4037 ADDITIONAL MATHEMATICS

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1 CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Level MARK SCHEME for May/June 03 series 4037 ADDITIONAL MATHEMATICS 4037/ Paer, maximum raw mark 80 This mark scheme is ublished as an aid to teachers and candidates, to indicate requirements of examination. It shows basis on which Examiners were instructed to award marks. It does not indicate details of discussions that took lace at an Examiners meeting before marking began, which would have considered accetability of alternative answers. Mark schemes should be read in conjunction with question aer and Princial Examiner Reort for Teachers. Cambridge will not enter into discussions about se mark schemes. Cambridge is ublishing mark schemes for May/June 03 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level comonents and some Ordinary Level comonents.

2 Page Mark Scheme Syllabus Paer Mark Scheme Notes Marks are of following three tyes: M A B Method mark, awarded for a valid method alied to roblem. Method marks are not lost for numerical errors, algebraic slis or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; formula or idea must be alied to secific roblem in hand, e.g. by substituting relevant quantities into formula. Correct alication of a formula without formula being quoted obviously earns M mark and in some cases an M mark can be imlied from a correct answer. Accuracy mark, awarded for a correct answer or intermediate ste correctly obtained. Accuracy marks cannot be given unless associated method mark is earned (or imlied). Accuracy mark for a correct result or statement indeendent of method marks. When a art of a question has two or more "method" stes, M marks are generally indeendent unless scheme secifically says orwise; and similarly when re are several B marks allocated. The notation DM or DB (or de*) is used to indicate that a articular M or B mark is deendent on an earlier M or B (asterisked) mark in scheme. When two or more stes are run toger by candidate, earlier marks are imlied and full credit is given. The symbol imlies that A or B mark indicated is allowed for work correctly following on from reviously incorrect results. Orwise, A or B marks are given for correct work only. A and B marks are not given for fortuitously "correct" answers or results obtained from incorrect working. Note: B or A means that candidate can earn or 0. B,, 0 means that candidate can earn anything from 0 to.

3 Page 3 Mark Schemee GCG CE O LEVEL May/June 03 Syllabus 4037 Paer The following abbreviations may bee used in a marm rk scheme or used on t scrits: AG Answer Given on question aerr (so extraa checking iss neededd to ensure that detailed working leading to result is valid) BOD Benefit clear) off DouD ubt (allowed when validity of a solution may not be absolutely CAO Correct Answer Only is allowed) (emhasisingg that at no "followw through" from a revious error ISW MR Ignore Subsequent Working Misread PA Prematuree AA roximation accurate) (resulting in basically correct workk that is insufficiently SOS See Or SoluS ution ( canc ndidate makes a betterr attemt at t same question) Penalties MR A enalty of MR is deducted from A or B marks when data of a question or art questionn are genuinely misread and object and difficulty of question remain unaltered. In thiss case all A anda B markss n become "follow through " marks. MRR is i not alied when candidate misrm reads his own figures this is regarded as an error inn acca curacy. OW, This is deducted from A or o B markss whew en essential working is omitted. PA S EX This is deducted from A or o B markss inn case of rematuree aa roximation. Occasionallyy used for ersistent slacknesss usually discussedd at a meeting. Alied too A or B marks when extra solutions aree offered to a articular eque uation. Again, thiss iss usually discussed at e meeting. Cambridge International Examinations 03

4 Page 4 Mark Scheme Syllabus Paer 8 3 m = or 5 soi 4 Y 3 = ir 5(X ) or Y 8 = ir 5(X 4) M or 8 = 4m + c and 3 = m + c subtracting/substituting to solve for m or c, condone one error or 3 = ir 5 + c or 8 = ir c y = (ir m) x + (ir c) or M or using ir m or ir c to find ir c or ir m, without furr error y = (ir m) (x ) + 3 or y = (ir m) (x 4) + 8 y = (5x ) or y = (5(x ) + 3) or y = (5(x 4) + 8) cao, isw M A ir m and c must be validly obtained (a) ( + ) ln 3 = ln 0.7 ln0.7 lg0.7 = or = ln3 lg3.3 cao M M A or + = log or 0.7 ln 3 = ln or = log or ln 3 = ln ln 3 allow M for = log correct answer only scores B3 5 6 (b) x y or 5 a =, b = 6, c = B3 B for each comonent 3 (a) (i) A and E B mark for each B for extra, B0 if or more extras (ii) C and D B mark for each B if extra, B0 if or more extras (b) 5 y B (, 0), (, 3), (3, 4) or B for two oints correct and joined or for three oints correct but clearly not joined 5 x

