Question 4:
4
Section B (36 marks)
7 Fig. 7 shows part of the curve y (cid:1) f(x), where f(x)= x 1+ x. The curve meets the x-axis at the
origin and at the point P.
y
P O x
Fig. 7
(i) Verify that the point P has coordinates ((cid:3)1, 0). Hence state the domain of the function f(x).
[2]
dy 2+3x
(ii) Show that = . [4]
dx 2 1+ x
(iii) Find the exact coordinates of the turning point of the curve. Hence write down the range of
the function. [4]
(iv) Use the substitution u (cid:1) 1(cid:2)x to show that
Û 0 Û 1( 3 1)
Ù x 1+ xdx =Ù u2 -u2 du.
ı ı
-1 0
Hence find the area of the region enclosed by the curve and the x-axis. [8]
4
8 Fig. 8 shows part of the curve y (cid:2) x cos 2x, together with a point Pat which the curve crosses the
x-axis.
y
O P x
Fig. 8
(i) Find the exact coordinates of P. [3]
(ii) Show algebraically that x cos 2xis an odd function, and interpret this result graphically. [3]
dy
(iii) Find .
dx [2]
(iv) Show that turning points occur on the curve for values of x which satisfy the equation
x tan 2x (cid:2) 1 . [2]
2
(v) Find the gradient of the curve at the origin.
Show that the second derivative of x cos 2x is zero when x (cid:2) 0. [4]
1p
Û4
(vi) Evaluate Ù xcos2xdx, giving your answer in terms of p. Interpret this result graphically.
ı
0 [6]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
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publisher will be pleased to make amends at the earliest possible opportunity.
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© OCR 2007 4753/01 June 07
PMT
4753/01
ADVANCED GCE
MATHEMATICS (MEI)
Methods for Advanced Mathematics (C3)
MONDAY 2 JUNE 2008 Morning
Time: 1hour30minutes
Additionalmaterials(enclosed): None
Additionalmaterials(required):
AnswerBooklet(8pages)
Graphpaper
MEIExaminationFormulaeandTables(MF2)
INSTRUCTIONSTOCANDIDATES
(cid:129) Writeyour name in capital letters, your Centre Number andCandidate Number in thespaces
providedontheAnswerBooklet.
(cid:129) Read each question carefully and make sure you know what you have to do before starting
youranswer.
(cid:129) Answerallthequestions.
(cid:129) Youarepermittedtouseagraphicalcalculatorinthispaper.
(cid:129) Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATIONFORCANDIDATES
(cid:129) Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestion.
(cid:129) Thetotalnumberofmarksforthispaperis72.
(cid:129) Youareadvisedthatananswermayreceivenomarksunlessyoushowsufficientdetailofthe
workingtoindicatethatacorrectmethodisbeingused.
Thisdocumentconsistsof4printedpages.
©OCR2008[M/102/2652] OCRisanexemptCharity [Turnover
PMT
2
SectionA(36marks)
1 Solvetheinequality |2x−1|≤ 3. [4]
2 Find(cid:3) xe3xdx. [4]
3 (i) Statethealgebraiccondition forthefunction f(x)tobean even function.
Whatgeometricalpropertydoesthegraphofanevenfunctionhave? [2]
(ii) Statewhetherthefollowing functionsareodd,evenorneither.
(A) f(x) = x2−3
(B) g(x) = sinx+cosx
1
(C) h(x) = [3]
x+x3
4
x
4 Showthat (cid:4) dx = 1ln6. [4]
x2+2 2
1
1
5 Showthatthecurvey = x2lnx hasastationarypointwhen x = √ . [6]
e
6 In achemicalreaction,themass mgramsofachemicalaftert minutesismodelled bytheequation
m = 20+30e −0.1t.
(i) Find theinitialmassofthechemical.
Whatisthemassofchemicalin thelong term? [3]
(ii) Find thetimewhen themassis30grams. [3]
(iii) Sketchthegraphofmagainstt. [2]
dy
7 Giventhatx2+xy+y2 = 12,find intermsofxandy. [5]
dx
©OCR2008 4753/01Jun08
PMT
3
SectionB(36marks)
1
8 Fig.8showsthecurvey = f(x),wheref(x) = ,for0 ≤ x ≤ 1π.
1+cosx 2
Pisthepointon thecurvewith x-coordinate 1π.
3
y
1 y=f(x)
P
1
2
x
O 1(cid:2) 1(cid:2)
3 2
Fig.8
(i) Find they-coordinateofP. [1]
(ii) Find f
(cid:6)(x).
HencefindthegradientofthecurveatthepointP. [5]
sinx 1
(iii) Showthatthederivativeof is . Hencefindtheexactareaoftheregionenclosed
1+cosx 1+cosx
bythecurvey = f(x),thex-axis,they-axisandthelinex = 1π. [7]
3
1
(iv) Showthatf −1(x)=arccos(cid:2) −1(cid:3). Statethedomainofthisinversefunction,andaddasketchof
x
y = f −1(x)toacopyofFig.8. [5]
[Question9isprintedoverleaf.]
©OCR2008 4753/01Jun08
PMT
4
(cid:4)
9 Thefunctionf(x)isdefinedby f(x) = 4−x2 for−2 ≤ x ≤ 2.
(cid:4)
(i) Show that the curve y = 4−x2 is a semicircle of radius 2, and explain why it is not the whole
ofthiscircle. [3]
Fig.9showsapointP(a, b)onthesemicircle. ThetangentatPisshown.
y
P(a,b)
x
–2 O 2
Fig.9
(ii) (A) UsethegradientofOPtofindthegradientofthetangentatPintermsofaandb.
(cid:4)
(B) Differentiate 4−x2 anddeducethevalueoff (cid:6)(a).
