PMT. Mark Scheme (Results) Summer Pearson Edexcel GCE in Further Pure Mathematics FP2 (6668/01)

Transcription

1 Mark (Results) Summer 04 Pearson Edecel GCE in Further Pure Mathematics FP (6668/0)

2 Edecel and BTEC Qualifications Edecel and BTEC qualifications come from Pearson, the world s leading learning company. We provide a wide range of qualifications including academic, vocational, occupational and specific programmes for employers. For further information, please visit our website at Our website subject pages hold useful resources, support material and live feeds from our subject advisors giving you access to a portal of information. If you have any subject specific questions about this specification that require the help of a subject specialist, you may find our Ask The Epert service helpful. Pearson: helping people progress, everywhere Our aim is to help everyone progress in their lives through education. We believe in every kind of learning, for all kinds of people, wherever they are in the world. We ve been involved in education for over 50 years, and by working across 70 countries, in 00 languages, we have built an international reputation for our commitment to high standards and raising achievement through innovation in education. Find out more about how we can help you and your students at: Summer 04 Publications Code UA0887 All the material in this publication is copyright Pearson Education Ltd 04

3 General Marking Guidance All candidates must receive the same treatment. Eaminers must mark the first candidate in eactly the same way as they mark the last. Mark schemes should be applied positively. Candidates must be rewarded for what they have shown they can do rather than penalised for omissions. Eaminers should mark according to the mark scheme not according to their perception of where the grade boundaries may lie. There is no ceiling on achievement. All marks on the mark scheme should be used appropriately. All the marks on the mark scheme are designed to be awarded. Eaminers should always award full marks if deserved, i.e. if the answer matches the mark scheme. Eaminers should also be prepared to award zero marks if the candidate s response is not worthy of credit according to the mark scheme. Where some judgement is required, mark schemes will provide the principles by which marks will be awarded and eemplification may be limited. Crossed out work should be marked UNLESS the candidate has replaced it with an alternative response.

4 EDEXCEL GCE MATHEMATICS 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.. 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. All A marks are correct answer only (cao.), unless shown, for eample, as A ft to indicate that previous wrong working is to be followed through. After a misread however, the subsequent A marks affected are treated as A ft, but manifestly absurd answers should never be awarded A marks. 5. For misreading which does not alter the character of a question or materially simplify it, deduct two from any A or B marks gained, in that part of the question affected. 6. If a candidate makes more than one attempt at any question: If all but one attempt is crossed out, mark the attempt which is NOT crossed out. If either all attempts are crossed out or none are crossed out, mark all the attempts and score the highest single attempt. 7. Ignore wrong working or incorrect statements following a correct answer.

5 General Principles for Further Pure Mathematics Marking (But note that specific mark schemes may sometimes override these general principles). Method mark for solving term quadratic:. Factorisation ( b c) ( p)( q), where pq c, leading to = ( a b c) ( m p)( n q), where pq c and mn a, leading to =. Formula Attempt to use the correct formula (with values for a, b and c).. Completing the square b Solving b c 0 : q c 0, q 0, leading to = Method marks for differentiation and integration:. Differentiation Power of at least one term decreased by. ( n n ). Integration Power of at least one term increased by. ( n n )

6 Use of a formula Where a method involves using a formula that has been learnt, the advice given in recent eaminers reports is that the formula should be quoted first. Normal marking procedure is as follows: Method mark for quoting a correct formula and attempting to use it, even if there are small errors in the substitution of values. Where the formula is not quoted, the method mark can be gained by implication from correct working with values, but may be lost if there is any mistake in the working. Eact answers Eaminers reports have emphasised that where, for eample, an eact answer is asked for, or working with surds is clearly required, marks will normally be lost if the candidate resorts to using rounded decimals.

