Specialist Mathematics Exams 1 and 2 - Specifications and sample ...

© VCAA 2016 – Version 3 – July 2016

VCE Specialist Mathematics 2016–2018

Written examinations 1 and 2 – End of year

Examination specifications

Overall conditions There will be two end-of-year examinations for VCE Specialist Mathematics – 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.

SPECMATH (SPECIFICATIONS)


Content The VCE Mathematics Study Design 2016–2018 (‘Specialist Mathematics Units 3 and 4’) is the document for the development of the examination. All outcomes in ‘Specialist Mathematics 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.

SPECMATH (SPECIFICATIONS)


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 Specialist Mathematics examinations:

• VCE Mathematics Study Design 2016–2018 (‘Specialist Mathematics Units 3 and 4’) • VCE Specialist Mathematics – Advice for teachers 2016–2018 (includes assessment advice) • VCE Exams Navigator • VCAA Bulletin

Advice During the 2016–2018 accreditation period for VCE Specialist Mathematics, 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 ESPECIALIST MATHEMATICS

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• Questionandanswerbookof11pages.• 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

Version3–July2016

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education Year

STUDENT NUMBER

Letter

SPECMATHEXAM1(SAMPLE) 2 Version3–July2016

THIS PAGE IS BLANK

Version3–July2016 3 SPECMATHEXAM1(SAMPLE)

TURN OVER

InstructionsAnswerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8

Question 1 (3marks)

a. Showthat 5 − i isasolutionoftheequation z i z z i3 25 4 4 5 4 0− − + − + =( ) . 1mark

b. Findallothersolutionsoftheequation z i z z i3 25 4 4 5 4 0− − + − + =( ) . 2marks


Question 2 (4marks)Giventherelation3x2+2xy + y2=11,findthegradientofthenormal tothegraphoftherelationatthepointinthefirstquadrantwherex=1.


TURN OVER

Question 3 (5marks)Acoffeemachinedispensesvolumesofcoffeethatarenormallydistributedwithmean240mLandstandarddeviation8mL.Themachinealsohastheoptionofaddingmilktoacupofcoffee,wherethevolumeofmilkdispensedisalsonormallydistributedwithmean10mLandstandarddeviation2mL.LettherandomvariableXrepresentthevolumeofcoffeethemachinedispensesandlettherandomvariableYrepresentthevolumeofmilkthemachinedispenses.XandYareindependentrandomvariables.

a. Findthemeanandvarianceofthevolumeofthecombineddrink,thatis,acupofcoffeewithmilk. 2marks

Asecondcoffeemachinealsodispensesvolumesofcoffeethatarenormallydistributed.Theownerhasbeentoldthatthemeanvolumeisagain240mL.Theownerisconcernedthatthesecondcoffeemachineis,onaverage,dispensinglesscoffeethanthefirst.Asampleof16cupsofcoffee(withnomilk)isdispensedanditisfoundthatthemeanvolumeofallcoffeesservedinthissampleis235mL.Assumethatthepopulationstandarddeviationof8mLisunchanged.

b. i. StateappropriatenullandalternativehypothesesforthevolumeVinthissituation. 1mark

ii. ThepvalueforthistestisgivenbytheexpressionPr(Z≤a),whereZhasthestandardnormaldistribution.

Findthevalueofaandhencedeterminewhetherthenullhypothesisshouldberejectedatthe0.05levelofsignificance. 2marks


Question 4 (4marks)Theregioninthefirstquadrantenclosedbythecoordinateaxes,thegraphwithequationy = e–x andthestraightlinex = awherea>0,isrotatedaboutthex-axistoformasolidofrevolution.

a. Expressthevolumeofthesolidofrevolutionasadefiniteintegral. 1mark

b. Calculatethevolumeofthesolidofrevolutionintermsofa. 1mark

c. Findtheexactvalueofaifthevolumeis518πcubicunits. 2marks


TURN OVER

Question 5 (4marks)Aflowerpotofmassmkilogramsisheldinequilibriumbytwolightropes,bothofwhichareconnectedtoaceiling.Thefirstropemakesanangleof30°totheverticalandhastension T1newtons.Thesecondropemakesanangleof60°totheverticalandhastensionT2newtons.

m kg

60°

30°

a. ShowthatTT

21

3= . 1mark

b. Thefirstropeisstrong,butthesecondropewillbreakifthetensioninitexceeds98N.

