MATHEMATICAL METHODS (CAS) METHODS (CAS) Written examination 2 Thursday 6 November 2014 Reading...
Transcript of MATHEMATICAL METHODS (CAS) METHODS (CAS) Written examination 2 Thursday 6 November 2014 Reading...
MATHEMATICAL METHODS (CAS)Written examination 2
Thursday 6 November 2014 Reading time: 3.00 pm to 3.15 pm (15 minutes) Writing time: 3.15 pm to 5.15 pm (2 hours)
QUESTION AND ANSWER BOOK
Structure of bookSection Number of
questionsNumber of questions
to be answeredNumber of
marks
1 22 22 222 5 5 58
Total 80
• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulers,aprotractor,set-squares,aidsforcurvesketching,oneboundreference,oneapprovedCAScalculator(memoryDOESNOTneedtobecleared)and,ifdesired,onescientificcalculator.Forapprovedcomputer-basedCAS,theirfullfunctionalitymaybeused.
• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orwhiteoutliquid/tape.
Materials supplied• Questionandanswerbookof22pageswithadetachablesheetofmiscellaneousformulasinthe
centrefold.• Answersheetformultiple-choicequestions.
Instructions• Detachtheformulasheetfromthecentreofthisbookduringreadingtime.• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice
questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.
• AllwrittenresponsesmustbeinEnglish.
At the end of the examination• Placetheanswersheetformultiple-choicequestionsinsidethefrontcoverofthisbook.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2014
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2014
STUDENT NUMBER
Letter
2014MATHMETH(CAS)EXAM2 2
SECTION 1–continued
Question 1ThepointP(4,–3)liesonthegraphofafunction f. Thegraphof f istranslatedfourunitsverticallyupandthenreflectedinthey-axis.ThecoordinatesofthefinalimageofPareA. (–4,1)B. (–4,3)C. (0,–3)D. (4,–6)E. (–4,–1)
Question 2Thelinearfunction f D R f x x: ,→ ( ) = −4 hasrange[–2,6).ThedomainDofthefunctionisA. [–2,6)B. (–2,2]C. RD. (–2,6]E. [–6,2]
Question 3Theareaoftheregionenclosedbythegraphof y x x x= +( ) −( )2 4 andthex-axisis
A. 1283
B. 203
C. 2363
D. 1483
E. 36
SECTION 1
Instructions for Section 1Answerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1,anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.
3 2014MATHMETH(CAS)EXAM2
SECTION 1–continuedTURN OVER
Question 4Let f beafunctionwithdomainRsuchthat ′( ) = ′( ) < ≠f f x x5 0 0 5 and when .At x=5,thegraphof f hasaA. localminimum.B. localmaximum.C. gradientof5.D. gradientof–5.E. stationarypointofinflection.
Question 5TherandomvariableXhasanormaldistributionwithmean12andstandarddeviation0.5.IfZhasthestandardnormaldistribution,thentheprobabilitythatXislessthan11.5isequaltoA. Pr(Z>–1)B. Pr(Z<–0.5)C. Pr(Z>1)D. Pr(Z≥0.5)E. Pr(Z <1)
Question 6
Thefunction f D R: → withrule f x x x x( ) = − −2 9 1683 2 willhaveaninversefunctionforA. D = R
B. D=(7,∞)
C. D=(–4,8)
D. D=(–∞,0)
E. D = − ∞
12
,
2014MATHMETH(CAS)EXAM2 4
SECTION 1–continued
