GCE. Mathematics (MEI) Mark Scheme for June Advanced GCE 4754A Applications of Advanced Mathematics (C4) Paper A

Transcription

1 GCE Mathematics (MEI) Advanced GCE 4754A Applications of Advanced Mathematics (C4) Paper A Mark Scheme for June 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 Any enquiries about publications should be addressed to: OCR Publications PO Box 55 Annesley NOTTINGHAM NG5 DL Telephone: Facsimile: publications@ocr.org.uk

3 4754A Mark Scheme June Section A x x x x ( x)( x) x x( x) ( x)( x) (3x ) ( x)( x) B x (x + )(x ) correct method for addition of fractions (3x ) or do not isw for incorrect x subsequent cancelling or x x( x) ( x ) x x ( x )( x ) 3x x ( x )( x ) (3x)( x) ( x )( x ) (3x ) ( x ) B correct method for addition of fractions (3x )(x + ) accept denominator as x²- or(x-)(x+) do not isw for incorrect subsequent cancelling (i) When x.5, y.8 A.5/{+.44+( )} (3 d.p.)* (ii) Explain that the area is an over-estimate. or The curve is below the trapezia, so the area is an over- estimate. This becomes less with more strips. or Greater number of strips improves accuracy so becomes less B E B B [] 4dp (.5 x 9.8) need evidence or use a diagram to show why (iii) V y dx ( x ) dx 3 ( xx /3) 3 B allow limits later x + x 3 /3 exact

4 4754A Mark Scheme June 3 y sin cos ½ sin x cos sin + cos x + (y) x + 4y * or x² +4y² (cos θ)² +4 (sinθcosθ)² cos ²θ +sin²θ * or cos θcos²θ- cos²θ (x+)/ cos θ -sin ²θ sin²θ (-x)/ y²sin²θcos²θ x x y² (-x²)/4 x²+ 4y²* or xcos θcos²θ sin²θ 4 x²cos θ -sin²θcos²θ+sin 4 θ y²sin²θcos²θ 4 4 x²+4y²cos θ-sin²θcos²θ+sin θ+4sin²θcos²θ (cos²θ+sin²θ)² * E E E E use of sin θ substitution use of sinθ for both correct use of double angle formulae correct squaring and use of sin²θ+cos²θ ½ ellipse correct intercepts

5 4754A Mark Scheme June 4 x 4x ( ) 4 dealing with 4 (or terms in 4, 4, etc). x (. x ( )...) 4 4 ( x x...) 8 8 x x correct binomial coefficients correct unsimplified expression for (+x/4) or (4+x) cao Valid for < x/4 < 4 < x < 4 B 5(i) 3 A B ( y)( y) y y Ay ( ) By ( ) ( y)( y) 3 A(y + ) + B(y ) y 3 3A A y 3 3B B (ii) dy x ( y )( y dx ) 3d y 3x dx ( y)( y) ( )d y 3x ( y) y ln(y ) ln(y + ) x 3 + c y 3 ln x c y 3 3 y x c x c x e e.e Ae y dx 3 * Bft B E substituting, equating coeffs or cover up separating variables ln(y ) ln(y + ) ft their A,B x 3 + c anti-logging including c www tan tan 45 6 tan( 45) tan tan45 tan tan tan tan tan + tan ( tan )( tan ) 3 tan + tan tan 4 tan tan (tan ) tan or or [7] oe using sin/cos multiplying up and expanding any correct one line equation solving quadratic for tan θ oe www - extra solutions in the range 3

6 4754A Mark Scheme June Section B ( ) 3 7(i) * AB AB (3 + + ) 33 m E accept surds 3 (ii) r 3 Angle is between and 3 cos 7.45 BB [6] oe and complete scalar product method(including cosine) for correct vectors 7.5º or better, accept.6 radians (iii) Meets plane of layer when ( +3) + (+) /5 3 8 r so meets layer at ( 8, 4, 4) [4] (iv) Normal to plane is 3 3 Angle is between and cos angle with layer 4. B complete method ft 9-theirθ accept radians 4

7 4754A Mark Scheme June 8(i) At A, y 4cos, / At B, cos, x-coord of A / sin / x-coord of B sin OA, AC + + ratio is ( ):( + ) * B B E for either A or B/C for both A and B/C (ii) d y 4sin d d x cos d dy d y/d dx d x /d 4sin cos At A, gradient 4sin( /) cos( /) B [4] either dx/dor dy/d www (iii) d y 4sin dx cos 4sin cos cos 4sin * (iv) cos 4sin Rcos( + R(cos cos sin sin ) Rcos Rsin R + 4 7, R 7 tan 4,.36 7 cos( +.36) cos( +.36) / , 5.9, (.6), 3.89, 6. E [] B [7] their dy/dx corr pairs accept 76.º,.33 radians inv cos (/ 7) ft their R for method - extra solutions in the range 5

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