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MATHEMATICAL METHODSWritten examination 2
Monday 4 June 2018 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.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2018
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2018
STUDENT NUMBER
Letter
2018MATHMETHEXAM2(NHT) 2
SECTION A – continued
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Question 1
Letf:R → R,f (x)=3–2cos π x4
.
TheperiodandrangeofthisfunctionarerespectivelyA. 4and[−2,2]
B. 8and[1,5]
C. 8πand[1,5]
D. 8πand[−2,2]
E. 12 and[−1,5]
SECTION A – Multiple-choice questions
Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrectforthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
3 2018MATHMETHEXAM2(NHT)
SECTION A – continuedTURN OVER
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Question 2Thediagrambelowshowspartofthegraphofapolynomialfunction.
x
y
–5 –4 –3 –2 –1 O 1 2 3 4 5
ApossibleruleforthisfunctionisA. y=(x+2)(x–1)(x–3)B. y=(x+2)2(x–1)(x–3)C. y=(x+2)2(x–1)(3–x)D. y=−(x−2)2(x–1)(3–x)E. y=−(x+2)(x–1)(x–3)
Question 3Adiscreterandomvariablehasabinomialdistributionwithameanof3.6andavarianceof1.98Thevaluesofn(thenumberofindependenttrials)andp(theprobabilityofsuccessineachtrial)areA. n=4andp=0.9B. n=5andp=0.72C. n=6andp=0.6D. n=8andp=0.45E. n=12andp=0.3
Question 4IfAandBareeventsfromasamplespacesuchthatPr(A)=0.6,Pr(B)=0.3andPr(A ∪ B)=0.7,then Pr(A ∩ B′)isequaltoA. 0.12B. 0.18C. 0.2D. 0.3E. 0.4
2018MATHMETHEXAM2(NHT) 4
SECTION A – continued
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Question 5Asetofthreenumbersthatcouldbethesolutionsofx3 + ax2+16x+84=0isA. {3,4,7}B. {–4,–3,7}C. {–2,–1,21}D. {–2,6,7}E. {2,6,7}
Question 6Thesumofthesolutionstotheequation 3 sin(2x)=–3cos(2x)forx ∈[0,2π]isequalto
A. π3
B. 76π
C. 113π
D. 133π
E. 143π
Question 7Sixballsnumberedfrom1to6areplacedinajar.Aballistakenrandomlyfromthejaranditsnumberisrecorded.Thisballisreturnedtothejar,andasecondballisthentakenrandomlyanditsnumberisrecorded.Thesumofthetworecordednumbersisthencalculated.Theprobabilitythatthesumofthetworecordednumbersis7,giventhatthefirstrecordednumberisodd,isequalto
A. 13
B. 14
C. 16
D. 112
E. 19
5 2018MATHMETHEXAM2(NHT)
SECTION A – continuedTURN OVER
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Question 8Partofthegraphofy=f(x)isshownbelow.
x
y
y = f (x)
–3 –2 –1 O 1 2 3 4
–3
–2
–1
1
Thegraphofy=f ′(x)isbestrepresentedby
x
x = 1
y = –1
y
O
x
x = 1y
Ox
x = 1y
O
x
x = 1
y = –1
y = 0
y = 0
y = 0
y
O
x
x = 1y
O
A. B.
C. D.
E.
2018MATHMETHEXAM2(NHT) 6
SECTION A – continued
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Question 9AcontinuousrandomvariableXhasanormaldistributionwithameanof40andastandarddeviationof5.ThecontinuousrandomvariableZhasthestandardnormaldistribution.Pr(–2<Z<1)isequaltoA. Pr(40<X<55)B. Pr(35<X<50)C. Pr(30<X<50)D. Pr(10<X<30)E. Pr(X>30)–Pr(X<45)
Question 10
Therangeofthefunction f R f x x x: , ,−
→ ( ) = − +
12
2 2 3 43 is
A. 4 2 4 2−( + ),
B. −
12
2,
C. 4 2 4 2−( + ,
D. −
12
2,
E. 4 2 4 2− + ,
Question 11Themaximaldomainofthefunctiong,whereg(x)=loge(–2x),isA. RB. R –
C. R+
D. [0,∞)E. (–∞,0]
Question 12
Theaveragevalueoff(x)=x2–2xovertheinterval[1,a]is133.