Findthemaximumvalueofmforwhichtheflowerpotwillremaininequilibrium. 3marks


Question 6 (3marks)

Evaluate cos2

2

34

π

π

∫ (2x)sin(2x)dx.

Question 7 (4marks)Solvethefollowingdifferentialequation dy

dxyx

= 2 fory,giventhatwhenx =1,y =–1.


TURN OVER

Question 8 (4marks)a. Writedownadefiniteintegralintermsof thatgivesthearclengthfrom = 0to = πforthe

curvedefinedparametricallyby 2marks

x=cos(2 )–3

y=sin(2 )+1

b. Hencefindthelengthofthisarc. 2marks


Question 9 (5marks)Considerthethreevectors

a i j k b i j k and c i j k where= − + = + + = + − ∈2 2, , .m m R

a. Findthevalue(s)ofmforwhich

b = 2 3. 2marks

b. Findthevalueofmsuchthat

a isperpendicularto

b. 1mark

c. i. Calculate3

c a.− 1mark

ii. Hence findavalueofmsuchthat

a, b and carelinearly dependent. 1mark


END OF QUESTION AND ANSWER BOOK

Question 10 (4marks)

a. Verifythat5 12 4

4

3

2 2x xx x

+ ++( ) canbewrittenas

1 3 2 142 2x x

xx

+ +−+

.1mark

b. Findanantiderivativeof 5 12 44

3

2 2x xx x

+ ++( )

. 3marks

S A M P L ESPECIALIST MATHEMATICS

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 6 6 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• Questionandanswerbookof23pages.• Formulasheet.• Answersheetformultiple-choicequestions.

Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice

questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.• Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.• 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.

©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016Version3–July2016

SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education Year

STUDENT NUMBER

Letter

SECTION A – continued


Question 1Acirclewithcentre(a,–2)andradius5unitshasequationx2–6x + y2 + 4y = b,whereaandbarerealconstants.ThevaluesofaandbarerespectivelyA. –3and38B. 3and12C. –3and–8D. –3and0E. 3and18

Question 2Themaximaldomainandrangeofthefunctionwithrule f x x( ) = − +−3 4 1

21sin ( ) π arerespectively

A. [–�,2�]and 0, 12

B. 0, 12

and[–�,2�]

C. −

32π π, 3

2and −

12

, 0

D. 0, 12

and[0,3�]

E. −

12

, 0 and[–�,2�]

SECTION A – Multiple-choice questions

Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenot drawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8

SECTION A – continuedTURN OVER


Question 3Thefeaturesofthegraphofthefunctionwithrule f x

x xx x

( ) = − +− −

2

24 3

6include

A. asymptotesatx=1andx=–2B. asymptotesatx=3andx=–2C. anasymptoteatx=1andapointofdiscontinuityatx=3D. anasymptoteatx=–2andapointofdiscontinuityat x=3E. anasymptoteatx=3andapointofdiscontinuityatx=–2

Question 4Thealgebraicfraction

7 54 92 2

xx x

−− +( ) ( )

couldbeexpressedinpartialfractionformas

A. A

xB

x−( )+

+4 92 2

B. Ax

Bx

Cx−

+−

++4 3 3

C. A

xBx Cx−( )

+++4 92 2

D. Ax

Bx

Cx Dx−

+−( )

+++4 4 92 2

E. Ax

Bx

Cx−

+−( )

++4 4 92 2

Question 5OnanArganddiagram,asetofpointsthatliesonacircleofradius2centredattheoriginisA. { : }z C zz∈ = 2

B. { : }z C z∈ =2 4

C. { :Re( ) Im( ) }z C z z∈ + =2 2 4

D. { : }z C z z z z∈ +( ) − −( ) =2 2 16

E. { : Re( ) Im( ) }z C z z∈ ( ) + ( ) =2 2 16

Question 6ThepolynomialP(z)hasrealcoefficients.FouroftherootsoftheequationP(z)=0 are z =0,z =1–2i, z =1+2i and z =3i.TheminimumnumberofrootsthattheequationP(z)=0couldhaveisA. 4B. 5C. 6D. 7E. 8



Question 7

–0.5� –0.4� –0.3� –0.2� –0.1� 0.1� 0.2� 0.3� 0.4� 0.5�

1

0.5

–0.5

–1

y

xO

Thedirection(slope)fieldforacertainfirst-orderdifferentialequationisshownabove.Thedifferentialequationcouldbe

