Question 4:
4
12 Use coordinate geometry to answer this question. Answers obtained from accurate drawing will
receive no marks.
Aand B are points with coordinates ((cid:1)1, 4) and (7, 8) respectively.
(i) Find the coordinates of the midpoint, M, of AB.
Show also that the equation of the perpendicular bisector of AB is y(cid:2)2x (cid:4) 12. [6]
(ii) Find the area of the triangle bounded by the perpendicular bisector, the y-axis and the line AM,
as sketched in Fig. 12. [6]
y
B
(7, 8) Not to
scale
A M
(–1, 4)
0 x
Fig. 12
13
y
x
Fig. 13
Fig. 13 shows a sketch of the curve y (cid:4) f(x), where f(x) (cid:4) x3 (cid:1) 5x(cid:2)2.
(i) Use the fact that x (cid:4) 2 is a root of f(x) (cid:4) 0 to find the exact values of the other two roots of
f(x) (cid:4) 0, expressing your answers as simply as possible. [6]
(ii) Show that f(x (cid:1) 3) (cid:4) x3 (cid:1) 9x2(cid:2)22x (cid:1) 10. [4]
(iii) Write down the roots of f.(x (cid:1) 3) (cid:4) 0. [2]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the
publisher will be pleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate
(UCLES), which is itself a department of the University of Cambridge.
© OCR 2007 4751/01 Jan 07
4
12 (i) Write 4x2 (cid:2) 24x(cid:3)27 in the form a (x (cid:2) b)2(cid:3)c. [4]
(ii) State the coordinates of the minimum point on the curve y (cid:1) 4x2 (cid:2) 24x(cid:3)27. [2]
(iii) Solve the equation 4x2 (cid:2) 24x(cid:3)27 (cid:1) 0. [3]
(iv) Sketch the graph of the curve y (cid:1) 4x2 (cid:2) 24x(cid:3)27. [3]
13 Acubic polynomial is given by f(x) (cid:1) 2x3 (cid:2) x2 (cid:2) 11x (cid:2) 12.
(i) Show that (x (cid:2) 3)(2x2(cid:3)5x(cid:3)4) (cid:1) 2x3 (cid:2) x2 (cid:2)11x (cid:2) 12.
Hence show that f(x) (cid:1) 0has exactly one real root. [4]
(ii) Show that x (cid:1) 2is a root of the equation f(x) (cid:1)(cid:2)22and find the other roots of this equation.
[5]
(iii) Using the results from the previous parts, sketch the graph ofy (cid:1) f(x). [3]
Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the
publisher will be pleased to make amends at the earliest possible opportunity.
OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate
(UCLES), which is itself a department of the University of Cambridge.
© OCR 2007 4751/01 June 07
PMT
2
SectionA(36marks)
1 MakevthesubjectoftheformulaE = 1mv2. [3]
2
3x2−7x+4
2 Factoriseand hencesimplify . [3]
x2−1
3 (i)
Writedownthevalueof(cid:1)1(cid:2)0
. [1]
4
−3
(ii) Find thevalueof16 2. [3]
4 Find,algebraically,thecoordinatesofthepointofintersectionofthelinesy=2x−5and6x+2y=7.
[4]
5 (i) Find thegradientoftheline4x+5y = 24. [2]
(ii) Alineparallelto4x+5y = 24passesthroughthepoint(0, 12). Findthecoordinatesofitspoint
ofintersectionwiththex-axis. [3]
6 When x3+kx+7 isdividedby (x−2),theremainderis3. Findthevalueofk. [3]
7 (i) Find thevalueof8C . [2]
3
(ii) Find thecoefficientofx3 inthebinomialexpansionof(cid:1)1− 1x(cid:2)8 . [2]
2
√ √ √
8 (i) Write 48+ 3 intheforma b,whereaandbareintegersand bisassmallaspossible. [2]
1 1
(ii) Simplify √ + √ . [3]
5+ 2 5− 2
9 (i) Provethat12 isafactorof3n2+6nforallevenpositiveintegersn. [3]
(ii) Determinewhether12isafactorof3n2+6nforallpositiveintegers n. [2]
©OCR2008 4751/01Jan08
PMT
3
SectionB(36marks)
10 (i)
y
x
O
Fig.10
1
Fig.10showsasketchofthegraphofy = .
x
1
Sketchthegraphofy= ,showingclearlythecoordinatesofanypointswhereitcrossesthe
x−2
axes. [3]
1
(ii) Find thevalueofxforwhich = 5. [2]
x−2
1
(iii) Find the x-coordinates of the points of intersection of the graphs of y = x and y = . Give
√ x−2
youranswersintheforma± b.
Show theposition ofthesepointsonyourgraph inpart(i). [6]
11 (i) Writex2−5x+8intheform(x−a)2+bandhenceshowthatx2−5x+8>0forallvaluesofx.
[4]
(ii) Sketchthegraphofy = x2−5x+8,showingthecoordinatesoftheturningpoint. [3]
(iii) Find thesetofvaluesofxforwhichx2−5x+8 > 14. [3]
(iv) Iff(x)=x2−5x+8,doesthegraphofy=f(x)−10crossthex-axis? Showhowyoudecide. [2]
[Question12isprintedoverleaf.]
©OCR2008 4751/01Jan08
PMT
4
12 Acirclehasequationx2+y2−8x−4y = 9.
(i) Show thatthecentreofthiscircleisC(4, 2)andfind theradiusofthecircle. [3]
(ii) Show thattheoriginliesinsidethecircle. [2]
(iii) Show that AB is a diameter of the circle, where A has coordinates (2, 7) and B has coordinates
(6, −3). [4]
(iv) FindtheequationofthetangenttothecircleatA.Giveyouranswerintheformy=mx+c. [4]
Permissiontoreproduceitemswherethird-partyownedmaterialprotectedbycopyrightisincludedhasbeensoughtandclearedwherepossible.Everyreasonable
efforthasbeenmadebythepublisher(OCR)totracecopyrightholders,butifanyitemsrequiringclearancehaveunwittinglybeenincluded,thepublisherwillbe
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OCRispartoftheCambridgeAssessmentGroup.CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),
whichisitselfadepartmentoftheUniversityofCambridge.
©OCR2008 4751/01Jan08
PMT
2
SectionA(36marks)
1 Solvetheinequality 3x−1 > 5−x. [2]
2 (i) Find thepointsofintersection oftheline2x+3y = 12with theaxes. [2]
(ii) Find alsothegradientofthisline. [2]
3 (i) Solvetheequation2x2+3x = 0. [2]
(ii) Find thesetofvaluesofk forwhichtheequation2x2+3x−k = 0 hasnorealroots. [3]
4 Given that n is a positive integer, write down whether the following statements are always true (T),
alwaysfalse(F)orcouldbeeithertrueorfalse(E).
