Mark Scheme (Results) Summer Pearson Edexcel Advanced Extension Award in Mathematics (9801/01)

Transcription

1 Mark Scheme (Results) Summer 05 Pearson Edexcel Advanced Extension Award in Mathematics (80/0)

2 Edexcel and BTEC Qualifications Edexcel and BTEC qualifications are awarded by Pearson, the UK s largest awarding body. We provide a wide range of qualifications including academic, vocational, occupational and specific programmes for employers. For further information visit our qualifications websites at or Alternatively, you can get in touch with us using the details on our contact us page at Pearson: helping people progress, everywhere Pearson aspires to be the world s leading learning company. 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 05 Publications Code UA040 All the material in this publication is copyright Pearson Education Ltd 05

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4 General Marking Guidance All candidates must receive the same treatment. Examiners must mark the first candidate in exactly 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. Examiners 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. Examiners should always award full marks if deserved, i.e. if the answer matches the mark scheme. Examiners 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 exemplification may be limited. When examiners are in doubt regarding the application of the mark scheme to a candidate s response, the team leader must be consulted. Crossed out work should be marked UNLESS the candidate has replaced it with an alternative response.

5 PEARSON EDEXCEL GCE MATHEMATICS General Instructions for Marking. The total number of marks for the paper is 00. 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) 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) d or dep dependent indep independent dp decimal places sf significant figures The answer is printed on the paper or ag- answer given or d The second mark is dependent on gaining the first mark 4. All A marks are correct answer only (cao.), unless shown, for example, 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.

6 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.

7 AEA June 05 Mark scheme Question Scheme Marks Notes y. (a) Both branches B Shape (Inc. cusp) 5 x B B () (b) x + 5 = so x = B or x 5 x 5 so x = (o.e.) A () [6]. (a) 0 so (x + ) is a factor Bcso () (b) x x5 x x xx. Position Asymptote and int with axes Correct equation with no ln Attempt to square. terms on RHS x x Prepare for final sq 0 x x (Accept x x o.e.) Acso 0x (x x ) Div attempt. At least correct 0 ( x)( x)( x ) terms of quadratic x = or A Correct factors and both roots Check : LHS = RHS = 0 so is not a solution B Must reject Check : LHS = 5 5 Attempts 0.5 in 4 RHS= 4.5 original or line Only award if (Only) solution is 0.5 A (8) check is in original [S- for treating 4 as + etc] [] cos x sin 78 LHS = sin x cos 78 Cot and tan to sin and cos cos x 78 sin x cos 78sec x sec 78 Use of cos(a + B) cos(x78) sin xcos x cos 78 sec78 sec x cos(x 78) sin x or cos(x78) cos(0 x) A x 780 x x = 4 x 7870 x x = x x x = 4 x x x = 44 ln5 x y > 0 & cusp on ve x-axis All required A A// [] Use of sinx and some cancelling Non-trig eqn in x Allow 0 + x Award A for each of these solutions found. Extras inside the range - i.e allow upto 4 answers. If more than 4 then deduct for each in range

8 Question Scheme Marks Notes 4. (a) 4 y Correct prep or y 4 dealing with 4 y y y... 8! 4! 4 y y y Allow o.e. for = A coefficients ALT (b) 5 4 y 4 4 y 4 y 4 y!! 5x x 5x 0 x [ x 4 ] 5 x... Let y 5x x so x x 45x (*) Acso () ALT 45xx 4x x ( a) 8 6 x x x (c) 5x x 4 so 5xx 4 x 5x 4 4 x x 5 4 x A () Clear use of bin for rd or 4 th terms. Condone missing st for the powers of 4 Some attempt to sub. for y Ignore higher order terms. Clearly attempt For st x Attempt factors or a correct equ Correct fact. Or cvs 4 x 4 and x 5x x 4 x4 x x or 4 A So x So < x < 4 5 or... So series is convergent for x A (6) (d) 4 5x x 45x... x = 8 64 = (o.e) A () [5] Solves nd eqn or correct interval A All correct for a suitable combined region A cso Some correct integration. Ignore limits Attempt both limits S+ for props of odd and even functions

