2017 mathematical methods-nht written examination 2€¦ · mathematical methods written...
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MATHEMATICAL METHODSWritten examination 2
Wednesday 7 June 2017 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.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2017
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2017
STUDENT NUMBER
Letter
2017MATHMETHEXAM2(NHT) 2
SECTION A – continued
Question 1Thegradientofalineperpendiculartothelinethatpassesthrough(3,0)and(0,–6)is
A. −12
B. –2
C. 12
D. 4
E. 2
Question 2
Thefunctionwithrule f x x( ) sin=
+2
41 hasperiod
A. π4
B. π2
C. π
D. 4π
E. 8π
Question 3Thesimultaneouslinearequationsmx+7y=12and7x + my = mhaveauniquesolutiononlyforA. m=7orm=–7B. m=12orm=3C. m ∈ R\{–7,7}D. m=4orm=3E. m ∈ R\{12,1}
SECTION A – Multiple-choice questions
Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
3 2017MATHMETHEXAM2(NHT)
SECTION A – continuedTURN OVER
Question 4
Thegraphofthefunction f R f x x:[ , ) ( ) ,0 413∞ → =, where isreflectedinthex-axisandthentranslated
fiveunitstotherightandsixunitsverticallydown.Whichoneofthefollowingistheruleofthetransformedgraph?
A. y x= − +4 5 613( )
B. y x= − + −4 5 613( )
C. y x= − + +4 5 613( )
D. y x= − − −4 5 613( )
E. y x= − +4 5 113( )
Question 5
Whichoneofthefollowingistheinversefunctionofthefunction f R f xx
: ( , ) , ( )−∞ → =−
+3 23
1?
A. f R f xx
− −−∞ → = −−
+1 123 4
13: ( , ) , ( )
( )
B. f R f xx
− −∞ → = −−
+1 121 4
31: ( , ) , ( )
( )
C. f R f xx
− −∞ → = −−
+1 121 4
13: ( , ) , ( )
( )
D. f R f xx
− −∞ → = − +1 121 4 3: ( , ) , ( )
E. f R R f xx
− + −→ = −−
+1 12
41
3: , ( )( )
Question 6
Let f D R f x xx
: , ( ) ,→ =−−
3 52
whereD isthemaximaldomainoff .
Whichofthefollowingaretheequationsoftheasymptotesofthegraphoff ?
A. x=2and y = 53
B. x=2andy=–3
C. x=–2andy=3
D. x=–3andy = 2
E. x=2andy=3
2017MATHMETHEXAM2(NHT) 4
SECTION A – continued
Question 7Thegraphofy = kx–2willnotintersectortouchthegraphofy = x2+3xwhenA. 3 2 2 3 2 2− < < +k
B. { : } { : }k k k k< − ∪ > +3 2 2 3 2 2
C. –5<k<11
D. 3 2 2 3 2 2− ≤ ≤ +k
E. k ∈ R+
Question 8Partofthegraphofafunctionf isshownbelow.
0
(–a, –a)
(a, a)y
x
Whichoneofthefollowingistheaveragevalueofthefunctionf overtheinterval[–a,a]?A. 0
B. 34a
C. 38a
D. a2
E. a4
5 2017MATHMETHEXAM2(NHT)
SECTION A – continuedTURN OVER
Question 9
Therangeofthefunction f R f x xe: , ( , ] , ( ) log ( )−
∪ → =
120 0 2 2 is
A. −( ]2 2 2 2log ( ), log ( )e e
B. −
log , log ( )e e
14
4
C. −∞( ], log ( )2 2e
D. R e e\ log ( ), log ( )−[ )2 2 2 2
E. R
Question 10Thetangenttothegraphofy=3sin(2x)–1isparalleltothelinewithequationy=3x+1atthepoints wherexisequalto
A. n n Zππ
± ∈6
,
B. −π π3 3, only
C. π π6
56
, only
D. n n Zππ
± ∈3
,
E. nπ,n ∈ Z
Question 11Abagcontainsfivebluemarblesandfourredmarbles.Asampleoffourmarblesistakenfromthebag,withoutreplacement.Theprobabilitythattheproportionofbluemarblesinthesampleisgreaterthan
12is
A. 12
B. 29
C. 514
D. 59
E. 2563
2017 MATHMETH EXAM 2 (NHT) 6
SECTION A – continued
Question 12The maximum temperature reached by the water heated in a kettle each time it is used is normally distributed with a mean of 95 °C and a standard deviation of 2 °C.When the kettle is used, the proportion of times that the maximum temperature reached by the water is greater than 98 °C is closest toA. 0.0671B. 0.0668C. 0.8669D. 0.9332E. 0.9342
Question 13The probability distribution for the discrete random variable X is defined by the following.
