Mark Scheme (Results) January 2011

Transcription

1 Mark (Results) January 0 GCE GCE Core Mathematics C (6664) Paper Edecel Limited. Registered in England and Wales No Registered Office: One90 High Holborn, London WCV 7BH

2 Edecel is one of the leading eamining 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, Edecel 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 Epert service helpful. Ask The Epert can be accessed online at the following link: January 0 Publications Code US0635 All the material in this publication is copyright Edecel Ltd 0

3 General Instructions for Marking. The total number of marks for the paper is 75.. The Edecel 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 0 Core Mathematics C 6664 Mark Question. (a) f( ) 4 3 = a + b Attempting f() or f( ). f() = a + b = 7 or 4 + a + b = 7 a + b = 3 (as required) AG * cso () (b) Attempting f( ) or f(). { } f( ) = a + b = 8 a + b = 4 Solving both equations simultaneously to get as far as a =... or b =... d Any one of a = 9orb = 6 Both a = 9andb = 6 cso (5) [7] Notes (a) for attempting either f() or f( ). for applying f(), setting the result equal to 7, and manipulating this correctly to give the result given on the paper as a + b = 3. Note that the answer is given in part (a). (b) : attempting either f( ) or f(). : correct underlined equation in a and b; eg a + b = 8 or equivalent, eg a + b = 4. d: an attempt to eliminate one variable from linear simultaneous equations in a and b. Note that this mark is dependent upon the award of the first method mark. : any one of a = 9orb = 6. : both a = 9andb = 6and a correct solution only. Alternative Method of Long Division: (a) for long division by ( ) to give a remainder in a and b which is independent of. for { Remainder = } b + a + 4 = 7 leading to the correct result of a + b = 3 (answer given.) (b) for long division by ( + ) to give a remainder in a and b which is independent of. for { Remainder = } b ( a 8) = 8 { a + b = 4}. Then d are applied in the same way as before. GCE Core Mathematics C (6664) January 0

5 . (a) 8 7 ( 8 7cosC) = cosc = 8 7 { Cˆ } (or equivalent) = Cˆ = awrt.64 cso (3) (b) Use of Area ABC = absin(their C), where a, b are any of 7, 8 or. = ( 7 8 ) sin C using the value of their C from part (a). ft c { = or } = awrt 7.9 (from angle of either.64 or 94. ) cso (3) [6] Notes cosc 7 = cosc (a) is also scored for = + ( ) or ( ) or cosc = or cosc = 7 8 st : Rearranged correctly to make cos C =... and numerically correct (possibly unsimplified). Award for any of cosc = or cosc = or cosc = or cosc = awrt SC: Also allow st for cosc = 8 or equivalent. Also note that the st can be implied for C ˆ = awrt.64 or ˆ C = awrt Special Case: cosc = or cosc = scores a SC: A0A nd : for awrt.64 cao c c A = or , B = or Note that ( ) ( ) (b) : alternative methods must be fully correct to score the. For any (or both) of the or the st ; their C can either be in degrees or radians. Candidates who use cosc = to give C = , can achieve the correct answer of awrt in part (b). These candidates will score A0cso, in part (b). Finding C = in part (a) and achieving awrt 7.9 with no working scores A0. Otherwise with no working in part (b), awrt 7.9 scores. Special Case: If the candidate gives awrt 7.9 from any of the below then award. c ( 7 ) sin(0.86 or ) = awrt 7.9, ( 8 c ) sin( or ) = awrt 7.9. Alternative: Hero s Formula: A = 3(3 )(3 8)(3 7) = awrt 7.9, where is attempt to apply A = s( s )( s 8)( s 7) and the first is for the correct application of the formula. GCE Core Mathematics C (6664) January 0

