Version.0 General Certificate of Education (A-level) January 0 Mathematics MFP (Specification 660) Further Pure Mark Scheme
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Key to mark scheme abbreviations M mark is for method m or dm mark is dependent on one or more M marks and is for method A mark is dependent on M or m marks and is for accuracy B mark is independent of M or m marks and is for method and accuracy E mark is for eplanation or ft or F follow through from previous incorrect result CAO correct answer only CSO correct solution only AWFW anything which falls within AWRT anything which rounds to ACF any correct form AG answer given SC special case OE or equivalent A, or (or 0) accuracy marks EE deduct marks for each error NMS no method shown PI possibly implied SCA substantially correct approach c candidate sf significant figure(s) dp decimal place(s) No Method Shown Where the question specifically requires a particular method to be used, we must usually see evidence of use of this method for any marks to be awarded. Where the answer can be reasonably obtained without showing working and it is very unlikely that the correct answer can be obtained by using an incorrect method, we must award full marks. However, the obvious penalty to candidates showing no working is that incorrect answers, however close, earn no marks. Where a question asks the candidate to state or write down a result, no method need be shown for full marks. Where the permitted calculator has functions which reasonably allow the solution of the question directly, the correct answer without working earns full marks, unless it is given to less than the degree of accuracy accepted in the mark scheme, when it gains no marks. Otherwise we require evidence of a correct method for any marks to be awarded.
MFP (a) y z Circle B P correct centre B through (0, 0) B (i) z correctly chosen BF ft if circle encloses (0, 0) (ii) z = 8 BF ft if centre misplotted (a) u r u = r Total r( r+ )( 4r+ ) ( r ) r( 4r+ 7) M 6 6 Correct epansion in any form, eg ( 4 4 7) 6 r r + r+ r r+ A = r r+ A AG ( ) Attempt to use method of differences M S00 = u00 u0 A = 6980 A CAO Total 6 πi (a) + i = i or +i = e 4 B ( ) ( ) i( + i) = i B AG Alternative method: ( ) + i = + i+ i + i B = i B (i) Substitute z = + i M Correct epansion A allow for correctly picking out either the real or the imaginary parts k = A (ii) β + γ = + i α = 4 B AG (iii) αβγ = ( + i) M allow if sign error βγ = AF ft incorrect k z 4z+ = 0 M Use of formula or ( z ) = AF No ft for real roots if error in k z = ± i AF NB allow marks for in whatever order they appear Total 4
MFP (cont) dy 4(a) sinh 8cosh d The B and M could be in reverse order if put in terms of e first ( e e ) ( e + e ) M0 if sinh and cosh in terms of e 8 = 0 M are interchanged e e 0 = 0 AF ft slips of sign ( )( ) e e + = 0 MAF ft provided quadratic factorises e E some indication of rejection needed = ln one stationary point Condone e = with statement provided AF 7 quadratic factorises Special Case If d y sinh 8cosh d = B0 For substitution in terms of e M leading to e = A Then M0 + b = 8 ln MAF for substitution into original equation 74 84 = ln 0 0 A CAO = 9 a A 4 AG; accept b= 9 a Total du = B d B (a) ( ) ( ) sin d = sin d M AA d = used AF ft sign error in d d A for each part of the integration by parts π + 4 m substitution of limits π 6 A 6 CAO Total 8 u
MFP (cont) 6(a) use of FB for sect ; d sect cost dt if done from first principles, allow B when sect is arrived at Use of cos t = sin t M d = sinttant dt A 4 AG + y = sin ttan t+ sin t MA sign error in d y A0 dt Use of + tan t = sec t m + y = tant AF ft sign error in d y dt π π [ ] tan t d t = lnsec t AF ft sign error in d y 0 0 dt = ln A 6 CAO Total 0 f k + f k 7(a) ( ) ( ) ( ) k + k k k = + + M k + = + k k k A for epansion of bracket k k k = = 7 A clearly shown k k = used Assume f( k) = M ( 7) Then f( k ) f( k) M( 7) M ( 7) f() 4 M ( 7) + = + M Not merely a repetition of part (a) = A clearly shown = + = = B Correct inductive process E 4 Total 7 (award only if all previous marks earned) 6
MFP (cont) 8(a)(i) 4 ( + i ) = 8 + i M πi If either r orθ is incorrect but the same = 8e A value in both (i) and (ii) allow A for either 4 ( + i ) or 4 ( i ) used but for θ only if it is given as π 6 (ii) ( ) πi 4 i = 8e A z 4=± 48 M taking square root z = 4± 4 i A AG (c)(i) πi πi +kπi +kπi z = e or z = e BF M for the ; ft incorrect 8, but no decimals for either, PI (ii) z = e, e πi 7πi πi 9 9 9, e πi 7πi πi 9 9 9 = e, e, e 0 A,,F Allow A for any roots not +/ each other Allow A for any roots not +/ each other Allow A for all 6 correct roots Deduct A for each incorrect root in the interval; ignore roots outside the interval ft incorrect r Radius BF clearly indicated; ft incorrect r allow B for correct points condone lines Plotting roots B, (d)(i) Sum of roots = 0 as coefficient of z = 0 E OE (ii) πi πi π 9 9 Use of, say, e + e = cos 9 M π cos = used 9 A π π π 7π cos + cos + cos + cos = 9 9 9 9 A AG Total 7 TOTAL 7 7