MATHEMATICAL METHODSWritten examination 1

Friday 1 June 2018 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

9 9 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2018

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2018

STUDENT NUMBER

Letter

2018MATHMETHEXAM1(NHT) 2

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THIS PAGE IS BLANK

3 2018MATHMETHEXAM1(NHT)

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Question 1 (4marks)

a. Let f x ex

x( ) =

−( )2 3.

Find f ′ (x). 2marks

b. Lety=(x+5) loge(x).

Find dydx

whenx=5. 2marks

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

2018MATHMETHEXAM1(NHT) 4

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Question 2 (4marks)Let f (x)=–x2+x+4andg(x)=x2–2.

a. Findg(f (3)). 2marks

b. Expresstherulefor f(g(x))intheformax4+bx2+c,wherea,bandcarenon-zerointegers. 2marks

5 2018MATHMETHEXAM1(NHT)

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Question 3 (2marks)

Evaluate e e dxx x0

1

∫ − − .

Question 4 (3marks)Solvelog3(t)–log3(t

2–4)=–1fort.

2018MATHMETHEXAM1(NHT) 6

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Question 5 (3marks)

Let h R R h xx

: { } , ( ) .+ ∪ → =+

−0 72

3

a. Statetherangeofh. 1mark

b. Findtheruleforh–1. 2marks

7 2018MATHMETHEXAM1(NHT)

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Question 6 (4marks)ThediscreterandomvariableX hastheprobabilitymassfunction

Pr( ){ , , }

X xkxk

xx= =∈=

0

1 4 63

otherwise

a. Showthat k = 112

. 2marks

b. FindE(X ). 1mark

c. EvaluatePr(X≥3X≥2). 1mark

2018MATHMETHEXAM1(NHT) 8

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Question 7 (9marks)

Let f R: ,02π

→ , f(x)=4cos(x)and g R: ,0

2π

→ ,g(x)=3sin(x).

a. Sketchthegraphoffandthegraphofgontheaxesprovidedbelow. 2marks

3

4

2

1

Ox

y

4π

2π

b. Letcbesuchthat f(c)=g(c),where c∈

02

, π .

Findthevalueofsin(c)andthevalueofcos(c). 3marks

Question 7 –continued

9 2018MATHMETHEXAM1(NHT)

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c. LetAbetheregionenclosedbythehorizontalaxis,thegraphoffandthegraphofg.

i. ShadetheregionAontheaxesprovidedinpart a.andalsolabelthepositionofconthehorizontalaxis. 1mark

ii. CalculatetheareaoftheregionA. 3marks

2018MATHMETHEXAM1(NHT) 10

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Question 8 (3marks)LetP̂betherandomvariablethatrepresentsthesampleproportionsofcustomerswhobringtheirownshoppingbagstoalargeshoppingcentre.Fromasampleconsistingofallcustomersonaparticularday,anapproximate95%confidenceintervalfortheproportionpofcustomerswhobringtheirownshoppingbagstothislarge

shoppingcentrewasdeterminedtobe4853

500005147

50000,

.

a. Findthevalueofp̂ thatwasusedtoobtainthisapproximate95%confidenceinterval. 1mark

b. Usethefactthat1 96 4925

. = tofindthesizeofthesamplefromwhichthisapproximate95%

confidenceintervalwasobtained. 2marks

11 2018MATHMETHEXAM1(NHT)

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Question 9 –continuedTURN OVER

Question 9 (8marks)Thediagrambelowshowsatrapeziumwithverticesat(0,0),(0,2),(3,2)and(b,0),wherebisarealnumberand0<b<2.

2

bx

y

3O

Onthesameaxesasthetrapezium,partofthegraphofacubicpolynomialfunctionisdrawn.Ithastheruley=ax(x–b)2,whereaisanon-zerorealnumberand0≤x≤b.

a. Atthelocalmaximumofthegraph,y=b.

Findaintermsofb. 3marks

2018MATHMETHEXAM1(NHT) 12

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Theareabetweenthegraphofthefunctionandthex-axisisremovedfromthetrapezium,asshowninthediagrambelow.

2

bx

y

3O

b. Showthattheexpressionfortheareaoftheshadedregionisb b+ −3 9

16

2squareunits. 3marks

Question 9 –continued

13 2018MATHMETHEXAM1(NHT)

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END OF QUESTION AND ANSWER BOOK

c. Findthevalueofbforwhichtheareaoftheshadedregionisamaximumandfindthismaximumarea. 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 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 ( )( ) = 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