4037 ADDITIONAL MATHEMATICS
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1 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 80 This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes must be read in conjunction with the question papers and the report on the examination. Cambridge will not enter into discussions or correspondence in connection with these mark schemes. Cambridge is publishing the mark schemes for the May/June 0 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.
2 Page Mark Scheme: Teachers version Syllabus Paper GCE O LEVEL May/June Mark Scheme Notes Marks are of the following three types: M A B Method mark, awarded for a valid method applied to the problem. Method marks are not lost for numerical errors, algebraic slips 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; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. Correct application of a formula without the formula being quoted obviously earns the M mark and in some cases an M mark can be implied from a correct answer. Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated method mark is earned (or implied). Accuracy mark for a correct result or statement independent of method marks. When a part of a question has two or more method steps, the M marks are generally independent unless the scheme specifically says otherwise; and similarly when there are several B marks allocated. The notation DM or DB (or dep*) is used to indicate that a particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier marks are implied and full credit is given. The symbol implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, 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 the candidate can earn or 0. B,, 0 means that the candidate can earn anything from 0 to. University of Cambridge International Examinations 0
3 Page Mark Scheme: Teachers version Syllabus Paper GCE O LEVEL May/June The following abbreviations may be used in a mark scheme or used on the scripts: AG BOD CAO ISW MR PA SOS Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid) Benefit of Doubt (allowed when the validity of a solution may not be absolutely clear) Correct Answer Only (emphasising that no follow through from a previous error is allowed) Ignore Subsequent Working Misread Premature Approximation (resulting in basically correct work that is insufficiently accurate) See Other Solution (the candidate makes a better attempt at the same question) Penalties MR A penalty of MR is deducted from A or B marks when the data of a question or part question are genuinely misread and the object and difficulty of the question remain unaltered. In this case all A and B marks then become follow through marks. MR is not applied when the candidate misreads his own figures this is regarded as an error in accuracy. OW, This is deducted from A or B marks when essential working is omitted. PA S EX This is deducted from A or B marks in the case of premature approximation. Occasionally used for persistent slackness usually discussed at a meeting. Applied to A or B marks when extra solutions are offered to a particular equation. Again, this is usually discussed at the meeting. University of Cambridge International Examinations 0
4 Page 4 Mark Scheme: Teachers version Syllabus Paper GCE O LEVEL May/June x ( k ) x ( k ) ( k+ 0) = 4( k + ) = 0 k = dy (or x ( k 0 ), x ( k ) dx = + + = + = ( k+ ) ( k+ )( k+ ) + k leading to k = ) x+ A = x + k+ 0 x+ k + (or ( ) ( ) [] for equating to zero and use of b = 4ac for solution for differentiation and attempt to equate to zero. for attempt to substitute in for x in terms of k, for y = 0 and for attempt at solution. for approach ( ) A= k+, A = k + for equating and attempt at solution ( ) k k + = +, leading to k = ) (or by completing the square y = ( x+ ( k+ ) ) ( k+ ) + ( k + ) ( ) k+ = k + leading to k = ) C a = (0) C 4 a = 6 a 9 (a) k =, m =, p = (b) (i) π (ii) There must be evidence of working without a calculator in all parts ( 4+ ) ( ) 4 (i) = + ( ) ( ) (ii) Area = ( 4 ) ( ) + + = 4+ BB [4] B B B [] for approach for equating last terms to zero and attempt to solve B for C a, B for 4 a C 9 for a relationship between the coefficients and attempt to solve B for each for attempt to rationalise and attempt to expand for attempt at area using surd form and attempt to expand (iii) Area = AC = ( 4+ ) + ( + ) =7 + 8 [6] for attempt at AC or AC in surd form, with attempt to expand University of Cambridge International Examinations 0
5 Page Mark Scheme: Teachers version Syllabus Paper GCE O LEVEL May/June (i) = 0 (ii) ( x )( x x+ 4) For ( x x 4) +, b < 4ac so only one real root of x = 0. 6 (i) lg y = ( x ) B [6] for substitution of x = 0. or attempt at long division attempt to obtain quadratic factor for correct quadratic factor for correct use of discriminant or solution of quadratic equation = 0, all correct with statement of root. B for gradient, for use of straight line equation (ii) Either b = + y = 0 x, x = 0 0 a = 00 Or lg y = lg a+ lg0 bx lg y = lg a+ bx, lg a = a = 00 b = B B [6] B for b = for use of powers of 0 correctly to obtain a for a for use of logarithms correctly to obtain a for a B for b = 0 = a 0 b Or ( ) ( ) 0 = a 0 b b =, a = 00 B, for simultaneous equations involving powers of 0 B for b =, for a = 00 7 (i) 4 C 6 = 00 B (ii) C C = (iii) 8 C C = 64 6 BB B BB B [7] B for 8 C4 or 6 C B for by 6 C or 8 C 4 B for 00 B for 8 C 6 or equivalent 8 B for 6 C or equivalent B for 64 University of Cambridge International Examinations 0
6 Page 6 Mark Scheme: Teachers version Syllabus Paper GCE O LEVEL May/June (i) y x 4 0 B B B B B for x = 0. B for x =. B for y = B for shape (ii) (, 9) B (iii) 0 y B B on shape from (i) x 4 B [7] B for a completely correct sketch θ 9 (i) OBA : θ + = π for using angles in an isosceles triangle (ii) π 9π = r r = for use of s = rθ π π (iii) Area = sin = (i) = 6 7 Magnitude =, unit vector 4 7 uuur uuur (ii) AC = AB uuur uuur uuur or AB+ BC = AB leading to uuur AC 6 = 0. uuur uuur uuur OC = OA + AC uuur uuur uuur uuur or OB OA = OC OB 4 leading to OC uuur = 6. (equivalent methods acceptable) [7] [7] for use of for use of method r θ or rs for subtraction r sinθ or other correct for attempt to find magnitude of their vector uuur for attempt to find AC may be part of a larger method uuur for attempt to find OC for each University of Cambridge International Examinations 0
7 Page 7 Mark Scheme: Teachers version Syllabus Paper GCE O LEVEL May/June (i) cosec x cos ecx = 0 ( θ )( θ ) cos ec + cos ec = 0 D for use of correct identity or attempt to get in terms of sin x D for attempt to solve leading to sin x =, x = 9., 60. o 80 their x (ii) tan y = 4 for attempt to get in terms of tan y =.4,.4 y =.7,.7, for dealing correctly with double angle 90 their y (iii) EITHER π π 4π z + =, 6 π π 4π π z = 6 6 π 7π z =, allow.7,.67 6, [] for dealing with order correctly and attempt to solve (i) dy 9x 4x dx = + d y when x =, = 0 dx tangent y =, A (0, ) D for differentiation and substitution of x = D for attempt at equation of tangent and coordinates of A (ii) B (0, ) At B, d y dx = normal y = x C (, 0) At D x + =, D (0, ) B B for B for attempt at normal and C, must be from differentiation and using correct point for attempt to obtain D, equating normal and tangent equations Area = 0, = 0 [0] for valid attempt at area University of Cambridge International Examinations 0
8 Page 8 Mark Scheme: Teachers version Syllabus Paper GCE O LEVEL May/June OR dy x x 9 dx = + When d y = 0, x =, dx P (, 4) Area = 8 x 6x + 9x dx x 9x = 8 x = = 4 4 B A,,0 D [0] for differentiation and equating to 0, can be using a product for attempt to solve for both x values for y coordinate B on y coordinate for area of rectangle for attempt to integrate each error D for application of limits University of Cambridge International Examinations 0
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