6 3. (a) ar = 750 and 4 ar = 6 (could be implied from later working in either (a) or (b)). B 3 6 r = 750 Correct answer from no working, ecept r = for special case below gains all three 5 marks. (b) a( 0.) = a = = ft () a 3750 (c) Applies correctly using both their a and their r <. Eg. r 0. So, S = 35 () [7] Notes (a) B: for both ar = 750 and 4 ar = 6 (may be implied from later working in either (a) or (b)). 4 : for eliminating a by either dividing ar = 6byar = 750or dividing ar = 750 by ar = 6, to achieve an equation in r or Note that r r = is M0. 3 r Note also that any of r = or r = { = 5} or = or = 3 { = 5 3 } are r 750 r 6 fine for the award of. α β 6 SC: ar = 750 and ar = 6 leading to r δ = or r 750 = { = 5} or = or = { = 5} where δ = β α and δ are fine for the award of. r δ 750 r δ 6 SC: ar = 750 and 5 ar = 6 leading to r = scores B0. (b) for inserting their r into either of their original correct equations of either ar = 750 or { a } = or ar = 6 or { a = } in both a and r. No slips allowed here for. 4 r r for either a = 3750 or a equal to the correct follow through result epressed either as 5 (3) (c) an eact integer, or a fraction in the form c where both c and d are integers, or correct to d awrt dp. a for applying correctly (only a slip in substituting r is allowed) using both their a r 3750 and their r <. Eg.. for In parts (a) or (b) or (c), the correct answer with no working scores full marks. GCE Core Mathematics C (6664) January 0 3

7 4. (a) Seeing and 5. (See note below.) B () (b) ( + )( 5) = 4 5 or B n n 3 M: + for any one term. 4 ( 4 5)d = 5{ + c } st at least two out of three terms 3 correctly ft. ft Substitutes 5 and (or limits from = (...) (... part(a)) into an integrated ) 3 function and subtracts, either way round. d = = Hence, Area = 36 Final answer must be 36, not 36 (6) [7] Notes (a) B: for and5., 0 and 5, 0 are acceptable for B. Also allow Note that ( ) ( ) ( 0, ) and ( 0, 5) generously for B. Note that if a candidate writes down that A:(5,0), B:(,0), (ie A and B interchanged,) then B0. Also allow values inserted in the correct position on the -ais of the graph. (b) B for 4 5 or If you believe that the candidate is applying the Way method then or would then be fine for B. st n n for an attempt to integrate meaning that + for at least one of the terms. Note that 5 5 is sufficient for. st at least two out of three terms correctly ft from their multiplied out brackets. nd for correct integration only and no follow through. Ignore the use of a ' + c'. Allow nd also for + 5. Note that + only counts as one integrated 3 term for the st mark. Do not allow any etra terms for the nd mark. nd : Note that this method mark is dependent upon the award of the first mark in part (b). Substitutes 5 and (and not if the candidate has stated = in part (a).) (or the limits the candidate has found from part(a)) into an integrated function and subtracts, either way round. 3 rd : For a final answer of 36, not 36. Note: An alternative method eists where the candidate states from the outset that 5 Area R = d is detailed in the Appendi. ( ) ( ) GCE Core Mathematics C (6664) January 0 4

8 5. (a) ; ( ) (b) 40 40! = + n 4 5 coefficients of and are p and q respectively. 4 4! b! b = 36 B Candidates should usually identify two terms as their p and q respectively. () Any one of Term or ! 40(39)(38)(37) Term : or C4 or or or !36! 4! Term ! 40(39)(38)(37)(36) : or C5 or or or !35! 5! Hence, q p = = = Term correct. (Ignore the label of p and/or q.) Both of them correct. (Ignore the label of p and/or q.) for oe oe cso Notes (a) B: for only b = 36. (b) The candidate may epand out their binomial series. At this stage no marks should be awarded until they start to identify either one or both of the terms that they want to focus on. Once they identify their terms then if one out of two of them (ignoring which one is p and which one is q) is correct then award. If both of the terms are identified correctly (ignoring which one is p and which one is q) then award the first ! 4 40(39)(38)(37) 4 4 Term = or C4( ) or or or 9390, 4 4!36! 4! ! 5 40(39)(38)(37)(36) 5 5 Term = or C5 ( ) or or or !35! 5! are fine for any (or both) of the first two marks in part (b). nd for stating q as p 9390 or equivalent. Note that q must be independent of. p (3) [4] Also note that 36 5 or 7. or any equivalent fraction is fine for the nd mark. SC: If candidate states Note that either 4!36! 5!35! p 5 q =, then award A0. 36 or 5!35! would be awarded. 4!36! GCE Core Mathematics C (6664) January 0 5