(i) 2n+1isan odd integer
(ii) 3n+1is an even integer
(iii) nisodd ⇒n2 isodd
(iv) n2 isodd ⇒ n3 iseven [3]
x+3
5 Makex thesubjectoftheequationy = . [4]
x−2
−1
6 (i) Find thevalueof(cid:1)1(cid:2) 2. [2]
25
(2x2y3(cid:2))5
(ii) Simplify . [3]
4y2(cid:2)
√
1 a+b 3
7 (i) Express √ intheform ,wherea,bandcareintegers. [2]
5+ 3 c
√
(ii) Expandandsimplify (cid:1)3−2 7(cid:2)2 . [3]
8 Find thecoefficientofx3 inthebinomialexpansionof(5−2x)5. [4]
9 Solvetheequationy2−7y+12 = 0.
Hencesolvetheequationx4−7x2+12 = 0. [4]
©OCR2008 4751/01Jun08
PMT
3
SectionB(36marks)
10 (i) Express x2−6x+2intheform(x−a)2−b. [3]
(ii) Statethecoordinatesoftheturning pointonthegraphofy = x2−6x+2. [2]
(iii) Sketch the graph of y = x2 −6x+2. You need not state the coordinates of the points where the
graphintersectsthex-axis. [2]
(iv) Solve the simultaneous equations y = x2 − 6x + 2 and y = 2x − 14. Hence show that the line
y = 2x−14isatangenttothecurvey = x2−6x+2. [5]
11 Youaregiventhatf(x) = 2x3+7x2−7x−12.
(i) Verifythatx = −4 isarootoff(x) = 0. [2]
(ii) Henceexpressf(x)in fullyfactorisedform. [4]
(iii) Sketchthegraphofy = f(x). [3]
(iv) Show thatf(x−4) = 2x3−17x2+33x. [3]
12 (i) Find theequationofthelinepassingthroughA(−1, 1)andB(3, 9). [3]
(ii) Show thattheequationoftheperpendicularbisectorofABis2y+x = 11. [4]
(iii) A circle has centre (5, 3), so that its equation is (x−5)2 +(y−3)2 = k. Given that the circle
passesthroughA,showthatk = 40. Showthatthecirclealso passesthroughB. [2]
(iv) Find the x-coordinates of the points where this circle crosses the x-axis. Give your answers in
surd form. [3]
©OCR2008 4751/01Jun08
PMT
ADVANCED SUBSIDIARY GCE
4751
MATHEMATICS (MEI)
Introduction to Advanced Mathematics (C1)
CandidatesanswerontheAnswerBooklet Friday 9 January 2009
OCRSuppliedMaterials: Morning
• 8pageAnswerBooklet
• InsertforQuestion13(inserted) Duration: 1hour30minutes
• MEIExaminationFormulaeandTables(MF2)
OtherMaterialsRequired:
None
*4751*
* 4 7 5 1 *
INSTRUCTIONSTOCANDIDATES
• Writeyournameclearlyincapitalletters,yourCentreNumberandCandidateNumberinthespacesprovided
ontheAnswerBooklet.
• Useblackink. Pencilmaybeusedforgraphsanddiagramsonly.
• Readeachquestioncarefullyandmakesurethatyouknowwhatyouhavetodobeforestartingyouranswer.
• Answerallthequestions.
• Donotwriteinthebarcodes.
• ThereisaninsertforuseinQuestion13.
• Youarenotpermittedtouseacalculatorinthispaper.
• 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.
No calculator can
be used for this
paper
©OCR2009[H/102/2647] OCRisanexemptCharity
2R–8G04 Turnover
PMT
2
SectionA(36marks)
1 Statethevalueofeach ofthefollowing.
(i) 2
−3
[1]
(ii) 90 [1]
2 Find the equation of the line passing through (−1, −9) and (3, 11). Give your answer in the form
y = mx+c. [3]
3 Solvetheinequality 7−x < 5x−2. [3]
4 Youaregiventhatf(x) = x4+ax−6 andthatx−2isafactoroff(x).
Find thevalueofa. [3]
5 (i) Find thecoefficientofx3 intheexpansionof(x2−3)(x3+7x+1). [2]
(ii) Find thecoefficientofx2 inthebinomialexpansionof(1+2x)7. [3]
3x+1
6 Solvetheequation = 4. [3]
2x
√
7 (i) Express125 5in theform5k. [2]
(ii)
Simplify(cid:0)4a3b5(cid:1)2
. [2]
8 Find therangeofvaluesofk forwhichtheequation 2x2+kx+18 = 0doesnothaverealroots. [4]
9 Rearrangey+5 = x(y+2)tomakey thesubjectoftheformula. [4]
√ √ √
10 (i) Express 75+ 48intheforma 3. [2]
√
14
(ii) Express √ intheformb+c d. [3]
3− 2
©OCR2009 4751Jan09
PMT
3
SectionB(36marks)
11
y
B (11, 4)
x
A (–1, 0)
Fig. 11
Fig.11showsthepointsAandB,which havecoordinates(−1, 0)and (11, 4)respectively.
(i) Show thattheequationofthecirclewithABasdiametermay bewrittenas
(x−5)2+(y−2)2 = 40. [4]
(ii) Findthecoordin√atesofthepointsofintersectionofthiscirclewiththey-axis. Giveyouranswer
intheforma± b. [4]
(iii) Find the equation of the tangent to the circle at B. Hence find the coordinates of the points of
intersectionofthistangentwith theaxes. [6]
12 (i) Findalgebraicallythecoordinatesofthepointsofintersectionofthecurvey=3x2+6x+10and
theliney = 2−4x. [5]
(ii) Write3x2+6x+10 intheforma(x+b)2+c. [4]
(iii) Henceorotherwise,showthatthegraph ofy = 3x2+6x+10isalwaysabovethex-axis. [2]
[Question13isprintedoverleaf.]
Turnover
©OCR2009 4751Jan09
PMT
4
13 Answerpart(i)ofthisquestionontheinsertprovided.
1
Theinsertshowsthegraphofy = .
x
(i) Ontheinsert,on thesameaxes,plotthegraphofy = x2−5x+5 for0 ≤ x ≤ 5. [4]
1
(ii) Show algebraically that the x-coordinates of the points of intersection of the curves y = and
x
y = x2−5x+5 satisfytheequationx3−5x2+5x−1 = 0. [2]
(iii) Giventhatx =1atoneofthepointsofintersectionofthecurves,factorisex3−5x2+5x−1into
alinearandaquadraticfactor.