A. dydx

x= ( )sin 2

B. dydx

x= ( )cos 2

C. dydx

y=

cos 1

2

D. dydx

y=

sin 1

2

E. dydx

x=

cos 1

2



Question 8Let f :[–π,2π] → R,wheref (x)=sin3(x).Usingthesubstitutionu=cos(x),theareaboundedbythegraphoffandthex-axiscouldbefoundbyevaluating

A. − −( )−∫ 1 22

u duπ

π

B. 3 1 21

1−( )

−∫ u du

C. − −( )∫3 1 20

u duπ

D. 3 1 21

1−( )

−

∫ u du

E. − −( )−∫ 1 2

1

1u du

Question 9

Letdydx

xx x

=+

+ +2

2 12 and(x0 ,y0)=(0,2).

UsingEuler’smethodwithastepsizeof0.1,thevalueofy1,correcttotwodecimalplaces,isA. 0.17B. 0.20C. 1.70D. 2.17E. 2.20



Question 10

Thecurvegivenbyy=sin–1(2x),where0≤x≤ 12,isrotatedaboutthey-axistoformasolidofrevolution.

Thevolumeofthesolidmaybefoundbyevaluating

A. π

π

41 2

0

2

−( )∫ cos( )y dy

B. π8

1 20

12

−( )∫ cos( )y dy

C. π

π

81 2

0

2

−( )∫ cos( )y dy

D. 18

1 20

2

−( )∫ cos( )y dy

π

E. π

π

π

81 2

2

2

−( )−

∫ cos( )y dy

Question 11Theanglebetweenthevectors3 6 2 2 2

i j k and i j k+ − − + ,correcttothenearesttenthofadegree,isA. 2.0°B. 91.0°C. 112.4°D. 121.3°E. 124.9°

Question 12Thescalarresoluteof

a i k= −3 inthedirectionof

b i j k= − −2 2 is

A. 810

B. 89

2 2

i j k− −( )

C. 8

D. 45

3

i k−( )

E. 83



Question 13

Thepositionvectorofaparticleattimetseconds,t ≥0,isgivenby

r i j + 5k( ) ( )t t t= − −3 6 .Thedirectionofmotionoftheparticlewhent=9is

A. − −6

i 18j + 5k

B. − −

i j

C. − −6

i j

D. − − +

i j k5

E. − − +13 5 108 45.

i j k

Question 14Thediagrambelowshowsarhombus,spannedbythetwovectors

a and

b .

�a

�b

ItfollowsthatA.

a b. = 0

B.

a = b

C.

a b a b+( ) −( ) =. 0

D.

a b a b+ = −

E. 2 2 0

a b+ =

Question 15A12kgmassmovesinastraightlineundertheactionofavariableforceF,sothatitsvelocityvms–1whenitisxmetresfromtheoriginisgivenby v x x= − +3 162 3 .TheforceFactingonthemassisgivenby

A. 12 3 32

2x x−

B. 12 3 162 3x x− +( )

C. 12 6 3 2x x−( )

D. 12 3 162 3x x− +

E. 12 3 3−( )x



Question 16

Theacceleration,a ms–2,ofaparticlemovinginastraightlineisgivenby a vve

=log ( )

,wherevisthe

velocityoftheparticleinms–1attimetseconds.Theinitialvelocityoftheparticlewas5ms–1.Thevelocityoftheparticle,intermsoft, isgivenbyA. v = e2t

B. v = e2t + 4

C. v e t e= +2 5log ( )

D. v e t e= +2 5 2(log )

E. v e t e= − +2 5 2(log )

Question 17A12kgmassissuspendedinequilibriumfromahorizontalceilingbytwoidenticallightstrings.Eachstringmakesanangleof60°withtheceiling,asshown.

