specialist mathematics exams 1 and 2 - specifications and sample
TRANSCRIPT
© 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)
© VCAA 2016 – Version 3 – July 2016 Page 2
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)
© VCAA 2016 – Version 3 – July 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 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
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
SPECMATHEXAM1(SAMPLE) 4 Version3–July2016
Question 2 (4marks)Giventherelation3x2+2xy + y2=11,findthegradientofthenormal tothegraphoftherelationatthepointinthefirstquadrantwherex=1.
Version3–July2016 5 SPECMATHEXAM1(SAMPLE)
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
SPECMATHEXAM1(SAMPLE) 6 Version3–July2016
Question 4 (4marks)Theregioninthefirstquadrantenclosedbythecoordinateaxes,thegraphwithequationy = e–x andthestraightlinex = awherea>0,isrotatedaboutthex-axistoformasolidofrevolution.
a. Expressthevolumeofthesolidofrevolutionasadefiniteintegral. 1mark
b. Calculatethevolumeofthesolidofrevolutionintermsofa. 1mark
c. Findtheexactvalueofaifthevolumeis518πcubicunits. 2marks
Version3–July2016 7 SPECMATHEXAM1(SAMPLE)
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
SPECMATHEXAM1(SAMPLE) 8 Version3–July2016
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.
Version3–July2016 9 SPECMATHEXAM1(SAMPLE)
TURN OVER
Question 8 (4marks)a. Writedownadefiniteintegralintermsof thatgivesthearclengthfrom = 0to = πforthe
curvedefinedparametricallyby 2marks
x=cos(2 )–3
y=sin(2 )+1
b. Hencefindthelengthofthisarc. 2marks
SPECMATHEXAM1(SAMPLE) 10 Version3–July2016
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
Version3–July2016 11 SPECMATHEXAM1(SAMPLE)
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
SPECMATHEXAM2(SAMPLE) 2 Version3–July2016
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
Version3–July2016 3 SPECMATHEXAM2(SAMPLE)
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
SECTION A – continued
SPECMATHEXAM2(SAMPLE) 4 Version3–July2016
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
SECTION A – continuedTURN OVER
Version3–July2016 5 SPECMATHEXAM2(SAMPLE)
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
SECTION A – continued
SPECMATHEXAM2(SAMPLE) 6 Version3–July2016
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
SECTION A – continuedTURN OVER
Version3–July2016 7 SPECMATHEXAM2(SAMPLE)
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
SECTION A – continued
SPECMATHEXAM2(SAMPLE) 8 Version3–July2016
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
Version3–July2016 9 SPECMATHEXAM2(SAMPLE)
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.
SPECMATHEXAM2(SAMPLE) 10 Version3–July2016
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
Version3–July2016 11 SPECMATHEXAM2(SAMPLE)
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
SPECMATHEXAM2(SAMPLE) 12 Version3–July2016
SECTION B – Question 2–continued
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
Version3–July2016 13 SPECMATHEXAM2(SAMPLE)
SECTION B – continuedTURN OVER
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
SPECMATHEXAM2(SAMPLE) 14 Version3–July2016
SECTION B – Question 3–continued
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
Version3–July2016 15 SPECMATHEXAM2(SAMPLE)
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
SPECMATHEXAM2(SAMPLE) 16 Version3–July2016
SECTION B – continued
e. SketchthegraphofNasafunctionoftontheaxesbelowfor0≤t≤15. 2marks
N
t
400
300
200
100
5 10 15O
Version3–July2016 17 SPECMATHEXAM2(SAMPLE)
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
SPECMATHEXAM2(SAMPLE) 18 Version3–July2016
SECTION B – Question 4–continued
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
Version3–July2016 19 SPECMATHEXAM2(SAMPLE)
SECTION B – continuedTURN OVER
d. Atwhatspeed,inmetrespersecond,doestheskierlandonthedown-slope?Giveyouranswercorrecttoonedecimalplace. 3marks
SPECMATHEXAM2(SAMPLE) 20 Version3–July2016
SECTION B – Question 5–continued
Question 5 (10marks)Thediagrambelowshowsparticlesofmass1kgand3kgconnectedbyalightinextensiblestringpassingoverasmoothpulley.ThetensioninthestringisT1newtons.
T1 T1
g 3 g
a. Letams–2betheaccelerationofthe3kgmassdownwards.
Findthevalueofa. 2marks
b. FindthevalueofT1. 1mark
Version3–July2016 21 SPECMATHEXAM2(SAMPLE)
SECTION B – continuedTURN OVER
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
SPECMATHEXAM2(SAMPLE) 22 Version3–July2016
SECTION B – Question 6–continued
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
Version3–July2016 23 SPECMATHEXAM2(SAMPLE)
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