0606 ADDITIONAL MATHEMATICS

Transcription

1 CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 04 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 May/June 04 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components.

2 Page Mark Scheme Syllabus Paper IGCSE May/June LHS sin cos + + for use of tan ( + ) + cos cos( + ) + ( + ) leading to sec D for attempt to obtain a single fraction D for use of + for finishing off Alternative solution: (- ) LHS + ( + )(-sin ) for use of tan cos cos( ) + for multiplication by ( ) ( ) + D D for use of + leading to sec Alternative solution: tan ( + ) + LHS + for finishing off for attempt to obtain a single fraction sin for use of tan + + cos cos( + ) + ( + ) D D for use of + leading to sec for finishing off Cambridge International Examinations 04

3 Page 3 Mark Scheme Syllabus Paper IGCSE May/June (i) a 4 b + c + 3 ( 3) for finding the modulus of either a or b + c for completion 4 5 λ + µ 7 3 4λ + µ 35 and 3 λ + µ 4 for equating like vectors and obtaining linear equations leading to λ 49, µ D D for solution of simultaneous equations for both 3 (a) (i) (iii) for each (b) (i) 0 4 k (4x 3) 4x + 8x 8 4x + x(8 4k) + 3k 8 0 b 4ac (8 4k) 6k k + 9 6(3k 8) b 4ac < 0, k 7k + < 0 critical values k 3, 4 3 < k < 4 D D for equating the line and the curve and attempt to obtain a quadratic equation in k D for use of b 4ac with k D for solution of a 3 term quadratic equation, dependent on both previous M marks for both critical values for the range 5 (i) dy xe dx x for e x, for x e x x e for k e x for x e (iii) e D D for correct use of limits for 6.8, allow exact value Cambridge International Examinations 04

4 Page 4 Mark Scheme Syllabus Paper IGCSE May/June (i) AB for at least 3 correct elements of a 3 matrix for all correct B for 7, for 5 3 (iii) 3 x 3 5 y for obtaining in matrix form x 5 y for pre-multiplying by B - x 0.5, y.5 for both 7 (i) y x ( + c) x + for each correct term when 5 5 x, y so + c leading to c for attempt to find + c, must have at least of the previous B marks Allow for c y x + x + When x, y 5 for using x in their (i) to find y d y 7 4 so gradient of normal dx Equation of normal y ( x ) 7 8 x + 34y 93 0 ( ) D for gradient of normal D for attempt at normal equation allow unsimplified ( fractions must not contain decimals) Cambridge International Examinations 04

5 Page 5 Mark Scheme Syllabus Paper IGCSE May/June (i) log p nlogv + logk for statement, but may be implied by later work. lnv lnp lgv lgp A,,0 for plotting a suitable graph for each error in points plotted Use of gradient n n.5 (allow.4 to.6) D D for equating numerical gradient to n (iii) Allow 3 to 6 D D for use of their graph or substitution into their equation. 9 (a) Distance travelled area under graph ( ) 480 for realising that area represents distance travelled and attempt to find area (b) for velocity of ms - for 0 Y t Y 6 for velocity of zero for their 6 to their 5 for velocity of ms - for 5 Y t Y 30 (c) (i) 6 v 4 t + When v 0, t 3 D for attempt at differentiation D for equating velocity to zero and attempt to solve 6 a ( t + ) for attempt at differentiation and equating to 0.5 with attempt to solve 0.5 ( t + ) 6 t 7 Cambridge International Examinations 04

6 Page 6 Mark Scheme Syllabus Paper IGCSE May/June (a) digit even numbers digit even numbers digit even numbers Total 8 (b) (i) 3M 5W 35 4M 4W 75 5M 3W 0 Total 40 for addition to obtain final answer, must be evaluated. or C 8 6M W 7M W or: as above, final for subtraction to get final answer Oldest man in, oldest woman out and vice versa 0 C 7 40 Alternative: man out woman in 6 men 4 women, for 0 C 7, for realising there are identical cases 6M W : 6 C 6 4 C 4 6 5M W : C5 4 C 36 4M 3W : 6 C 4 4 C M 4W : 6 C 3 4 C 4 0 Total 0 There are identical cases to consider, so 40 ways in all. All separate cases correct for for realising there are identical cases, which have integer values Cambridge International Examinations 04

7 Page 7 Mark Scheme Syllabus Paper IGCSE May/June (a) 5 sin x + 3cosx 0 tanx 0.6 x 49, 39 x 74.5, 64.5 D, In each case the last A mark is for a second correct solution and no extra solutions within the range for use of tan D for dealing with x correctly for each Alternatives: sin( x + 3 ) 0 or cos( x 59 ) 0 for either, then mark as above (b) cot y + 3cosecy 0 (cosec y ) + 3cosecy 0 for use of correct identity cos ec y + 3cos ecy 0 ( cosecy )(cosecy + ) 0 One valid solution cos ecy, sin y y 0, 330, for attempt to factorise a 3 term quadratic equation for each Alternative: cos y sin y sin y leads to sin y 3sin y 0 and sin y only cos y for use of cot y and sin y cos ecy sin y for attempt to factorise a 3 term quadratic equation y 0, 330 (c) 3 cos( z +.) cos( z +.) 3 ( z +.) z 4.4, , 5.44, 7.4 for correct order of operations to end up with 0.84 radians or better for one of 5.44 or 7.4 (or better) for each valid solution Cambridge International Examinations 04