written examinations 1 and 2 – end of year...differentiate 4− x with respect to x. 1 mark b. if...
Post on 30-Dec-2019
2 Views
Preview:
TRANSCRIPT
© VCAA 2016 – Version 2 – April 2016
VCE Mathematical Methods 2016–2018
Written examinations 1 and 2 – End of year
Examination specifications
Overall conditions There will be two end-of-year examinations for VCE Mathematical Methods – examination 1 and examination 2.
The examinations will be sat at a time and date to be set annually by the Victorian Curriculum and Assessment Authority (VCAA). VCAA examination rules will apply. Details of these rules are published annually in the VCE and VCAL Administrative Handbook.
Examination 1 will have 15 minutes reading time and 1 hour writing time. Students are not permitted to bring into the examination room any technology (calculators or software) or notes of any kind.
Examination 2 will have 15 minutes reading time and 2 hours writing time. Students are permitted to bring into the examination room an approved technology with numerical, graphical, symbolic and statistical functionality, as specified in the VCAA Bulletin and the VCE Exams Navigator. One bound reference may be brought into the examination room. This may be a textbook (which may be annotated), a securely bound lecture pad, a permanently bound student-constructed set of notes without fold-outs or an exercise book. Specifications for the bound reference are published annually in the VCE Exams Navigator.
A formula sheet will be provided with both examinations.
The examinations will be marked by a panel appointed by the VCAA.
Examination 1 will contribute 22 per cent to the study score. Examination 2 will contribute 44 per cent to the study score.
MATHMETH (SPECIFICATIONS)
© VCAA 2016 – Version 2 – April 2016 Page 2
Content The VCE Mathematics Study Design 2016–2018 (‘Mathematical Methods Units 3 and 4’) is the document for the development of the examination. All outcomes in ‘Mathematical Methods Units 3 and 4’ will be examined.
All content from the areas of study, and the key knowledge and skills that underpin the outcomes in Units 3 and 4, are examinable.
Examination 1 will cover all areas of study in relation to Outcome 1. The examination is designed to assess students’ knowledge of mathematical concepts, their skill in carrying out mathematical algorithms without the use of technology, and their ability to apply concepts and skills.
Examination 2 will cover all areas of study in relation to all three outcomes, with an emphasis on Outcome 2. The examination is designed to assess students’ ability to understand and communicate mathematical ideas, and to interpret, analyse and solve both routine and non-routine problems.
Format
Examination 1
The examination will be in the form of a question and answer book.
The examination will consist of short-answer and extended-answer questions. All questions will be compulsory.
The total marks for the examination will be 40.
A formula sheet will be provided with the examination. The formula sheet will be the same for examinations 1 and 2.
All answers are to be recorded in the spaces provided in the question and answer book.
Examination 2
The examination will be in the form of a question and answer book.
The examination will consist of two sections.
Section A will consist of 20 multiple-choice questions worth 1 mark each and will be worth a total of 20 marks.
Section B will consist of short-answer and extended-answer questions, including multi-stage questions of increasing complexity, and will be worth a total of 60 marks.
All questions will be compulsory. The total marks for the examination will be 80.
A formula sheet will be provided with the examination. The formula sheet will be the same for examinations 1 and 2.
Answers to Section A are to be recorded on the answer sheet provided for multiple-choice questions.
Answers to Section B are to be recorded in the spaces provided in the question and answer book.
MATHMETH (SPECIFICATIONS)
© VCAA 2016 – Version 2 – April 2016 Page 3
Approved materials and equipment
Examination 1
• normal stationery requirements (pens, pencils, highlighters, erasers, sharpeners and rulers)
Examination 2
• normal stationery requirements (pens, pencils, highlighters, erasers, sharpeners and rulers) • an approved technology with numerical, graphical, symbolic and statistical functionality • one scientific calculator • one bound reference
Relevant references The following publications should be referred to in relation to the VCE Mathematical Methods examinations:
• VCE Mathematics Study Design 2016–2018 (‘Mathematical Methods Units 3 and 4’) • VCE Mathematical Methods – Advice for teachers 2016–2018 (includes assessment advice) • VCE Exams Navigator • VCAA Bulletin
Advice During the 2016–2018 accreditation period for VCE Mathematical Methods, examinations will be prepared according to the examination specifications above. Each examination will conform to these specifications and will test a representative sample of the key knowledge and skills from all outcomes in Units 3 and 4.