9 6. (a) At { } At { } y =.5, y = 0.30 (only) At least one y-ordinate correct. B =.75, y = 0.4 (only) Both y-ordinates correct. B () Outside brackets 0.5 or 8 B aef (b) { ( )} For structure of {...} ; 0.5 ; their their 0.4 Correct epression inside brackets which all must be multiplied by their outside constant. { = (.54) 8 } = awrt 0.3 awrt 0.3 (4) (c) Area of triangle = 0. = 0. B Area( S ) = "0.375" 0. = 0.75 ft (3) [9] GCE Core Mathematics C (6664) January 0 6

10 Notes (b) B for using 0.5 or or equivalent. 8 requires the correct{...} bracket structure. This is for the first bracket to contain first y- ordinate plus last y-ordinate and the second bracket to be the summation of the remaining y- ordinates in the table. No errors (eg. an omission of a y-ordinate or an etra y-ordinate or a repeated y-ordinate) are allowed in the second bracket and the second bracket must be multiplied by. Only one copying their their 0.4 bracket. (c) error is allowed here in the ( ) ft for the correct bracket {...} following through candidate s y-ordinates found in part (a). for answer of awrt 0.3. Bracketing mistake: Unless the final answer implies that the calculation has been done correctly then award A0A0 for either ( 0.38 their 0.30 their 0.4 ) (nb: yielding final answer of.05) so that the 0.5 is only multiplied by 0.5 or 0.5 ( ) ( 0.38 their 0.30 their 0.4 ) (nb: yielding final answer of.975) so that the ( ) is multiplied by 0.5. Need to see trapezium rule answer only (with no working) gains no marks. Alternative: Separate trapezia may be used, and this can be marked equivalently. (See appendi.) B for the area of the triangle identified as either 0. or 0.. May be identified on the diagram. for part (b) answer 0. only or part (b) answer their attempt at 0. only. (Strict attempt!) ft for correctly following through part (b) answer 0.. This is also dependent on the answer to (b) being greater than 0.. Note: candidates may round answers here, so allow ft if they round their answer correct to dp. GCE Core Mathematics C (6664) January 0 7

11 7. (a) 3sin 7sin cos 4 + = ; 0 < 360 3sin 7sin ( sin ) 4 GCE Core Mathematics C (6664) January = (b) { } 4sin + 7sin + 3 = 0 AG * cso () (4sin + 3)(sin + ) = 0 Valid attempt at factorisation and sin = sin =, sin = Both sin = 4 4 and sin =. ( α = ) = or = Either ( 80 + α ) or ( 360 α ) d = , = Both awrt 8.6 and awrt 3.4 { } sin = = B Notes (a) for a correct method to change cos into sin (must use cos = sin ). Note that applying cos = sin, scores M0. for obtaining the printed answer without error (ecept for implied use of zero.), although the equation at the end of the proof must be = 0. Solution just written only as above would score. (b) st for a valid attempt at factorisation, can use any variable here, s, y, or sin, and an attempt to find at least one of the solutions. Alternatively, using a correct formula for solving the quadratic. Either the formula must be stated correctly or the correct form must be implied by the substitution. st for the two correct values of sin. If they have used a substitution, a correct value of their s or their y or their. nd for solving sin = k, 0 < k < and realising a solution is either of the form ( 80 α ) + or ( 360 α ) where α = sin ( k). Note that you cannot access this mark from sin = = 70. Note that this mark is dependent upon the st mark awarded. nd for both awrt 8.6 and awrt 3.4 B for 70. If there are any EXTRA solutions inside the range 0 < 360 and the candidate would otherwise score FULL MARKS then withhold the final ba mark (the fourth mark in this part of the question). Also ignore EXTRA solutions outside the range 0 < 360. Working in Radians: Note the answers in radians are = , , If a candidate works in radians then mark part (b) as above awarding the nd for both awrt 4.0 and awrt 5.4 and the B for awrt 4.7 or 3 π. If the candidate would then score FULL MARKS then withhold the final ba mark (the fourth mark in this part of the question.) No working: Award B for 70 seen without any working. Award M0A0 for awrt 8.6 and awrt 3.4 seen without any working. Award M0A0A0 for any one of awrt 8.6 or awrt 3.4 seen without any working. (5) [7]

