Mark Scheme (Results) June GCE Core Mathematics C1 (6663) Paper 1

Transcription

1 Mark (Results) June 0 GCE Core Mathematics C (666) Paper

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 Examiners Report 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: June 0 Publications Code UA07654 All the material in this publication is copyright Edexcel Ltd 0

3 EDEXCEL GCE MATHEMATICS General Instructions for Marking. 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.. Abbreviations These are some of the traditional marking abbreviations that will appear in the mark schemes and can be used if you are using the annotation facility on epen. 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 GCE Core Mathematics C (666) June 0

4 . Question June 0 Core Mathematics C 666 Mark (a) 5 (or ± 5 ) B (b) 5 = or 5 =5 or better M 5 or (or ± ) A 5 5 (a) Give B for 5 or ± 5 Anything else is B0 (including just 5) (b) M: Requires reciprocal OR 5 = 5 Accept,,,, for M () () Correct answer with no working ( or notation errors in working) scores both marks i.e. M A MA0 for - without +5 5 GCE Core Mathematics C (666) June 0

5 . (a) (b) ( = ) dy = 0x x or 0x 4 dx x M A A () 6 6 x x x x + 7x + = + 7x 6 M A A + C B (4) 7 n n (a) M: Attempt to differentiate x x (for any of the terms) 4 4 i.e. ax or ax, where a is any non-zero constant or the 7 differentiated to give 0 is sufficient evidence for M st A: One correct (non-zero) term, possibly unsimplified. nd A: Fully correct simplified answer. n (b) M: Attempt to integrate x x 6 (i.e. ax or ax or ax, where a is any non-zero constant). st A: Two correct terms, possibly unsimplified. nd A: All three terms correct and simplified. 6 x Allow correct equivalents to printed answer, e.g. + 7x or x + 7x x x 6 x Allow or 7x B: + C appearing at any stage in part (b) (independent of previous work) n+ 6 GCE Core Mathematics C (666) June 0

6 . Mid-point of PQ is (4, ) B 0 6 PQ: m =, = 9 ( ) 5 B 5 Gradient perpendicular to PQ ( ) m M 5 y = ( x 4) M 5x y = 0 or y 5x + = 0 or multiples e.g. 0x 6y = 0 A (5) 5 B: correct midpoint. B: correct numerical expression for gradient need not be simplified st M: Negative reciprocal of their numerical value for m nd M: Equation of a line through their (4, ) with any gradient except 0 or. If the 4 and are the wrong way round the nd M mark can still be given if a correct formula (e.g. y y = m( x x) ) is seen, otherwise M0. If (4, ) is substituted into y = mx + c to find c, the nd M mark is for attempting this. A: Requires integer form with an = zero (see examples above) GCE Core Mathematics C (666) June 0

7 4. Either y = 4x + 4 x Or = 4 4y y M x + 4(4 4x + x or ) x x x = = 4( ) or 4y y (4 4y + y y = 4 ( ) ) = M x 6x + 5 = 0 y + 4y 5 = 0 Correct terms A ( x )( x 5) = 0, x =. ( y 5)( y + ) = 0, y =... M x = x = 5 5 y = y = A 5 y = y = x = x = 5 M A st M: Squaring to give or 4 terms (need a middle term) nd M: Substitute to give quadratic in one variable (may have just two terms) rd M: Attempt to solve a term quadratic. 4 th M: Attempt to find at least one y value (or x value). (The second variable) This will be by substitution or by starting again. If y solutions are given as x values, or vice-versa, penalise accuracy, so that it is possible to score M MA M A0 M A0. Non-algebraic solutions: No working, and only one correct solution pair found (e.g. x = 5, y = ): M0 M0 A0 M A0 M A0 No working, and both correct solution pairs found, but not demonstrated: M0 M0 A0 M A M A Both correct solution pairs found, and demonstrated: Full marks are possible (send to review) (7) 7 GCE Core Mathematics C (666) June 0 4

8 5. = k + (a) ( ) 5 a B () (b) ( = ) 5(5k + ) (c) a + M = 5k + 8 (*) A cso (i) = 5(5k + 8) + ( = 5k 9) a M = r a = k + (5k + ) + (5k + 8) + (5k + 9) M r = 56k + 4 A cao (ii) = 6(6k + 9) (or explain each term is divisible by 6) A ft (a) 5k + must be seen in (a) to gain the mark (b) st M: Substitutes their a into 5 a + - note the answer is given so working must be seen. (c) st M: Substitutes their a into 5 a + or uses 5k +9 nd M: for their sum k+ a + a+ a4 - must see evidence of four terms with plus signs and must not be sum of AP st A: All correct so far nd Aft: Limited ft previous answer must be divisible by 6 ( eg 56k + 4). This is dependent on second M mark in (c) Allow 56 k + 4 = 6k + 9 without explanation. No conclusion is 6 needed. () (4) 7 GCE Core Mathematics C (666) June 0 5