Show thatonly oneofthethreerootsofx3−5x2+5x−1 = 0 isrational. [5]
Permissiontoreproduceitemswherethird-partyownedmaterialprotectedbycopyrightisincludedhasbeensoughtandclearedwherepossible.Everyreasonable
efforthasbeenmadebythepublisher(OCR)totracecopyrightholders,butifanyitemsrequiringclearancehaveunwittinglybeenincluded,thepublisherwillbe
pleasedtomakeamendsattheearliestpossibleopportunity.
OCRispartoftheCambridgeAssessmentGroup.CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),
whichisitselfadepartmentoftheUniversityofCambridge.
©OCR2009 4751Jan09
5
4
3
2
1
– | 5 | – | 4 | – | 3 | – | 2 | – | 1 | 0 | 1 | 2 | 3 | 4 | 5
– | 1
– | 2
– | 3
– | 4
– | 5
PMT
ADVANCED SUBSIDIARY GCE
4751
MATHEMATICS (MEI)
Introduction to Advanced Mathematics (C1)
CandidatesanswerontheAnswerBooklet Wednesday 20 May 2009
OCRSuppliedMaterials: Afternoon
• 8pageAnswerBooklet
• MEIExaminationFormulaeandTables(MF2) Duration: 1hour30minutes
OtherMaterialsRequired:
None
*4751*
* 4 7 5 1 *
INSTRUCTIONSTOCANDIDATES
• Writeyournameclearlyincapitalletters,yourCentreNumberandCandidateNumberinthespacesprovided
ontheAnswerBooklet.
• Useblackink. Pencilmaybeusedforgraphsanddiagramsonly.
• Readeachquestioncarefullyandmakesurethatyouknowwhatyouhavetodobeforestartingyouranswer.
• Answerallthequestions.
• Donotwriteinthebarcodes.
• Youarenotpermittedtouseacalculatorinthispaper.
• 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.
No calculator can
be used for this
paper
©OCR2009[H/102/2647] OCRisanexemptCharity
2R–8K22 Turnover
PMT
2
SectionA(36marks)
1 A line has gradient −4 and passes through the point (2, 6). Find the coordinates of its points of
intersectionwiththeaxes. [4]
2 Makeathesubjectoftheformulas = ut+ 1at2. [3]
2
3 Whenx3−kx+4isdividedbyx−3,theremainderis1. Usetheremaindertheoremtofindthevalue
ofk. [3]
4 Solvetheinequality x(x−6) > 0. [2]
5 (i) Calculate5C . [2]
3
(ii) Find thecoefficientofx3 intheexpansionof(1+2x)5. [2]
6 Provethat,whennisaninteger,n3−nisalwayseven. [3]
7 Find thevalueofeachofthefollowing.
(i) 52×5 −2 [2]
3
(ii) 1002 [1]
√
48
8 (i) Simplify √ . [2]
2 27
√
(ii) Expandandsimplify (cid:0)5−3 2(cid:1)2 . [3]
9 (i) Expressx2+6x+5intheform(x+a)2+b. [3]
(ii) Writedownthecoordinatesoftheminimumpointonthegraphofy = x2+6x+5. [2]
10 Find therealrootsoftheequation x4−5x2−36 = 0byconsidering itasaquadraticequationinx2.
[4]
©OCR2009 4751Jun09
PMT
3
SectionB(36marks)
11
y
A
(0, 3)
B (6, 1)
x
O
Fig. 11
Fig.11showsthelinejoining thepointsA(0, 3)andB(6, 1).
(i) Find theequationofthelineperpendiculartoABthatpassesthrough theorigin,O. [2]
(ii) Find thecoordinatesofthepointwherethisperpendicularmeetsAB. [4]
√
9 10
(iii) Show thattheperpendiculardistanceofABfromtheorigin is . [2]
10
√
(iv) Find thelengthofAB,expressingyouranswerin theforma 10. [2]
(v) Find theareaoftriangleOAB. [2]
12 (i) Youaregiventhatf(x) = (x+1)(x−2)(x−4).
(A) Show thatf(x)= x3−5x2+2x+8. [2]
(B) Sketchthegraphofy = f(x). [3]
3
(C) Thegraphofy = f(x)istranslated by (cid:16) (cid:17).
0
Stateanequationfortheresulting graph. Youneednotsimplifyyouranswer.
Find thecoordinatesofthepointatwhichtheresulting graphcrossesthey-axis. [3]
(ii) Showthat3isarootofx3−5x2+2x+8 =−4. Hencesolvethisequationcompletely,givingthe
otherrootsinsurdform. [5]
13 Acirclehasequation(x−5)2+(y−2)2 = 20.
(i) Statethecoordinatesofthecentreand theradiusofthiscircle. [2]
(ii) State,withareason,whetherornotthiscircleintersectsthey-axis. [2]
(iii) Findtheequationofthelineparalleltotheliney=2xthatpassesthroughthecentreofthecircle.
[2]
(iv) Show that the line y = 2x + 2 is a tangent to the circle. State the coordinates of the point of
contact. [5]
©OCR2009 4751Jun09
PMT
ADVANCED SUBSIDIARY GCE
4751
MATHEMATICS (MEI)
Introduction to Advanced Mathematics (C1)
QUESTIONPAPER
CandidatesansweronthePrintedAnswerBook Monday 11 January 2010
OCRSuppliedMaterials: Morning
• PrintedAnswerBook4751
• MEIExaminationFormulaeandTables(MF2) Duration: 1hour30minutes
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*4751*
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INSTRUCTIONSTOCANDIDATES
TheseinstructionsarethesameonthePrintedAnswerBookandtheQuestionPaper.
• Writeyournameclearlyincapitalletters,yourCentreNumberandCandidateNumberinthespacesprovided
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• Write your answer to each question in the space provided in the Printed Answer Book. If you need
morespaceforananswerusea4-pageanswerbook; labelyouranswerclearly. WriteyourCentreNumber
andCandidateNumberonthe4-pageanswerbookandattachitsecurelytothePrintedAnswerBook.
• Useblackink. Pencilmaybeusedforgraphsanddiagramsonly.
• Readeachquestioncarefullyandmakesurethatyouknowwhatyouhavetodobeforestartingyouranswer.
• Answerallthequestions.
• Donotwriteinthebarcodes.
• Youarenotpermittedtouseacalculatorinthispaper.
• Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATIONFORCANDIDATES
ThisinformationisthesameonthePrintedAnswerBookandtheQuestionPaper.