12 8. (a) Graph of y 7, = and solving ( ) y ( 0, ) = 0 At least two of the three criteria correct. (See notes below.) All three criteria correct. (See notes below.) B B (b) y { } 4y + 3 = 0 { ( y 3)( y ) 0 = or ( )( ) = 0} Forming a quadratic {using " " 7 y = }. y { } 4y + 3 = 0 y = 3, y = or 7 = 3, 7 = Both y = 3 and y =. { } O 7 = 3 log 7 = log3 A valid method for solving log3 or = or = log7 3 7 = k where k > 0, k log 7 d = or awrt 0.56 = 0 = 0 stated as a solution. B Notes (a) BB0: Any two of the following three criteria below correct. BB: All three criteria correct. Criteria number : Correct shape of curve for 0. Criteria number : Correct shape of curve for < 0. Criteria number 3: (0, ) stated or marked on the y-ais. Allow (, 0) rather than (0, ) if marked in the correct place on the y-ais. () (6) [8] GCE Core Mathematics C (6664) January 0 9

13 (b) st is an attempt to form a quadratic equation {using " y " = 7.} st mark is for the correct quadratic equation of y 4y + 3{ = 0}. Can use any variable here, eg: y, or = 0. Writing ( 7 ) 4( 7 ) 3 0 Award M0A0 for seeing ( ) ( 7 ) 4( 7 ) + 3= 0. + = is also sufficient for Allow for { } + = by itself without seeing y 4y 3{ 0} + = or st mark for both y = 3 and y = or both 7 = 3 and 7 =. Do not give this accuracy mark for both = 3and =, unless these are recovered in later working by candidate applying logarithms on these. Award for 7 = 3and 7 = written down with no earlier working. 3 rd ln k d for solving 7 = k, k > 0, k to give either ln 7 = ln k or = or = log7 k. ln 7 d is dependent upon the award of. nd for or awrt B is for the solution of = 0, from any working. GCE Core Mathematics C (6664) January 0 0

14 = (a) C, C( 3, 6) (b) AG Correct method (no errors) for finding the mid-point of AB giving ( 3, 6) ( 8 3) + ( 6) or ( 8 3) + ( 6) or Applies distance formula in order to find the radius. ( 3) + ( 6) or ( 3) + ( 6) Correct application of formula. ( ± 3) + ( y ± 6) = k, ( 3) + ( y 6) = 50 ( or ( 50 ) ( ) ) or 5 k is a positive value. ( 3) + ( y 6) = 50 (Not 7.07 ) (c) {For ( 0, 7 ), } ( ) ( ) 7 6 Gradient of radius = or (d) { } = 50, {so the point lies on C.} B This must be seen in part (d). 7 Gradient of tangent = Using a perpendicular gradient method. y 7 = 7( 0) y 7 = (their gradient)( 0) y = y = or y = 77 7 cao (4) [0] Notes 8 (a) Alternative method: C +, + or 8 C 8 +, + (b) You need to be convinced that the candidate is attempting to work out the radius and not the diameter of the circle to award the first. Therefore allow st generously for ( 8) + ( ) 8 + or 4 Correct answer in (b) with no working scores full marks. Award st for ( 8 ) + ( ) (c) B awarded for correct verification of ( ) ( ) ( ) ( ) + = with no errors Also to gain this mark candidates need to have the correct equation of the circle either from part (b) or re-attempted in part (c). They cannot verify (0, 7) lies on C without a correct C. Also a candidate could either substitute = 0 in C to find y = 7 or substitute y = 7 in C to find = 0.. B* B () (4) () GCE Core Mathematics C (6664) January 0

15 (d) nd mark also for the complete method of applying 7 = (their gradient)(0) + c, finding c. Note: Award nd M0 in (d) if their numerical gradient is either 0 or. Alternative: For first two marks (differentiation): dy ( 3) + ( y 6) = 0 (or equivalent) scores B. d st for substituting both = 0 and y = 7 to find a value for d, which must contain both d and y. (This M mark can be awarded generously, even if the attempted differentiation is not implicit.) Alternative: (0 3)( 3) + (7 6)( y 6) = 50 scores B which leads to y = y GCE Core Mathematics C (6664) January 0

