Post on 17-Apr-2018
MATHEMATICAL METHODSWritten examination 2
Thursday 9 November 2017 Reading time: 11.45 am to 12.00 noon (15 minutes) Writing time: 12.00 noon to 2.00 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• Questionandanswerbookof22pages• 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.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2017
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2017
STUDENT NUMBER
Letter
2017MATHMETHEXAM2 2
SECTION A – continued
Question 1Letf :R → R,f (x)=5sin(2x)–1.TheperiodandrangeofthisfunctionarerespectivelyA. πand[−1,4]B. 2πand[−1,5]C. πand[−6,4]D. 2πand[−6,4]E. 4πand[−6,4]
Question 2Partofthegraphofacubicpolynomialfunctionfandthecoordinatesofitsstationarypointsareshownbelow.
O 5
50
–50
–6
(–3, 36)
x
y
5 400,3 27
−
f′(x)<0fortheintervalA. (0,3)
B. ( , ) ,−∞ − ∪( )5 0 3
C. ( , ) ,−∞ − ∪ ∞
3 5
3
D. −
3 5
3,
E. −
40027
36,
SECTION A – Multiple-choice questions
Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrectforthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
3 2017MATHMETHEXAM2
SECTION A – continuedTURN OVER
Question 3Aboxcontainsfiveredmarblesandthreeyellowmarbles.Twomarblesaredrawnatrandomfromtheboxwithoutreplacement.Theprobabilitythatthemarblesareofdifferentcoloursis
A. 58
B. 35
C. 1528
D. 1556
E. 3028
Question 4Let fandgbefunctionssuchthat f (2)=5, f (3)=4,g(2)=5,g(3)=2andg(4)=1.Thevalueof f (g(3))isA. 1B. 2C. 3D. 4E. 5
Question 5The95%confidenceintervalfortheproportionofferryticketsthatarecancelledontheintendeddeparturedayiscalculatedfromalargesampletobe(0.039,0.121).ThesampleproportionfromwhichthisintervalwasconstructedisA. 0.080B. 0.041C. 0.100D. 0.062E. 0.059
2017MATHMETHEXAM2 4
SECTION A – continued
Question 6Partofthegraphofthefunctionf isshownbelow.Thesamescalehasbeenusedonbothaxes.
x
y
Thecorrespondingpartofthegraphoftheinversefunctionf −1isbestrepresentedby
x
y
x
y
x
y
x
y
x
yA. B.
D. E.
C.
Question 7Theequation(p–1)x2 + 4x=5–p hasnorealrootswhenA. p2–6p+6<0B. p2–6p+1>0C. p2–6p–6<0D. p2–6p+1<0E. p2–6p+6>0
5 2017MATHMETHEXAM2
SECTION A – continuedTURN OVER
Question 8Ify=ab–4x+2,wherea>0,thenxisequalto
A. 14
2b ya− −( )log ( )
B. 14
2b ya− +( )log ( )
C. b ya− +
log ( )1
42
D. b ya42− −log ( )
E. 14
2b ya+ −( )log ( )
Question 9Theaveragerateofchangeofthefunctionwiththerulef (x)=x2–2xovertheinterval[1,a], wherea>1,is8.ThevalueofaisA. 9
B. 8
C. 7
D. 4
E. 1 2+
Question 10AtransformationT:R2 → R2withruleT
xy
xy
=
2 0
0 13
mapsthegraphof y x= +
3 2
4sin π
ontothegraphof
A. y=sin(x + π)
B. y x= −
sin π
2
C. y=cos(x + π)
D. y=cos(x)
E. y x= −
cos π
2
2017MATHMETHEXAM2 6
SECTION A – continued
Question 11Thefunctionf :R → R,f (x)=x3 + ax2 + bxhasalocalmaximumatx=–1andalocalminimumatx=3.ThevaluesofaandbarerespectivelyA. –2and–3B. 2and1C. 3and–9D. –3and–9E. –6and–15
Question 12Thesumofthesolutionsof sin( )2 3
2x = overtheinterval[–π,d]is–π.
ThevalueofdcouldbeA. 0
B. 6π
C. 34π
D. 76π
E. 32π
Question 13Let h R h x
x: ( , ) , ( ) .− → =
−1 1
11
Whichoneofthefollowingstatementsabouthisnottrue?A. h(x)h(–x)=–h(x2)B. h(x)+h(–x)=2h(x2)C. h(x)–h(0)=xh(x)D. h(x)–h(–x)=2xh(x2)E. (h(x))2=h(x2)
Question 14
TherandomvariableXhasthefollowingprobabilitydistribution,where0 13
< <p .
x –1 0 1
Pr(X=x) p 2p 1–3p
ThevarianceofXisA. 2p(1–3p)B. 1–4pC. (1–3p)2
D. 6p–16p2
E. p(5–9p)
7 2017MATHMETHEXAM2
SECTION A – continuedTURN OVER
Question 15ArectangleABCDhasverticesA(0,0),B(u,0),C(u, v)andD(0,v),where(u, v)liesonthegraphof y=−x3+8,asshownbelow.
y
8D C(u, v)
A B 2x
Themaximumareaoftherectangleis
A. 23
B. 6 23
C. 16D. 8
E. 3 23
Question 16ForrandomsamplesoffiveAustralians,P̂ istherandomvariablethatrepresentstheproportionwholiveinacapitalcity.