Question 7
y
x
(–2, 3)
O–4
Theruleofthefunctionwhosegraphisshownaboveis
A. y = – 32
|x|+3
B. y = 23
|x +3|+2
C. y = 23
|2 + x|+3
D. y = – 32
|2 – x|+3
E. y = –32
|x+2|+3
Question 8
If f x dx( ) =∫ 61
4,then 5 2
1
4−( )∫ f x dx( ) isequalto
A. 3B. 4C. 5D. 6E. 16
5 2014MATHMETH(CAS)EXAM2
SECTION 1–continuedTURN OVER
Question 9Theinverseofthefunction f R R f x
x: ,+ → ( ) = +
1 4 is
A. f –1:(4,∞)→ R f xx
− ( ) =−( )
12
14
B. f –1:R+ → R f xx
− ( ) = +12
1 4
C. f –1:R+ → R f x x− ( ) = +( )1 24
D. f –1:(–4,∞)→ R f xx
− ( ) =+( )
12
14
E. f –1:(–∞,4)→ R f xx
− ( ) =−( )
12
14
Question 10Whichoneofthefollowingfunctionssatisfiesthefunctionalequation f f x x( )( ) = foreveryrealnumberx?A. f x x( ) = 2
B. f x x( ) = 2
C. f x x( ) = 2
D. f x x( ) = − 2
E. f x x( ) = −2
Question 11Abagcontainsfiveredmarblesandfourbluemarbles.Twomarblesaredrawnfromthebag,withoutreplacement,andtheresultsarerecorded.Theprobabilitythatthemarblesaredifferentcoloursis
A. 2081
B. 518
C. 49
D. 4081
E. 59
2014MATHMETH(CAS)EXAM2 6
SECTION 1–continued
Question 12
ThetransformationT R R: 2 2→ withrule
Txy
xy
=
−
+ −
1 00 2
12
mapsthelinewithequation x y− =2 3 ontothelinewithequationA. x + y = 0B. x+4y = 0C. –x – y=4D. x+4y=–6E. x – 2y = 1
Question 13Thedomainofthefunctionh,where h x xa( ) = ( )cos log ( ) andaisarealnumbergreaterthan1,ischosensothat h isaone-to-onefunction.Whichoneofthefollowingcouldbethedomain?
A. a a−
π π2 2,
B. (0,p)
C. 1 2, aπ
D. a a−
π π2 2,
E. a a−
π π2 2,
Question 14IfXisarandomvariablesuchthatPr Pr ,X a X b>( ) = >( ) =5 8 and then Pr X X< <( )5 8 is
A. ab
B. a bb
−−1
C. 11−−ba
D. abb1−
E. ab−−
11
7 2014MATHMETH(CAS)EXAM2
SECTION 1–continuedTURN OVER
Question 15Zoehasarectangularpieceofcardboardthatis8cmlongand6cmwide.Zoecutssquaresofsidelengthxcentimetresfromeachofthecornersofthecardboard,asshowninthediagrambelow.
8 cm
6 cm
x cm
Zoeturnsupthesidestoformanopenbox.
ThevalueofxforwhichthevolumeoftheboxisamaximumisclosesttoA. 0.8B. 1.1C. 1.6D. 2.0E. 3.6
Question 16ThecontinuousrandomvariableX,withprobabilitydensityfunctionp(x),hasmean2andvariance5.
Thevalueof x p x dx2 ( )−∞
∞
∫ isA. 1B. 7C. 9D. 21E. 29
Question 17Thesimultaneouslinearequations ax–3y=5 and 3x – ay = 8 – a haveno solution forA. a=3B. a=–3C. botha=3anda=–3D. a ∈ R\{3}E. a ∈ R\[–3,3]
2014MATHMETH(CAS)EXAM2 8
SECTION 1–continued
Question 18Thegraphof y = kx–4 intersectsthegraphof y = x2 + 2x attwodistinctpointsforA. k=6B. k>6ork < –2C. –2≤k≤6D. 6 2 3 6 2 3− ≤ ≤ +kE. k = –2
Question 19
JakeandAnitaarecalculatingtheareabetweenthegraphof y x= andthey-axisbetweeny=0andy=4.Jakeusesapartitioning,showninthediagrambelow,whileAnitausesadefiniteintegraltofindtheexactarea.
y
x
4
3
2
1
O
y x=
ThedifferencebetweentheresultsobtainedbyJakeandAnitaisA. 0
B. 223
C. 263
D. 14
E. 35
9 2014MATHMETH(CAS)EXAM2
SECTION 1–continuedTURN OVER
Question 20Thegraphofafunction,h,isshownbelow.