x –1 0 1 2
Pr(X = x) a 3b 2a b
The mean and variance of the distribution, respectively, areA. a + 2b and 3a + 4bB. a + 2b and 3a + 4b – a2 – 4ab – 4b2
C. 3a + 2b and 3a + 4b – 9a2 – 12ab – 4b2
D. 3a + 2b and 3a + 4bE. 3a + 5b and a + b
Question 14
2a
a0 2a 3a 4a 5a
y
x
The rule of the function with the graph shown above could be
A. y a xa
a=
+cos π
2
B. y a xa
a=
+2 sin π
C. y a xa
a= −
+cos π
2
D. y aax a a= −
+sin ( )π
E. y a xa
a= −
+cos π
7 2017 MATHMETH EXAM 2 (NHT)
SECTION A – continuedTURN OVER
Question 15The graph of f where f (x) = ekx, k > 0, touches the graph of its inverse function f –1 at exactly one point, as shown below.
0
y = f (x)
y = f –1(x)
y
x
The value of k must beA. e
B. e2
C. 1
D. 12e
E. 1e
Question 16
The area under the graph of y = f (x), between x = 1 and x = 4, where f (x) >12
over this interval, is eight units.The area under the graph of y = 2 f (x) – 1 over the same interval isA. 15 units.B. 13 units.C. 10 units.D. 7 units.E. 3 units.
2017MATHMETHEXAM2(NHT) 8
END OF SECTION A
Question 17Afunctionfsatisfiestherelationf(x2)=f(x)+f(x+2).Apossibleruleforf isA. f x x( ) = + 2
B. f (x)=x + 2
C. f (x)=log10(x–1)
D. f x x( ) = −( )12
12
E. f xx
( ) =−1
1
Question 18Letf (x)=xmeax,wherea andmarenon-zerorealconstants.If(x+2)isafactoroff′(x),thenwhichoneofthefollowingmustbetrue?A. m = 2B. m=–2C. m=2–aD. m = 2aE. m=–2a
Question 19ThefirstdigitofeachhousenumberinalargeaddressdatabaseisarandomvariableD.Thepossiblevalues
ofDare1,2,…,9and Pr( ) log .D dd
= = +
10 11
TheprobabilitythatDisgreaterthan8,giventhatDisgreaterthan7,is
A. 1–log10(9)
B. 1 91 8
10
10
−−log ( )log ( )
C. 1 92 9 8
10
10 10
−− −
log ( )log ( ) log ( )
D. log1089
E. log1098
Question 20Considerthefunctionf :[2,∞)→ R,f (x)=x4+2(a–4)x2–8ax+1,wherea ∈ R.Themaximalsetofvaluesofaforwhichtheinversefunctionf –1existsisA. (–9,∞)B. (–∞,1)C. [–9,1]D. [–8,∞)E. (–∞,–8]
9 2017MATHMETHEXAM2(NHT)
TURN OVER
CONTINUES OVER PAGE
2017MATHMETHEXAM2(NHT) 10
SECTION B – Question 1–continued
Question 1 (11marks)Thetemperature,T °C,inanofficeiscontrolled.Foraparticularweekday,thetemperatureattimet,wheretisthenumberofhoursaftermidnight,isgivenbythefunction
T t t t( ) sin ( ) , .= + −
≤ ≤19 6
128 0 24π
a. Whatarethemaximumandminimumtemperaturesintheoffice? 2marks
b. Whatisthetemperatureintheofficeat6.00am? 1mark
c. Mostofthepeopleworkingintheofficearriveat8.00am.