Giventhat ( ) 1Pr 0 ,243
= =P̂ then ( )Pr 0.6 ,>P̂ correcttofourdecimalplaces,is
A. 0.0453B. 0.3209C. 0.4609D. 0.5390E. 0.7901
2017MATHMETHEXAM2 8
SECTION A – continued
Question 17Thegraphofafunctionf,wheref(−x)=f(x),isshownbelow.
y
xa b c d
Thegraphhasx-interceptsat(a,0),(b,0),(c,0)and(d,0)only.Theareaboundbythecurveandthex-axisontheinterval[a,d]is
A. ( )d
af x dx∫
B. f x dx f x dx f x dxa
b
c
b
c
d( ) ( ) ( )∫ ∫ ∫− +
C. 2 f x dx f x dxa
b
b
c( ) ( )∫ ∫+
D. 2 2f x dx f x dxa
b
b
b c( ) ( )∫ ∫−
+
E. f x dx f x dx f x dxa
b
c
b
d
c( ) ( ) ( )∫ ∫ ∫+ +
Question 18LetXbeadiscreterandomvariablewithbinomialdistributionX~Bi(n, p).Themeanandthestandarddeviationofthisdistributionareequal.Giventhat0<p<1,thesmallestnumberoftrials,n,suchthatp ≤0.01isA. 37B. 49C. 98D. 99E. 101
9 2017MATHMETHEXAM2
END OF SECTION ATURN OVER
Question 19Aprobabilitydensityfunctionfisgivenby
f xx k x k
( )cos( ) ( )
=+ < < +
1 10 elsewhere
where0<k<2.ThevalueofkisA. 1
B. 3 1
2π −
C. π−1
D. 1
2π −
E. 2π
Question 20Thegraphsof : 0, , ( ) cos( ) and : 0, , ( ) 3 sin( )
2 2πf R f x x g R g x x → = → =
π areshownbelow.
ThegraphsintersectatB.
π2
π2
3,
y x= 3sin( )
y = cos(x)
B1
O
y
xA
TheratiooftheareaoftheshadedregiontotheareaoftriangleOABisA. 9:8
B. 3 1 38
− : π
C. 8 3 – 3 : 3π
D. 3 1 34
− : π
E. 1 38
: π
2017MATHMETHEXAM2 10
SECTION B – Question 1–continued
Question 1 (11marks)Let f :R → R, f (x)=x3 –5x. Partofthegraphof f isshownbelow.
x
y
10
5–5
–10
–5
O
5
a. Findthecoordinatesoftheturningpoints. 2marks
b. A(−1,f(−1))andB(1,f(1))aretwopointsonthegraphof f.
i. FindtheequationofthestraightlinethroughAandB. 2marks
ii. FindthedistanceAB. 1mark
SECTION B
Instructions for Section BAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
11 2017MATHMETHEXAM2
SECTION B – continuedTURN OVER
Letg:R → R,g(x)=x3 –kx, k ∈ R+.
c. LetC(–1,g(−1))andD(1,g(1))betwopointsonthegraphofg.
i. FindthedistanceCDintermsofk. 2marks
ii. FindthevaluesofksuchthatthedistanceCDisequaltok+1. 1mark
d. Thediagrambelowshowspartofthegraphsofgandy=x.Thesegraphsintersectatthepointswiththecoordinates(0,0)and(a,a).
y y = g(x)
y = x(a, a)
(0, 0) x
i. Findthevalueofaintermsofk. 1mark
ii. Findtheareaoftheshadedregionintermsofk. 2marks
2017 MATHMETH EXAM 2 12
SECTION B – Question 2 – continued
Question 2 (12 marks)Sammy visits a giant Ferris wheel. Sammy enters a capsule on the Ferris wheel from a platform above the ground. The Ferris wheel is rotating anticlockwise. The capsule is attached to the Ferris
wheel at point P. The height of P above the ground, h, is modelled by ( ) 65 – 55cos ,15
th t π =
where t is the time in minutes after Sammy enters the capsule and h is measured in metres.
Sammy exits the capsule after one complete rotation of the Ferris wheel.
platform
a capsule
h
P
a. State the minimum and maximum heights of P above the ground. 1 mark
b. For how much time is Sammy in the capsule? 1 mark
c. Find the rate of change of h with respect to t and, hence, state the value of t at which the rate of change of h is at its maximum. 2 marks
13 2017MATHMETHEXAM2
SECTION B – Question 2–continuedTURN OVER
AstheFerriswheelrotates,astationaryboatatB, onanearbyriver,firstbecomesvisibleatpointP1.Bis500mhorizontallyfromtheverticalaxisthroughthecentreC oftheFerriswheelandangleCBO=θ,asshownbelow.