• The number of marks is given in brackets [ ] at the end of each question or part question on the Question
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• You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicatethatacorrectmethodisbeingused.
• Thetotalnumberofmarksforthispaperis72.
• ThePrintedAnswerBookconsistsof12pages. TheQuestionPaperconsistsof4pages. Anyblankpages
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©OCR2010[H/102/2647] OCRisanexemptCharity
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PMT
2
AnswerallquestionsonthePrintedAnswerBookprovided.
SectionA(36marks)
r
a+b
1 Rearrangetheformulac = tomakeathesubject. [3]
2
5x−3
2 Solvetheinequality <x+5. [3]
2
3 (i) Findthecoordinatesofthepointwheretheline5x+2y =20intersectsthex-axis. [1]
(ii) Findthecoordinatesofthepointofintersectionofthelines5x+2y =20andy =5−x. [3]
4 (i) Describe fullythetransformationwhichmapsthe curve y= x2 ontothecurvey =(x+4)2. [2]
(ii) Sketchthegraphofy = x2−4. [2]
−1
5 (i) Findthevalueof144 2. [2]
√
1 4 a+b 7
(ii) Simplify √ + √ . Giveyouranswer intheform . [3]
5+ 7 5− 7 c
6 Youaregiventhatf(x)=(x+1)2(2x−5).
(i) Sketchthegraphofy = f(x). [3]
(ii) Expressf(x) intheformax3+bx2+cx+d. [2]
7 Whenx3+2x2+5x+k isdividedby(x+3), theremainderis6. Findthevalueofk. [3]
5 3
8 Findthebinomialexpansionof(cid:18)x+ (cid:19) ,simplifyingtheterms. [4]
x
9 Provethattheliney =3x−10doesnotintersectthecurvey =x2−5x+7. [5]
©OCR2010 4751Jan10
PMT
3
SectionB(36marks)
10
y
C
D
B
x
O
A
Fig. 10
Fig.10 shows a trapezium ABCD. The coordinates of its vertices are A(−2, −1), B(6, 3), C(3, 5)
andD(−1, 3).
(i) VerifythatthelinesABandDCareparallel. [3]
(ii) Provethatthetrapeziumisnotisosceles. [3]
(iii) The diagonalsofthe trapeziummeetatM.Findtheexactcoordinatesof M. [4]
(iv) Showthatneitherdiagonalofthetrapeziumbisectstheother. [3]
11 A circlehasequation(x−3)2+(y+2)2 = 25.
(i) Statethecoordinatesofthecentreofthiscircleanditsradius. [2]
(ii) VerifythatthepointAwithcoordinates(6, −6)liesonthiscircle. ShowalsothatthepointBon
thecircleforwhichABisadiameterhascoordinates(0, 2). [3]
(iii) Findtheequationofthetangenttothe circleatA. [4]
(iv) AsecondcircletouchestheoriginalcircleatA.Itsradiusis10anditscentreisatC,whereBAC
is a straight line. Find the coordinates of C and hence write down the equation of this second
circle. [3]
[Question12isprintedoverleaf.]
Turnover
©OCR2010 4751Jan10
PMT
4
12 The curve with equation y = 1x(10−x) is used to model the arch of a bridge over a road, where x
5
and y are distances in metres, with the origin as shown in Fig.12.1. The x-axis represents the road
surface.
y
B
x
O A
Fig. 12.1
(i) Statethevalueofx atA,where thearchmeetstheroad. [1]
(ii) Usingsymmetry,orotherwise,statethevalueofx atthemaximumpointBofthegraph.
Hence findtheheightofthearch. [2]
(iii) Fig.12.2 shows a lorry which is 4m high and 3m wide, with its cross-section modelled as a
rectangle. Find the value of d when the lorry is in the centre of the road. Hence show that the
lorrycanpassthroughthisarch. [3]
y
B
3 m
4 m
x
O A
d
Fig. 12.2
(iv) Another lorry, also modelled as having a rectangular cross-section, has height 4.5m and just
touches the arch when it is in the centre of the road. Find the width of this lorry, giving your
answerinsurdform. [5]
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OCRispartoftheCambridgeAssessmentGroup;CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocalExaminationsSyndicate(UCLES),whichisitselfadepartment
oftheUniversityofCambridge.
©OCR2010 4751Jan10
PMT
ADVANCED SUBSIDIARY GCE
4751
MATHEMATICS (MEI)
Introduction to Advanced Mathematics (C1)
QUESTIONPAPER
CandidatesansweronthePrintedAnswerBook
Monday 24 May 2010
OCRSuppliedMaterials:
Afternoon
• PrintedAnswerBook4751
• MEIExaminationFormulaeandTables(MF2)
Duration: 1hour30minutes
OtherMaterialsRequired:
None
INSTRUCTIONSTOCANDIDATES
TheseinstructionsarethesameonthePrintedAnswerBookandtheQuestionPaper.
• Writeyournameclearlyincapitalletters,yourCentreNumberandCandidateNumberinthespacesprovided
onthePrintedAnswerBook.
• ThequestionsareontheinsertedQuestionPaper.
• WriteyouranswertoeachquestioninthespaceprovidedinthePrintedAnswerBook. Additionalpaper
maybeusedifnecessarybutyoumustclearlyshowyourCandidateNumber,CentreNumberandquestion
number(s).
• Useblackink. Pencilmaybeusedforgraphsanddiagramsonly.
• Readeachquestioncarefullyandmakesurethatyouknowwhatyouhavetodobeforestartingyouranswer.
• Answerallthequestions.
• Donotwriteinthebarcodes.
• Youarenotpermittedtouseacalculatorinthispaper.
• Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATIONFORCANDIDATES
ThisinformationisthesameonthePrintedAnswerBookandtheQuestionPaper.
• The number of marks is given in brackets [ ] at the end of each question or part question on the Question
Paper.
• You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicatethatacorrectmethodisbeingused.
• Thetotalnumberofmarksforthispaperis72.
• ThePrintedAnswerBookconsistsof12pages. TheQuestionPaperconsistsof4pages. Anyblankpages
areindicated.
INSTRUCTIONTOEXAMSOFFICER/INVIGILATOR
• DonotsendthisQuestionPaperformarking;itshouldberetainedinthecentreordestroyed.