The following sample examinations provide an indication of the types of questions teachers and students can expect until the current accreditation period is over.
Answers to multiple-choice questions are provided at the end of examination 2.
Answers to other questions are not provided.
S A M P L EMATHEMATICAL METHODS
Written examination 1
Day Date Reading time: *.** to *.** (15 minutes) Writing time: *.** to *.** (1 hour)
QUESTION AND ANSWER BOOK
Structure of bookNumber of questions
Number of questions to be answered
Number of marks
10 10 40
• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpenersandrulers.
• StudentsareNOTpermittedtobringintotheexaminationroom:anytechnology(calculatorsorsoftware),notesofanykind,blanksheetsofpaperand/orcorrectionfluid/tape.
Materials supplied• Questionandanswerbookof13pages.• Formulasheet.• Workingspaceisprovidedthroughoutthebook.
Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.• AllwrittenresponsesmustbeinEnglish.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016
Version2–April2016
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education Year
STUDENT NUMBER
Letter
MATHMETHEXAM1(SAMPLE) 2 Version2–April2016
THIS PAGE IS BLANK
Version2–April2016 3 MATHMETHEXAM1(SAMPLE)
TURN OVER
Question 1 (3marks)a. Differentiate 4 − x withrespecttox. 1mark
b. If f x xx
( )sin ( )
= ,find f ′ π2
. 2marks
InstructionsAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegiven,unlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmustbeshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
MATHMETHEXAM1(SAMPLE) 4 Version2–April2016
Question 2 (3marks)Ontheaxesbelow,sketchthegraphof f :R\{–1}→R, f(x)= 2
41
−+x
.
Labeleachaxisinterceptwithitscoordinates.Labeleachasymptotewithitsequation.
y
Ox
Version2–April2016 5 MATHMETHEXAM1(SAMPLE)
TURN OVER
Question 3 (4marks)
a. Findanantiderivativeof 12 1 3x −( )
withrespecttox.2marks
b. Thefunctionwithrule g (x)hasderivativeg′(x)=sin(2πx).
Giventhatg (1)=1π,findg (x). 2marks
MATHMETHEXAM1(SAMPLE) 6 Version2–April2016
Question 4 (3marks)LetXbetherandomvariablethatrepresentsthenumberoftelephonecallsthatDanielreceivesonanygivendaywithprobabilitydistributiongivenbythetablebelow.
x 0 1 2 3
Pr(X=x) 0.2 0.2 0.5 0.1
a. FindthemeanofX. 2marks
b. WhatistheprobabilitythatDanielreceivesonlyonetelephonecalloneachofthreeconsecutivedays? 1mark
Question 5 (3marks)Thegraphsof y=cos(x)and y=asin(x),whereaisarealconstant,haveapointofintersection
at x = π3.
a. Findthevalueofa. 2marks
b. Ifx∈[0,2π],findthex-coordinateoftheotherpointofintersectionofthetwographs. 1mark
Version2–April2016 7 MATHMETHEXAM1(SAMPLE)
TURN OVER
Question 6 (5marks)a. Solvetheequation2log3(5)–log3(2)+log3(x)=2forx. 2marks
b. Solve3e t=5+8e–tfort. 3marks
MATHMETHEXAM1(SAMPLE) 8 Version2–April2016
Question 7 (4marks)Astudentperformsanexperimentinwhichacomputerisusedtosimulatedrawingarandomsampleofsizenfromalargepopulation.Theproportionofthepopulationwiththecharacteristicofinteresttothestudentisp.
a. LettherandomvariableP̂representthesampleproportionobservedintheexperiment.
If p = 15,findthesmallestintegervalueofthesamplesizesuchthatthestandarddeviationof
P̂islessthanorequalto1
100 . 2marks
Eachof23studentsinaclassindependentlyperformstheexperimentdescribedaboveandeachstudentcalculatesanapproximate95%confidenceintervalforpusingthesampleproportionsfortheirsample.Itissubsequentlyfoundthatexactlyoneofthe23confidenceintervalscalculatedbytheclassdoesnotcontainthevalueofp.
b. Twooftheconfidenceintervalscalculatedbytheclassareselectedatrandomwithoutreplacement.