9 6. (a) p =, q = or (b) x, B, B 6 x 6x x + = 4x + x ( ) M Aft x = 4, y = 90: C = 90 C = 6 M A 4 " 6" y = x + x + their A (5) 7 (a) Accept any equivalent answers, e.g. p = 0.5, q = 4/ (b) st M: Attempt to integrate x n x n+ (for either term) st A: ft their p and q, but terms need not be simplified (+C not required for this mark) nd M: Using x = 4 and y = 90 to form an equation in C. nd A: cao rd A: answer as shown with simplified correct coefficients and powers but follow through their value for C If there is a 'restart' in part (b) it can be marked independently of part (a), but marks for part (a) cannot be scored for work seen in (b). Numerator and denominator integrated separately: First M mark cannot be awarded so only mark available is second M mark. So out of 5 marks. () GCE Core Mathematics C (666) June 0 6

10 7. (a) Discriminant: b 4ac = ( k + ) 4k or equivalent M A (b) ( k + ) 4k = k + k + 9 = ( k + ) + 8 M A (c) For real roots, b 4ac 0 or b 4ac > 0 or ( k + ) + 8> 0 M ( ) k + 0 for all k, so b 4ac > 0, so roots are real for all k (or A cso equiv.) (a) M: attempt to find discriminant substitution is required If formula b 4ac is seen at least of a, b and c must be correct If formula b 4ac is not seen all of a, b and c must be correct Use of b + 4ac is M0 A: correct unsimplified (b) M: Attempt at completion of square (see earlier notes) A: both correct (no ft for this mark) (c) M: States condition as on scheme or attempts to explain that their ( k + ) + 8 is greater than 0 A: The final mark (Acso) requires ( k + ) 0 and conclusion. We will allow ( k + ) > 0 ( or word positive) also allow b 4ac 0and conclusion. () () () 6 GCE Core Mathematics C (666) June 0 7

11 8. (a) 5 y 4 Shape through (0, 0) B (, 0) B 5 0 x (.5, ) B () (b) y x 5 Shape B (0, 0) and (6, 0) B (, ) B () (c) y Shape, not through (0, 0) M x Minimum in 4 th quadrant A ( p, 0) and (6 p, 0) B ( p, ) B (4) 0 (a) B: U shaped parabola through origin B: (,0) stated or labelled on x axis B: (.5, -) or equivalent e.g. (/, -) (b) B: Cap shaped parabola in any position B: through origin (may not be labelled) and (6,0) stated or 6 labelled on x - axis B: (,) shown (c) M: U shaped parabola not through origin A: Minimum in 4 th quadrant (depends on M mark having been given) B: Coordinates stated or shown on x axis B: Coordinates stated Note: If values are taken for p, then it is possible to give MAB0B0 even if there are several attempts. (In this case all minima should be in fourth quadrant) GCE Core Mathematics C (666) June 0 9

12 9. (a) Series has 50 terms B S = (50)( + 00) = 550 or S = (50)( ) = 550 M A () (b) (i) 00 k B (ii) Sum: ( k + 00) or k + k k k k M A 5000 = 50 + k (*) A cso (4) (c) 50 th term = a + ( n ) d = (k+ ) + 49"(k+ )" Or k + 49(k) () M = 00 k +48 = 00 k + 48 A () 9 (a) B for seeing attempt to use n = 50 or n = 50 stated M for attempt to use na ( + l) or n( a+ ( n ) d) with a = and values for other variables (Using n = 00 may earn B0 MA0) (b) M for use of a = k and d = k or l = 00 with their value for n, could be numerical or even letter n in correct formula for sum. A: Correct formula with n = 00/k A: NB Answer is printed so no slips should have appeared in working (c) M for use of formula a+ 49d with a = k + and with d obtained from difference of terms A: Requires this simplified answer GCE Core Mathematics C (666) June 0 0

13 0. (a) y 0 x Shape (cubic in this orientation) Touching x-axis at Crossing at on x-axis Intersection at 9 on y-axis B B B B (4) (b) y = ( x + )( x + 6x + 9) = x + 7x + 5x + 9 or equiv. (possibly unsimplified) B Differentiates their polynomial correctly may be unsimplified M d y = x + 4x + 5 dx (*) A cso (c) dy At x = 5: = = 0 dx B At x = 5: y = 6 B y (" 6") = "0"( x ( 5)) or y = 0x + c with (-5, - 6 ) used to find c M y = 0 x + 84 A (d) Parallel: x + 4x+ 5 = "0" M ( x )( x + 5) = 0 x = M x = A (a) Crossing at is B0. Touching at is B0 (b) M: This needs to be correct differentiation here A: Fully correct simplified answer. (c) M: If the 5 and -6 are the wrong way round or omitted the M mark can still be given if a correct formula is seen, (e.g. y y = m( x x) ) otherwise M0. m should be numerical and not 0 or infinity and should not have involved negative reciprocal. (d) st M: Putting the derivative expression equal to their value for gradient nd M: Attempt to solve quadratic (see notes) This may be implied by correct answer. () (4) () 4 GCE Core Mathematics C (666) June 0

14 Further copies of this publication are available from Edexcel Publications, Adamsway, Mansfield, Notts, NG8 4FN Telephone Fax Order Code UA07654 June 0 For more information on Edexcel qualifications, please visit Pearson Education Limited. Registered company number 8788 with its registered office at Edinburgh Gate, Harlow, Essex CM0 JE