No calculator can
be used for this
paper
©OCR2010[H/102/2647] OCRisanexemptCharity
4R–0B27 Turnover
PMT
2
SectionA(36marks)
1 Find the equation of the line which is parallel to y = 3x+1 and which passes through the point with
coordinates(4, 5). [3]
2 (i) Simplify(5a2b)3×2b4. [2]
(ii)
Evaluate(cid:0)1(cid:1)−1
. [1]
16
(iii)
Evaluate(16)3
2. [2]
√
y−5
3 Makey thesubjectof theformulaa = . [3]
c
4 Solvethefollowinginequalities.
(i) 2(1−x)>6x+5 [3]
(ii) (2x−1)(x+4)<0 [2]
√ √ √
5 (i) Express 48+ 27inthe forma 3. [2]
√ √
5 2 b+c 2
(ii) Simplify √ . Giveyour answerintheform . [3]
3− 2 d
6 Youaregiventhat
• thecoefficientofx3 intheexpansionof (5+2x2)(x3+kx+m)is29,
• whenx3+kx+misdividedby(x−3), theremainderis59.
Findthevaluesofk andm. [5]
7 Expand(cid:0)1+ 1x(cid:1)4 ,simplifyingthecoefficients. [4]
2
8 Express5x2+20x+6intheforma(x+b)2+c. [4]
9 Show thatthefollowingstatementisfalse.
x−5=0 ⇔ x2 =25 [2]
©OCR2010 4751Jun10
PMT
3
SectionB(36marks)
10 (i) Solve,byfactorising, theequation2x2−x−3=0. [3]
(ii) Sketchthegraphofy = 2x2−x−3. [3]
(iii) Showthattheequationx2−5x+10=0hasnorealroots. [2]
(iv) Find the x-coordinates of the points of intersecti√on of the graphs of y = 2x2 − x − 3 and
y = x2−5x+10. Give youranswerintheforma± b. [4]
11
y
A
B
x
O
Fig. 11
Fig.11showsthelinethroughthepointsA(−1, 3)andB(5, 1).
(i) FindtheequationofthelinethroughA andB. [3]
(ii) Show that the area of the triangle bounded by the axes and the line through A and B is
32 squareunits. [2]
3
(iii) Showthattheequationoftheperpendicular bisectorofABisy= 3x−4. [3]
(iv) A circle passing through A and B has its centre on the line x = 3. Find the centre of the circle
andhencefindtheradiusandequationofthecircle. [4]
12 Youaregiventhatf(x)=x3+6x2−x−30.
(i) Use thefactortheoremtofindarootoff(x)=0andhencefactorisef(x)completely. [6]
(ii) Sketchthegraphofy = f(x). [3]
1
(iii) The graphof y =f(x)istranslatedby(cid:16) (cid:17).
0
Showthattheequationofthetranslatedgraphmaybewrittenas
y =x3+3x2−10x−24. [3]
©OCR2010 4751Jun10
PMT
ADVANCED SUBSIDIARY GCE
4751
MATHEMATICS (MEI)
Introduction to Advanced Mathematics (C1)
QUESTIONPAPER
Candidatesanswerontheprintedanswerbook.
Monday 10 January 2011
OCRsuppliedmaterials:
Morning
• Printedanswerbook4751
• MEIExaminationFormulaeandTables(MF2)
Duration: 1hour30minutes
Othermaterialsrequired:
None
INSTRUCTIONSTOCANDIDATES
Theseinstructionsarethesameontheprintedanswerbookandthequestionpaper.
• Thequestionpaperwillbefoundinthecentreoftheprintedanswerbook.
• Writeyourname, centrenumberandcandidate number in thespacesprovided on the printed answer book.
Pleasewriteclearlyandincapitalletters.
• Writeyouranswertoeachquestioninthespaceprovidedintheprintedanswerbook. Additionalpaper
may be used if necessary but you must clearly show your candidate number, centre number and question
number(s).
• Useblackink. Pencilmaybeusedforgraphsanddiagramsonly.
• Readeachquestioncarefully. Makesureyouknowwhatyouhavetodobeforestartingyouranswer.
• Answerallthequestions.
• Donotwriteinthebarcodes.
• Youarenotpermittedtouseacalculatorinthispaper.
• Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATIONFORCANDIDATES
Thisinformationisthesameontheprintedanswerbookandthequestionpaper.
• Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestiononthequestionpaper.
• You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicatethatacorrectmethodisbeingused.
• Thetotalnumberofmarksforthispaperis72.
• Theprintedanswerbookconsistsof12pages. Thequestionpaperconsistsof4pages. Anyblankpagesare
indicated.
INSTRUCTIONTOEXAMSOFFICER/INVIGILATOR
• Donotsendthisquestionpaperformarking;itshouldberetainedinthecentreordestroyed.
No calculator can
be used for this
paper
©OCR2011[H/102/2647] OCRisanexemptCharity
2R–0I08 Turnover
PMT
2
SectionA(36marks)
1 Findtheequationofthelinewhichisparalleltoy=5x−4andwhichpassesthroughthepoint(2, 13).
Give youranswerintheformy= ax+b. [3]
2 (i) Writedownthevalueofeachofthefollowing.
(A) 4 −2 [1]
(B) 90 [1]
4
(cid:16) 64 (cid:17)3
(ii) Findthevalueof . [2]
125
(cid:0)3xy4(cid:1)3
3 Simplify . [3]
6x5y2
4 Solvetheinequality5−2x < 0. [2]
5 ThevolumeV ofaconewithbaseradiusr andslantheightlisgivenbytheformula
p
V = 1πr2 l2−r2.
3
Rearrangethisformulatomakelthe subject. [4]
6 Find the first 3 terms, in ascending powers of x, of the binomial expansion of (2−3x)5, simplifying
eachterm. [4]
81
7 (i) Express p intheform3k. [2]
3
p p
5+ 3 a+b 3
(ii) Express p intheform ,wherea,bandcareintegers. [3]
5− 3 c
8 Findthecoordinatesofthepointof intersectionofthelinesx+2y = 5andy =5x−1. [3]
©OCR2011 4751 Jan11
PMT
3
9 Fig.9showsatrapeziumABCD,withthelengthsincentimetresofthree ofitssides.
D x+ 2 C
2x
A B
3x+ 6
Fig. 9
Thistrapeziumhasarea140cm2.
(i) Showthatx2+2x−35=0. [2]
(ii) Hence findthelengthofside ABof thetrapezium. [3]
10 Selectthebeststatementfrom
P ⇒ Q
P ⇐ Q
P ⇔ Q
noneoftheabove
todescribetherelationshipbetweenPandQineachof thefollowingcases.