(C) Show thatyouranswersto parts(A)and(B)areequivalent. [6]
Thefunctiong(x)isdefined byg(x) = 3f(x−2),for0 ≤ x ≤ 4.
(iii) Describe a sequence of two transformations that would map the curve y = f(x) onto the curve
y = g(x).
Hencesketchthecurvey = g(x). [6]
(iv) Show thatify = g(x)then9x2+y2 = 36x. [3]
Permissiontoreproduceitemswherethird-partyownedmaterialprotectedbycopyrightisincludedhasbeensoughtandclearedwherepossible.Everyreasonable
efforthasbeenmadebythepublisher(OCR)totracecopyrightholders,butifanyitemsrequiringclearancehaveunwittinglybeenincluded,thepublisherwillbe
pleasedtomakeamendsattheearliestpossibleopportunity.
OCRispartoftheCambridgeAssessmentGroup.CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),
whichisitselfadepartmentoftheUniversityofCambridge.
©OCR2008 4753/01Jun08
PMT
4753/01
ADVANCED GCE
MATHEMATICS (MEI)
Methods for Advanced Mathematics (C3)
MONDAY 2 JUNE 2008 Morning
Time: 1hour30minutes
Additionalmaterials(enclosed): None
Additionalmaterials(required):
AnswerBooklet(8pages)
Graphpaper
MEIExaminationFormulaeandTables(MF2)
INSTRUCTIONSTOCANDIDATES
(cid:129) Writeyour name in capital letters, your Centre Number andCandidate Number in thespaces
providedontheAnswerBooklet.
(cid:129) Read each question carefully and make sure you know what you have to do before starting
youranswer.
(cid:129) Answerallthequestions.
(cid:129) Youarepermittedtouseagraphicalcalculatorinthispaper.
(cid:129) Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATIONFORCANDIDATES
(cid:129) Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestion.
(cid:129) Thetotalnumberofmarksforthispaperis72.
(cid:129) Youareadvisedthatananswermayreceivenomarksunlessyoushowsufficientdetailofthe
workingtoindicatethatacorrectmethodisbeingused.
Thisdocumentconsistsof4printedpages.
©OCR2008[M/102/2652] OCRisanexemptCharity [Turnover
PMT
2
SectionA(36marks)
1 Solvetheinequality |2x−1|≤ 3. [4]
2 Find(cid:3) xe3xdx. [4]
3 (i) Statethealgebraiccondition forthefunction f(x)tobean even function.
Whatgeometricalpropertydoesthegraphofanevenfunctionhave? [2]
(ii) Statewhetherthefollowing functionsareodd,evenorneither.
(A) f(x) = x2−3
(B) g(x) = sinx+cosx
1
(C) h(x) = [3]
x+x3
4
x
4 Showthat (cid:4) dx = 1ln6. [4]
x2+2 2
1
1
5 Showthatthecurvey = x2lnx hasastationarypointwhen x = √ . [6]
e
6 In achemicalreaction,themass mgramsofachemicalaftert minutesismodelled bytheequation
m = 20+30e −0.1t .
(i) Find theinitialmassofthechemical.
Whatisthemassofchemicalin thelong term? [3]
(ii) Find thetimewhen themassis30grams. [3]
(iii) Sketchthegraphofmagainstt. [2]
dy
7 Giventhatx2+xy+y2 = 12,find intermsofxandy. [5]
dx
©OCR2008 4753/01Jun08
PMT
3
SectionB(36marks)
1
8 Fig.8showsthecurvey = f(x),wheref(x) = ,for0 ≤ x ≤ 1π.
1+cosx 2
Pisthepointon thecurvewith x-coordinate 1π.
3
y
1 y=f(x)
P
1
2
x
O 1(cid:2) 1(cid:2)
3 2
Fig.8
(i) Find they-coordinateofP. [1]
(ii) Find f
(cid:6)(x).
HencefindthegradientofthecurveatthepointP. [5]
sinx 1
(iii) Showthatthederivativeof is . Hencefindtheexactareaoftheregionenclosed
1+cosx 1+cosx
bythecurvey = f(x),thex-axis,they-axisandthelinex = 1π. [7]
3
1
(iv) Showthatf −1(x)=arccos(cid:2) −1(cid:3). Statethedomainofthisinversefunction,andaddasketchof
x
y = f −1(x)toacopyofFig.8. [5]
[Question9isprintedoverleaf.]
©OCR2008 4753/01Jun08
PMT
4
(cid:4)
9 Thefunctionf(x)isdefinedby f(x) = 4−x2 for−2 ≤ x ≤ 2.
(cid:4)
(i) Show that the curve y = 4−x2 is a semicircle of radius 2, and explain why it is not the whole
ofthiscircle. [3]
Fig.9showsapointP(a, b)onthesemicircle. ThetangentatPisshown.
y
P(a,b)
x
–2 O 2
Fig.9
(ii) (A) UsethegradientofOPtofindthegradientofthetangentatPintermsofaandb.
(cid:4)
(B) Differentiate 4−x2 anddeducethevalueoff (cid:6)(a).
(C) Show thatyouranswersto parts(A)and(B)areequivalent. [6]
Thefunctiong(x)isdefined byg(x) = 3f(x−2),for0 ≤ x ≤ 4.
(iii) Describe a sequence of two transformations that would map the curve y = f(x) onto the curve
y = g(x).
Hencesketchthecurvey = g(x). [6]
(iv) Show thatify = g(x)then9x2+y2 = 36x. [3]
Permissiontoreproduceitemswherethird-partyownedmaterialprotectedbycopyrightisincludedhasbeensoughtandclearedwherepossible.Everyreasonable
efforthasbeenmadebythepublisher(OCR)totracecopyrightholders,butifanyitemsrequiringclearancehaveunwittinglybeenincluded,thepublisherwillbe
pleasedtomakeamendsattheearliestpossibleopportunity.