10
8
6
4
2
(6, 10)
(1, 4) (11, 4)
2 4 6 8 10 12
y = h(x)
y
x
TheaveragevalueofhisA. 4B. 5C. 6D. 7E. 10
Question 21ThetrapeziumABCDisshownbelow.ThesidesAB,BCandDAareofequallength,p.ThesizeoftheacuteangleBCDisxradians.
D C
A B
p p
p
x
Theareaofthetrapeziumisamaximumwhenthevalueofxis
A. 12
B. 6
C. 4
D. 3
E. 512
2014MATHMETH(CAS)EXAM2 10
END OF SECTION 1
Question 22JohnandRebeccaareplayingdarts.Theresultofeachoftheirthrowsisindependentoftheresultofany
otherthrow.TheprobabilitythatJohnhitsthebullseyewithasinglethrowis14 .Theprobabilitythat
Rebeccahitsthebullseyewithasinglethrowis12 .JohnhasfourthrowsandRebeccahastwothrows.
TheratiooftheprobabilityofRebeccahittingthebullseyeatleastoncetotheprobabilityofJohnhittingthebullseyeatleastonceisA. 1:1B. 32:27C. 64:85D. 2:1E. 192:175
11 2014MATHMETH(CAS)EXAM2
SECTION 2 –continuedTURN OVER
Question 1 (7marks)Thepopulationofwombatsinaparticularlocationvariesaccordingtotherule
n t t( ) = +
1200 400
3cos π ,wherenisthenumberofwombatsandtisthenumberofmonthsafter
1March2013.
a. Findtheperiodandamplitudeofthefunctionn. 2marks
b. Findthemaximumandminimumpopulationsofwombatsinthislocation. 2marks
c. Findn(10). 1mark
d. Overthe12monthsfrom1March2013,findthefractionoftimewhenthepopulationofwombatsinthislocationwaslessthann(10). 2marks
SECTION 2
Instructions for Section 2Answerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
2014MATHMETH(CAS)EXAM2 12
SECTION 2 – Question 2–continued
Question 2 (13marks)On1January2010,TasmaniaJoneswaswalkingthroughanice-coveredregionofGreenlandwhenhefoundalargeicecylinderthatwasmadeathousandyearsagobytheVikings.Astatuewasinsidetheicecylinder.Thestatuewas1mtallanditsbasewasatthecentreofthebaseofthecylinder.
1 m
h metres
d metres
Thecylinderhadaheightofhmetresandadiameterofdmetres.TasmaniaJonesfoundthatthevolumeofthecylinderwas216m3.Atthattime,1January2010,thecylinderhadnotchangedinathousandyears.ItwasexactlyasitwaswhentheVikingsmadeit.
a. Writeanexpressionforh intermsofd. 2marks
13 2014MATHMETH(CAS)EXAM2
SECTION 2 – Question 2–continuedTURN OVER
b. Showthatthesurfaceareaofthecylinderexcludingthebase,Ssquaremetres,isgivenbythe
rule S dd
= +π 2
4864 . 1mark
TasmaniafoundthattheVikingsmadethecylindersothatS isaminimum.
c. FindthevalueofdforwhichSisaminimumandfindthisminimumvalueofS. 2marks
d. FindthevalueofhwhenSisaminimum. 1mark
2014MATHMETH(CAS)EXAM2 14
SECTION 2 – Question 2–continued
On1January2010,TasmaniabelievedthatduetorecenttemperaturechangesinGreenland,theiceofthecylinderhadjuststartedmelting.Therefore,hedecidedtoreturnon1Januaryeachyeartomeasuretheicecylinder.Heobservesthatthevolumeoftheicecylinderdecreasesbyaconstantrateof10m3peryear.Assumethatthecylindricalshapeisretainedandd = 2h atthebeginningandasthecylindermelts.
