GCE. Mathematics (MEI) Mark Scheme for June Advanced GCE 4753 Methods for Advanced Mathematics (C3) Oxford Cambridge and RSA Examinations

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1 GCE Mathematics (MEI) Advanced GCE 4753 Methods for Advanced Mathematics (C3) Mark Scheme for June 00 Oxford Cambridge and RSA Examinations

2 OCR (Oxford Cambridge and RSA) is a leading UK awarding body, providing a wide range of qualifications to meet the needs of pupils of all ages and abilities. OCR qualifications include AS/A Levels, Diplomas, GCSEs, OCR Nationals, 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. 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 00 Any enquiries about publications should be addressed to: OCR Publications PO Box 5050 Annesley NOTTINGHAM NG5 0DL Telephone: Facsimile: publications@ocr.org.uk

3 4753 Mark Scheme June 00 Section A /6 /6 cosd x sin = sin 0 3 = /3 cao fg( x) x gf( x) x k sin, k > 0, k 3 k = ()/ or better soi from correctly-shaped graphs (i.e. without intercepts) or for u = cos u d u condone 90 in limit 3 or for sin u 3 so: sin : B0, sin : M0B0, 3sin : M0B0, /3 sin : M0 but must indicate which is which bod gf if negative x values are missing - [4] graph of x only graph of x V shape with (, 0) and (0, ) labelled V shape with (0, ) labelled (0, ) 3(i) / y(3 x ) dy / dx (3 x ). 6x = ( + ) / / chain rule ½ u / o.e., but must be 3 can isw here (ii) / y x(3 x ) dy / dx x..(3 x ) 6x * / ft [4] product rule ft their dy/dx from (i) common denominator or factoring (+ ) / www must show this step for

4 4753 Mark Scheme June 00 4 p = 00/x = 00 x dp/dx = 00x = 00/x dp/dt = dp/dx dx/dt dx/dt = 0 When x = 50, dp/dx =( 00/50 ) dp/dt = = 0.4 dep cao [6] attempt to differentiate 00x o.e. o.e. soi soi substituting x = 50 into their dp/dx dep nd o.e. e.g. decreasing at 0.4 condone poor notation if chain rule correct or x = t P = 00/x = 00/( t) dp/dt = 00( t) 0 = 000/(50 + 0t) When t = 0, dp/dt = 000/50 = y 3 = xy x 3y dy/dx = x dy/dx + y x 3y dy/dx x dy/dx = y x (3y x) dy/dx = y x dy/dx = (y x)/(3y x) * 3y dy/dx x dy/dx + y x collecting terms in dy/dx only must show x dy/dx + y on one side TP when dy/dx = 0 y x = 0 y = x (x) 3 = x.x x 8x 3 = x x = /8 *(or 0) [7] or x = /8 and dy/dx = 0 y = ¼ or (/4) 3 = (/8)(/4) (/8) or verifying e.g. /64 = /64 or x = /8 y 3 = (/8)y /64 verifying that y = ¼ is a solution (must show evidence*) dy/dx = (¼ (/8))/( ) = 0 *just stating that y = ¼ is M0 E0 6 f(x) = + sin = y x y x = + sin 3y sin 3y = (x )/ 3y = arcsin [(x )/] x y arcsin so x 3 f ( x) arcsin 3 Range of f is to 3 x 3 [6] attempt to invert must be y = or f (x) = or (x )/ must be x, not y or f(x) at least one step attempted, or reasonable attempt at flow chart inversion (or any other variable provided same used on each side) condone < s for allow unsupported correct answers; to 3 is A0 7 (A) True, (B) True, (C) False Counterexample, e.g. + ( ) = 0 B,,0

5 4753 Mark Scheme June 00 8(i) When x =, y = 3 ln + = 0 [] (ii) dy 3 x dx x At R, dy 3 0 x dx x 3 + x x = 0 (3 x)( + x) = 0 x =.5, (or ) y = 3 ln = (3 s.f.) d y 3 dx x When x =.5, d y/dx (= 0/3) < 0 max cao cao ft [9] d/dx (ln x) = /x re-arranging into a quadratic = 0 factorising or formula or completing square substituting their x ft their dy/dx on equivalent work www don t need to calculate 0/3 SC for x =.5 unsupported, SC3 if verified but condone rounding errors on (iii) Let u = ln x, du/dx = /x dv/dx =, v = x ln xdx xln x x. dx x x ln x. dx = x ln x x + c parts condone no c allow correct result to be quoted (SC3).05 (3ln ) A xxx dx.05 3 lnx x x 3 = = 0.33 ( s.f.) ft dep cao [7] correct integral and limits (soi) 3 3their' x ln xx' x x 3 substituting correct limits dep st 3

6 4753 Mark Scheme June 00 9(i) (0, ½ ) allow y = ½, but not (x =) ½ or ( ½,0) [] nor P = / x x x x (ii) dy (e )e e.e x dx ( e ) x e x ( e ) When x = 0, dy/dx = e 0 /(+e 0 ) = ½ x (iii) A = e d x 0 x e ln( e x ) 0 or let u = + e x, du/dx = e x e A = / du = ln u u ln( e ) ln e ln * e x x x x (iv) e e e e g( x) x x e e g( x) x x e e Rotational symmetry of order about O x x x x x x (v)(a) e e e e e e gx ( )..( ) x x x x e e e e x e.( ) x x e e x x x e.e e f( x) x x x x e(ee ) e (B) Translation 0 / (C) Rotational symmetry [of order ]about P ft [4] [5] [6] Quotient or product rule correct expression condone missing bracket cao mark final answer follow through their derivative correct integral and limits (soi) k ln( + e x ) k = ½ or v = e x, dv/dx = e x o.e. [½ ln u] or [½ ln (v + )] substituting correct limits www substituting x for x in g(x) completion www taking out ve must be clear must have rotational about O, order (oe) combining fractions (correctly) translation in y direction up ½ unit dep translation used o.e. condone omission of 80/order product rule: dy x x x x x e.e ( )(e ) e ( e ) dx x e from (udv vdu)/v SC x ( e ) condone no dx allow missing dx s or incompatible limits, but penalise missing brackets not g(x) g(x). Condone use of f for g. allow shift, move in correct direction for. 0 alone is SC. / 4

7 OCR (Oxford Cambridge and RSA Examinations) Hills Road Cambridge C EU OCR Customer Contact Centre 4 9 Qualifications (General) Telephone: Facsimile: general.qualifications@ocr.org.uk For staff training purposes and as part of our quality assurance programme your call may be recorded or monitored Oxford Cambridge and RSA Examinations is a Company Limited by Guarantee Registered in England Registered Office; Hills Road, Cambridge, C EU Registered Company Number: OCR is an exempt Charity OCR (Oxford Cambridge and RSA Examinations) Head office Telephone: Facsimile: OCR 00

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