ThevalueofaisA. 2
B. 3
C. 103
D. 5
E. 163
7 2018MATHMETHEXAM2(NHT)
SECTION A – continuedTURN OVER
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Question 13Thefunctionfhasthepropertyf(2x)=(f(x))2–2forallrealnumbersx.Apossibleruleforthefunctionf(x)is
A. 1
42x +B. cos(x)
C. 2loge(x2+1)
D. ex + e–x
E. x2
Question 14
Thegraphofthefunctionf isobtainedfromthegraphofthefunctiongwithrule g x x( ) cos= −
3
6π
bya
dilationofafactorof12fromthex-axis,areflectioninthey-axis,atranslationof
π6unitsinthe
negativexdirectionandatranslationof4unitsinthenegativeydirection,inthatorder.Theruleoff is
A. f x x( ) cos= − −
−
32 3
4π
B. f x x( ) cos( )= − −32
4
C. f x x( ) cos( )= − −32
4
D. f x x( ) cos= − −
−3
2 34π
E. f x x( ) cos= − +
−
32 3
4π
Question 15
If f x dx( )−∫ = −
3
28 and f x dx( ) =∫ 10
2
3,thevalueof f x dx( )
−∫ 3
3is
A. 2B. –2C. –18D. 18E. 0
Question 16Letf:R+→ R,f (x)=–loge(x)andg :R → R,g(x)=x2+1.Thedomainandrangeoff(g(x))arerespectivelyA. RandR+ ∪{0}B. RandR –
C. [1,∞)andR+ ∪{0}D. R+andR+ ∪{0}E. RandR – ∪{0}
2018MATHMETHEXAM2(NHT) 8
SECTION A – continued
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Question 17IfF(x)isanantiderivativeoff (x)andF(4)=–6,thenF(8)isequaltoA. f ′(8)+6
B. –6+f ′(4)
C. ∫48
f x dx( )
D. − +( )∫ 64
8f x dx( )
E. − + ∫64
8f x dx( )
Question 18Considerthegraphsoffandgbelow,whichhavethesamescale.
x
x = 1
y = 2
y
O
y = f (x)
y = g(x)
x
x = –2
y = –2
y
O
IfTtransformsthegraphoffontothegraphofg,then
A. T R Rxy
xy
T: ,2 2 1 00 1
34
→
=
+
−−
B. T R Rxy
xy
T: ,2 2 1 00 1
34
→
=
−
+
−−
C. T R Rxy
xy
T: ,2 2 1 00 1
30
→
= −
+−
D. T R Rxy
xy
T: ,2 2 2 00 1
→
=
−−
E. T R Rxy
xy
T: ,2 2 1 00 2
→
=
−−
9 2018MATHMETHEXAM2(NHT)
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END OF SECTION ATURN OVER
Question 19Aboxcontains20000marblesthatareeitherblueorred.Therearemorebluemarblesthanredmarbles.Randomsamplesof100marblesaretakenfromthebox.Eachrandomsampleisobtainedbysamplingwithreplacement.Ifthestandarddeviationofthesamplingdistributionfortheproportionofbluemarblesis0.03,thenthenumberofbluemarblesintheboxisA. 11000B. 16000C. 17000D. 18000E. 19000
Question 20Letfbeaone-to-onedifferentiablefunctionsuchthatf(3)=7,f(7)=8,f ′(3)=2andf ′(7)=3.Thefunctiongisdifferentiableandg(x)=f –1(x)forallx.g′(7)isequalto
A. 12
B. 2
C. 16
D. 18
E. 13
2018MATHMETHEXAM2(NHT) 10
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SECTION B – Question 1–continued
Question 1 (9marks)Letf:R → R,f (x)=x4–4x–8.
a. Givenf (x)=(x–2)(x3 + ax2 + bx + c),finda,bandc. 1mark
b. Findtwoconsecutiveintegersmandnsuchthatasolutiontof (x)=0isintheinterval(m,n),wherem<n<0. 2marks
SECTION B
Instructions for Section BAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
11 2018MATHMETHEXAM2(NHT)
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SECTION B – continuedTURN OVER
Thediagrambelowshowspartofthegraphoffandastraightlinedrawnthroughthepoints(0,–8)and(2,0).AsecondstraightlineisdrawnparalleltothehorizontalaxisandittouchesthegraphoffatthepointQ.ThetwostraightlinesintersectatthepointP.