OCRispartoftheCambridgeAssessmentGroup.CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),
whichisitselfadepartmentoftheUniversityofCambridge.
©OCR2008 4753/01Jun08
PMT
ADVANCED GCE
4753/01
MATHEMATICS (MEI)
Methods for Advanced Mathematics (C3)
CandidatesanswerontheAnswerBooklet Thursday 15 January 2009
OCRSuppliedMaterials: Morning
• 8pageAnswerBooklet
• Graphpaper Duration: 1hour30minutes
• MEIExaminationFormulaeandTables(MF2)
OtherMaterialsRequired:
None
*475301*
* 4 7 5 3 0 1 *
INSTRUCTIONSTOCANDIDATES
• Writeyournameclearlyincapitalletters,yourCentreNumberandCandidateNumberinthespacesprovided
ontheAnswerBooklet.
• Useblackink. Pencilmaybeusedforgraphsanddiagramsonly.
• Readeachquestioncarefullyandmakesurethatyouknowwhatyouhavetodobeforestartingyouranswer.
• Answerallthequestions.
• Donotwriteinthebarcodes.
• Youarepermittedtouseagraphicalcalculatorinthispaper.
• Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATIONFORCANDIDATES
• Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestion.
• You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicatethatacorrectmethodisbeingused.
• Thetotalnumberofmarksforthispaperis72.
• Thisdocumentconsistsof4pages. Anyblankpagesareindicated.
©OCR2009[M/102/2652] OCRisanexemptCharity
3R–8H16 Turnover
PMT
2
SectionA(36marks)
1 Solvetheinequality |x−1|< 3. [3]
2 (i) Differentiatexcos2xwithrespecttox. [3]
(ii) Integratexcos2xwith respecttox. [4]
3 Giventhatf(x) = 1ln(x−1)andg(x) = 1+e2x,showthatg(x)istheinverseoff(x). [3]
2
2√
4 Find theexactvalueofã 1+4xdx,showingyourworking. [5]
0
5 (i) Statetheperiod ofthefunctionf(x) = 1+cos2x,wherexisindegrees. [1]
(ii) State a sequence of two geometrical transformations which maps the curve y = cosx onto the
curvey = f(x). [4]
(iii) Sketchthegraphofy = f(x)for−180 ◦ < x < 180 ◦ . [3]
6 (i) Disprovethefollowingstatement.
1 1
‘Ifp > q,then < .’ [2]
p q
(ii) Stateaconditiononpandqsothatthestatementistrue. [1]
2 2
7 Thevariablesx andysatisfytheequation x3 +y3 = 5.
1
dy y
(i) Show that = −(cid:16) (cid:17)3 . [4]
dx x
Bothx and yarefunctionsoft.
dy dx
(ii) Find thevalueof whenx = 1,y = 8 and = 6. [3]
dt dt
©OCR2009 4753/01Jan09
3
SectionB(36marks)
8 Fig.8 shows the curve y = x2 − 1lnx. P is the point on this curve with x-coordinate 1, and R is the
8
point(cid:0)0, −7(cid:1).
8
y
P
Q
x
O
R (0,–7
8
(
PMT
Fig. 8
(i) Find thegradientofPR. [3]
dy
(ii) Find . HenceshowthatPRisatangentto thecurve. [3]
dx
(iii) Find theexactcoordinatesoftheturning pointQ. [5]
(iv) Differentiatexlnx−x.
Hence, or otherwise, show that the area of the region enclosed by the curve y = x2 − 1lnx, the
8
x-axisandthelinesx = 1and x = 2 is 59 − 1ln2. [7]
24 4
[Question9isprintedoverleaf.]
Turnover
©OCR2009 4753/01Jan09
PMT
4
1
9 Fig.9showsthecurvey = f(x),wheref(x) = p .
2x−x2
Thecurvehasasymptotesx = 0andx = a.
y
x
O a
Fig. 9
(i) Find a. Hencewritedownthedomain ofthefunction. [3]
dy x−1
(ii) Show that = .
dx 3
(cid:0)2x−x2(cid:1)2
Hence find the coordinates of the turning point of the curve, and write down the range of the
function. [8]
1
Thefunctiong(x)isdefined byg(x) = p .
1−x2
(iii) (A) Show algebraicallythatg(x)isanevenfunction.
(B) Show thatg(x−1) = f(x).
(C) Henceprovethatthecurvey = f(x)issymmetrical,andstateitslineofsymmetry. [7]
Permissiontoreproduceitemswherethird-partyownedmaterialprotectedbycopyrightisincludedhasbeensoughtandclearedwherepossible.Everyreasonable
efforthasbeenmadebythepublisher(OCR)totracecopyrightholders,butifanyitemsrequiringclearancehaveunwittinglybeenincluded,thepublisherwillbe
pleasedtomakeamendsattheearliestpossibleopportunity.
OCRispartoftheCambridgeAssessmentGroup.CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),
whichisitselfadepartmentoftheUniversityofCambridge.
©OCR2009 4753/01Jan09
PMT
ADVANCED GCE
4753/01
MATHEMATICS (MEI)
Methods for Advanced Mathematics (C3)
CandidatesanswerontheAnswerBooklet Friday 5 June 2009
OCRSuppliedMaterials: Afternoon
• 8pageAnswerBooklet
• Graphpaper Duration: 1hour30minutes
• MEIExaminationFormulaeandTables(MF2)
OtherMaterialsRequired:
None
*475301*
* 4 7 5 3 0 1 *
INSTRUCTIONSTOCANDIDATES
• Writeyournameclearlyincapitalletters,yourCentreNumberandCandidateNumberinthespacesprovided
ontheAnswerBooklet.