Findtheprobabilitythatexactlyoneoftheselectedconfidenceintervalsdoesnotcontainthevalueofp. 2marks
Version2–April2016 9 MATHMETHEXAM1(SAMPLE)
TURN OVER
Question 8 (4marks)Acontinuousrandomvariable,X,hasaprobabilitydensityfunctiongivenby
f xe x
x
x
( ) = ≥
<
−15
0
0 0
5
ThemedianofXism.
a. Determinethevalueofm. 2marks
b. Thevalueofmisanumbergreaterthan1.
Find Pr .X X m< ≤( )1 2marks
MATHMETHEXAM1(SAMPLE) 10 Version2–April2016
Question 9 (4marks)Partofthegraphof f :R+→R,f (x)=xloge(x)isshownbelow.
1 3O
y
x
a. Findthederivativeofx2loge(x). 1mark
b. Useyouranswertopart a.tofindtheareaoftheshadedregionintheformaloge(b)+c,wherea,bandcarenon-zerorealconstants. 3marks
Version2–April2016 11 MATHMETHEXAM1(SAMPLE)
Question 10–continuedTURN OVER
Question 10 (7marks)ThelinecontainingthepointsMandNintersectsthex-axisatthepointMwithcoordinates(6,0).Thelineisalsoatangenttothegraphofy=ax2+bxatthepointQwithcoordinates(2,4),asshownbelow.
O
N
Q (2, 4)
M (6, 0)
y
x
a. Ifaandbarenon-zerorealnumbers,findthevaluesofaandb. 3marks
MATHMETHEXAM1(SAMPLE) 12 Version2–April2016
Question 10–continued
b. ThelinecontainingthepointsUandVintersectsthecoordinateaxesatthepointsUandVwithcoordinates(u,0)and(0,v),respectively,whereuandvarepositiverealnumbersand52 ≤u≤6,asshownbelow.
TherectangleOPQRhasavertexatQontheline.ThecoordinatesofQare(2,4),asshownbelow.
O
R
P
V (0, v)
Q (2, 4)
U (u, 0)
y
x
i. Findanexpressionforvintermsofu. 1mark
Version2–April2016 13 MATHMETHEXAM1(SAMPLE)
END OF QUESTION AND ANSWER BOOK
ii. Findtheminimumtotalshadedareaandthevalueofuforwhichtheareaisaminimum. 2marks
iii. Findthemaximumtotalshadedareaandthevalueofuforwhichtheareaisamaximum. 1mark
S A M P L EMATHEMATICAL METHODS
Written examination 2Day Date
Reading time: *.** to *.** (15 minutes) Writing time: *.** to *.** (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• Questionandanswerbookof20pages.• Formulasheet.• Answersheetformultiple-choicequestions.
Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice
questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.• Unlessotherwiseindicated,thediagramsinthisbookarenot drawntoscale.• 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.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016
Version2–April2016
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education Year
STUDENT NUMBER
Letter
MATHMETHEXAM2(SAMPLE) 2 Version2–April2016
SECTION A – 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 2Thelinearfunctionf:D → R,f(x)=4–xhasrange[–2,6).ThedomainDofthefunctionisA. [–2,6)B. (–2,2]C. RD. (–2,6]E. [–6,2]
Question 3Thefunctionwithrulef(x)=–3tan(2π x)hasperiod
A. 2π
B. 2
C. 12
D. 14
E. 2π
SECTION A – Multiple-choice questions
Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
Version2–April2016 3 MATHMETHEXAM2(SAMPLE)
SECTION A – continuedTURN OVER
Question 4Ifx + aisafactorof7x3 + 9x2–5ax,wherea R\{0},thenthevalueofaisA. –4B. –2C. –1D. 1E. 2
Question 5
Thefunctiong:[–a,a]→ R,g (x)= sin 26
x −
π hasaninversefunction.Themaximumpossiblevalueofais
A. π12
B. 1
C. π6
D. π4
E. π2
Question 6
If f x dx( ) =∫ 61
4,then 5 2
1
4−( )∫ f x dx( ) isequalto
A. 3B. 4C. 5D. 6E. 16
Question 7ForeventsAandB,Pr (A ∩ B)=p,Pr (A′ ∩ B)= p − 1
8 andPr (A ∩ B′)=35p.