(i) P: WXYZisaquadrilateralwith4equalsides
Q: WXYZisasquare
(ii) P: nisanoddinteger
Q: (n+1)2 isanoddinteger
(iii) P: nisgreater than1andnisaprime number
p
Q: n isnotaninteger [3]
SectionB(36marks)
11 ThepointsA(−1, 6),B(1, 0)andC(13, 4)arejoinedbystraightlines.
(i) ProvethatthelinesABandBCareperpendicular. [3]
(ii) FindtheareaoftriangleABC. [3]
(iii) AcirclepassesthroughthepointsA,BandC.JustifythestatementthatACisadiameterofthis
circle. Findtheequationofthiscircle. [6]
(iv) FindthecoordinatesofthepointonthiscirclethatisfurthestfromB. [1]
Turnover
©OCR2011 4751 Jan11
PMT
4
12 (i) Youaregiventhatf(x)= (2x−5)(x−1)(x−4).
(A) Sketchthegraphof y =f(x). [3]
(B) Showthatf(x)=2x3−15x2+33x−20. [2]
(ii) Youaregiventhatg(x)=2x3−15x2+33x−40.
(A) Showthatg(5)=0. [1]
(B) Expressg(x) astheproductof alinearandquadraticfactor. [3]
(C) Henceshow thattheequationg(x)= 0hasonlyonerealroot. [2]
(iii) Describe fullythetransformationthatmapsy =f(x)ontoy =g(x). [2]
13
y
x
Fig. 13
Fig.13showsthecurvey =x4−2.
(i) Findtheexactcoordinatesofthepointsofintersectionofthiscurvewiththe axes. [3]
(ii) Find the exact coordinates of the points of intersection of the curve y = x4 −2 with the curve
y = x2. [5]
(iii) Showthatthecurvesy =x4−2andy =kx2 intersectforallvaluesofk. [2]
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 4751 Jan11
PMT
ADVANCED SUBSIDIARY GCE
4751
MATHEMATICS (MEI)
Introduction to Advanced Mathematics (C1)
QUESTIONPAPER
Candidatesanswerontheprintedanswerbook.
Wednesday 18 May 2011
OCRsuppliedmaterials:
Morning
• Printedanswerbook4751
• MEIExaminationFormulaeandTables(MF2)
Duration: 1hour30minutes
Othermaterialsrequired:
None
INSTRUCTIONSTOCANDIDATES
Theseinstructionsarethesameontheprintedanswerbookandthequestionpaper.
• Thequestionpaperwillbefoundinthecentreoftheprintedanswerbook.
• Writeyourname, centrenumberandcandidate number in thespacesprovided on the printed answer book.
Pleasewriteclearlyandincapitalletters.
• Writeyouranswertoeachquestioninthespaceprovidedintheprintedanswerbook. Additionalpaper
may be used if necessary but you must clearly show your candidate number, centre number and question
number(s).
• Useblackink. Pencilmaybeusedforgraphsanddiagramsonly.
• Readeachquestioncarefully. Makesureyouknowwhatyouhavetodobeforestartingyouranswer.
• Answerallthequestions.
• Donotwriteinthebarcodes.
• Youarenotpermittedtouseacalculatorinthispaper.
• Finalanswersshouldbegiventoadegreeofaccuracyappropriatetothecontext.
INFORMATIONFORCANDIDATES
Thisinformationisthesameontheprintedanswerbookandthequestionpaper.
• Thenumberofmarksisgiveninbrackets[]attheendofeachquestionorpartquestiononthequestionpaper.
• You are advised that an answer may receive no marks unless you show sufficient detail of the working to
indicatethatacorrectmethodisbeingused.
• Thetotalnumberofmarksforthispaperis72.
• Theprintedanswerbookconsistsof12pages. Thequestionpaperconsistsof4pages. Anyblankpagesare
indicated.
INSTRUCTIONTOEXAMSOFFICER/INVIGILATOR
• Donotsendthisquestionpaperformarking;itshouldberetainedinthecentreordestroyed.
No calculator can
be used for this
paper
©OCR2011[H/102/2647] OCRisanexemptCharity
3R–0J29 Turnover
PMT
2
SectionA(36marks)
1 Solvetheinequality6(x+3)> 2x+5. [3]
2 A line has gradient 3 and passes through the point (1, −5). The point (5, k) is on this line. Find the
valueofk. [2]
−1
3 (i) Evaluate (cid:0) 9 (cid:1) 2. [2]
16
(cid:0)2ac2(cid:1)3×9a2c
(ii) Simplify . [3]
36a4c12
4 ThepointP(5, 4)isonthecurvey= f(x). StatethecoordinatesoftheimageofPwhenthegraphof
y =f(x)istransformedtothegraphof
(i) y = f(x−5), [2]
(ii) y = f(x)+7. [2]
5 Findthecoefficientofx4 inthebinomialexpansionof(5+2x)6. [4]
6 Expand(2x+5)(x−1)(x+3),simplifyingyouranswer. [3]
7 Find the discriminant of 3x2 +5x+2. Hence state the number of distinct real roots of the equation
3x2+5x+2=0. [3]
1−2x
8 Makex thesubjectoftheformulay = . [4]
x+3
9 A line L is parallel to the line x+2y = 6 and passes through the point (10, 1). Find the area of the
regionboundedbythelineLandtheaxes. [5]
10 Factorise n3 + 3n2 + 2n. Hence prove that, when n is a positive integer, n3 + 3n2 + 2n is always
divisibleby6. [3]
©OCR2011 4751 Jun11
7
6
5
4
3
2
1
PMT
4
13
y
x
Fig. 13
Fig.13showsthecircle withequation(x−4)2+(y−2)2 =16.
(i) Writedowntheradiusofthecircleandthecoordinatesofitscentre. [2]
(ii) Find the x-coordinates of the points where the circle crosses the x-axis. Give your answers in
surdform. [4]
p p
(iii) Show that the point A(cid:0)4+2 2, 2+2 2(cid:1) lies on the circle and mark point A on the copy of
Fig.13.
SketchthetangenttothecircleatA andtheothertangentthatisparalleltoit.
Findtheequationsofboththesetangents. [7]
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 4751 Jun11
Friday 13 January 2012 – Morning
AS GCE MATHEMATICS (MEI)
4751 Introduction to Advanced Mathematics (C1)
QUESTION PAPER
Candidates answer on the Printed Answer Book.
OCR supplied materials:
(cid:129) Printed Answer Book 4751
(cid:129) MEI Examination Formulae and Tables (MF2)
Other materials required:
None
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 not permitted to use a 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 12 pages. The Question Paper consists of 4 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 [H/102/2647] OCR is an exempt Charity
DC (NH/SW) 44312/4 Turn over
*4733060112*
PMT
Duration: 1 hour 30 minutes
No calculator can
be used for this
paper
6
4
4
y | = | g(x | )
2 | 2
–4 | –4 | –2 | –2 | 0 | 2 | 4 | 6 | 6 | 8
–2
–
4
–6
PMT
3
8 Express 5x2 + 15x + 12 in the form a(x + b)2 + c.
Hence state the minimum value of y on the curve y = 5x2 + 15x + 12. [5]
9 Complete each of the following by putting the best connecting symbol (⇔, ⇐ or ⇒) in the box. Explain
your choice, giving full reasons.