• Useblackink. Pencilmaybeusedforgraphsanddiagramsonly.
• Readeachquestioncarefullyandmakesurethatyouknowwhatyouhavetodobeforestartingyouranswer.
• Answerallthequestions.
• Donotwriteinthebarcodes.
• Youarepermittedtouseagraphicalcalculatorinthispaper.
• Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATIONFORCANDIDATES
• Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestion.
• You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicatethatacorrectmethodisbeingused.
• Thetotalnumberofmarksforthispaperis72.
• Thisdocumentconsistsof4pages. Anyblankpagesareindicated.
©OCR2009[M/102/2652] OCRisanexemptCharity
2R–9A07 Turnover
PMT
2
SectionA(36marks)
1π
1
Evaluateã6
sin3xdx. [3]
0
2 A radioactive substance decays exponentially, so that its mass M grams can be modelled by the
equationM = Ae −kt,wheret isthetimein years,andAandk arepositiveconstants.
(i) Aninitialmassof100gramsofthesubstancedecaysto 50gramsin 1500 years. FindAandk.
[5]
(ii) Thesubstancebecomessafewhen99%ofitsinitialmasshasdecayed. Findhowlongitwilltake
beforethesubstancebecomessafe. [3]
3 Sketchthecurvey = 2arccosxfor−1 ≤ x ≤ 1. [3]
4 Fig.4 shows a sketch of the graph of y = 2|x − 1|. It meets the x- and y-axes at (a, 0) and (0, b)
respectively.
y
b
x
O a
Fig. 4
Find thevaluesofaand b. [3]
5 Theequationofacurveisgivenbye2y = 1+sinx.
dy
(i) Bydifferentiatingimplicitly,find intermsofx andy. [3]
dx
(ii) Find anexpressionfory intermsofx,anddifferentiateitto verifytheresultin part(i). [4]
x+1
6 Giventhatf(x) = ,showthatff(x)= x.
x−1
Hencewritedowntheinversefunctionf
−1(x).
Whatcanyoudeduceaboutthesymmetryofthecurve
y = f(x)? [5]
©OCR2009 4753/01Jun09
PMT
3
7 (i) Show that
(A) (x−y)(x2+xy+y2) = x3−y3,
(B) (x+ 1y)2+ 3y2 = x2+xy+y2 . [4]
2 4
(ii) Henceprovethat,forallrealnumbersxandy,ifx > ythen x3 > y3. [3]
SectionB(36marks)
8 Fig.8showstheliney = x andpartsofthecurvesy = f(x)andy = g(x),where
f(x) = e x−1, g(x) = 1+lnx.
The curves intersect the axes at the points A and B, as shown. The curves and the line y = x meet at
thepointC.
y
y=x
C
B
y= f(x)
O
x
A
y= g(x)
Fig. 8
(i) Find theexactcoordinatesofAand B.Verify thatthecoordinatesofCare(1, 1). [5]
(ii) Provealgebraicallythatg(x)istheinverseoff(x). [2]
1
(iii) Evaluateã f(x)dx,givingyouranswerintermsofe. [3]
0
(iv) Useintegration by partsto findã lnxdx.
1 1
Henceshowthatã g(x)dx = . [6]
e−1 e
(v) Find theareaoftheregion enclosedbythelinesOAandOB,andthearcsACand BC. [2]
Turnover
©OCR2009 4753/01Jun09
PMT
4
x2
9 Fig.9showsthecurvey = .
3x−1
Pisaturningpoint,andthecurvehasaverticalasymptotex = a.
y
P
x=a
x
O
Fig. 9
(i) Writedownthevalueofa. [1]
dy x(3x−2)
(ii) Show that = . [3]
dx (3x−1)2
(iii) Find theexactcoordinatesoftheturning pointP.
Calculatethegradientofthecurvewhenx=0.6andx=0.8,andhenceverifythatPisaminimum
point. [7]
x2 1 1
(iv) Using thesubstitutionu = 3x−1,showthat ä dx = ä (cid:16)u+2+ (cid:17)du.
3x−1 27 u
Hence find the exact area of the region enclosed by the curve, the x-axis and the lines x = 2 and
3
x = 1. [7]
CopyrightInformation
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oftheUniversityofCambridge.
©OCR2009 4753/01Jun09
PMT
ADVANCED GCE
4753/01
MATHEMATICS (MEI)
Methods for Advanced Mathematics (C3)
CandidatesanswerontheAnswerBooklet Wednesday 20 January 2010
OCRSuppliedMaterials: Afternoon
• 8pageAnswerBooklet
• MEIExaminationFormulaeandTables(MF2) Duration: 1hour30minutes
OtherMaterialsRequired:
None
*475301*
* 4 7 5 3 0 1 *
INSTRUCTIONSTOCANDIDATES
• Writeyournameclearlyincapitalletters,yourCentreNumberandCandidateNumberinthespacesprovided
ontheAnswerBooklet.
• Useblackink. Pencilmaybeusedforgraphsanddiagramsonly.
• Readeachquestioncarefullyandmakesurethatyouknowwhatyouhavetodobeforestartingyouranswer.
• Answerallthequestions.
• Donotwriteinthebarcodes.
• Youarepermittedtouseagraphicalcalculatorinthispaper.
• Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATIONFORCANDIDATES
• Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestion.
• You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicatethatacorrectmethodisbeingused.
• Thetotalnumberofmarksforthispaperis72.
• Thisdocumentconsistsof4pages. Anyblankpagesareindicated.