x
y
O
P Q
y = f (x)
2
–8
c. i. Findtheequationofthelinethrough(0,–8)and(2,0). 1mark
ii. StatetheequationofthelinethroughthepointsPandQ. 1mark
iii. StatethecoordinatesofthepointsPandQ. 2marks
d. AtransformationT R R Txy
xy
d: ,2 2
0→
=
+
isappliedtothegraphoff.
i. FindthevalueofdforwhichPistheimageofQ. 1mark
ii. Let(m′,0)and(n′,0)betheimagesof(m,0)and(n,0)respectively,underthetransformationT,wheremandnaredefinedinpart b.
Findthevaluesofm′andn′. 1mark
2018MATHMETHEXAM2(NHT) 12
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SECTION B – Question 2–continued
Question 2 (18marks)Rebecca’sRoboticsmanufacturesthreetypesofcomponentsforrobots:sensors,motorsandcontrollers.Themanufacturingprocessesforeachtypeofcomponentareindependent.Itisknownthat8%ofallofthesensorsmanufacturedaredefective.
a. Arandomsampleoffivesensorsisselected.
Find,correcttofourdecimalplaces,theprobabilitythat
i. exactlytwooftheseselectedsensorsaredefective 2marks
ii. exactlytwooftheseselectedsensorsaredefective,giventhatatmosttwosensorsinthesamplearedefective. 2marks
b. Arandomsampleof50sensorsisselectedanditisfoundthattheproportionofdefectivesensorsinthissampleis0.08
Determineanapproximate90%confidenceintervalfortheproportionofdefectivesensors,correcttofourdecimalplaces. 2marks
Aholeisdrilledintoeachmotor.Thedepthoftheholeisnormallydistributedwithameanof20mmandastandarddeviationof0.3mm.
c. Whatistheprobabilitythat,forarandomlyselectedmotor,thedepthoftheholeisgreaterthan20.6mm?Giveyouranswercorrecttofourdecimalplaces. 1mark
13 2018MATHMETHEXAM2(NHT)
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SECTION B – Question 2–continuedTURN OVER
Thedepthoftheholedrilledintoamotormustbewithin0.5mmofthemean,otherwisethemotorisdefective.
d. Whatistheprobabilitythatamotorisdefective,correcttofourdecimalplaces? 2marks
e. Rebeccadeliversanorderforfivesensorsandfivemotors.
Whatistheprobabilitythattheordercontainsexactlytwodefectivecomponents?Giveyouranswercorrecttothreedecimalplaces. 3marks
f. Aknobisattachedtoeachcontroller.Theheightofaknobisnormallydistributedwithameanof30mm.Iftheknobonacontrollerhasaheightgreaterthan30.4mmorlessthan29.6mm,thenthecontrollerisdefective.
Rebeccawantstoensurethatlessthan2%ofallcontrollersmanufacturedaredefective.
Whatisthemaximumstandarddeviationoftheheightofaknob,inmillimetres,thatcanbeattachedtoacontrollersothatlessthan2%ofcontrollersaredefective?Giveyouranswercorrecttotwodecimalplaces. 2marks
2018MATHMETHEXAM2(NHT) 14
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SECTION B – continued
Theweight,w,ingrams,ofcontrollersismodelledbythefollowingprobabilitydensityfunction.
C ww w w
( )( ) ( )
=− − ≤ ≤
3640000
330 290 290 330
0
2
elsewhere
g. Determinethemeanweight,ingrams,ofthecontrollers. 2marks
h. Determinetheprobabilitythatarandomlyselectedcontrollerweighslessthanthemeanweightofthecontrollers.Giveyouranswercorrecttofourdecimalplaces. 2marks
15 2018MATHMETHEXAM2(NHT)
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SECTION B – continuedTURN OVER
CONTINUES OVER PAGE
2018MATHMETHEXAM2(NHT) 16
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SECTION B – Question 3–continued
Question 3 (13marks)Thefrontofabuildinghasalengthof80mandaheightof20m.Onthefrontofthebuildingisaglasspanelthatliesbetweentwoboundarycurves,asshownbytheshadedregioninthediagrambelow.Theboundarycurvesoftheregionaredefinedovertheinterval[0,80]withtherules
y x
y x
1
2
52 10
15
254 10
10
=
+
=
+
sin
sin
wherexisthehorizontaldistance,inmetres,andyistheverticaldistance,inmetres,measuredrelativetoanorigin,O,atthebottomleftcornerofthefrontofthebuilding.