500 m
C
y
Bθ
P1
Ox
d. Findθindegrees,correcttotwodecimalplaces. 1mark
PartofthepathofPisgivenbyy x x= − + ∈ −3025 62 5 55 55, [ , ], wherexandyareinmetres.
e. Finddydx. 1mark
2017MATHMETHEXAM2 14
SECTION B – continued
AstheFerriswheelcontinuestorotate,theboatatB isnolongervisiblefromthepointP2(u,v)onwards.ThelinethroughBandP2istangenttothepathofP,whereangleOBP2=α.
500 m
α
y
B
O
P2(u, v)
C
x
f. FindthegradientofthelinesegmentP2Bintermsofuand,hence,findthecoordinatesof P2,correcttotwodecimalplaces. 3marks
g. Findαindegrees,correcttotwodecimalplaces. 1mark
h. Henceorotherwise,findthelengthoftime,tothenearestminute,duringwhichtheboatat B isvisible. 2marks
2017MATHMETHEXAM2 16
SECTION B – Question 3–continued
Question 3 (19marks)ThetimeJenniferspendsonherhomeworkeachdayvaries,butshedoessomehomework everyday.ThecontinuousrandomvariableT,whichmodelsthetime,t,inminutes,thatJenniferspends eachdayonherhomework,hasaprobabilitydensityfunctionf,where
f t
t t
t t( )
( )
( )=
− ≤ <
− ≤ ≤
1625
20 20 45
1625
70 45 70
0 elsewhere
a. Sketchthegraphoffontheaxesprovidedbelow. 3marks
150
125
200 40 60 80 100
y
t
b. FindPr(25≤T≤55). 2marks
c. FindPr(T≤25|T≤55). 2marks
17 2017MATHMETHEXAM2
SECTION B – Question 3–continuedTURN OVER
d. FindasuchthatPr(T≥a)=0.7,correcttofourdecimalplaces. 2marks
e. TheprobabilitythatJenniferspendsmorethan50minutesonherhomeworkonanygivenday
is 825
. Assumethattheamountoftimespentonherhomeworkonanydayisindependentof
thetimespentonherhomeworkonanyotherday.
i. FindtheprobabilitythatJenniferspendsmorethan50minutesonherhomeworkonmorethanthreeofsevenrandomlychosendays,correcttofourdecimalplaces. 2marks
ii. FindtheprobabilitythatJenniferspendsmorethan50minutesonherhomeworkonatleasttwoofsevenrandomlychosendays,giventhatshespendsmorethan50minutesonherhomeworkonatleastoneofthosedays,correcttofourdecimalplaces. 2marks
2017MATHMETHEXAM2 18
SECTION B – continued
LetpbetheprobabilitythatonanygivendayJenniferspendsmorethandminutesonherhomework.Letqbetheprobabilitythatontwoorthreedaysoutofsevenrandomlychosendaysshespendsmorethandminutesonherhomework.
f. Expressqasapolynomialintermsofp. 2marks
g. i. Findthemaximumvalueofq,correcttofourdecimalplaces,andthevalueofpforwhichthismaximumoccurs,correcttofourdecimalplaces. 2marks
ii. Findthevalueofd forwhichthemaximumfoundinpart g.i.occurs,correcttothenearestminute. 2marks
2017MATHMETHEXAM2 20
SECTION B – Question 4–continued
Question 4 (18marks)Letf :R → R :f (x)=2x +1–2.Partofthegraphoff isshownbelow.
4
3
2
1
O–1
–2–3
–4
–4 –3 –2 –1 1 2 3 4
y
x
a. ThetransformationT R R Txy
xy
cd
: ,2 2→
=
+
mapsthegraphofy=2
xontothegraphoff.
Statethevaluesofcandd. 2marks
b. Findtheruleanddomainforf −1,theinversefunctionoff. 2marks
c. Findtheareaboundedbythegraphsoffandf −1. 3marks
21 2017MATHMETHEXAM2
SECTION B – Question 4–continuedTURN OVER
d. Partofthegraphsoffandf −1areshownbelow.
O
f
f –1
y
x–2
–2
Findthegradientof fandthegradientof f −1 at x=0. 2marks
Thefunctionsofgk ,wherek ∈ R+,aredefinedwithdomainRsuchthatgk(x)=2ekx–2.
e. Findthevalueofksuchthatgk(x)=f(x). 1mark
f. Findtherulefortheinversefunctionsgk–1ofgk ,wherek ∈ R+. 1mark
2017MATHMETHEXAM2 22
END OF QUESTION AND ANSWER BOOK
g. i. Describethetransformationthatmapsthegraphofg1ontothegraphofgk. 1mark
ii. Describethetransformationthatmapsthegraphofg1–1ontothegraphofgk
–1. 1mark
h. ThelinesL1andL2arethetangentsattheorigintothegraphsofgkandgk–1respectively.
Findthevalue(s)ofkforwhichtheanglebetweenL1andL2is30°. 2marks
i. Letpbethevalueofkforwhichgk(x)=gk−1(x)hasonlyonesolution.
i. Findp. 2marks
ii. LetA(k)betheareaboundedbythegraphsofgkandgk–1forallk>p.
StatethesmallestvalueofbsuchthatA(k)<b. 1mark
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 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 ( )( ) = 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 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