©OCR2010[M/102/2652] OCRisanexemptCharity
2R–9F25 Turnover
PMT
2
SectionA(36marks)
1 Solvetheequatione2x−5ex =0. [4]
2 The temperature T in degrees Celsius of water in a glass t minutes after boiling is modelled by the
equation T = 20+be −kt, where b and k are constants. Initially the temperature is 100 ◦ C, and after
◦
5minutesthetemperatureis60 C.
(i) Findbandk. [4]
◦
(ii) Findatwhattimethe temperaturereaches50 C. [2]
p
dy
3 (i) Giventhaty = 31+3x2,use thechainruletofind intermsofx. [3]
dx
dy
(ii) Giventhaty3 =1+3x2,useimplicitdifferentiationtofind intermsofxandy. Showthatthis
dx
resultisequivalenttotheresultinpart(i). [4]
4 Evaluatethefollowingintegrals,givingyour answersinexactform.
1
2x
ä
(i) dx. [3]
x2+1
0
1
2x
ä
(ii) dx. [5]
x+1
0
5 The curves in parts (i) and (ii) have equations of the form y = a +bsincx, where a, b and c are
constants. For eachcurve,findthevaluesofa, bandc.
(i) y
3
x
p
–3
[2]
(ii) y
2
1
x
1p
2 [2]
©OCR2010 4753/01Jan10
PMT
3
6 Writedowntheconditionsforf(x) tobeanoddfunctionandforg(x)tobeanevenfunction.
Henceprovethat,iff(x)isoddandg(x) iseven,thenthecompositefunctiongf(x)iseven. [4]
7 Giventhatarcsinx =arccosy,provethatx2+y2 = 1. [Hint: letarcsinx = θ.] [3]
SectionB(36marks)
8 Fig.8showspartofthecurvey =xcos3x.
Thecurvecrossesthex-axisatO,PandQ.
y
x
O P Q
Fig. 8
(i) FindtheexactcoordinatesofPandQ. [4]
(ii) Findtheexactgradientofthecurve atthepointP.
Showalsothattheturningpointsofthecurveoccurwhenxtan3x = 1. [7]
3
(iii) Find the area of the region enclosed by the curve and the x-axis between O and P, giving your
answerinexactform. [6]
[Question9isprintedoverleaf.]
Turnover
©OCR2010 4753/01Jan10
PMT
4
2x2−1
9 Fig.9showsthecurvey =f(x),wheref(x)= forthedomain0≤x ≤2.
x2+1
y
x
O Ö2 2
2
–1
Fig. 9
6x
(i) Showthatf ′(x)= ,andhencethatf(x)isanincreasingfunctionforx >0. [5]
(x2+1)2
(ii) Findtherange off(x). [2]
6−18x2
(iii) Giventhatf
′′(x)=
,findthemaximumvalueof f
′(x).
[4]
(x2+1)3
Thefunctiong(x)istheinversefunctionoff(x).
(iv) Writedownthedomainandrangeofg(x). Addasketchofthecurvey=g(x)toacopyofFig.9.
[4]
r
x+1
(v) Showthatg(x)= . [4]
2−x
CopyrightInformation
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whoseworkisusedinthispaper. Toavoidtheissueofdisclosureofanswer-relatedinformationtocandidates,allcopyrightacknowledgementsarereproducedintheOCRCopyright
AcknowledgementsBooklet. Thisisproducedforeachseriesofexaminations,isgiventoallschoolsthatreceiveassessmentmaterialandisfreelyavailabletodownloadfromourpublic
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oftheUniversityofCambridge.
©OCR2010 4753/01Jan10
PMT
ADVANCED GCE
4753/01
MATHEMATICS (MEI)
Methods for Advanced Mathematics (C3)
CandidatesanswerontheAnswerBooklet Friday 11 June 2010
OCRSuppliedMaterials: Morning
• 8pageAnswerBooklet
• MEIExaminationFormulaeandTables(MF2) Duration: 1hour30minutes
OtherMaterialsRequired:
• Scientificorgraphicalcalculator
*475301*
* 4 7 5 3 0 1 *
INSTRUCTIONSTOCANDIDATES
• Writeyournameclearlyincapitalletters,yourCentreNumberandCandidateNumberinthespacesprovided
ontheAnswerBooklet.
• Useblackink. Pencilmaybeusedforgraphsanddiagramsonly.
• Readeachquestioncarefullyandmakesurethatyouknowwhatyouhavetodobeforestartingyouranswer.
• Answerallthequestions.
• Donotwriteinthebarcodes.
• Youarepermittedtouseagraphicalcalculatorinthispaper.
• Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATIONFORCANDIDATES
• Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestion.
• You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicatethatacorrectmethodisbeingused.
• Thetotalnumberofmarksforthispaperis72.
• Thisdocumentconsistsof4pages. Anyblankpagesareindicated.
©OCR2010[M/102/2652] OCRisanexemptCharity
3R–0A12 Turnover
......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... ......... .........
xcm
PMT
3
SectionB(36marks)
8 Fig.8showsthecurvey =3lnx+x−x2.
The curve crosses the x-axis at P and Q, and has a turning point at R. The x-coordinate of Q is
approximately2.05.
y
R
x
O P Q
Fig. 8
(i) VerifythatthecoordinatesofPare(1, 0). [1]
(ii) FindthecoordinatesofR, givingthey-coordinatecorrectto3significantfigures.
d2y
Find ,andusethistoverifythatRisamaximumpoint. [9]
dx2
(iii) Findã lnxdx.
Hence calculate the area of the region enclosed by the curve and the x-axis between P and Q,
givingyouranswerto2significantfigures. [7]
[Question9isprintedoverleaf.]