e. WritedownanexpressionforVintermsofh. 1mark
f. Find dhdt
intermsofh. 3marks
g. Findtherateatwhichtheheightofthecylinderwillbedecreasingwhenthetopofthestatueisjustexposed. 1mark
15 2014MATHMETH(CAS)EXAM2
SECTION 2–continuedTURN OVER
h. Findtheyearinwhichthetopofthestatuewilljustbeexposed.(Assumethatthemeltingstartedon1January2010.) 2marks
2014MATHMETH(CAS)EXAM2 16
SECTION 2 – Question 3–continued
Question 3 (11marks)Inacontrolledexperiment,Juantooksomemedicineat8pm.Theconcentrationofmedicineinhisbloodwasthenmeasuredatregularintervals.TheconcentrationofmedicineinJuan’sblood
ismodelledbythefunction c t tet
( ) = −52
32 ,t ≥0,wherecistheconcentrationofmedicineinhis
blood,inmilligramsperlitre,thoursafter8pm.Partofthegraphofthefunctioncisshownbelow.
c
0.5
O t
a. WhatwasthemaximumvalueoftheconcentrationofmedicineinJuan’sblood,inmilligramsperlitre,correcttotwodecimalplaces? 1mark
b. i. Findthevalueoft,inhours,correcttotwodecimalplaces,whentheconcentrationofmedicineinJuan’sbloodfirstreached0.5milligramsperlitre. 1mark
ii. FindthelengthoftimethattheconcentrationofmedicineinJuan’sbloodwasabove0.5milligramsperlitre.Expresstheanswerinhours,correcttotwodecimalplaces. 2marks
17 2014MATHMETH(CAS)EXAM2
SECTION 2–continuedTURN OVER
c. i. Whatwasthevalueoftheaveragerateofchangeoftheconcentrationofmedicinein
Juan’sbloodovertheinterval 23
3,
?Expresstheanswerinmilligramsperlitre
perhour,correcttotwodecimalplaces. 2marks
ii. Attimest1andt2 ,theinstantaneousrateofchangeoftheconcentrationofmedicinein
Juan’sbloodwasequaltotheaveragerateofchangeovertheinterval 23
3,
.
Findthevaluesoft1andt2 ,inhours,correcttotwodecimalplaces. 2marks
Aliciatookpartinasimilarcontrolledexperiment.However,sheusedadifferentmedicine.Theconcentrationofthisdifferentmedicinewasmodelledbythefunction n t Ate kt( ) = − ,t ≥0, whereAandk ∈ R+.
d. IfthemaximumconcentrationofmedicineinAlicia’sbloodwas0.74milligramsperlitreatt=0.5hours,findthevalueofA,correcttothenearestinteger. 3marks
2014MATHMETH(CAS)EXAM2 18
SECTION 2 – Question 4–continued
Question 4 (14marks)Patriciaisagardenerandsheownsagardennursery.Shegrowsandsellsbasilplantsandcorianderplants.Theheights,incentimetres,ofthebasilplantsthatPatriciaissellingaredistributednormallywithameanof14cmandastandarddeviationof4cm.Thereare2000basilplantsinthenursery.
a. Patriciaclassifiesthetallest10percentofherbasilplantsassuper.
Whatistheminimumheightofasuperbasilplant,correcttothenearestmillimetre? 1mark
Patriciadecidesthatsomeofherbasilplantsarenotgrowingquicklyenough,sosheplanstomovethemtoaspecialgreenhouse.Shewillmovethebasilplantsthatarelessthan9cminheight.
b. HowmanybasilplantswillPatriciamovetothegreenhouse,correcttothenearestwholenumber? 2marks
19 2014MATHMETH(CAS)EXAM2
SECTION 2 – Question 4–continuedTURN OVER
Theheightsofthecorianderplants,xcentimetres,followtheprobabilitydensityfunction h x( ),where
h xx x
( ) =
< <
π π100 50
0 50
0
sin
otherwise
c. Statethemeanheightofthecorianderplants. 1mark
Patriciathinksthatthesmallest15percentofhercorianderplantsshouldbegivenanewtypeofplantfood.
d. Findthemaximumheight,correcttothenearestmillimetre,ofacorianderplantifitistobegiventhenewtypeofplantfood. 2marks
Patriciaalsogrowsandsellstomatoplantsthatsheclassifiesaseithertallorregular.Shefindsthat20percentofhertomatoplantsaretall.Acustomer,Jack,selectsntomatoplantsatrandom.
e. LetqbetheprobabilitythatatleastoneofJack’sntomatoplantsistall.