(i) n3 + 1 is an odd integer n is an even integer [2]
(ii) (x − 3)(x − 2) > 0 x > 3 [2]
Section B (36 marks)
10 Point A has coordinates (4, 7) and point B has coordinates (2, 1).
(i) Find the equation of the line through A and B. [3]
(ii) Point C has coordinates (−1, 2). Show that angle ABC = 90° and calculate the area of triangle ABC. [5]
(iii) Find the coordinates of D, the midpoint of AC.
Explain also how you can tell, without having to work it out, that A, B and C are all the same distance
from D. [3]
11 You are given that f(x) = 2x3 − 3x2 − 23x + 12.
(i) Show that x = −3 is a root of f(x) = 0 and hence factorise f(x) fully. [6]
(ii) Sketch the curve y = f(x). [3]
(iii) Find the x-coordinates of the points where the line y = 4x + 12 intersects y = f(x). [4]
12 A circle has equation (x − 2)2 + y2 = 20.
(i) Write down the radius of the circle and the coordinates of its centre. [2]
(ii) Find the points of intersection of the circle with the y-axis and sketch the circle. [3]
(iii) Show that, where the line y = 2x + k intersects the circle,
5x2 + (4k − 4) x + k2 − 16 = 0. [3]
(iv) Hence find the values of k for which the line y = 2x + k is a tangent to the circle. [4]
© OCR 2012 4751 Jan12
PMT
4
THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible
opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
department of the University of Cambridge.
© OCR 2012 4751 Jan12
Wednesday 16 May 2012 – Morning
AS GCE MATHEMATICS (MEI)
4751 Introduction to Advanced Mathematics (C1)
QUESTION PAPER
Candidates answer on the Printed Answer Book.
OCR supplied materials:
(cid:129) Printed Answer Book 4751
(cid:129) MEI Examination Formulae and Tables (MF2)
Other materials required:
None
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 not permitted to use a 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 12 pages. The Question Paper consists of 4 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 [H/102/2647] OCR is an exempt Charity
DC (NH/SW) 44315/3 Turn over
*4715620612*
PMT
Duration: 1 hour 30 minutes
No calculator can
be used for this
paper
PMT
2
Section A (36 marks)
1 Find the equation of the line with gradient −2 which passes through the point (3, 1). Give your answer in the
form y = ax + b.
Find also the points of intersection of this line with the axes. [3]
2 Make b the subject of the following formula.
a = 2 b2c [3]
3
1 −2
3 (i) Evaluate . [2]
5
8 2
(ii) Evaluate 3. [2]
27
4 Factorise and hence simplify the following expression.
x2 − 9
x2 + 5x + 6 [3]
10(cid:2) 6(cid:3)3
5 (i) Simplify . [3]
24
1 1
(ii) Simplify + . [2]
4 − 5 4 + 5
6 (i) Evaluate 5C . [1]
3
(ii) Find the coefficient of x3 in the expansion of (3 − 2x)5. [4]
7 Find the set of values of k for which the graph of y = x2 + 2kx + 5 does not intersect the x-axis. [4]
8 The function f(x) = x4 + bx + c is such that f(2) = 0. Also, when f(x) is divided by x + 3, the remainder is 85.
Find the values of b and c. [5]
9 Simplify (n + 3)2 − n2. Hence explain why, when n is an integer, (n + 3)2 − n2 is never an even number.
Given also that (n + 3)2 − n2 is divisible by 9, what can you say about n? [4]
© OCR 2012 4751 Jun12
PMT
3
Section B (36 marks)
10
y
A
D
B
x
C
Fig. 10
Fig. 10 is a sketch of quadrilateral ABCD with vertices A (1, 5), B (−1, 1), C (3, −1) and D (11, 5).
(i) Show that AB = BC. [3]
(ii) Show that the diagonals AC and BD are perpendicular. [3]
(iii) Find the midpoint of AC. Show that BD bisects AC but AC does not bisect BD. [5]
11 A cubic curve has equation y = f(x). The curve crosses the x-axis where x = − 1, −2 and 5.
2
(i) Write down three linear factors of f(x). Hence find the equation of the curve in the form
y = 2x3 + ax2 + bx + c. [4]
(ii) Sketch the graph of y = f(x). [3]
0
(iii) The curve y = f(x) is translated by . State the coordinates of the point where the translated curve
–8
intersects the y-axis. [1]
3
(iv) The curve y = f(x) is translated by to give the curve y = g(x).
0
Find an expression in factorised form for g(x) and state the coordinates of the point where the curve
y = g(x) intersects the y-axis. [4]
[Question 12 is printed overleaf.]
Turn over
© OCR 2012 4751 Jun12
7
6
5
4
3
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Monday 14 January 2013 – Morning
AS GCE MATHEMATICS (MEI)
4751/01 Introduction to Advanced Mathematics (C1)
QUESTION PAPER
Candidates answer on the Printed Answer Book.
OCR supplied materials:
• Printed Answer Book 4751/01
• MEI Examination Formulae and Tables (MF2)
Other materials required:
None
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the Printed Answer Book and the Question Paper.
• The Question Paper will be found in the centre of the Printed Answer Book.
• Write your name, centre number and candidate number in the spaces provided on the
Printed Answer Book. Please write clearly and in capital letters.
• 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).
• Use black ink. HB pencil may be used for graphs and diagrams only.
• Read each question carefully. Make sure you know what you have to do before starting your
answer.
• Answer all the questions.
• Do not write in the bar codes.
• You are not permitted to use a calculator in this paper.
• 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.
• The number of marks is given in brackets [ ] at the end of each question or part question on
the Question Paper.
• 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.
• The total number of marks for this paper is 72.
• The Printed Answer Book consists of 12 pages. The Question Paper consists of 4 pages. Any
blank pages are indicated.
INSTRUCTION TO EXAMS OFFICER / INVIGILATOR
• 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.