9 Question Scheme Marks Notes 5. (a) 6 f( x) x x 6 f ( x) x or quotient rule Some correct diff (b) g(x) 8 (Accept y 8 (c) y 0 x 6 y =0 x = 8 8 a = 4 and b = (o.e.) AA ( 4, ) is OK (4) 8 or (, ] o.e.) Bft ft their value of b Values are not required. Graphs should not go beyond 4. Penalise once only S+ for values B B B () () Cor posit of g(x) [not crossing axes] Cor posit of g - (x) [not crossing axes] Intersection at roughly correct pt And labels 5 (d) y = g(x) gives x xy6 0 TQ in x (needn t = 0) Attempt formula or y y 64 x complete square. Leading to x =... A A for + or + or - Consideration of + and and take + Must have chosen + x x 64 g ( x) A S+ for good reason 8 8 Domain [x] (o.e.) [Not g ( x) ] Bft ft their (b) or b (5) (e) Simpler to do g(x) = x leading to x 6 x Write down a correct eqn (ft their g - ) and attempt to simplify to quad or quartic x 8 Solving T quad or T quartic x or 8 A () S+ for reason for - [6]

10 Question Scheme Marks Notes 6 (a) Attempt correct scalar product so, the lines are perpendicular 4 A () Correct calc and conclusion (b) 05 and 4 (or + = + ) Form suitable eqns 4, A In rd equation: LHS and RHS = so skew () Check and comment Solve i,j Should get: and Check: LHS = RHS = Solve i,k Get no solution e.g. 5 = 5 (c) Let R be a point on L then AR 8 5 and AR is to so 0 7 OX i j k Forming vector AR Attempt suitable scalar product. Must see some products A Solve for A Allow coord form (5) (d) 4, Attempt vector AX 4 AX, so AX 7 6 (Allow (e) Let M be midpoint of AB AM AX (unit vector in direction of AM 4 4 So AB 4 or AM (f) OB OA or ) A () L ) Suitable strategy A Correct calcs ( unit vector in direction of L ) Strategy 4 4 OB XAXB cos 0 cos Acao () [] A (5) o.e. Attempt a correct scalar product o.e. and set = to AX BX cos

11 Question Scheme Marks Notes 7 (a) dx x sec sec tan d sec tan A, Correct integral I d or sec cot d, = cosec cot d tan Prep to integrate 6 For coseco.e. A, = cosec, = (*) Changing limits 4 Acso and cso (5) (b) J cosec cot cosec d Suitable st step Condone sign slips J cosec cot cosec [ cot ] d Use of + cot J cosec cot cosec d J [Allow cosec cot d ] Correctly dealing A with nd int (c) Use of cosec d ln cosec cot B J... so, J ln cosec +cot cosec cot V x dx [= K ] B x Int of cosec Must come from their working (*) A Identify J and cso (6) Correct integral + limits (Ignore ) 5 sec K sec cot d (o.e.) Changes to int in 4 K cosec cot d cosec cot cot cosec d 4 cot cosec d cot cosec d cot cosec d Attempt int by parts Split nd int Award after nd K cosec cot J K K = f() +J - nk coseccot J 4 4 ln K 4 ln 7 so V ln A A Acso (8) [] Only award A marks once all the integration is completed Including Questions Mark Awarding of S and T marks, S For a fully correct solution that is succinct or includes an S+ point 4-7 S For a fully correct solution that is succinct or includes an S+ point 4-7 S For a fully correct solution that is succinct but has an S point 4-7 S For a fully correct solution that is slightly laboured but includes an S+ point 4-7 S For a score of n - but solution is otherwise succinct or contains an S+ point Maximum of 6 S marks ALL T For at least half marks on every question

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