2017 mathematical methods written examination 1...2017 mathmeth exam 1 12 end of question and answer...
TRANSCRIPT
MATHEMATICAL METHODSWritten examination 1
Wednesday 8 November 2017 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
9 9 40
• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpenersandrulers.
• StudentsareNOTpermittedtobringintotheexaminationroom:anytechnology(calculatorsorsoftware),notesofanykind,blanksheetsofpaperand/orcorrectionfluid/tape.
Materials supplied• Questionandanswerbookof12pages• 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 2
THIS PAGE IS BLANK
3 2017MATHMETHEXAM1
TURN OVER
Question 1 (4marks)
a. Letf:(–2,∞)→R,f(x)= xx + 2
.
Differentiatefwithrespecttox. 2marks
b. Letg(x)=(2–x3)3.
Evaluateg′ (1). 2marks
InstructionsAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegiven,unlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmustbeshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
2017 MATHMETH EXAM 1 4
Question 2 (4 marks)Let y = x loge(3x).
a. Find dydx
. 2 marks
b. Hence, calculate (log ( ) ) .e x dx1
23 1∫ + Express your answer in the form loge(a), where a is a
positive integer. 2 marks
5 2017MATHMETHEXAM1
TURN OVER
Question 3 (4marks)Letf:[–3,0]→R,f (x)=(x+2)2(x–1).
a. Showthat(x+2)2(x–1)=x3+3x2–4. 1mark
b. Sketchthegraphoffontheaxesbelow.Labeltheaxisinterceptsandanystationarypointswiththeircoordinates. 3marks
6
4
2
O–1–2–3–4 1 2
–2
–4
–6
x
y
2017MATHMETHEXAM1 6
Question 4 (2marks)Inalargepopulationoffish,theproportionofangelfishis
14.
LetP̂ betherandomvariablethatrepresentsthesampleproportionofangelfishforsamplesofsizendrawnfromthepopulation.
FindthesmallestintegervalueofnsuchthatthestandarddeviationofP̂ islessthanorequalto1
100.
7 2017MATHMETHEXAM1
TURN OVER
Question 5 (4marks)ForJactologontoacomputersuccessfully,Jacmusttypethecorrectpassword.Unfortunately,Jachasforgottenthepassword.IfJactypesthewrongpassword,Jaccanmakeanotherattempt.Theprobabilityofsuccessonanyattemptis2
5.Assumethattheresultofeachattemptisindependent
oftheresultofanyotherattempt.Amaximumofthreeattemptscanbemade.
a. WhatistheprobabilitythatJacdoesnotlogontothecomputersuccessfully? 1mark
b. CalculatetheprobabilitythatJaclogsontothecomputersuccessfully.Expressyouranswerintheforma
b,whereaandbarepositiveintegers. 1mark
c. CalculatetheprobabilitythatJaclogsontothecomputersuccessfullyonthesecondoronthethirdattempt.Expressyouranswerintheform c
d,wherecanddarepositiveintegers. 2marks
2017MATHMETHEXAM1 8
Question 6 (3marks)
Let tan( ) sin( ) cos( ) sin( ) cos( ) .θ θ θ θ θ− −( ) +( ) =1 3 3 0( )a. Stateallpossiblevaluesoftan(θ ). 1mark
b. Hence,findallpossiblesolutionsfor tan( ) sin ( ) cos ( ) ,θ θ θ− −( ) =1 3 02 2( ) where0≤θ≤π. 2marks
9 2017MATHMETHEXAM1
TURN OVER
Question 7 (5marks)
Let f R f x x: [ , ) , ( ) .0 1∞ → = +
a. Statetherangeoff. 1mark
b. Letg:(–∞,c]→R,g(x)=x2+4x+3,wherec<0.
i. Findthelargestpossiblevalueofcsuchthattherangeofgisasubsetofthedomainoff. 2marks
ii. Forthevalueofcfoundinpart b.i.,statetherangeoff(g(x)). 1mark
c. Leth:R→R,h(x)=x2+3.
Statetherangeof f(h(x)). 1mark
2017MATHMETHEXAM1 10
Question 8 (5marks)
ForeventsAandBfromasamplespace,Pr and PrA B B A| | .( ) = ( ) =15
14Let Pr ( ) .A B p∩ =
a. FindPr(A)intermsofp. 1mark
b. FindPr ( )′∩ ′A B intermsofp. 2marks
c. GiventhatPr ( ) ,A B∪ ≤15statethelargestpossibleintervalforp. 2marks
11 2017MATHMETHEXAM1
Question 9 –continuedTURN OVER
Question 9 (9marks)Thegraphof f R f x xx: , , ( )(0 1 1[ ]→ = – isshownbelow.
y x= )1 – x
y
x0 1
a. Calculatetheareabetweenthegraphoffandthex-axis. 2marks
b. Forxintheinterval(0,1),showthatthegradientofthetangenttothegraphoffis1 32− x
x. 1mark
2017MATHMETHEXAM1 12
END OF QUESTION AND ANSWER BOOK
Theedgesoftheright-angled triangleABCarethelinesegmentsACandBC,whicharetangenttothegraphoff,andthelinesegmentAB,whichispartofthehorizontalaxis,asshownbelow.LetθbetheanglethatACmakeswiththepositivedirectionofthehorizontalaxis,where45°≤θ<90°.
y x= )1 – x
y
x0A B
C
θ
c. FindtheequationofthelinethroughBandCintheformy=mx+c,forθ=45°. 2marks
d. FindthecoordinatesofCwhenθ=45°. 4marks
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 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