marks

d. Apassengercatchesatrainatthisstationonfiveseparateoccasions.

Whatistheprobabilitythatthetrainarrivesbefore thescheduledtimeonexactlyfouroftheseoccasions? 2marks

9 2017MATHMETHEXAM1(NHT)

TURN OVER

Question 6 (3marks)Atalargesportingarenathereareanumberoffoodoutlets,includingacafe.

a. Thecafeemploysfivemenandfourwomen.Fourofthesepeoplearerosteredatrandomtoworkeachday.LetP̂representthesampleproportionofmenrosteredtoworkonaparticularday.

i. ListthepossiblevaluesthatP̂cantake. 1mark

ii. FindPr(P̂=0). 1mark

b. Thereareover80000spectatorsatasportingmatchatthearena.FiveinnineofthesespectatorssupporttheGoannasteam.Asimplerandomsampleof2000spectatorsisselected.

WhatisthestandarddeviationofthedistributionofP̂,thesampleproportionofspectatorswhosupporttheGoannasteam? 1mark

2017MATHMETHEXAM1(NHT) 10

Question 7 (6marks)Let f :R →R,where f (x)=2x3+1,andletg:R→R,whereg (x)=4–2x.

a. i. Findg (f (x)). 1mark

ii. Find f (g (x))andexpressitintheformk–m(x–d )3,wherem,kanddareintegers. 2marks

b. ThetransformationT:R2 →R2withruleTxy a

xy

bc

=

+

1 00

, wherea,bandcare

integers,mapsthegraphof y=g (f (x))ontothegraphof y=f (g (x)).

Findthevaluesofa,bandc. 3marks

11 2017MATHMETHEXAM1(NHT)

TURN OVER

CONTINUES OVER PAGE

2017MATHMETHEXAM1(NHT) 12

Question 8 –continued

Question 8 (8marks)Theruleforafunction f isgivenby f x x( ) = + −2 3 1,where f isdefinedonitsmaximaldomain.

a. Findthedomainandruleoftheinversefunction f –1. 2marks

b. Solve f (x)=f–1(x). 2marks

13 2017MATHMETHEXAM1(NHT)

END OF QUESTION AND ANSWER BOOK

c. Let g D R g x x c: , ( ) ,→ = + −2 1 whereDisthemaximaldomainofgandcisarealnumber.

i. Forwhatvalue(s)ofcdoesg (x)=g–1(x)havenorealsolutions? 2marks

ii. Forwhatvalue(s)ofcdoesg (x)=g–1(x)haveexactlyonerealsolution? 2marks

MATHEMATICAL METHODS

Written examination 1

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 dx n x c nn n=

++ ≠ −+∫ 1 1 1

1 ,

ddx

ax b an ax bn n( )+( ) = +( ) −1 ( ) ( ) ( ) ,ax b dx a n ax b c nn n+ =

++ + ≠ −+∫ 1 1 1

1

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 rule ddx

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 ) ( )= −1approximate confidence interval

,p zp p

np z

p pn

−−( )

+−( )

1 1ˆ ˆ ˆˆˆ ˆ

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 ) ( )= −1approximate confidence interval

,p zp p

np z

p pn

−−( )

+−( )

1 1ˆ ˆ ˆˆˆ ˆ

2017 Mathematical Methods 1InstructionsFormula sheet