2019 Mathematical Methods-nht Written examination 2€¦ · 5 2019 MATHMETH EXAM 2 (NHT) SECTION A...
Transcript of 2019 Mathematical Methods-nht Written examination 2€¦ · 5 2019 MATHMETH EXAM 2 (NHT) SECTION A...
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MATHEMATICAL METHODSWritten examination 2
Monday 3 June 2019 Reading time: 10.00 am to 10.15 am (15 minutes) Writing time: 10.15 am to 12.15 pm (2 hours)
QUESTION AND ANSWER BOOK
Structure of bookSection Number of
questionsNumber of questions
to be answeredNumber of
marks
A 20 20 20B 5 5 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• Questionandanswerbookof26pages• 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.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2019
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2019
STUDENT NUMBER
Letter
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2019MATHMETHEXAM2(NHT) 2
SECTION A – continued
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Question 1Themaximaldomainofthefunctionwithrule f (x)=x2+loge(x)isA. RB. (0,∞)C. [0,∞)D. (−∞,0)E. [1,∞)
Question 2Thediagrambelowshowsonecycleofacircularfunction.
5
4
3
2
1
O–1
–2
–3
1 2 3 4
y
x
Theamplitude,periodandrangeofthisfunctionarerespectivelyA. 3,2and[−2,4]
B. 3,π2 and[−2,4]
C. 4,4and[0,4]
D. 4,π4 and[−2,4]
E. 3,4and[−2,4]
SECTION A – Multiple-choice questions
Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrectforthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
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3 2019MATHMETHEXAM2(NHT)
SECTION A – continuedTURN OVER
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Question 3Ifx + aisafactorof8x3–14x2 – a2x,wherea ∈ R\{0},thenthevalueofaisA. 7B. 4C. 1D. –2E. –1
Question 4
Thegraphofthefunction f :D → R, f x xx
( ) = −+
2 34
,whereDisthemaximaldomain,hasasymptotesA. x =–4,y = 2
B. x y= = −32
4,
C. x y= −4 =, 32
D. x y= =32
2,
E. x =2,y =1
Question 5ConsidertheprobabilitydistributionforthediscreterandomvariableXshowninthetablebelow.
x −1 0 1 2 3
Pr(X = x) b b b35
− b 35b
ThevalueofE(X) is
A. 7665
B. 1
C. 0
D. 2
13
E. 8665
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2019MATHMETHEXAM2(NHT) 4
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Question 6Let f:[0,∞)→ R, f(x)= x2+1.
Theequation f f x( )( ) = 18516
hasrealsolution(s)
A. x = ± 134
B. x =134
C. x = ± 132
D. x =32
E. x = ± 32
Question 7
Ifmxdx= ∫ 2
1
3,thenthevalueofemis
A. loge(9)
B. −9
C. 19
D. 9
E. −19
Question 8Thesimultaneouslinearequations2y +(m–1)x =2andmy + 3x = khaveinfinitelymanysolutionsforA. m =3andk =−2B. m =3andk = 2C. m =3andk =4D. m =–2andk =−2E. m =−2andk = 3
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5 2019MATHMETHEXAM2(NHT)
SECTION A – continuedTURN OVER
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Question 9Atthestartofaparticularweek,Kimhasthreeredapplesandtwogreenapples.Sheeatsoneappleeveryday.OnMonday,TuesdayandWednesdayofthatweek,sherandomlyselectsanappletoeat.Inthisthree-dayperiod,theprobabilitythatKimdoesnoteatanappleofthesamecolouronanytwoconsecutivedaysis
A. 1
10
B. 15
C. 3
10
D. 25
E. 625
Question 10
Let f betheprobabilitydensityfunction f R f x kx x x x: , , ( ) ( )( )( ).0 23
2 1 3 2 3 2
→ = + − +
Thevalueofkis
A. 308405
B. −308405
C. −405308
D. 405308
E. 960133
Question 11Thefunction f :D → R, f (x)= 5x3+10x2+1willhaveaninversefunctionforA. D = R
B. D =(−2,∞)
C. D = −∞
, 12
D. D =(–∞,–1]
E. D =[0,∞)
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2019 MATHMETH EXAM 2 (NHT) 6
SECTION A – continued
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Question 12The transformation T : R2 → R2, which maps the graph of y x= − + −2 1 3 onto the graph of y x= , has rule
A. Txy
xy
=
−
+
−−
12
0
0 1
13
B. Txy
xy
=
−
+
−
12
0
0 1
13
C. Txy
xy
=
−
+ −
12
0
0 1
13
D. Txy
xy
= −
+ −
2 00 1
13
E. Txy
xy
= −
+
−
2 00 1
13
Question 13The graph of f (x) = x3 – 6(b – 2)x2 + 18x + 6 has exactly two stationary points forA. 1 < b < 2
B. b = 1
C. b = ±4 62
D. 4 62
4 62
−≤ ≤
+b
E. b b<−
>+4 6
24 6
2or
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7 2019MATHMETHEXAM2(NHT)
SECTION A – continuedTURN OVER
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Question 14If2 loge(x)–loge(x+2)=loge(y),thenxisequalto
A. y y y+ +2 8
2
B. y y y± +2 8
2
C. y y y± −2 8
2
D. − ± −1 4 7
2y
E. − + −1 4 7
2y
Question 15Theareaboundedbythegraphofy=f(x),thelinex=2,thelinex=8andthex-axis,asshadedinthediagrambelow,is3 loge(13).