60° 60°

12 kg

Themagnitude,innewtons,ofthetensionineachstringisequaltoA. 6 g

B. 12g

C. 24 g

D. 4 3 g

E. 8 3 g

Question 18GiventhatXisanormalrandomvariablewithmean10andstandarddeviation8,andthatYisanormalrandomvariablewithmean3andstandarddeviation2,andXandYareindependentrandomvariables,therandomvariableZdefinedbyZ = X–3YwillhavemeanμandstandarddeviationσgivenbyA. μ=1,σ = 28B. μ=19,σ = 2

C. μ=1,σ = 2 7D. μ=19,σ=14E. μ=1,σ=10


END OF SECTION ATURN OVER

Question 19ThemeanstudyscoreforalargeVCEstudyis30withastandarddeviationof7.Aclassof20studentsmaybeconsideredasarandomsampledrawnfromthiscohort.Theprobabilitythattheclassmeanforthegroupof20exceeds32isA. 0.1007B. 0.3875C. 0.3993D. 0.6125E. 0.8993

Question 20AtypeIerrorwouldoccurinastatisticaltestwhereA. H0isacceptedwhenH0isfalse.B. H1isacceptedwhenH1istrue.C. H0 isrejectedwhenH0istrue.D. H1isrejectedwhenH1istrue.E. H0 isacceptedwhenH0istrue.


SECTION B – Question 1–continued

Question 1 (10marks)Considerthefunctionf :[0,3)→ R,wheref (x)=–2+2sec

π x6

.

a. Evaluatef (2). 1mark

Letf –1betheinversefunctionoff.

b. Ontheaxesbelow,sketchthegraphsoffand f –1,showingtheirpointsofintersection. 2marks

1

O 1 2 3 4

2

3

4

y

x

SECTION B

Instructions for Section BAnswerallquestionsinthespacesprovided.Unlessotherwisespecified,anexactanswerisrequiredtoaquestion.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.Taketheacceleration due to gravitytohavemagnitudegms–2,whereg=9.8


SECTION B – continuedTURN OVER

c. Theruleforf –1canbewrittenasf –1(x)=karccos2

2x +

.

Findtheexactvalueofk. 2marks

LetAbethemagnitudeoftheareaenclosedbythegraphsoffandf –1.

d. WriteadefiniteintegralexpressionforAandevaluateitcorrecttothreedecimalplaces. 2marks

e. i. Writedownadefiniteintegralintermsofxthatgivesthearclengthofthegraphof ffromx = 0 to x =2. 2marks

ii. Evaluatethisdefiniteintegralcorrecttothreedecimalplaces. 1mark



Question 2 (9marks)

Letu = 12

32

+ i .

a. i. Expressuinpolarform. 2marks

ii. Henceshowthatu6=1. 1mark

iii. Plotallrootsofz6–1=0ontheArganddiagrambelow,labellinguandwwhere w =–u. 2marks

Im(z)

Re(z)1 2 3–3 –2 –1

2

1

–1

–2

O



b. i. Drawandlabelthesubsetofthecomplexplanegivenby S z z= ={ }: 1 ontheArganddiagrambelow. 1mark

Im(z)

Re(z)1 2 3–3 –2 –1

2

1

–1

–2

O

ii. DrawandlabelthesubsetofthecomplexplanegivenbyT z z u z u= − = +{ }: ontheArganddiagramabove. 1mark

iii. FindthecoordinatesofthepointsofintersectionofSandT. 2marks



Question 3 (11marks)Thenumberofmobilephones,N,ownedinacertaincommunityaftertyearsmaybemodelledbyloge(N)=6–3e–0.4t,t≥0.

a. Verifybysubstitutionthatloge(N)=6–3e–0.4tsatisfiesthedifferentialequation

1 0 4 2 4 0NdNdt

Ne+ − =. log ( ) . 2marks

b. Findtheinitialnumberofmobilephonesownedinthecommunity.Giveyouranswercorrecttothenearestinteger. 1mark

c. Usingthismathematicalmodel,findthelimitingnumberofmobilephonesthatwouldeventuallybeownedinthecommunity.Giveyouranswercorrecttothenearestinteger. 2marks


SECTION B – Question 3–continuedTURN OVER

Thedifferentialequationinpart a.canalsobewrittenintheformdNdt =0.4N(6–loge(N )).

d. i. Findd Ndt

2

2 intermsofNandloge(N ). 2marks

ii. ThegraphofNasafunctionofthasapointofinflection.