7 .(a) (b) (b) Way n rr4 r r4 n r r4 r r r n n n n4 r 4 n n4 7 n n4 n n n n nn n n n n n n n7n 5 n n 4 * 7 n n4 7 n4n nn4 7nn44n84 n n4 7n 49n844n84 n n 4 n7n 5 n n 4 * Correct partial fractions. Can be seen in (b) give B for that. Attempts at least the first terms and at least the last terms as shown.(may be implied by later work) Must start at and end at n : Identifies their four fractions that do not cancel. If all terms are positive this mark is lost. A: Correct four fractions Attempt to combine at least fractions, of which have a function of n in the denominator and epands the numerator. As a minimim, the product of linear factors must be epanded in the numerator. cso Must be factorised. If worked with r instead of n throughout, deduct last mark only. Attempt to combine at least fractions, of which have a function of n in the denominator and epands the numerator. Min as above cso B () A A (5) Total 6 A

8 and attempt to solve correctly May be solved as an 80.. inequality, 6 Both (ie critical values seen) A and attempt to solve correctly May be 70.. solved as an inequality Both (critical values seen) Accept awrt.67 A, 6 Must be strict inequalities. Accept awrt.67 A either correct, A both correct. But give AA0 if both correct apart from seen somewhere in the final answers. Give AA0 if both correct and etra intervals seen A, A (6) Total 6 If no algebra seen (implies a calculator solution) no marks. With algebra: Squaring and reaching a quartic = 0 Attempt to factorise and obtain at least one solution for. Coefficient of 4 and constant term correct for their quartic. A Any correct values A All 4 correct values Final A marks as above Accept set notation for the final A marks.,,,6 not,

9 . y 8 e dy y 8e 8e e d y 8e e 8e e 4 dy : k 8e e A: Correct differentiation : Correct use of the product rule dy k 8e e 8 e e K A: Correct second derivative with e e or e f(0) May only appear in the epansion B 7 f(0),f (0) y Alternative Methods: e... or!! y9... Attempt both f(0)andf (0) with their derivatives found above : Uses the correct Maclaurin series with their values. Accept or! in term A: Correct epression A A Acso (8) Total 8 : Subst corrrect epansion B: for... A: for bracket 9 9 : Binomial epansion up to at least the squared term, or! With squared term 9 9! 9 9 A: Correct epansion ie contents of bracket correct B A A Remove all brackets y By implicit differentiation: For the first 4 marks (rest as first method) y 8 e dy dy d y A y e A + y =e d : Combine terms and obtain a term quadratic A: Correct epression with or without y = A

10 4.(a) (b) 6 Ignore any imaginary parts included in cos 6 Re[(cos i sin ) ] their epansion (cos isin ) c 6cis5ci s 0ci s 5ci s 6ci s i s Attempt to epand correctly or only show real terms (May be implied) Often seen with powers of i simplified. If is n seen, but becomes i n s n (oe) later, deduct the final A mark of (a) even if no further errors cos6 c 5c s 5c s s : Attempt to identify real parts. These M marks may be awarded together A A: Correct epression 6 4 c 5 c ( c ) 5 c ( c ) ( c ) Correct use of s c in all their sine terms cos 6 c 5c 5c 5 c ( c c ) 4 6 ( c c c ) 6 4 cos 6 cos 48cos 8cos * ( cos6 must be seen somewhere) Acso (5) 6 4 : Uses part (a) to obtain an equation 64cos 96cos 6cos 0 in cos 6 cos 6 0cos 6 or 0.5 A: Correct underlined equation A 5 7 cos6 6,, : Valid attempt to solve cos6 k, k leading to 5 7,, Can be degrees A correct answers A rd correct answer with no etras within the range, ignore etras outside the range. Must be radians Answers in degrees or decimal answers score A0A0 AA (5) Total 0

11 5.(a) d y dy 0y 7e m m0 0 m... Form and solve the au equation m i A y e AcosBsin y= not needed May be seen with i i or e y A Be instead of A y ke, y ke, y ke y twice ke and attempt to differentiate e kk0k 7e k A y e AcosBsin i i or y Ae Be e (b) 0, y0 A Must be and have y =...Ignore any attempts to change the second form. (But see note at end about marking (b)) ft, so y = their CF + their PI Bft (NB A on e- PEN) (6) Uses 0, y 0 in an attempt to find A : Attempt to differentiate using the y e AcosBsin product rule, with A or their value of A e Bcos Asin A: Correct derivative, with A or their value of A A 0, y 0 B 0 : Uses 0, y 0 and their value of A in an attempt to find B A A: B = 0 ye cos oe cao and cso A (6) Total Alternative for (b) using i i y Ae Be e 0, y0 to get an equation in A and B y' i Ae i i i Be e 0, y 0 AB 0 from real parts and A B 0 from imaginary parts So A B May come from the real part of their derivative instead : Attempt differentiation using chain rule A: Correct differentiation : Uses 0, y 0 and equates imaginary parts to obtain a second equation for A and B and attempts to solve their equations A: A B A A i i e e y e A: Ignore any attempts to change. A Some may change the second form in (a) before proceeding to (b). If their changed form is correct, all marks for (b) are available; if their changed form is incorrect only M marks are available.