–10 –8 –6 –4 –2
O 2 4 6 8 10
y
y = f (x)x
Thevalueof 3 24
10f x dx( )−∫ is
A. 3 loge(13)B. 9 loge(13)C. 6 loge(39)D. loge(13)E. 9 loge(11)
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2019MATHMETHEXAM2(NHT) 8
SECTION A – continued
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Question 16Partsofthegraphsofy = f (x)andy = g(x)areshownbelow.
O
y = g(x)
y = f (x)y
x
Thecorrespondingpartofthegraphofy = g(f (x))isbestrepresentedby
O
y
xO
y
x
O
y
x
O
y
x
O
y
x
A. B.
C. D.
E.
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9 2019MATHMETHEXAM2(NHT)
SECTION A – continuedTURN OVER
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Question 17Thegraphofthefunctiongisobtainedfromthegraphofthefunction f withrule f x x( ) cos( )= −
38bya
dilationoffactor4πfromthey-axis,adilationoffactor
43fromthex-axis,areflectioninthey-axisanda
translationof 32unitsinthepositiveydirection,inthatorder.
Therangeandtheperiodofgarerespectively
A. −
13
73
2, and
B. −
13
73
, and 8
C. −
73
13
, and 2
D. −
73
13
, and 8
E. −
43
43 2
2, and π
Question 18Partofthegraphofthefunction f,where f (x)= 8 – 2x−1,isshownbelow.ItintersectstheaxesatthepointsAandB. ThelinesegmentjoiningAandBisalsoshownonthegraph.
y
A
xBO
y = f (x)
Theareaoftheshadedregionis
A. 16 −152log ( )e
B. 17 −15
2 2log ( )e
C. 72
15916log ( )e
−
D. 16 152 2
−log ( )e
E. 17
2 215
log ( )e−
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2019MATHMETHEXAM2(NHT) 10
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END OF SECTION A
Question 19Arandomsampleofcomputeruserswassurveyedaboutwhethertheusershadplayedaparticularcomputergame.Anapproximate95%confidenceintervalfortheproportionofcomputeruserswhohadplayedthisgamewascalculatedfromthisrandomsampletobe(0.6668,0.8147).ThenumberofcomputerusersinthesampleisclosesttoA. 5B. 33C. 135D. 150E. 180
Question 20Let f (x)=(ax + b)5andletgbetheinversefunctionof f.Giventhat f (0)=1,whatisthevalueofg ′(1)?
A. 5a
B. 1
C. 1
5a
D. 5a(a+1)4
E. 0
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11 2019MATHMETHEXAM2(NHT)
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TURN OVER
CONTINUES OVER PAGE
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2019MATHMETHEXAM2(NHT) 12
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SECTION B – Question 1–continued
Question 1 (9marks)Partsofthegraphsof f (x)=(x–1)3(x+2)3andg(x)=(x–1)2(x+2)3areshownontheaxesbelow.
y
x
y = f (x)
y = g(x)
8
O–2 –1 1 2
–8
Thetwographsintersectatthreepoints,(–2,0),(1,0)and(c,d).Thepoint(c,d)isnotshowninthediagramabove.
a. Findthevaluesofc andd. 2marks
b. Findthevaluesofxsuchthat f (x)>g(x). 1mark
SECTION B
Instructions for Section BAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
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13 2019MATHMETHEXAM2(NHT)
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SECTION B – continuedTURN OVER
c. Statethevaluesofxforwhich
i. f ′(x)>0 1mark
ii. g′(x)>0 1mark
d. Showthat f (1+m)=f (–2–m)forallm. 1mark
e. Findthevaluesofhsuchthatg(x + h)=0hasexactlyonenegativesolution. 2marks
f. Findthevaluesofksuchthat f (x)+k=0hasnosolutions. 1mark
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2019MATHMETHEXAM2(NHT) 14
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SECTION B – Question 2–continued
Question 2 (14marks)Thewindspeedataweathermonitoringstationvariesaccordingtothefunction
v t t( ) sin= +
20 16
14π
wherevisthespeedofthewind,inkilometresperhour(km/h),andtisthetime,inminutes,after 9am.