Findthevaluesofthecoordinatesofthispoint.GivethevalueoftcorrecttoonedecimalplaceandthevalueofNcorrecttothenearestinteger. 2marks


SECTION B – continued

e. SketchthegraphofNasafunctionoftontheaxesbelowfor0≤t≤15. 2marks

N

t

400

300

200

100

5 10 15O


SECTION B – Question 4–continuedTURN OVER

Question 4 (10marks)Askieracceleratesdownaslopeandthenskisupashortskijump,asshownbelow.Theskierleavesthejumpataspeedof12ms–1andatanangleof60°tothehorizontal.Theskierperformsvariousgymnastictwistsandlandsonastraight-linesectionofthe45°down-slopeTsecondsafterleavingthejump.LettheoriginOofacartesiancoordinatesystembeatthepointwheretheskierleavesthejump,with

i aunitvectorinthepositivexdirectionand

j aunitvectorinthepositiveydirection.Displacementsaremeasuredinmetresandtimeinseconds.

45°60°

y

xskijump

down-slope

O

a. Showthattheinitialvelocityoftheskierwhenleavingthejumpis6 6 3

i j+ . 1mark



b. Theaccelerationoftheskier, tsecondsafterleavingtheskijump,isgivenby

��

� �r( ) i jt t g t= − − −( )0 1 0 1. . ,0≤t≤T

Showthatthepositionvectoroftheskier, tsecondsafterleavingthejump,isgivenby

r i jt t t t gt t( ) = −

+ − +

6 1

606 3 1

2160

3 2 3 ,0≤t≤T 3marks

c. ShowthatT g= +( )12 3 1 . 3marks



d. Atwhatspeed,inmetrespersecond,doestheskierlandonthedown-slope?Giveyouranswercorrecttoonedecimalplace. 3marks



Question 5 (10marks)Thediagrambelowshowsparticlesofmass1kgand3kgconnectedbyalightinextensiblestringpassingoverasmoothpulley.ThetensioninthestringisT1newtons.

T1 T1

g 3 g

a. Letams–2betheaccelerationofthe3kgmassdownwards.

Findthevalueofa. 2marks

b. FindthevalueofT1. 1mark



The3kgmassisplacedonasmoothplaneinclinedatanangleof °tothehorizontal.ThetensioninthestringisnowT2newtons.

3 gT2

T2

g

° θ

c. When °=30°,theaccelerationofthe1kgmassupwardsisbms–2.

Findthevalueofb. 3marks

d. Forwhatangle °willthe3kgmassbeatrestontheplane?Giveyouranswercorrecttoonedecimalplace. 2marks

e. Whatangle °willcausethe3kgmasstoaccelerateuptheplaneatg4

1 32

2−

−ms ? 2marks



Question 6 (10marks)Acertaintypeofcomputer,oncefullycharged,isclaimedbythemanufacturertohaveμ=10hourslifetimebeforearechargeisneeded.Whenchecked,arandomsampleofn=25suchcomputersisfoundtohaveanaveragelifetimeofx =9.7hoursandastandarddeviationofs=1hour.Todecidewhethertheinformationgainedfromthesampleisconsistentwiththeclaimμ=10,astatisticaltestistobecarriedout.Assumethatthedistributionoflifetimesisnormalandthats isasufficientlyaccurateestimateofthepopulation(oflifetimes)standarddeviationσ.

a. WritedownsuitablehypothesesH0andH1totestwhetherthemeanlifetimeislessthanthatclaimedbythemanufacturer. 2marks

b. Findthepvalueforthistest,correcttothreedecimalplaces. 2marks

c. StatewithareasonwhetherH0shouldberejectedornotrejectedatthe5%levelofsignificance. 1mark

LettherandomvariableX denotethemeanlifetimeofarandomsampleof25computers,assumingμ=10.

d. FindC *suchthatPr * .X C< =( ) =µ 10 0 05.Giveyouranswercorrecttothreedecimalplaces. 2marks


e. i. Ifthemeanlifetimeofallcomputersisinfactμ =9.5hours,findPr * .X C> =( )µ 9 5 ,givingyouranswercorrecttothreedecimalplaces,whereC*isyouranswertopart d. 2marks

ii. Doestheresultinpart e.i.indicateatypeIortypeIIerror?Explainyouranswer. 1mark

END OF QUESTION AND ANSWER BOOK

SPECMATH EXAM 2 (SAMPLE – ANSWERS)


Answers to multiple-choice questions

Question Answer

1 B

2 B

3 D

4 D

5 D

6 B

7 B

8 B

9 E

10 C

11 C

12 E

13 B

14 C

15 A

16 D

17 D

18 E

19 A

20 C

Specialist Mathematics Exams 1 and 2 - Specifications and sample ...

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