12 6.(a) 4i z 8i w i z i Method :Substituting z i at the start 4 i i 8i w i i i Substitutes for z : Attempt to epand numerator and denominator 4 i i 8i w i i i A: Correct epression 8 i 8 4 i : Multiplies numerator and. denominator by the conjugate of 4 i 4i their denom. No epansion needed A: Uses correct conjugate (not ft) 40i 8 cso Award only if final answer is 6 correct and follows correct working NB: The B mark appears first on e-pen but will be awarded last A A B (6) Method : if they proceed without y = (substitution may happen anywhere in the working) i z8i 4i iy 8i w Substitutes for z i i i iy i 4 i4 ii y8i y y y y y i i i i i Attempt to epand numerator and denominator i Correct epression A 44y 4y48 i y y i y y i y y i : Multiplies numerator and denominator by the conjugate of their denom. No epansion needed. A: Uses correct conjugate. (not ft) 6 6y y8 00yi 8 8y 44y cso Correct answer using y = 40i 8 Award only if final answer is B 6 correct and follows correct working NB: The B mark appears first on e-pen but will be awarded last (6) A

13 NB: The order of awarding the marks here has changed from the original mark scheme, but they must still be entered on e-pen by their descriptors (M or A) Identifies u and v (Real and 8 40 imaginary parts) 6(b) u, v st M 6 6 May be implied by their working mark on and may be in terms of and y Substitutes for their u and v in the given equation. May be in terms of and y. May have a, b, c instead of their values (which may be chosen by the candidate if unable to do (a)) Combines to form a single correct fraction k = 5 or k =5 may (but need not) be seen eplicitly e-pen d nd M mark on e-pen A st A mark on e-pen A nd A mark on e-pen (4) Total 0

14 7.(a) dv v y y dy Way 4 Correct derivative B 4 d y d y dv y dv dv dy Or y y 6 4 d 4 4 y v y y y dv 4 4 dv v y y 6 : Correct use of the chain rule A: Correct equation d: Substitutes to obtain an equation in v and. A: Correct completion with no errors seen A da 4 Way dy 4 y v v dv Correct derivative B dy dy dv 4 d v : Correct use of the chain rule v dv A: Correct equation A 4 v dv v v 4 4 v dv dv v v v d: Substitutes to obtain an equation in v and. A: Correct completion with no errors seen Way (Working in reverse) v y dv dy y d d d 4 d : Correct use of chain rule y dy A: Correct epression for dv/ : Substitutes correctly for d v 4 dy y y 6 and v in equation (II) to obtain a D.E. in terms of and y only. A: Correct completion to obtain equation (I) with no errors seen 4 B: Correct derivative B d A A da

15 7(b) I e - e ln v 6 6 c 6 c y... y y c 6 4 d : e and attempt integration. If not correct, ln must be seen. A: : vtheir I 6 their I A: Correct equation with or without + c Include the constant, then substitute for y and attempt to rearrange to y =... or y =... with the constant treated correctly Or equivalent A da dd dep on both M marks of (b) A (6) Total

16 8.(a) Alt for the diff marks r tan rcos tan cos States or implies rcos : Attempt to differentiate cos sin, cos sin rcos or rsin A d A: Correct derivative : Attempt to differentiate using sec cos tan sin d product rule (dep on first ) A: correct (unsimplified) differentiation Set their derivative = 0 and attempt to 0 tan... d solve for d (Dependent on second M mark above), 4 r Both A NB: Use of rsin can score M0A0A0 ma (5) r d tan d (b) tan tan tan tan sec tan sec d lnsec tan 4 Use of limits needed r d and r tan No Epands and uses the correct identity Correct epression Need not be simplified, no limits needed. : Attempt to integrate at least one trig term integrated. Dependent on the second M mark A: Correct integration. Need not be simplified or include limits. A da R ln sec tan ln sec tan 4 4 Substitutes and their and subtracts 4 (Dependent on previous method marks in (b)) R ln Cao and cso A d (7) Total

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