Whatisthetemperatureintheofficewhentheyarrive? 1mark
SECTION B
Instructions for Section BAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
11 2017MATHMETHEXAM2(NHT)
SECTION B – continuedTURN OVER
d. Forhowmanyhoursofthedayisthetemperaturegreaterthanorequalto19°C? 2marks
e. Whatistheaveragerateofchangeofthetemperatureintheofficebetween8.00amandnoon? 2marks
f. i. FindT ′(t). 1mark
ii. Atwhattimeofthedayisthetemperatureintheofficedecreasingmostrapidly? 2marks
2017MATHMETHEXAM2(NHT) 12
SECTION B – Question 2–continued
Question 2 (13marks)Letf :R → R,wheref(x)=(x–2)2(x–5).
a. Findf ′(x). 1mark
b. Forwhatvaluesofxisf ′(x)<0? 1mark
c. i. Findthegradientofthelinesegmentjoiningthepointsonthegraphofy = f (x) wherex=1andx=5. 1mark
ii. Showthatthemidpointofthelinesegmentinpart c.i.alsoliesonthegraphofy = f (x). 2marks
13 2017MATHMETHEXAM2(NHT)
SECTION B – Question 2–continuedTURN OVER
iii. Findthevaluesofxforwhichthegradientofthetangenttothegraphofy = f (x)isequaltothegradientofthelinesegmentjoiningthepointsonthegraphwherex=1andx=5. 2marks
Letg :R → R,whereg (x)=(x–2)2(x–a),wherea ∈ R.
d. ThecoordinatesofthestationarypointsofgareP(2,0)andQ(p(a+1),q(a–2)3),where pandqarerationalnumbers.
Findthevaluesofpandq. 2marks
2017MATHMETHEXAM2(NHT) 14
SECTION B – continued
e. Showthatthegradientofthetangenttothegraphofy = g (x)atthepoint(a,0)ispositivefora ∈ R\{2}. 1mark
f. i. Findthecoordinatesofanotherpointwherethetangenttothegraphofy = g (x)isparalleltothetangentatx = a. 2marks
ii. Hence,findthedistancebetweenthispointandpointQ whena>2. 1mark
15 2017 MATHMETH EXAM 2 (NHT)
SECTION B – Question 3 – continuedTURN OVER
Question 3 (18 marks)A company supplies schools with whiteboard pens.The total length of time for which a whiteboard pen can be used for writing before it stops working is called its use-time.There are two types of whiteboard pens: Grade A and Grade B.The use-time of Grade A whiteboard pens is normally distributed with a mean of 11 hours and a standard deviation of 15 minutes.
a. Find the probability that a Grade A whiteboard pen will have a use-time that is greater than 10.5 hours, correct to three decimal places. 1 mark
The use-time of Grade B whiteboard pens is described by the probability density function
f xx e x
x
( )( )( )
=− − ≤ ≤
576
12 x 1
0
0 126
otherwise
where x is the use-time in hours.
b. Determine the expected use-time of a Grade B whiteboard pen. Give your answer in hours, correct to two decimal places. 2 marks
c. Determine the standard deviation of the use-time of a Grade B whiteboard pen. Give your answer in hours, correct to two decimal places. 2 marks
2017MATHMETHEXAM2(NHT) 16
SECTION B – Question 3–continued
d. FindtheprobabilitythatarandomlychosenGradeBwhiteboardpenwillhaveause-timethatisgreaterthan10.5hours,correcttofourdecimalplaces. 2marks
Aworkeratthecompanyfindstwoboxesofwhiteboardpensthatarenotlabelled,butknowsthatoneboxcontainsonlyGradeAwhiteboardpensandtheotherboxcontainsonlyGradeBwhiteboardpens.Theworkerdecidestorandomlyselectawhiteboardpenfromoneoftheboxes.Iftheselectedwhiteboardpenhasause-timethatisgreaterthan10.5hours,thentheboxthatitcamefromwillbelabelledGradeAandtheotherboxwillbelabelledGradeB.Otherwise,theboxthatitcamefromwillbelabelledGradeBandtheotherboxwillbelabelledGradeA.