IfAandBareindependent,thenthevalueofpisA. 0
B. 14
C. 38
D. 12
E. 35
MATHMETHEXAM2(SAMPLE) 4 Version2–April2016
SECTION A – continued
Question 8Whichoneofthefollowingfunctionssatisfiesthefunctionalequation 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 9Abagcontainsfiveredmarblesandfourbluemarbles.Twomarblesaredrawnfromthebag,withoutreplacement,andtheresultsarerecorded.Theprobabilitythatthemarblesaredifferentcoloursis
A. 2081
B. 518
C. 49
D. 4081
E. 59
Question 10
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
Version2–April2016 5 MATHMETHEXAM2(SAMPLE)
SECTION A – continuedTURN OVER
Question 11Ifthetangenttothegraphofy = eax,a≠0,atx = cpassesthroughtheorigin,thencisequaltoA. 0
B. 1a
C. 1
D. a
E. −1a
Question 12Thesimultaneouslinearequations 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]
Question 13Itisknownthat26%ofthe19-year-oldsinaregiondonothaveadriver’slicence.Ifarandomsampleoften19-year-oldsfromtheregionistaken,theprobability,correcttofourdecimalplaces,thatmorethanhalfofthemwillnothaveadriver’slicenceisA. 0.0239B. 0.0904C. 0.2600D. 0.9096E. 0.9761
Question 14Anopinionpollsterreportedthatforarandomsampleof574votersinatown,76%indicatedapreferenceforretainingthecurrentcouncil.Anapproximate90%confidenceintervalfortheproportionofthetotalvotingpopulationwithapreferenceforretainingthecurrentcouncilcanbefoundbyevaluating
A. 0 76 0 76 0 24574
0 76 0 24574
. . . , . .−
× ×
0.76 +
B. 0 76 1 65 0 76 0 24574
0 76 65 0 76 0 24574
. . . . , . . .−
× ×
+1.
C. 0 76 2 58 0 76 0 24574
0 76 0 24574
. . . . , . .−
× ×
0.76 + 2.58
D. 436 1 96 0 76 0 24 574 0 76 0 24 574− × × × ×( ). . . , . .436 +1.96
E. 0 76 2 0 76 0 24 574 0 76 0 24 574. . . , . .− × × × ×( )0.76 + 2
MATHMETHEXAM2(SAMPLE) 6 Version2–April2016
SECTION A – continued
Question 15Thecubicfunctionf :R → R,f(x)=ax3–bx2 + cx,wherea,bandcarepositiveconstants,hasnostationarypointswhen
A. c ba
>2
4
B. c ba
<2
4
C. c<4b2a
D. c ba
>2
3
E. c ba
<2
3
Question 16Apartofthegraphofg :R → R,g (x)=x2–4isshownbelow.
A
B
y
xaO
TheareaoftheregionlabelledAisthesameastheareaoftheregionlabelledB.TheexactvalueofaisA. 0
B. 6
C. 6
D. 12E. 2 3
Version2–April2016 7 MATHMETHEXAM2(SAMPLE)
SECTION A – continuedTURN OVER
Question 17Theequationx3–9x2+15x + w=0hasonlyonesolutionforxwhenA. –7<w < 25B. w≤–7C. w≥25D. w<–7orw > 25E. w>1
Question 18Acubicfunctionhastheruley = f(x).Thegraphofthederivativefunction f ʹcrossesthex-axisat(2,0)and(–3,0).Themaximumvalueofthederivativefunctionis10.Thevalueofxforwhichthegraphofy = f(x)hasalocalmaximumisA. –2B. 2C. –3D. 3
E. -12
Question 19ButterfliesofaparticularspeciesdieTdaysafterhatching,whereTisanormallydistributedrandomvariablewithameanof120daysandastandarddeviationofdays.If,fromapopulationof2000newlyhatchedbutterflies,150areexpectedtodieinthefirst90days,thenthevalueofisclosesttoA. 7days.B. 13days.C. 17days.D. 21days.E. 37days.
MATHMETHEXAM2(SAMPLE) 8 Version2–April2016
END OF SECTION A
Question 20Thegraphsof y = f (x)andy = g (x)areshownbelow.
y
O x
y = f (x)
y = g (x)
Thegraphofy = f (g (x))isbestrepresentedby
y
Ox
y
O x
y
O x
y
O x
y
O x
A. B.
C. D.
E.