Turnover
©OCR2010 4753/01Jun10
PMT
4
e2x
9 Fig.9showsthecurvey =f(x),wheref(x)= . The curvecrossesthe y-axisatP.
1+e2x
y
P
x
O
Fig. 9
(i) FindthecoordinatesofP. [1]
dy
(ii) Find ,simplifyingyouranswer.
dx
Hence calculate thegradientofthecurveatP. [4]
(iii) Show that the area ofthe regionenclosedby y = f(x), the x-axis, the y-axisandtheline x =1 is
1+e2
1ln(cid:18) (cid:19). [5]
2 2
ex−e −x
Thefunctiong(x)isdefinedbyg(x)= 1(cid:18) (cid:19).
2 ex+e−x
(iv) Provealgebraicallythatg(x)isanoddfunction.
Interpretthisresultgraphically. [3]
(v) (A) Showthatg(x)+1 =f(x).
2
(B) Describethetransformationwhichmapsthecurvey =g(x)ontothecurvey =f(x).
(C) Whatcanyouconclude aboutthesymmetryofthecurvey =f(x)? [6]
CopyrightInformation
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whoseworkisusedinthispaper. Toavoidtheissueofdisclosureofanswer-relatedinformationtocandidates,allcopyrightacknowledgementsarereproducedintheOCRCopyright
AcknowledgementsBooklet. Thisisproducedforeachseriesofexaminations,isgiventoallschoolsthatreceiveassessmentmaterialandisfreelyavailabletodownloadfromourpublic
website(www.ocr.org.uk)aftertheliveexaminationseries.
IfOCRhasunwittinglyfailedtocorrectlyacknowledgeorclearanythird-partycontentinthisassessmentmaterial,OCRwillbehappytocorrectitsmistakeattheearliestpossibleopportunity.
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OCRispartoftheCambridgeAssessmentGroup;CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),whichisitselfadepartment
oftheUniversityofCambridge.
©OCR2010 4753/01Jun10
PMT
ADVANCED GCE
4753/01
MATHEMATICS (MEI)
Methods for Advanced Mathematics (C3)
QUESTIONPAPER
Candidatesanswerontheprintedanswerbook.
Wednesday 19 January 2011
OCRsuppliedmaterials:
Afternoon
• Printedanswerbook4753/01
• MEIExaminationFormulaeandTables(MF2)
Duration: 1hour30minutes
Othermaterialsrequired:
• Scientificorgraphicalcalculator
INSTRUCTIONSTOCANDIDATES
Theseinstructionsarethesameontheprintedanswerbookandthequestionpaper.
• Thequestionpaperwillbefoundinthecentreoftheprintedanswerbook.
• Write your name, centre number and candidate number in the spaces provided on the printed
answerbook. Pleasewriteclearlyandincapitalletters.
• Write your answer to each question in the space provided in the printed answer book.
Additional paper may be used if necessary but you must clearly show your candidate number,
centrenumberandquestionnumber(s).
• Useblackink. Pencilmaybeusedforgraphsanddiagramsonly.
• Read each question carefully. Make sure you know what you have to do before starting your
answer.
• Answerallthequestions.
• Donotwriteinthebarcodes.
• Youarepermittedtouseascientificorgraphicalcalculatorinthispaper.
• Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATIONFORCANDIDATES
Thisinformationisthesameontheprintedanswerbookandthequestionpaper.
• The number of marks is given in brackets [ ] at the end of each question or part question on the
questionpaper.
• You are advised that an answer may receive no marks unless you show sufficient detail of the
workingtoindicatethatacorrectmethodisbeingused.
• Thetotalnumberofmarksforthispaperis72.
• Theprintedanswerbookconsistsof12pages. Thequestionpaperconsistsof8pages. Anyblank
pagesareindicated.
INSTRUCTIONTOEXAMSOFFICER/INVIGILATOR
• Donotsendthisquestionpaperformarking;itshouldberetainedinthecentreordestroyed.
©OCR2011[M/102/2652] OCRisanexemptCharity
3R–0I24 Turnover
PMT
2
SectionA(36marks)
p
dy
1 Giventhaty = 3 1+x2,find . [4]
dx
2 Solvetheinequality|2x+1|≥4. [4]
3 The area of a circular stain is growing at a rate of 1mm2 per second. Find the rate of increase of its
radiusataninstantwhenitsradiusis2mm. [5]
4 UsethetriangleinFig.4toprovethatsin2θ+cos2θ=1. Forwhatvaluesofθisthisproofvalid?
[3]
C
q
A B
Fig. 4
5 (i) Onasinglesetofaxes,sketchthecurvesy =ex−1andy =2e −x. [3]
(ii) Findtheexactcoordinatesofthepointofintersectionofthesecurves. [5]
6 A curve isdefinedbytheequation(x+y)2 =4x. Thepoint(1, 1)liesonthiscurve.
dy 2
Bydifferentiatingimplicitly,showthat = −1.
dx x+y
Henceverifythatthe curvehasastationarypointat(1, 1). [4]
©OCR2011 4753/01 Jan11
PMT
3
7 Fig.7showsthecurvey=f(x),wheref(x)=1+2arctanx,x∈>. Thescalesonthex-andy-axesare
the same.
y
1
x
O
Fig. 7
(i) Findtherange off, givingyour answerintermsofπ. [3]
(ii) Findf −1(x),andaddasketchofthecurvey =f −1(x)tothecopyofFig.7. [5]
Turnover
©OCR2011 4753/01 Jan11
PMT
4
SectionB (36Marks)
8 (i) Use thesubstitutionu =1+x toshowthat
1 b
x3 1
ä dx =ä (cid:16)u2−3u+3− (cid:17)du,
1+x u
0 a
whereaandb aretobefound.