e. Findtheprobability,correcttothreedecimalplaces,thattheworkerlabelstheboxesincorrectly. 2marks
f. Findtheprobability,correcttothreedecimalplaces,thatthewhiteboardpenselectedwasGradeB,giventhattheboxeshavebeenlabelledincorrectly. 2marks
17 2017MATHMETHEXAM2(NHT)
SECTION B – continuedTURN OVER
Asawhiteboardpenages,itstipmaydrytothepointwherethewhiteboardpenbecomes defective(unusable).Thecompanyhasstockthatistwoyearsoldand,atthatage,itisknownthat5%ofGradeAwhiteboardpenswillbedefective.
g. AschoolpurchasesaboxofGradeAwhiteboardpensthatistwoyearsoldandaclassof26studentsisthefirsttousethem.
Ifeverystudentreceivesawhiteboardpenfromthisbox,findtheprobability,correcttofourdecimalplaces,thatatleastonestudentwillreceiveadefectivewhiteboardpen. 2marks
h. LetP̂ AbetherandomvariableofthedistributionofsampleproportionsofdefectiveGradeAwhiteboardpensinboxesof100.Theboxescomefromthestockthatistwoyearsold.
Find(P̂ A>0.04|P̂ A<0.08).Giveyouranswercorrecttofourdecimalplaces.Donotuseanormalapproximation. 3marks
i. Aboxof100GradeAwhiteboardpensthatistwoyearsoldisselectedanditisfoundthatsixofthewhiteboardpensaredefective.
Determinea90%confidenceintervalforthepopulationproportionfromthissample,correcttotwodecimalplaces. 2marks
2017MATHMETHEXAM2(NHT) 18
SECTION B – Question 4–continued
Question 4 (18marks)
Let f R f x x x:[ , ] , ( ) .0 16 4→ = −
a. Findthevalueofxatwhichfhasamaximumandstatethemaximumvalue. 2marks
b. Sketchthegraphofy = f (x)ontheaxesbelow. 1mark
0
2
4
6
2 4 6 8 10 12 14 16
y
x
19 2017MATHMETHEXAM2(NHT)
SECTION B – Question 4–continuedTURN OVER
c. i. FindtheareaAoftheregionenclosedbythegraphoffandthex-axis. 1mark
ii. LetXbethepoint(16,0).TrianglescanbeformedwithverticesO,XandC(c,f (c)), whereCisapointonthegraphofy = f (x).
FindthevalueofcforwhichtriangleOCXhasmaximumareaandfindthismaximumarea. 2marks
2017MATHMETHEXAM2(NHT) 20
SECTION B – Question 4–continued
Letaandbbetwovaluesinthedomainoffsuchthatf (b)=f (a),andassumeb > a.
d. Showthatb a= −( )42. 1mark
e. Arectangleisformedwithvertices(a,0),(b,0),(b,f (b))and(a,f (a)).
i. Findtheareaofthisrectangleintermsofa. 2marks
ii. Findthevaluesofaandbforwhichtherectanglehasmaximumarea. 3marks
iii. Findthevalueofthemaximumareaintheform m np
,wherem,nandparepositiveintegers. 2marks
21 2017MATHMETHEXAM2(NHT)
END OF QUESTION AND ANSWER BOOK
f. AtrapeziumODBXisformedwithverticesatO,X(16,0),B(b,f (b))andD(a,f (a)),asshowninthediagrambelow,whereDBisparalleltoOX.
O X(16, 0)
D(a, f (a)) B(b, f (b))
y
x
i. Findthevalueofa forwhichthetrapeziumhasmaximumareaATandfindthismaximumarea. 3marks
ii. FindAAT ,whereAistheareaoftheregionenclosedbythegraphoffandthex-axis. 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 ( )( ) = 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