Version2–April2016 9 MATHMETHEXAM2(SAMPLE)
SECTION B – 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 B
Instructions for Section BAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
MATHMETHEXAM2(SAMPLE) 10 Version2–April2016
SECTION B – Question 2–continued
Question 2 (10marks)Asolidblockintheshapeofarectangularprismhasabaseofwidthxcentimetres.Thelengthofthebaseistwo-and-a-halftimesthewidthofthebase.
52x cm
x cm
h cm
Theblockhasatotalsurfaceareaof6480cm2.
a. Showthatiftheheightoftheblockishcentimetres,h = 6480 57
2− xx
. 2marks
Version2–April2016 11 MATHMETHEXAM2(SAMPLE)
SECTION B – continuedTURN OVER
b. Thevolume,Vcubiccentimetres,oftheblockisgivenbyV(x)= 5 6480 514
2x x( ).−
GiventhatV (x)>0andx >0,findthepossiblevaluesofx. 3marks
c. Find dVdx
,expressingyouranswerintheform dVdx
= ax2 + b,whereaandbarerealnumbers. 3marks
d. Findtheexactvaluesofxandhiftheblockistohavemaximumvolume. 2marks
MATHMETHEXAM2(SAMPLE) 12 Version2–April2016
SECTION B – Question 3–continued
Question 3 (20marks)FullyFitisaninternationalcompanythatownsandoperatesmanyfitnesscentres(gyms)inseveralcountries.Ithasmorethan100000membersworldwide.AteveryoneofFullyFit’sgyms,eachmemberagreestohavetheirfitnessassessedeverymonthbyundertakingasetofexercisescalledS.IfsomeonecompletesSinlessthanthreeminutes,theyareconsideredfit.
a. IthasbeenfoundthattheprobabilitythatanymemberofFullyFitwillcompleteSinlessthan
threeminutesis58.Thisisindependentofanyothermember.Arandomsampleof20FullyFit
membersistaken.Forasampleof20members,letXbetherandomvariablethatrepresentsthenumberofmemberswhocompleteSinlessthanthreeminutes.
i. FindPr(X≥10)correcttofourdecimalplaces. 2marks
ii. FindPr(X≥15|X≥10)correcttothreedecimalplaces. 3marks
Version2–April2016 13 MATHMETHEXAM2(SAMPLE)
SECTION B – Question 3–continuedTURN OVER
Forsamplesof20members,P̂istherandomvariableofthedistributionofsampleproportionsofpeoplewhocompleteSinlessthanthreeminutes.
iii. FindtheexpectedvalueandvarianceofP̂. 3marks
iv. Findtheprobabilitythatasampleproportionlieswithintwostandarddeviationsof58.
Giveyouranswercorrecttothreedecimalplaces.Donotuseanormalapproximation. 3marks
v. FindPr(P̂≥34 |P̂≥
58 ).Giveyouranswercorrecttothreedecimalplaces.Donotuse
anormalapproximation. 2marks
MATHMETHEXAM2(SAMPLE) 14 Version2–April2016
SECTION B – Question 3–continued
b. PaulaisamemberofFullyFit’sgyminSanFrancisco.ShecompletesSeverymonthasrequired,butotherwisedoesnotattendregularlyandsoherfitnesslevelvariesovermanymonths.Paulafindsthatifsheisfitonemonth,theprobabilitythatsheisfitthenextmonth
is34 ,andifsheisnotfitonemonth,theprobabilitythatsheisnotfitthenextmonthis
12 .
IfPaulaisnotfitinoneparticularmonth,whatistheprobabilitythatsheisfitinexactlytwoofthenextthreemonths? 2marks
c. WhenFullyFitsurveyedallitsgymsthroughouttheworld,itwasfoundthatthetimetakenbymemberstocompleteanotherexerciseroutine,T,isacontinuousrandomvariableWwithaprobabilitydensityfunctiong,asdefinedbelow.