1
x3
ä
Hence evaluate dx,givingyouranswerinexactform. [7]
1+x
0
Fig.8showsthecurvey =x2ln(1+x).
y
x
O
Fig. 8
dy
(ii) Find .
dx
Verifythattheoriginisastationarypointofthecurve. [5]
(iii) Using integration by parts, and the result of part (i), find the exact area enclosed by the curve
y = x2ln(1+x),thex-axisandthelinex =1. [6]
©OCR2011 4753/01 Jan11
PMT
5
1
9 Fig.9showsthecurvey=f(x),wheref(x)= ,−1π<x< 1π,togetherwithitsasymptotesx= 1π
cos2x 2 2 2
andx =−1π.
2
y
x
– 1p O 1p
2 2
Fig. 9
sinx 1
(i) Use thequotientruletoshow thatthederivative of is . [3]
cosx cos2x
(ii) Findtheareaboundedbythecurvey =f(x),thex-axis,they-axisandtheline x = 1π. [3]
4
Thefunctiong(x)isdefinedbyg(x)= 1f(cid:0)x+ 1π(cid:1).
2 4
(iii) Verifythatthecurvesy =f(x)andy =g(x)crossat(0, 1). [3]
(iv) State a sequence of two transformations such that the curve y = f(x) is mapped to the curve
y = g(x).
OnthecopyofFig.9,sketchthecurvey=g(x),indicatingclearlythecoordinatesoftheminimum
pointandtheequationsoftheasymptotestothecurve. [8]
(v) Use your result from part (ii) to write down the area bounded by the curve y = g(x), the x-axis,
they-axisandthelinex = −1π. [1]
4
©OCR2011 4753/01 Jan11
PMT
8
CopyrightInformation
OCRiscommittedtoseekingpermissiontoreproduceallthird-partycontentthatitusesinitsassessmentmaterials. OCRhasattemptedtoidentifyandcontactallcopyrightholders
whoseworkisusedinthispaper. Toavoidtheissueofdisclosureofanswer-relatedinformationtocandidates,allcopyrightacknowledgementsarereproducedintheOCRCopyright
AcknowledgementsBooklet.Thisisproducedforeachseriesofexaminationsandisfreelyavailabletodownloadfromourpublicwebsite(www.ocr.org.uk)aftertheliveexaminationseries.
IfOCRhasunwittinglyfailedtocorrectlyacknowledgeorclearanythird-partycontentinthisassessmentmaterial,OCRwillbehappytocorrectitsmistakeattheearliestpossibleopportunity.
ForqueriesorfurtherinformationpleasecontacttheCopyrightTeam,FirstFloor,9HillsRoad,CambridgeCB21GE.
OCRispartoftheCambridgeAssessmentGroup;CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),whichisitselfadepartment
oftheUniversityofCambridge.
©OCR2011 4753/01 Jan11
PMT
ADVANCED GCE
4753/01
MATHEMATICS (MEI)
Methods for Advanced Mathematics (C3)
QUESTIONPAPER
Candidatesanswerontheprintedanswerbook.
Monday 20 June 2011
OCRsuppliedmaterials:
Morning
• Printedanswerbook4753/01
• MEIExaminationFormulaeandTables(MF2)
Duration: 1hour30minutes
Othermaterialsrequired:
• Scientificorgraphicalcalculator
INSTRUCTIONSTOCANDIDATES
Theseinstructionsarethesameontheprintedanswerbookandthequestionpaper.
• Thequestionpaperwillbefoundinthecentreoftheprintedanswerbook.
• Write your name, centre number and candidate number in the spaces provided on the printed
answerbook. Pleasewriteclearlyandincapitalletters.
• Write your answer to each question in the space provided in the printed answer book.
Additional paper may be used if necessary but you must clearly show your candidate number,
centrenumberandquestionnumber(s).
• Useblackink. Pencilmaybeusedforgraphsanddiagramsonly.
• Read each question carefully. Make sure you know what you have to do before starting your
answer.
• Answerallthequestions.
• Donotwriteinthebarcodes.
• Youarepermittedtouseascientificorgraphicalcalculatorinthispaper.
• Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATIONFORCANDIDATES
Thisinformationisthesameontheprintedanswerbookandthequestionpaper.
• The number of marks is given in brackets [ ] at the end of each question or part question on the
questionpaper.
• You are advised that an answer may receive no marks unless you show sufficient detail of the
workingtoindicatethatacorrectmethodisbeingused.
• Thetotalnumberofmarksforthispaperis72.
• Theprintedanswerbookconsistsof16pages. Thequestionpaperconsistsof4pages. Anyblank
pagesareindicated.
INSTRUCTIONTOEXAMSOFFICER/INVIGILATOR
• Donotsendthisquestionpaperformarking;itshouldberetainedinthecentreordestroyed.
©OCR2011[M/102/2652] OCRisanexemptCharity
2R–1B23 Turnover
PMT
2
SectionA(36marks)
1 Solvetheequation|2x−1| =|x|. [4]
2 Given that f(x) = 2lnx and g(x) = ex, find the composite function gf(x), expressing your answer as
simplyaspossible. [3]
lnx
3 (i) Differentiate ,simplifyingyouranswer. [4]
x2
lnx 1
(ii) Usingintegrationbyparts, show that ä dx =− (1+lnx)+c. [4]
x2 x
4 Theheighthmetresof atreeaftert yearsismodelledbytheequation
h =a−be −kt,
wherea,b andk arepositive constants.