a. Whatistheamplitudeandtheperiodofv(t)? 2marks
b. Whatarethemaximumandminimumwindspeedsattheweathermonitoringstation? 1mark
c. Findv(60),correcttofourdecimalplaces. 1mark
d. Findtheaveragevalueofv(t)forthefirst60minutes,correcttotwodecimalplaces. 2marks
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15 2019MATHMETHEXAM2(NHT)
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SECTION B – Question 2–continuedTURN OVER
Asuddenwindchangeoccursat10am.Fromthatpointintime,thewindspeedvariesaccordingtothenewfunction
v t t k1 28 18
7( ) sin ( )
= +−
π
wherev1isthespeedofthewind,inkilometresperhour,tisthetime,inminutes,after9amand k ∈ R+.Thewindspeedafter9amisshowninthediagrambelow.
50
40
30
20
10
O 60 120
wind speed (km/h)
time (min)
e. Findthesmallestvalueofk,correcttofourdecimalplaces,suchthatv(t)andv1(t)areequalandarebothincreasingat10am. 2marks
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2019MATHMETHEXAM2(NHT) 16
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SECTION B – continued
f. Anotherpossiblevalueofk wasfoundtobe31.4358 Usingthisvalueofk,theweathermonitoringstationsendsasignalwhenthewindspeedis
greaterthan38km/h.
i. Findthevalueoftatwhichasignalisfirstsent,correcttotwodecimalplaces. 1mark
ii. Findtheproportionofonecycle,tothenearestwholepercent,forwhichv1>38. 2marks
g. Let f x x( ) sin= +
20 16
14π and g x x k( ) sin ( ) .= +
−
28 18
7π
ThetransformationTxy
ab
xy
cd
=
+
00
mapsthegraphof f ontothegraphofg.
Statethevaluesofa,b,candd,intermsofkwhereappropriate. 3marks
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17 2019MATHMETHEXAM2(NHT)
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SECTION B – Question 3–continuedTURN OVER
Question 3 (14marks)ConcertsattheMathslandConcertHallbeginLminutesafterthescheduledstartingtime.Lisarandomvariablethatisnormallydistributedwithameanof10minutesandastandarddeviationoffourminutes.
a. Whatproportionofconcertsbeginbeforethescheduledstartingtime,correcttofour decimalplaces? 1mark
b. Findtheprobabilitythataconcertbeginsmorethan15minutesafterthescheduled startingtime,correcttofourdecimalplaces. 1mark
Ifaconcertbeginsmorethan15minutesafterthescheduledstartingtime,thecleanerisgivenanextrapaymentof$200.Ifaconcertbeginsupto15minutesafterthescheduledstartingtime,thecleanerisgivenanextrapaymentof$100.Ifaconcertbeginsatorbeforethescheduledstartingtime,thereisnoextrapaymentforthecleaner.LetCbetherandomvariablethatrepresentstheextrapaymentforthecleaner,indollars.
c. i. Usingyourresponsesfrompart a.andpart b.,completethefollowingtable,correcttothreedecimalplaces. 1mark
c 0 100 200
Pr(C=c)
ii. Calculatetheexpectedvalueoftheextrapaymentforthecleaner,tothenearestdollar. 1mark
iii. CalculatethestandarddeviationofC,correcttothenearestdollar. 1mark
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2019MATHMETHEXAM2(NHT) 18
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SECTION B – Question 3–continued
TheownersoftheMathslandConcertHalldecidetoreviewtheiroperation.Theystudyinformationfrom1000concertsatothersimilarvenues,collectedasasimplerandomsample.Thesamplevalueforthenumberofconcertsthatstartmorethan15minutesafterthescheduledstartingtimeis43.
d. i. Findthe95%confidenceintervalfortheproportionofconcertsthatbeginmorethan 15minutesafterthescheduledstartingtime.Givevaluescorrecttothreedecimalplaces. 1mark
ii. Explainwhythisconfidenceintervalsuggeststhattheproportionofconcertsthatbeginmorethan15minutesafterthescheduledstartingtimeattheMathslandConcertHallisdifferentfromtheproportionatthevenuesinthesample. 1mark
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19 2019 MATHMETH EXAM 2 (NHT)
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SECTION B – continuedTURN OVER
The owners of the Mathsland Concert Hall decide that concerts must not begin before the scheduled starting time. They also make changes to reduce the number of concerts that begin after the scheduled starting time. Following these changes, M is the random variable that represents the number of minutes after the scheduled starting time that concerts begin. The probability density function for M is
f x xx
x( ) ( )= +
≥
<
82
0
0
0
3
where x is the time, in minutes, after the scheduled starting time.
