Wednesday 23 January 2013 – Morning

1 Wednesday 23 January 2013 MorningA2 GCE MATHEMATICS (MEI)4753/01 Methods for Advanced Mathematics (C3)QUESTION PAPER*4733990113*INSTRUCTIONS TO CANDIDATEST hese 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 permitted to use a scientific or graphical calculator in this paper.

2 Final answers should be given to a degree of accuracy appropriate to the FOR CANDIDATESThis 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 16 pages. The Question Paper consists of 4 pages. Any blank pages are 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 is an exempt CharityTurn over OCR 2013 [M/102/2652]DC (AC/SW) 63784/5 Candidates answer on the Printed Answer supplied materials: Printed Answer Book 4753/01 MEI Examination Formulae and Tables (MF2)Other materials required: Scientific or graphical calculatorDuration: 1 hour 30 minutes24753/01 Jan13 OCR 2013 Section A (36 marks)1 (i) Given that sinyx2ex=-, find ddxy.

3 [3] (ii) Hence show that the curve sinyx2ex=- has a stationary point when actanrx221=. [3]2 A curve has equation xyx2422+=. (i) By differentiating implicitly, find ddxy in terms of x and y. [3] (ii) Hence find the exact coordinates of the stationary points of the curve. [You need not determine their nature.] [3]3 Express x1311 in the form xab1-, where a and b are to be determined. [2]4 The temperature i C of water in a container after t minutes is modelled by the equationabekti=--, where a, b and k are positive constants. The initial and long-term temperatures of the water are 15 C and 100 C respectively. After 1 minute, the temperature is 30 C. (i) Find a, b and k. [6] (ii) Find how long it takes for the temperature to reach 80 C. [2]5 The driving force F newtons and velocity 1kmvs- of a car at time t seconds are related by the equation Fv25=. (i) Find dvdF. [2] (ii) Find dtdF when v = 50 and dtdv = [3] 6 Evaluate ()dx x10321+-xy, giving your answer as an exact fraction.

4 [5]7 (i) Disprove the following statement: 32n+ is prime for all integers n0H. [2] (ii) Prove that no number of the form 3n (where n is a positive integer) has 5 as its final digit. [2]34753/01 Jan13 Turn over OCR 2013 Section B (36 marks)8 Fig. 8 shows parts of the curves f()yx= and g()yx=, where f()tanx x= and g()1f()xx41r=+ = f(x)y = g(x)41r-41r21rFig. 8 (i) Describe a sequence of two transformations which maps the curve f()yx= to the curve g()yx=. [4] It can be shown that g()sincossinxx xx2=+. (ii) Show that g()()sincosxx x22=+l. Hence verify that the gradient of g()yx= at the point (),141r is the same as that of f()yx= at the origin. [7] (iii) By writing tancossinxxx= and using the substitution cosux=, show that ()xxfd=uu1d014121ryy. Evaluate this integral exactly. [4] (iv) Hence find the exact area of the region enclosed by the curve g()yx=, the x-axis and the lines x41r= and x21r= . [2] 44753/01 Jan13 OCR 20139 Fig.

5 9 shows the line yx= and the curve f()yx=, where f()(1)xex21=-. The line and the curve intersect at the origin and at the point P(a, a).yxP(a, a)y = f(x)y = xOFig. 9 (i) Show that 12aea=+. [1] (ii) Show that the area of the region enclosed by the curve, the x-axis and the line xa= is a21. Hence find, in terms of a, the area enclosed by the curve and the line yx=. [6] (iii) Show that the inverse function of f()x is g()x, where g()n(12).xxl=+ Add a sketch of g()yx= to the copy of Fig. 9. [5] (iv) Find the derivatives of f()x and g()x. Hence verify that ()()aa1gf=ll. Give a geometrical interpretation of this result. [7]Copyright InformationOCR 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.

6 This is produced for each series of examinations and is freely available to download from our public website ( ) after the live examination 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 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 23 January 2013 MorningA2 GCE MATHEMATICS (MEI)4753/01 Methods for Advanced Mathematics (C3)PRINTED ANSWER BOOKINSTRUCTIONS TO CANDIDATEST hese 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.

7 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 permitted to use a scientific or graphical calculator in this paper. Final answers should be given to a degree of accuracy appropriate to the FOR CANDIDATESThis 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.

8 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 16 pages. The Question Paper consists of 4 pages. Any blank pages are indicated.*475301*OCR is an exempt CharityTurn over OCR 2013 [M/102/2652]DC (AC) 63785/3 Candidates answer on this Printed Answer supplied materials: Question Paper 4753/01 (inserted) MEI Examination Formulae and Tables (MF2)Other materials required: Scientific or graphical calculator*4734000113*Duration: 1 hour 30 minutes2 OCR 2013 Section A (36 marks)1 (i)1 (ii)3 Turn over OCR 20132 (i)2 (ii)4 OCR 201335 Turn over OCR 20134 (i)6 OCR 20134 (ii)7 Turn over OCR 20135 (i)5 (ii)8 OCR 201369 Turn over OCR 20137 (i)7 (ii)10 OCR 2013 Section B (36 marks)8 (i)8 (ii)(answer space continued on next page)11 Turn over OCR 20138 (ii)(continued)12 OCR 20138 (iii)8 (iv)13 Turn over OCR 20139 (i)9 (ii)14 OCR 20139 (iii)yxP(a, a)y = f(x)y = xO15 Turn over OCR 20139 (iv)(answer space continued on next page) Oxford Cambridge and RSA Examinations GCEM athematics (MEI) Advanced GCE Unit 4753.

9 Methods for Advanced Mathematics Mark Scheme for January 2013 OCR (Oxford Cambridge and RSA) is a leading UK awarding body, providing a wide range of qualifications to meet the needs of candidates of all ages and abilities. OCR qualifications include AS/A Levels, Diplomas, GCSEs, Cambridge Nationals, Cambridge Technicals, Functional Skills, Key Skills, Entry Level qualifications, NVQs and vocational qualifications in areas such as IT, business, languages, teaching/training, administration and secretarial skills. It is also responsible for developing new specifications to meet national requirements and the needs of students and teachers. OCR is a not-for-profit organisation; any surplus made is invested back into the establishment to help towards the development of qualifications and support, which keep pace with the changing needs of today s society. This mark scheme is published as an aid to teachers and students, to indicate the requirements of the examination.

10 It shows the basis on which marks were awarded by examiners. It does not indicate the details of the discussions which took place at an examiners meeting before marking commenced. All examiners are instructed that alternative correct answers and unexpected approaches in candidates scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes should be read in conjunction with the published question papers and the report on the examination. OCR will not enter into any discussion or correspondence in connection with this mark scheme. OCR 2013 4753/01 Mark Scheme January 2013 1 Annotations Annotation Meaning and BOD Benefit of doubt FT Follow through ISW Ignore subsequent working M0, M1 Method mark awarded 0, 1 A0, A1 Accuracy mark awarded 0, 1 B0, B1 Independent mark awarded 0, 1 SC Special case ^ Omission sign MR Misread Highlighting Other abbreviations in mark scheme Meaning E1 Mark for explaining U1 Mark for correct units G1 Mark for a correct feature on a graph M1 dep* Method mark dependent on a previous mark.