5 Page 5 Mark Scheme Syllabus Paer 4 (i) uuur uuur uuur OC = OA + AC or uuur uuur uuur uuur OB OA = 3( OC OA) soi uuur uuur uuur ± ( 8i 9j) o.e. or OC = OA + OB 3 3 4i j + ( ir(8i 9j) ) o.e. or 3 (4i j) + (i 30j) 3 3 0i 4j cao (ii) OC = ir0 + ir( 4) soi j isw 6 ( i j ) or ( i ) 5 AX = 45 AX = 3 5 ( x) ir 45 soi 5 ( 5 + ) = ( x) ir 45 or better Correctly divide ir equation by ir 5 or ir 45 and rationalise denominator comletion to www B B M A M A FT B B M M M A uuur or 3AC = 3( c 4) i + 3( c + ) j o.e. soi or 3(c 4) = ir 8 and 3(c + ) = ir ( 9) condone OC = ir0 + ir(4) FT ir xi + yj o.e. may be imlied by 3 5 may be seen later may be imlied by e.g. summation of rectangle and two triangles or correctly multily both sides of ir equation by ir 5 or ir 45 and obtain a rational coefficient of x soi answer only does not score

6 Page 6 Mark Scheme Syllabus Paer 6 (i) arc AB = r π chord AB = r with justification and summation and comletion to given answer B B 3 + π r (ii) r =.7 π ir r sin π 3 awrt 4.6 B M3 A must be seen; accet awrt.7 may be imlied for examle or M for ir r π or and M for ir r sin π o.e. 3 or and M for Area Sector Area triangle attemted 7 (i) k( 3 5x) 5 (3 5x) or better, isw M A (ii) x (ir cos x) + (ir x) sin x x cos x + x sin x isw M A clearly alies correct form of roduct rule (iii) Quotient rule attemt: Product rule attemt: d ( tan x) = sec x d ( e x) e x + = clearly alies correct form of quotient rule x x ( + e )( ir sec x) ( ir e ) tan x x ( + e ) ( + e x )sec x e x ( + e ) x tan x isw B B M A d ( tan x) = sec x d x ( + e ) = e x ( + e x tan x (ir e x ( + e x ) ) + ( + e x ) (ir sec x) tan x ( e x ( + e x ) ) + ( + e x ) (sec x) )

7 Page 7 Mark Scheme Syllabus Paer 8 (i) 6 y = ( x+ 6) + 6 o.e. soi M or y 6 = ( x ) y = x + 5 isw A (ii) Use of m m = y 6 = (ir )(x ) or better, isw M A FT or y = (ir ) x + c, c = ir 0, isw (iii) ( + 6 ) + ( y ) = 0 x o.e. Substitute y = ir ( x + 0) Solve ir quadratic (0, 0) and (4, ) o.e. only B M* M de* A or (x ) + (y 6) = ( 0 ) o.e. or ( 80 ) + ( ) + ( y 6) ) x = 0 or identifying one oint by insection from length equation and testing it in equation of BC or vice versa or identifying second oint by insection from length equation and testing it in equation of BC or vice versa answer only does not score 9 (a) 4 = k + c and 6 = 9 k + c o.e. c = 5 k = 9 M A A for two equations in k and c; may be unsimlified; condone one sli in one equation (b) (i) 79. or rot to 4 or more sf B (ii) e x + 5e x 4(= 0) or (e x ) + 5e x 4(= 0) o.e. factorise ir 3 term quadratic M M condone one error, but must be three terms or correct/correct ft use of formula or comleting square e x = 3 x = ln 3 or.(0) or.0986 rot to 3 or more sf as only answer from fully correct working A A ignore e x = 8 do not allow final mark if value given from e x = 8 if M0M0 n SC if e x = 3 is seen www and leads to x = ln3 or.(0) or.0986 rot to 3 or more sf

8 Page 8 Mark Scheme Syllabus Paer 0 (a) (i) y B shae; cosine curve ends must be aroaching a turning oint x B B B be centred on y = clear intent to have min at and max at 4 cycles (ii) 3 B (iii) 80 B (b) cosec x = soi sin x B or + tan x = cos x sin x = cos x or B or cosec x = + soi o.e. B or + or better

9 Page 9 Mark Scheme Syllabus Paer (i) dy 4 = 3 3( x 4) o.e. isw = (ir )(x 4) ir ( 5) o.e. = (x 4) 5 o.e. isw B + B M A if M0 n SC for (x 4) 5 + one or term (ii) Verifies = 0 when x = 3 and x = 5 3 or solves ( 4) = to obtain 3 and 5 x Shows that x = 3 y = 8 and x = 5 y = 6 M A if M0 n SC for verifying or correctly solving to find one x coordinate and showing that it gives rise to corresonding y coordinate (iii) x = 5 (=) > 0 min or x = 3 (= ) < 0 max Both correct cao M A or, using first derivative e.g. x min at x = 5 or x max at x = 3 (iv) 3x ( x 4) ( + c) o.e. isw B + B may be unsimlified (v) ir 3(6) 3(5) (6 4) (5 4) to 3 or more sf or or 6 cao 8 8 M A

4037 ADDITIONAL MATHEMATICS

4037 ADDITIONAL MATHEMATICS UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Level MARK SCHEME for the May/June 0 question paper for the guidance of teachers 407 ADDITIONAL MATHEMATICS 407/ Paper, maximum raw mark