x
y
O
a. Findthetotalareaoftheglasspanel,insquaremetres,correcttotwodecimalplaces. 2marks
LetDbetheverticaldistancebetweentheupperandlowerboundarycurves.
b. FindtheminimumvalueofD,inmetres,andthevalue(s)ofxwherethisminimumoccurs. 3marks
17 2018MATHMETHEXAM2(NHT)
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SECTION B – continuedTURN OVER
c. WhatistheaveragevalueofD,inmetres,correcttotwodecimalplaces? 2marks
Theboundarycurvesovertheinterval[0,80]aregeneralisedto
c x a x
c x a x
1
22
1015
1010
( ) sin
( ) sin
=
+
=
+
wherea ∈ R+.
d. Theboundarycurvesdonotintersectfora ∈(0,p).
Findthemaximalvalueofp. 3marks
e. Findthevalueofaforwhichtheareaoftheglasspanelisamaximum.Alsostatethemaximumarea,insquaremetres,correcttotwodecimalplaces. 3marks
2018MATHMETHEXAM2(NHT) 18
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SECTION B – Question 4–continued
Question 4 (20marks)Letf:(0,∞)→ R,f (x)=x–xloge(x).Partofthegraphof f isshownbelow.
1O 2 3ex
y
a. Findthevaluesofxforwhich
i. –1<f ′(x)<– 12
2marks
ii. 12<f ′(x)<1 1mark
19 2018MATHMETHEXAM2(NHT)
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SECTION B – Question 4–continuedTURN OVER
b. i. Findtheequationofthetangenttothegraphoffatthepoint(a,f (a))intheform y=mx + c. 1mark
ii. Findthecoordinatesofthepointofintersectionofthetangenttothegraphoff at x=a
andthetangenttothegraphoff at xa
=1. 2marks
iii. Hence,findthecoordinatesofthepointofintersectionofthetangentstothegraphoff at
x=eand xe
=1.Expresseachcoordinateintermsofe. 1mark
c. i. Foravalueofb > e,thetangenttofatthepoint(b,f(b))andthetangenttofatthepoint(2,f(2))intersectthex-axisatthesamepoint.
Findthevalueofb. 2marks
ii. Ifthetangenttofatthepoint(p,f(p)),where1<p<e,andthetangenttofatthe point(q,f(q)),whereq > e,intersectonthex-axis,showthatpq=q p. 2marks
2018MATHMETHEXAM2(NHT) 20
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SECTION B – Question 4–continued
d. Findtheequationofthetangenttothegraphoffatthepointwhere x e=12. 1mark
e. Partofthegraphoff,withthetangenttothegraphatPwhere x e=12,isshownbelow.
Eisthepointcorrespondingtothex-axisinterceptofthistangent. Fisthepointonthistangentwherey=1. Gisthepointcorrespondingtothelocalmaximumofthegraphoff. Histhepoint(1,0). Qisthepoint(e,0).
1
O 1 eH
G
Q
F
E x
y
1 12 2( , ( ))P e f e
i. FindthecoordinatesofthepointsEandF. 2marks
ii. FindtheareaofthequadrilateralEFGH. 2marks
iii. FindtheareaofthetriangleQGH. 1mark
21 2018MATHMETHEXAM2(NHT)
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END OF QUESTION AND ANSWER BOOK
iv. Findanapproximationfortheareaoftheshadedregionbycalculatingtheaverageoftheareasfoundinpart e.ii.andpart e.iii. 1mark
v. Findtheerroroftheapproximationobtainedinpart e.iv.asapercentageoftheactualarea.Giveyouranswercorrecttotwodecimalplaces. 2marks
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 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