0606 ADDITIONAL MATHEMATICS

Transcription

1 PAPA CAMBRIDGE CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 0 series 00 ADDITIONAL MATHEMATICS 00/ Paper, maximum raw mark 0 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 0 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components.

2 Page Mark Scheme Syllabus Paper cos cos A + ( + sin A) ( + sin A) A + + sin A + sin ( + sin A) ( + sin A) ( + sin A) A M M DM M for obtaining a single fraction, correctly M for expansion of ( ) + sin A and use of identity DM for factorisation and + sin A factor cancelling of ( ) sec A A A for use of final answer A sec and Alternative: ( sin A) + sin + ( + sin A)( sin A) ( sin A) + sin A sin A sin A ( ) sin A + sin A sin A + A M M M M for multiplying first term by sin A sin A M for expansion of ( sin A)( + sin A) and use of identity M for simplification of the terms sec A A A for use of final answer A sec and (a) (i) B (i) B (b) (i) (iii) 9 B B B Cambridge International Examinations 0

3 Page Mark Scheme Syllabus Paper (i) B B for shape B B for y (must have a graph) B B for x 0. and (must have a graph) Maximum point occurs when y M M for obtaining the value of y at the maximum point, by either completing the square, differentiation, use of discriminant or symmetry. so k > A Must have the correct sign for A Ignore any upper limits a sinx dx 0 cosx a 0 d x cos a cos a 0. a, a 9 B,B M A M A B for kcosx only, B for cosx only M for correct substitution of the correct limits into their result A for correct equation M for correct method of solution of equation of the form cos ma k A allow 0.9, must be a radian answer (i) x y leads to x + y B, B DB y B for, B for, B for dealing with indices correctly to obtain given answer x y 7 9 can be written as x + y 0 B B y B for either 7 or B for x + y seen Solving x + y and x + y 0 leads to x, y M A M for solution of their simultaneous equations, must both be linear A for both, allow equivalent fractions only Cambridge International Examinations 0

4 Page Mark Scheme Syllabus Paper (a) YX and ZY B,B B for each, must be in correct order, (b) 9 B A, M M for pre-multiplication by A - 9 B,B B for, B for 9 or 7 DM A DM for attempt at matrix multiplication A allow in either form Alternative method: a c b d 9 M M for a complete method to obtain equations Leads to a c, b d 9 a + c, b + d Solutions give matrix A,,0 M for each incorrect equation M for solution to find unknowns 9 or 7 A A for a correct, final matrix Cambridge International Examinations 0

5 Page Mark Scheme Syllabus Paper 7 (i) θ sin or θ, 0. or better + cosθ M M for a complete method to find either θ or θ θ.9 or better A Answer given. or using areas 7 sinθ oe sin θ 0.99, θ. or. 9 M A M for using the area of the triangle in different forms A for choosing the correct angle. Arc length (.9) (.97 or.7) M A M for correct attempt at a minor or major arc length A for correct major arc length, allow unsimplified Perimeter + (.9). 7 A A for.7 or better (iii) Area (.9) + sin. 9 M,M M for correct attempt to find area of major sector 7., 7., awrt 79 A M for correct attempt to find area of triangle, using any method Alternative:.9 sin. 9 Area ( ) M for attempt at area of circle area of minor sector M for area of triangle Cambridge International Examinations 0

6 Page Mark Scheme Syllabus Paper (a) (i) 70 B 0 B (iii) Starts with either a or a : ways B allow unevaluated Does not start with either a or a : 9 ways (i.e. starts with or ) B allow unevaluated Total B must be evaluated Alternative : Ends with a, starts with a, or : 7 ways Ends with a, starts with a, or : 7 ways Total B B B Alternative : ( ) or ( P ) ( P ) 0 P B B B for correct expression seen, allow P notation Alternative : P P P or B B Allow P notation here, for B (b) With twins : C ( 0) B Without twins: C ( 00) B Total: 9 B Alternative: C 9 ( C ) B,B B B for C...,, B for C Cambridge International Examinations 0

7 Page 7 Mark Scheme Syllabus Paper 9 (i) 000 h or r h 000 r A rh + r B 000 A r + r r M A M for substitution of h or rh into their equation for A A Answer given da r dr r d A When 0, r 000 dr B, B M B for each term correct M for equating to zero and attempt to find r leading to A 9, 90 d A 000 +, d r r which, is positive so a minimum. M A B M for substitution of their r to obtain A. A for 90 or awrt 9 B for a complete correct method and conclusion. Cambridge International Examinations 0

8 Page Mark Scheme Syllabus Paper 0 (i) 0 i + j Velocity ( i + j) M A M for ( i + j) Alternative : 0 i + j 0 + M M for working from given answer to obtain the given speed Showing that one vector is a multiple of the other, hence same direction A A for a completely correct method Alternative : +, k, so k i + j, Velocity ( ) Velocity 0 i + j M A M for attempt to obtain the multiple and apply to the direction vector A for a completely correct method Alternative : Use of trig: tan α, α 7. Velocity cos 7. i + sin 7. j M M for reaching this stage Velocity 0 i + j A A for a completely correct method Position vector ( 0i + j) or 0 i + 9j B Allow either form for B (iii) ( 0 i 9j) + ( 0i + j)t + oe M A M for their ( ii) ( 0 i + j)t ( 0 i + j) ( t + ) A correct answer only + or (iv) ( 0 i j) + ( i + 0j)t + oe B (v) 0 + 0t 0 t or 9 + t + 0t M M for equating like vectors t. or : 0 A A Allow for t. Position vector i + j DM DM for use of t to obtain position vector A A cao Cambridge International Examinations 0

9 Page 9 Mark Scheme Syllabus Paper (a) ( tan x + ) 0 tan x tan x 0, x 0, 0 tan x, x 0. B,B B B for each, must be from correct work (b) ( sin y) sin y 0 sin y + sin y 0 ( sin y ) (sin y + ) 0 sin y, y 0, 0 M A,A M for use of correct identity and attempt to solve resulting term quadratic equation. sin y, y 70 A (c) cos z M M for dealing with sec correctly and obtaining or.0 z z or 0.7 or better A z M M for obtaining a second equation z their oe z or. or better A Cambridge International Examinations 0