MATHEMATICAL METHODSWritten examination 1
Wednesday 2 November 2016 Reading time: 9.00 am to 9.15 am (15 minutes) Writing time: 9.15 am to 10.15 am (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.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2016
STUDENT NUMBER
Letter
2016MATHMETHEXAM1 2
THIS PAGE IS BLANK
3 2016MATHMETHEXAM1
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Question 1 (4marks)
a. Let y xx
=+
cos( )2 2
.
Find dydx. 2marks
b. Let f (x)=x2e5x.
Evaluate f ′(1). 2marks
InstructionsAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegiven,unlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmustbeshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
2016MATHMETHEXAM1 4
Question 2 (3marks)
Let f R: , ,−∞
→
12
where f x x( ) = −1 2 .
a. Find f ′(x). 1mark
b. Findtheangleθfromthepositivedirectionofthex-axistothetangenttothegraphof f atx=–1,measuredintheanticlockwisedirection. 2marks
5 2016MATHMETHEXAM1
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Question 3 (5marks)
Let f :R\{1} →R,where f xx
( ) = +−
2 31.
a. Sketchthegraphof f .Labeltheaxisinterceptswiththeircoordinatesandlabelanyasymptoteswiththeappropriateequation. 3marks
–3 –2 –1 O 1 2 3
4
5
3
2
1
–1
–2
–3
–4
–5
y
x
b. Findtheareaenclosedbythegraphof f ,thelinesx=2andx=4,andthex-axis. 2marks
2016MATHMETHEXAM1 6
Question 4 (3marks)Apaddockcontains10taggedsheepand20untaggedsheep.Fourtimeseachday,onesheepisselectedatrandomfromthepaddock,placedinanobservationareaandstudied,andthenreturnedtothepaddock.
a. Whatistheprobabilitythatthenumberoftaggedsheepselectedonagivendayiszero? 1mark
b. Whatistheprobabilitythatatleastonetaggedsheepisselectedonagivenday? 1mark
c. Whatistheprobabilitythatnotaggedsheepareselectedoneachofsixconsecutivedays?
Expressyouranswerintheformab
c
,wherea,bandcarepositiveintegers. 1mark
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CONTINUES OVER PAGE
2016MATHMETHEXAM1 8
Question 5–continued
Question 5 (11marks)Let f :(0,∞)→R,where f(x)=loge(x)andg:R→R,whereg (x)=x2+1.
a. i. Findtheruleforh,where h x f g x( ) ( )= ( ). 1mark
ii. Statethedomainandrangeofh. 2marks
iii. Showthat h x h x f g x( ) ( ) ( ) .+ − = ( )( )2 2marks
iv. Findthecoordinatesofthestationarypointofhandstateitsnature. 2marks
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b. Letk:(–∞,0]→R,wherek (x)=loge(x2+1).
i. Findtherulefork–1. 2marks
ii. Statethedomainandrangeofk–1. 2marks
2016MATHMETHEXAM1 10
Question 6 (5marks)Let f :[–π,π]→R,where f (x)=2sin(2x)–1.
a. Calculatetheaveragerateofchangeof f between x = −π3and x = π
6. 2marks
b. Calculatetheaveragevalueof f overtheinterval − ≤ ≤π π3 6
x . 3marks
11 2016MATHMETHEXAM1
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Question 7 (3marks)Acompanyproducesmotorsforrefrigerators.Therearetwoassemblylines,LineAandLineB.5%ofthemotorsassembledonLineAarefaultyand8%ofthemotorsassembledonLineBarefaulty.Inonehour,40motorsareproducedfromLineAand50motorsareproducedfromLineB.Attheendofanhour,onemotorisselectedatrandomfromallthemotorsthathavebeenproducedduringthathour.
a. Whatistheprobabilitythattheselectedmotorisfaulty?Expressyouranswerintheform1b,
wherebisapositiveinteger. 2marks
b. Theselectedmotorisfoundtobefaulty.
WhatistheprobabilitythatitwasassembledonLineA?Expressyouranswerintheform1c,
wherecisapositiveinteger. 1mark
2016MATHMETHEXAM1 12
Question 8–continued
Question 8 (6marks)LetXbeacontinuousrandomvariablewithprobabilitydensityfunction
f xx x xe( ) = − ( ) < ≤
4 0 10
logelsewhere
Partofthegraphof f isshownbelow.Thegraphhasaturningpointat xe
=1 .
0 1
y
x1e
a. Showbydifferentiationthat
xk
k xk
e2 1log ( ) −( )
isanantiderivativeof xk–1loge(x),wherekisapositiverealnumber. 2marks
13 2016MATHMETHEXAM1
END OF QUESTION AND ANSWER BOOK
b. i. CalculatePr .Xe
>
12marks
ii. Hence,explainwhetherthemedianofXisgreaterthanorlessthan1e ,giventhat
e > 52
. 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 2016
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2016
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