e. Calculate the expected value of M. 2 marks
f. i. Find the probability that a concert now begins more than 15 minutes after the scheduled starting time. 1 mark
ii. Find the probability that each of the next nine concerts begins no more than 15 minutes after the scheduled starting time and the 10th concert begins more than 15 minutes after the scheduled starting time. Give your answer correct to four decimal places. 2 marks
iii. Find the probability that a concert begins up to 20 minutes after the scheduled starting time, given that it begins more than 15 minutes after the scheduled starting time. Give your answer correct to three decimal places. 2 marks
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2019 MATHMETH EXAM 2 (NHT) 20
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SECTION B – Question 4 – continued
Question 4 (13 marks)A mining company has found deposits of gold between two points, A and B, that are located on a straight fence line that separates Ms Pot’s property and Mr Neg’s property. The distance between A and B is 4 units.The mining company believes that the gold could be found on both Ms Pot’s property and Mr Neg’s property.The mining company initially models the boundary of its proposed mining area using the fence line and the graph of
f : [0, 4] → R, f (x) = x(x – 2)(x – 4)
where x is the number of units from point A in the direction of point B and y is the number of units perpendicular to the fence line, with the positive direction towards Ms Pot’s property. The mining company will only mine from the boundary curve to the fence line, as indicated by the shaded area below.
y
xA
B
42
Ms Pot’s property
Mr Neg’s property
fence line
4
2
O
–2
–4
a. Determine the total number of square units that will be mined according to this model. 2 marks
The mining company offers to pay Mr Neg $100 000 per square unit of his land mined and Ms Pot $120 000 per square unit of her land mined.
b. Determine the total amount of money that the mining company offers to pay. 1 mark
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21 2019MATHMETHEXAM2(NHT)
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SECTION B – Question 4–continuedTURN OVER
Theminingcompanyreviewsitsmodeltousethefencelineandthegraphof
p R p x x xa
x:[ , ] , ( ) ( )0 4 4 41
4→ = − ++
−
wherea>0.
c. Findthevalueofaforwhichp(x)=f(x)forallx. 1mark
d. Solvep′(x)=0forxintermsofa. 2marks
MrNegdoesnotwanthispropertytobeminedfurtherthan4unitsmeasuredperpendicularfromthefenceline.
e. Findthesmallestvalueofa,correcttothreedecimalplaces,forthisconditiontobemet. 2marks
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2019 MATHMETH EXAM 2 (NHT) 22
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SECTION B – continued
f. Find the value of a for which the total area of land mined is a minimum. 3 marks
g. The mining company offers to pay Ms Pot $120 000 per square unit of her land mined and Mr Neg $100 000 per square unit of his land mined.
Determine the value of a that will minimise the total cost of the land purchase for the mining company. Give your answer correct to three decimal places. 2 marks
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23 2019MATHMETHEXAM2(NHT)
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SECTION B – continuedTURN OVER
CONTINUES OVER PAGE
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2019MATHMETHEXAM2(NHT) 24
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SECTION B – Question 5–continued
Question 5 (10marks)
Let f R R f x e g R R g x xx
e: , ( ) : , ( ) log ( ).→ = → =
+2 2and
a. Findg−1(x). 1mark
b. FindthecoordinatesofpointA,wherethetangenttothegraphof f at Aisparalleltothegraphofy=x. 2marks
c. Showthattheequationofthelinethatisperpendiculartothegraphofy=xandgoesthrough pointAisy=−x+2 loge (2)+2. 1mark
LetBbethepointofintersectionofthegraphsofgandy = −x+2 loge (2)+2,asshowninthediagrambelow.
y = f (x) y = x
y = g(x)
B
A
y
x
y = –x + 2loge(2) + 2O
d. DeterminethecoordinatesofpointB. 1mark
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25 2019MATHMETHEXAM2(NHT)
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SECTION B – Question 5–continuedTURN OVER
e. Theshadedregionbelowisenclosedbytheaxes,thegraphsof f andg,andtheline y = −x+2 loge (2)+2.
y = f (x) y = x
y = g(x)
B
y
x
y = –x + 2loge(2) + 2O
A
Findtheareaoftheshadedregion. 2marks
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2019MATHMETHEXAM2(NHT) 26
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END OF QUESTION AND ANSWER BOOK
Let p R R p x e q R R q xk
xkxe: , ( ) : , ( ) log ( ).→ = → =+and 1
f. Thegraphsofp,qandy = xareshowninthediagrambelow.Thegraphsofpandqtouchbutdonotcross.
y = p(x) y = x
y = q(x)
y
xO
Findthevalueofk. 2marks
g. Findthevalueofk, k >0,forwhichthetangenttothegraphofpatitsy-interceptandthetangenttothegraphofqatitsx-interceptareparallel. 1mark
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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 2019
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2019
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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
=
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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