g w
w w
w w( )
( )
=
− +≤ ≤
+< ≤
3 64256
3
29128
3 5
0
3 1
elsewhere
i. FindE(W)correcttofourdecimalplaces. 2marks
ii. Inarandomsampleof200FullyFitmembers,howmanymemberswouldbeexpectedtotakemorethanfourminutestocompleteT?Giveyouranswertothenearestinteger. 2marks
Version2–April2016 15 MATHMETHEXAM2(SAMPLE)
SECTION B – continuedTURN OVER
d. Fromarandomsampleof100members,itwasfoundthatthesampleproportionofpeoplewhospentmorethantwohoursperweekinthegymwas0.6
Findanapproximate95%confidenceintervalforthepopulationproportioncorrespondingtothissampleproportion.Givevaluescorrecttothreedecimalplaces. 1mark
MATHMETHEXAM2(SAMPLE) 16 Version2–April2016
SECTION B – Question 4–continued
Question 4 (8marks)TheshadedregioninthediagrambelowistheplanofaminesitefortheBlackPossumminingcompany.Alldistancesareinkilometres.Twooftheboundariesoftheminesiteareintheshapeofthegraphsofthefollowingfunctions.
f :R → R,f (x)=ex g:R+ → R,g (x)=loge(x)
y
x–3 –2 –1 1 2 3
–3
–2
–1
O
1
2
3
x = 1
y = f (x) y = g (x)
y = –2
a. i. Evaluate f x dx( ) .−∫ 2
0
1mark
ii. Hence,orotherwise,findtheareaoftheregionboundedbythegraphofg,thex-axisandy-axis,andtheliney =–2. 1mark
Version2–April2016 17 MATHMETHEXAM2(SAMPLE)
SECTION B – Question 4–continuedTURN OVER
iii. Findthetotalareaoftheshadedregion. 1mark
b. Theminingengineer,Victoria,decidesthatabettersiteforthemineistheregionboundedbythegraphofgandthatofanewfunctionk:(–∞,a)→ R,k (x)=–loge(a – x),whereaisapositiverealnumber.
i. Find,intermsofa,thex-coordinatesofthepointsofintersectionofthegraphsof gandk. 2marks
ii. Hence,findthesetofvaluesofaforwhichthegraphsofgandkhavetwodistinctpointsofintersection. 1mark
MATHMETHEXAM2(SAMPLE) 18 Version2–April2016
SECTION B – continued
c. Forthenewminesite,thegraphsofgandkintersectattwodistinctpoints,AandB.ItisproposedtostartminingoperationsalongthelinesegmentAB,whichjoinsthetwopointsofintersection.
Victoriadecidesthatthegraphofkwillbesuchthatthex-coordinateofthemidpointofAB is 2 .
Findthevalueofainthiscase. 2marks
Version2–April2016 19 MATHMETHEXAM2(SAMPLE)
SECTION B – Question 5–continuedTURN OVER
Question 5 (15marks)Let f:R → R, f (x)=(x–3)(x–1)(x 2+3)andg:R → R,g (x)=x 4–8x.
a. Expressx4–8x intheformx x a x b c−( ) + +( )( )2 . 2marks
b. Describethetranslationthatmapsthegraphofy = f(x)ontothegraphofy = g(x). 2marks
c. Findthevaluesofdsuchthatthegraphofy = f(x + d )has i. onepositivex-axisintercept 1mark
ii. twopositivex-axisintercepts. 1mark
d. Findthevalueofn forwhichtheequationg (x)= nhasonesolution. 1mark
MATHMETHEXAM2(SAMPLE) 20 Version2–April2016
END OF QUESTION AND ANSWER BOOK
e. Atthepoint u g u, ( ) ,( ) thegradientofy = g (x)ismandatthepoint v g v, ( ) ,( ) thegradientis–m,wheremisapositiverealnumber.
i. Findthevalueof u3 + v3. 2marks
ii. Finduandvif u + v=1. 1mark
f. i. Findtheequationofthetangenttothegraphofy = g(x)atthepoint p g p, ( )( ). 2marks
ii. Findtheequationsofthetangentstothegraphofy = g(x)thatpassthroughthepoint
withcoordinates 32
12, −
. 3marks
MATHMETH EXAM 2 (SAMPLE – ANSWERS)
© VCAA 2016 – Version 2 – April 2016
Answers to multiple-choice questions
Question Answer
1 A
2 D
3 C
4 E
5 A
6 A
7 C
8 E
9 E
10 C
11 B
12 B
13 A
14 B
15 D
16 E
17 D
18 B
19 D
20 E
top related