Mark Scheme (Results) January 2011

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1 Mark (Results) January 011 GCE GCE Core Mathematics C (6666) Paper 1 Edexcel Limited. Registered in England and Wales No Registered Office: One90 High Holborn, London WC1V 7BH

2 Edexcel is one of the leading examining and awarding bodies in the UK and throughout the world. We provide a wide range of qualifications including academic, vocational, occupational and specific programmes for employers. Through a network of UK and overseas offices, Edexcel s centres receive the support they need to help them deliver their education and training programmes to learners. For further information, please call our GCE line on , our GCSE team on , or visit our website at If you have any subject specific questions about the content of this Mark that require the help of a subject specialist, you may find our Ask The Expert service helpful. Ask The Expert can be accessed online at the following link: January 011 Publications Code UA061 All the material in this publication is copyright Edexcel Ltd 011

3 General Instructions for Marking 1. The total number of marks for the paper is 75.. The Edexcel Mathematics mark schemes use the following types of marks: M marks: method marks are awarded for knowing a method and attempting to apply it, unless otherwise indicated. A marks: Accuracy marks can only be awarded if the relevant method (M) marks have been earned. B marks are unconditional accuracy marks (independent of M marks) should not be subdivided. 3. Abbreviations These are some of the traditional marking abbreviations that will appear in the mark schemes. bod benefit of doubt ft follow through the symbol will be used for correct ft cao correct answer only cso - correct solution only. There must be no errors in this part of the question to obtain this mark isw ignore subsequent working awrt answers which round to SC: special case oe or equivalent (and appropriate) dep dependent indep independent dp decimal places sf significant figures The answer is printed on the paper The second mark is dependent on gaining the first mark

4 January 011 Core Mathematics C 6666 Mark 1. x cos x cos x xsin xdx= + dx sin x =... + π π [... ] = A1 0 A1 A1 [6]. ( ) At 3 di t 16ln dt A1 di 16ln dt = ln0.5= ln A1 t = ( ) 3 [5] GCE Core Mathematics C (6666) January 011 1

5 3. (a) (b) A B = + x 1 3x+ x 1 3x+ 5 ( )( ) ( ) ( ) 5= A 3x+ + B x 1 x 1 5= 5A A= 1 A1 5 x 5= B B = A1 (3) dx ( )( ) = dx x 1 3x+ x 1 3x+ = ln x 1 ln 3x+ + C ft constants A1ft A1ft ( ) ( ) ( ) (3) (c) 5 1 dx dy ( x 1)( 3x ) = + y ln ( x 1) ln ( 3x ) ln y ( C) K( x 1) Using ( ) + = + A1 y = depends on first two Ms in (c) dep 3x + K, 8 8 = depends on first two Ms in (c) dep 8 6( x 1) y = A1 (6) 3x + [1] GCE Core Mathematics C (6666) January 011

6 . (a) ( ) AB = i + j k i 3j + k = 3i+ 5j 3k A1 () (b) = λ ( ) or r = i + j k+ λ ( 3i+ 5j 3k ) r i j k i j k A1ft () (c) AC = i + pj k ( i 3j + k ) ( p ) = i j 6k or CA B1 1 3 AC. AB = p = p = 0 Leading to p = 6 A1 () (d) ( 1) ( 6 3) ( ) ( 6) AC = = AC = 6 accept awrt 6.8 A1 () [10] GCE Core Mathematics C (6666) January 011 3

7 5. 3 = 1 x (a) ( x) 3 B A1 1 x = 1+ ( ) x + x + x = 1+ 3 x+ x + x +... ( ) x = + x+ x + x +... A1 (5) 16 8 f x = a+ bx + x+ x + x a b Coefficient of x; + = 0 ( 3a+ b= 0) 7a 3b 9 Coefficient of x ; + = ( 9a+ b= 3) A1 either correct A Leading to a= 1, b= 3 A1 (5) (b) ( ) ( ) 3 (c) Coefficient of 7a 7b = A1ft 7 = 16 cao A1 (3) 3 x is ( ) [13] GCE Core Mathematics C (6666) January 011

8 6. (a) dx 1 =, dt t dy t dt = dy t dx = A1 Using mm = 1, at t = 3 1 m = 18 A1 1 y 7= ( x ln3) 18 A1 (6) (b) x= ln t t = e x B1 x y = e A1 (3) V = π e dx x (c) ( ) ( x ) ( x e d e e x x = + ) dx x x e e = + x A1 ln x x e e π + x = π ( ln ) ( 8 + ln ) ln ( 36 ln ) = π + A1 Alternative to (c) using parameters ( ) d V = π t x dt dt ( ) 1 3 t dt = t t+ dt t t t = t + lnt A1 The limits are t = and t = t ln t π 6 3 ln 8 ln t π + = ( + ) ( + ) ( 36 ln ) = π + A1 (6) [15] (6) GCE Core Mathematics C (6666) January 011 5

9 7. (a) x= 3 y = awrt B1 x= 5 y = awrt or 1 B1 6 () B1 A1ft (b) I + + ( + ) or 0.53 A1 () dx du B1 1 1 dx = ( u ) du + ( x 1) u 8 = du u A1 = u 8lnu A1 x= u = 5, x= 5 u = 6 B1 (c) ( u ) [ u 8ln u] 6 ( 1 8ln 6) ( 10 8ln 5) = 5 5 = + 8ln A1 6 (8) [1] GCE Core Mathematics C (6666) January 011 6

10 Further copies of this publication are available from Edexcel Publications, Adamsway, Mansfield, Notts, NG18 FN Telephone Fax Order Code UA061 January 011 For more information on Edexcel qualifications, please visit Edexcel Limited. Registered in England and Wales no Registered Office: One90 High Holborn, London, WC1V 7BH

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