Findtheminimumvalueofnsothatq isgreaterthan0.95. 2marks
2014MATHMETH(CAS)EXAM2 20
SECTION 2–continued
Inanothersectionofthenursery,acraftsmanmakesplantpots.Thepotsareclassifiedassmoothorrough.Thecraftsmanfinisheseachpotbeforestartingonthenext.Overaperiodoftime,itisfoundthatifoneplantpotissmooth,theprobabilitythatthenextoneissmoothis0.7,whileifoneplantpotisrough,theprobabilitythatthenextoneisroughisp,where0<p <1.Thevalueofpstaysfixedforaweekatatime,butcanvaryfromweektoweek.Thefirstpotmadeeachweekisalwaysasmoothpot.
f. i. Find,intermsofp,theprobabilitythatthethirdpotmadeinagivenweekissmooth. 2marks
ii. Inoneparticularweek,theprobabilitythatthethirdpotmadeissmoothis0.61.
Calculatethevalueofpinthisweek. 2marks
g. If,inanotherweek,p=0.8,findtheprobabilitythatthefifthpotmadethatweekissmooth. 2marks
21 2014MATHMETH(CAS)EXAM2
SECTION 2 – Question 5–continuedTURN OVER
Question 5 (13marks)
Let and f R R f x x x x g R R g x x x: , : , .→ ( ) = −( ) −( ) +( ) → ( ) = −3 1 3 82 4
a. Expressx4 – 8x intheform x x a x b c−( ) + +( )( )2 . 2marks
b. Describethetranslationthatmapsthegraphof y f x= ( ) ontothegraphof y g x= ( ) . 1mark
c. Findthevaluesofdsuchthatthegraphof y f x d= +( ) has i. onepositivex-axisintercept 1mark
ii. twopositivex-axisintercepts. 1mark
d. Findthevalueofn forwhichtheequation g x n( ) = hasonesolution. 1mark
2014MATHMETH(CAS)EXAM2 22
END OF QUESTION AND ANSWER BOOK
e. Atthepoint u g u, ( ) ,( ) thegradientof y g x= ( ) ismandatthepoint v g v, ( ) ,( ) thegradientis–m,wheremisapositiverealnumber.
i. Findthevalueof u3 + v3. 2marks
ii. Finduandvif u + v=1. 1mark
f. i. Findtheequationofthetangenttothegraphof y g x= ( ) atthepoint p g p, ( )( ) . 1mark
ii. Findtheequationsofthetangentstothegraphof y g x= ( ) thatpassthroughthepointwithcoordinates 3
212, −
. 3marks
MATHEMATICAL METHODS (CAS)
Written examinations 1 and 2
FORMULA SHEET
Directions to students
Detach this formula sheet during reading time.
This formula sheet is provided for your reference.
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2014
MATHMETH (CAS) 2
THIS PAGE IS BLANK
3 MATHMETH (CAS)
END OF FORMULA SHEET
Mathematical Methods (CAS)Formulas
Mensuration
area of a trapezium: 12a b h+( ) volume of a pyramid:
13Ah
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 triangle: 12bc Asin
volume of a cone: 13
2π r h
Calculusddx
x nxn n( ) = −1
x dx
nx c nn n=
++ ≠ −+∫ 1
111 ,
ddxe aeax ax( ) =
e dx a e cax ax= +∫ 1
ddx
x xelog ( )( ) = 1 1x dx x ce= +∫ 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 rule: ddxuv u dv
dxv dudx
( ) = + quotient rule: ddx
uv
v dudx
u dvdx
v
=
−
2
chain rule: dydx
dydududx
= approximation: f x h f x h f x+( ) ≈ ( ) + ′( )
ProbabilityPr(A) = 1 – Pr(A′) Pr(A ∪ B) = Pr(A) + Pr(B) – Pr(A ∩ B)
Pr(A|B) = Pr
PrA BB∩( )( ) transition matrices: Sn = Tn × S0
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