16 0. (a) (b) V = = + 4 (5 ) 4 (5 0 ) ± α ± β ± γ 3, where α, β, γ 0 3 So, V = V = At least two of their epanded terms dv differentiated correctly. d = cao (4) = 0 Sets their d V from part (a) = 0 d = 0 4(3 5)( 5) = 0 { ( ) } 5 5 { As 0 < < 5} = = or 3 3 = awrt.67 = ( )( ) Substitute candidate s value of V = where 0 < < 5 into a formula for V. d 000 So, V = = 74 = Either 000 or or awrt 74. (4) (c) d V = Differentiates their d V d d d V d 5 d V 5 When =, = d 3 d V d V = 40 < 0 V is a maimum d d = 40 and < 0 or negative and maimum. cso () [0] Notes (a) st for a three term cubic in the form ± α ± β ± γ 3. Note that an un-combined ± α ± λ ± µ ± γ 3, α, λ, µ, γ 0 is fine for the st. st for either or nd for any two of their epanded terms differentiated correctly. NB: If epanded epression is divided by a constant, then the nd can be awarded for at least two terms are correct. Note for un-combined ± λ ± µ, ± λ ± µ counts as one term differentiated correctly. nd for , cao. Note: See appendi for those candidates who apply the product rule of differentiation. GCE Core Mathematics C (6664) January 0 3

17 (b) Note you can mark parts (b) and (c) together. Ignore the etra solution of = 5( andv = 0 ). Any etra solutions for V inside found for values inside the range of, then award the final A0. (c) is for their d V d differentiated correctly (follow through) to give d V d. d V for all three of = 40 and < 0 or negative and maimum. d Ignore any second derivative testing on = 5 for the final accuracy mark. Alternative Method: Gradient Test: for finding the gradient either side of their -value from part (b) where 0 < < 5. for both gradients calculated correctly to the near integer, using > 0 and < 0 respectively or a correct sketch and maimum. (See appendi for gradient values.) GCE Core Mathematics C (6664) January 0 4

18 Aliter 4 (b) Way ( )( 5) 4 5 or = + Can be implied by later working. B n n 3 M: + for any one term. 4 ( 4 5)d = + + 5{ + c } st any two out of three terms ft 3 correctly ft. Substitutes 5 and (or limits = (...) (... from part(a)) into an integrated ) d 3 function and subtracts, either way round = 3 3 Hence, Area = 36 (6) GCE Core Mathematics C (6664) January 0 5

19 Aliter 6 (b) Way 0.5 and a divisor of on all B terms inside brackets One of first and last ordinates, two of the middle ordinates which is equivalent to: inside brackets ignoring the denominator of. Correct epression inside 0.5 ; { ( ) ( 0.38 their 0.30 their 0.4 )} brackets if was to be factorised out. { = (.54) 8 } = awrt 0.3 awrt 0.3 (4) GCE Core Mathematics C (6664) January 0 6

20 Aliter 0 (a) Way Aliter 0 (a) Way3 Product Rule Method: u = 4 v = (5 ) du dv = 4 = (5 ) ( ) d d ± (their u )(5 ) ± (4 )(their v') dy A correct attempt at differentiating d 4(5 ) 4 ()(5 ) ( ) any one of either u or v correctly. d = + Both d u and d v correct d d dy 4(5 ) 8 (5 ) d = 4(5 ) 8 (5 ) (4) u = 4 v = du dv = 4 = 0 + d d ( ) ± (their u ) their(5 ) ± (4 )(their v') A correct attempt at differentiating dy 4(5 0 ) 4 ( 0 ) any one of either u or their v d d = correctly. Both d u and d v correct d d dv d = Note: The candidate needs to use a complete product rule method in order for you to award the the first mark here. The second method mark is dependent on the first method mark awarded. (4) GCE Core Mathematics C (6664) January 0 7

21 Aliter 0 (c) Way Gradient Test Method: dv d = + Helpful table! dv d GCE Core Mathematics C (6664) January 0 8

22 8 (b) Method of trial and improvement Helpful table: y = 7 4( 7 ) For a full method of trial and improvement by trialing f (value between 0. and ) = value and f (value between and ) = value Any one of these values correct to sf or truncated sf. Both of these values correct to sf or truncated sf. A method to confirm root to dp by finding by trialing f (value between 0.56and ) = value and f (value between and 0.565) = value Both values correct to sf or truncated sf and the confirmation that the root is = 0.56 (only) = 0 B (6) Note: If a candidate goes from 7 = 3 with no working to = then give implied. GCE Core Mathematics C (6664) January 0 9

23

24 Further copies of this publication are available from Edecel Publications, Adamsway, Mansfield, Notts, NG8 4FN Telephone Fa Order Code US0635 January 0 For more information on Edecel qualifications, please visit Edecel Limited. Registered in England and Wales no Registered Office: One90 High Holborn, London, WCV 7BH