This paper has been pre modified for carrier language
© OCR 2013 [H/102/2647] OCR is an exempt Charity
DC (LEG) 62008/3 Turn over
*4733950113*
PMT
Duration: 1 hour 30 minutes
No calculator can
be used for this
paper
PMT
2
Section A (36 marks)
1 Find the value of each of the following.
-2
5
(i) d n [2]
3
3
(ii) 814 [2]
_4x 5 yi 3
2 Simplify . [3]
_2xy 2i#_8x 10 y 4i
3 A circle has diameter d, circumference C, and area A. Starting with the standard formulae for a circle, show
that Cd = kA, finding the numerical value of k. [3]
4 Solve the inequality 5x 2- 28x- 12G 0. [4]
5 You are given that f(x)= x 2+ kx+ c.
Given also that f(2)= 0 and f(-3)= 35, find the values of the constants k and c. [4]
6
5
6 The binomial expansion of d2x+ n has a term which is a constant. Find this term. [4]
x
7 (i) Express 48+ 75 in the form a b, where a and b are integers. [2]
7+ 2 5 a+ b 5
(ii) Simplify , expressing your answer in the form , where a, b and c are integers. [3]
7+ 5 c
8 Rearrange the equation 5c+ 9t = a(2c+ t) to make c the subject. [4]
9 You are given that f(x)= (x+ 2) 2 (x- 3).
(i) Sketch the graph of y= f(x). [3]
(ii) State the values of x which satisfy f(x+ 3)= 0. [2]
© OCR 2013 4751/01 Jan13
PMT
3
Section B (36 marks)
10 (i) Points A and B have coordinates (–2, 1) and (3, 4) respectively. Find the equation of the perpendicular
bisector of AB and show that it may be written as 5x+ 3y= 10. [6]
(ii) Points C and D have coordinates (–5, 4) and (3, 6) respectively. The line through C and D has equation
4y= x+ 21. The point E is the intersection of CD and the perpendicular bisector of AB. Find the
coordinates of point E. [3]
(iii) Find the equation of the circle with centre E which passes through A and B. Show also that CD is a
diameter of this circle. [5]
11 (i) Express x 2- 5x+ 6 in the form _x- ai 2 - b. Hence state the coordinates of the turning point of the
curve y= x 2- 5x+ 6. [4]
(ii) Find the coordinates of the intersections of the curve y= x 2- 5x+ 6 with the axes and sketch this
curve. [4]
(iii) Solve the simultaneous equations y = x 2- 5x+ 6 and x+ y= 2. Hence show that the line x+ y= 2
is a tangent to the curve y= x 2- 5x+ 6 at one of the points where the curve intersects the axes. [4]
12 You are given that f(x)= x 4- x 3+ x 2+ 9x- 10.
(i) Show that x= 1 is a root of f(x)= 0 and hence express f(x) as a product of a linear factor and a cubic
factor. [3]
(ii) Hence or otherwise find another root of f(x)= 0. [2]
(iii) Factorise f(x), showing that it has only two linear factors. Show also that f(x)= 0 has only two real
roots. [5]
© OCR 2013 4751/01 Jan13
PMT
4
THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.
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whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
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opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
department of the University of Cambridge.
© OCR 2013 4751/01 Jan13
Monday 13 May 2013 – Afternoon
AS GCE MATHEMATICS (MEI)
4751/01 Introduction to Advanced Mathematics (C1)
QUESTION PAPER
Candidates answer on the Printed Answer Book.
OCR supplied materials:
• Printed Answer Book 4751/01
• MEI Examination Formulae and Tables (MF2)
Other materials required:
None
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the Printed Answer Book and the Question Paper.
• The Question Paper will be found in the centre of the Printed Answer Book.
• Write your name, centre number and candidate number in the spaces provided on the
Printed Answer Book. Please write clearly and in capital letters.
• 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).
• Use black ink. HB pencil may be used for graphs and diagrams only.
• Read each question carefully. Make sure you know what you have to do before starting your
answer.
• Answer all the questions.
• Do not write in the bar codes.
• You are not permitted to use a calculator in this paper.
• 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.
• The number of marks is given in brackets [ ] at the end of each question or part question on
the Question Paper.
• 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.
• The total number of marks for this paper is 72.
• The Printed Answer Book consists of 12 pages. The Question Paper consists of 4 pages. Any
blank pages are indicated.
INSTRUCTION TO EXAMS OFFICER / INVIGILATOR
• 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 2013 [H/102/2647] OCR is an exempt Charity
DC (NH/JG) 65475/2 Turn over
*4715620613*
PMT
Duration: 1 hour 30 minutes
No calculator can
be used for this
paper
PMT
2
Section A (36 marks)
1 Find the equation of the line which is perpendicular to the line y= 2x- 5 and which passes through the
point _4,1i. Give your answer in the form y= ax+ b. [3]
2 Find the coordinates of the point of intersection of the lines y= 3x- 2 and x+ 3y= 1. [4]
-2
3 (i) Evaluate _0.2i . [2]
3
(ii) Simplify _16a 12i4. [3]
4 Rearrange the following formula to make r the subject, where r 20.
V= 1 rr 2 (a+ b) [3]
3
5 You are given that f(x)= x 5+ kx- 20. When f(x) is divided by (x- 2), the remainder is 18. Find the
value of k. [3]
6 Find the coefficient of x 3 in the binomial expansion of _2- 4xi 5 . [4]
7 (i) Express 125 5 in the form 5k. [2]
38
(ii) Simplify 10+ 7 5+ , giving your answer in the form a+ b 5. [3]
1- 2 5
8 Express 3x 2- 12x+ 5 in the form a_x- bi 2 - c. Hence state the minimum value of y on the curve
y= 3x 2- 12x+ 5. [5]
9 n- 1, n and n+ 1 are any three consecutive integers.
(i) Show that the sum of these integers is always divisible by 3. [1]
(ii) Find the sum of the squares of these three consecutive integers and explain how this shows that the
sum of the squares of any three consecutive integers is never divisible by 3. [3]
© OCR 2013 4751/01 Jun13
PMT
3
Section B (36 marks)
2 2
10 The circle _x- 3i +_y- 2i = 20 has centre C.
(i) Write down the radius of the circle and the coordinates of C. [2]
(ii) Find the coordinates of the intersections of the circle with the x- and y-axes. [5]
(iii) Show that the points A_1,6i and B_7,4i lie on the circle. Find the coordinates of the midpoint of AB.
Find also the distance of the chord AB from the centre of the circle. [5]
11 You are given that f(x)= _2x- 3i_x+ 2i_x+ 4i.
(i) Sketch the graph of y= f(x). [3]
(ii) State the roots of f(x- 2)= 0. [2]
(iii) You are also given that g(x)= f(x)+ 15.
(A) Show that g(x)= 2x 3+ 9x 2- 2x- 9. [2]
(B) Show that g(1)= 0 and hence factorise g(x) completely. [5]
[Question 12 is printed overleaf.]
© OCR 2013 4751/01 Jun13 Turn over
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