0606 ADDITIONAL MATHEMATICS

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1 CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 0 series 0606 ADDITIONAL MATHEMATICS 0606/ 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 should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the October/November 0 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components.

2 Page Mark Scheme Syllabus Paper IGCSE October/November 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. Cambridge International Examinations 0

3 Page Mark Scheme Syllabus Paper IGCSE October/November a, b, c,, [] for each Using b 4ac, 9 4 (k + 4k + 8k 5 0 D for any use of b 4ac D for solution of their quadratic in k 5 k, for critical value(s, not necessary To be below the x-axis 5 k < [4] for 5 k < only dy Or: ( k + x dx dy when 0, x dx k + ( 9 9 y ( k ( k + ( k + To lie under the x-axis, y < ( k + + k + < 4 k + k + ( ( ( k ( 0 leading to 9 4(k + or equivalent then as for previous method for a complete method to this point. Cambridge International Examinations 0

4 Page 4 Mark Scheme Syllabus Paper IGCSE October/November sin θ cosθ cosθ sin θ + sinθ + sin cosθ ( + sin θ + cos cosθ ( + sin θ θ + cos ( + sinθ θ θ for dealing with the fractions, denominator must be correct, be generous with numerator + sinθ cos cos θ ( + sinθ ( + sinθ θ ( + sinθ D D for expansion and use of cos θ + sin θ for attempt to factorise [4] for obtaining final answer correctly Alternative solution: + ( + for dealing with the fractions sec θ + tanθ + tan θ + sec θ + tanθ ( D D for expansion and use of tan θ + sec θ D for attempt to factorise for obtaining final answer correctly 4 (i n (A If elements listed for (i, then they must be correct elements to get leading to n (A. If they are not listed and correct answer given then. (ii n (B 4 If elements listed for (ii, then they must be correct elements leading to n (B 4 to get. If they are not listed and correct answer given then. (iii A B {60, 40, 00, 40, 600 } Follow through on any sets listed in (i and (ii. Do not allow any repetitions. (iv A B {60, 40 } Follow through on any sets listed in (i and (ii. Cambridge International Examinations 0

5 Page 5 Mark Scheme Syllabus Paper IGCSE October/November (i 9x cosx (+c,, [] for 9x, for or cosx for cosx Condone omission of + c (ii 9 x cosx π π 9 9π cosπ π 8 π + cos π, [] for correct use of limits in their answer to (i for each term a b 6 f for substitution of x into f (x leading to a + 4b 8 0 for correct equation in any form f( f( 8a b ( a + 4 b for attempt to substitute x or x into f(x and use f( ±f( or f( ±f( for a correct equation in any form leading to 0a + 4b or equivalent 5 a, b D [6] D (on both previous M marks for attempt to solve simultaneous equations to obtain either a or b for both correct Cambridge International Examinations 0

6 y x Page 6 Mark Scheme Syllabus Paper IGCSE October/November (a (i 60 (ii 0 (b (i 94 (ii (iii 94 ( C C ( C C (i.e. 94 M W M 4W 4 for terms, at least of which must be correct in terms of C notation or evaluated. (i.e [] for any pair (must be evaluated for final answer 8 Or: 4M W C 4 C 40 8 Or: 5M W C 5 C 4 Or: 6M 8 C for terms, at least of which must be correct in terms of C notation or evaluated. for any pair (must be evaluated Total 67 for final answer 8 (i for correct shape for (, 0 or seen on graph for (, 0 or seen on graph for (0, 6 or 6 seen on graph or in a table [4] (ii 5, 4, [] for each 5 5 (iii k > or < k ( Cambridge International Examinations 0

7 Page 7 Mark Scheme Syllabus Paper IGCSE October/November x + 9 (a ( x n x + + 4x A,, 0 [] for differentiation of a correct product for each error (b (i dy dx ( x + x( x + x +, for differentiation of a quotient involving ( x + ( x + ( x + x + 4 ( x + ( ( x + x D [4] all correct unsimplified D for attempt to simplify for correct simplification to obtain the given answer Or: d y x x dx 0x ( x + + ( + ( ( x + ( ( x + x x + 4 ( x + (ii ( + c (iii x + 0x x , D, [] [] for differentiation of a product x + involving ( all correct unsimplified D for attempt to simplify for correct simplification to obtain the given answer x x for or 5 5 x + x + correct only, allow unsimplified. Condone omission of + c for correct application of limits in their answer to (b(ii Cambridge International Examinations 0

8 Page 8 Mark Scheme Syllabus Paper IGCSE October/November (i 0 or 4.47 (ii Grad AB, grad line y 4 (x (y x + 6 (iii Coords of C (x, y and BC 0 (x + (y 4 0 or Coords of C (x, y and AC 40 (x + + (y 40 Need intersection with y x + 6, leads to 5x 0x 5 0 or leads to 5y 40y 0 giving x, and y 0, 8 Or, using vector approach: 4 AB OC OC , D [] D, [6],, for attempt at a perp gradient for attempt at straight line equation, must be perpendicular and passing through B. allow unsimplified for attempt to obtain relationship using an appropriate length and the point (, 4 or (, for a correct equation D for attempt to solve with y x + 6 and obtain a quadratic equation in terms of one variable only for attempt to solve quadratic for each pair May be implied for correct approach for each element correct for each element correct Cambridge International Examinations 0

9 Page 9 Mark Scheme Syllabus Paper IGCSE October/November (a (i 4 (ii A 6 9, [] for any correct elements for all correct (iii B is the inverse matrix of A , [] Follow through on their A (b det C x(x ( (x x + det C x x + for attempt to obtain det C for this correct quadratic expression from a correct det C b 4ac < 0, 4 8 < 0 D D for use of discriminant or attempt to solve using the formula, or attempt to complete the square in order to show there are no real roots. No real solutions (so det C 0 [4] for correct reasoning or statement that there are no real roots. Cambridge International Examinations 0

10 Page 0 Mark Scheme Syllabus Paper IGCSE October/November (a (i f( 0 99, f(8 9 Min point at (0, or when y range y 99 [] for substitution of either x 0 or x 8, may be seen on diagram May be implied from final answer, may be seen on diagram Must have for, do not allow x (ii x 0 or equivalent Allow any domain which will make f a one-one function Assume upper and lower bound when necessary. (b (i ( g or x + x n 4 x + g 4 g e [] for complete method to find the form inverse function, must involve ln or lg if appropriate. May still be in terms of y. must be in terms of x (ii gh(x g(n5x gh(x 4e n5x for correct order for correct expression 4e n5x 0x 8, x [] for correct solution from correct working Or h(x g (8 n5x n5 for correct order for correct equation leading to x for correct solution from correct working Cambridge International Examinations 0

4037 ADDITIONAL MATHEMATICS

4037 ADDITIONAL MATHEMATICS CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Level MARK SCHEME for the October/November 0 series 07 ADDITIONAL MATHEMATICS 07/ Paper, maximum raw mark 80 This mark scheme is published as an aid to