(i) Given that the long-term height of the tree is 10.5 metres, and the initial height is 0.5 metres,
findthevaluesofaandb. [3]
(ii) Givenalsothatthe treegrowstoaheightof 6metresin8years,findthevalueofk,givingyour
answercorrectto2decimalplaces. [3]
p dy 2x(5x+1)
5 Giventhaty =x2 1+4x,show that = p . [5]
dx 1+4x
p
6 A curve isdefinedbytheequationsin2x+cosy = 3.
(i) VerifythatthepointP(cid:0)1π, 1π(cid:1)liesonthecurve. [1]
6 6
dy
(ii) Find intermsofx andy.
dx
Hence findthegradientofthe curveatthepointP. [5]
7 (i) Multiplyout(3n+1)(3n−1). [1]
(ii) Hence provethatifn isapositiveintegerthen32n−1isdivisibleby8. [3]
©OCR2011 4753/01 Jun11
PMT
3
SectionB(36marks)
8
y
x
Fig. 8
1
Fig.8showsthecurvey =f(x),wheref(x)= .
ex+e−x+2
(i) Showalgebraicallythatf(x)isanevenfunction, andstatehowthispropertyrelatestothe curve
y = f(x). [3]
(ii) Findf
′(x).
[3]
ex
(iii) Showthatf(x)= . [2]
(ex+1)2
(iv) Hence, using the substitutionu = ex+1, or otherwise, find the exact area enclosed by the curve
y = f(x), thex-axis,andthe linesx =0andx =1. [5]
(v) Show that there is only one point of intersection of the curves y = f(x) and y = 1ex, and find its
4
coordinates. [5]
[Question9isprintedoverleaf.]
Turnover
©OCR2011 4753/01 Jun11
PMT
4
9 Fig.9 shows the curve y = f(x). The endpoints of the curve are P(−π, 1) and Q(π, 3), and
f(x)=a+sinbx,whereaandbareconstants.
y
Q (p, 3)
3 y= f(x)
2
P (–p, 1)
1
x
–p p
Fig. 9
(i) UsingFig.9,showthata =2andb= 1. [3]
2
(ii) Findthegradientofthecurve y =f(x)atthepoint(0, 2).
Showthatthereisnopointonthecurveatwhichthegradientisgreaterthanthis. [5]
(iii) Findf
−1(x),andstateitsdomainandrange.
Writedownthegradientofy =f −1(x)atthepoint(2, 0). [6]
(iv) Findtheareaenclosedbythecurvey =f(x),thex-axis,the y-axisandthelinex =π. [4]
CopyrightInformation
OCRiscommittedtoseekingpermissiontoreproduceallthird-partycontentthatitusesinitsassessmentmaterials. OCRhasattemptedtoidentifyandcontactallcopyrightholders
whoseworkisusedinthispaper. Toavoidtheissueofdisclosureofanswer-relatedinformationtocandidates,allcopyrightacknowledgementsarereproducedintheOCRCopyright
AcknowledgementsBooklet.Thisisproducedforeachseriesofexaminationsandisfreelyavailabletodownloadfromourpublicwebsite(www.ocr.org.uk)aftertheliveexaminationseries.
IfOCRhasunwittinglyfailedtocorrectlyacknowledgeorclearanythird-partycontentinthisassessmentmaterial,OCRwillbehappytocorrectitsmistakeattheearliestpossibleopportunity.
ForqueriesorfurtherinformationpleasecontacttheCopyrightTeam,FirstFloor,9HillsRoad,CambridgeCB21GE.
OCRispartoftheCambridgeAssessmentGroup;CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),whichisitselfadepartment
oftheUniversityofCambridge.
©OCR2011 4753/01 Jun11
Friday 20 January 2012 – Afternoon
A2 GCE MATHEMATICS (MEI)
4753/01 Methods for Advanced Mathematics (C3)
QUESTION PAPER
Candidates answer on the Printed Answer Book.
OCR supplied materials:
(cid:129) Printed Answer Book 4753/01
(cid:129) MEI Examination Formulae and Tables (MF2)
Other materials required:
(cid:129) Scientific or graphical calculator
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the Printed Answer Book and the Question Paper.
(cid:129) The Question Paper will be found in the centre of the Printed Answer Book.
(cid:129) Write your name, centre number and candidate number in the spaces provided on the
Printed Answer Book. Please write clearly and in capital letters.
(cid:129) Write your answer to each question in the space provided in the Printed Answer
Book. Additional paper may be used if necessary but you must clearly show your
candidate number, centre number and question number(s).
(cid:129) Use black ink. HB pencil may be used for graphs and diagrams only.
(cid:129) Read each question carefully. Make sure you know what you have to do before starting
your answer.
(cid:129) Answer all the questions.
(cid:129) Do not write in the bar codes.
(cid:129) You are permitted to use a scientific or graphical calculator in this paper.
(cid:129) Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
This information is the same on the Printed Answer Book and the Question Paper.
(cid:129) The number of marks is given in brackets [ ] at the end of each question or part question
on the Question Paper.
(cid:129) You are advised that an answer may receive no marks unless you show sufficient detail
of the working to indicate that a correct method is being used.
(cid:129) The total number of marks for this paper is 72.
(cid:129) The Printed Answer Book consists of 16 pages. The Question Paper consists of 8 pages.
Any blank pages are indicated.
INSTRUCTION TO EXAMS OFFICER / INVIGILATOR
(cid:129) Do not send this Question Paper for marking; it should be retained in the centre or
recycled. Please contact OCR Copyright should you wish to re-use this document.
© OCR 2012 [M/102/2652] OCR is an exempt Charity
DC (NH/SW) 55316/6 R Turn over
*4733100112*
PMT
Duration: 1 hour 30 minutes
y = | 3
f(x)
y = (cid:3)x | (cid:3) | 2 | 2
1