0606 ADDITIONAL MATHEMATICS

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1 CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME f the May/June 05 series 0606 ADDITIONAL MATHEMATICS 0606/ 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 ds not indicate the details of the discussions that took place at an Examiners meeting befe 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 Rept f Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes f the May/June 05 series f most Cambridge IGCSE, Cambridge International A and AS Level components and some Cambridge O Level components. IGCSE is the registered trademark of Cambridge International Examinations.

2 PagP ge MarM k Schemee CamC mbridgee IGCSE May/June 05 Syllabus 0606 Paper Abbreviations awrt caoo depp FTT isww rott SCC soii ww www answers which round to crectt answer only dependent followw through afterr errr igne subsequent wkw king equivalent rounded truncatedd Special Casee seenn r implied without wrong wking (i) A C B 5 B,,,,0 crectly placed in Venn diad gram;,,, 6 crectly placed;,, 0, 7, 9, 0 crectlyy placp ced;, 5 crectlyy placed (ii) ft cc rrect crect ft their (i), provided nonn n-zero ( iii) {, 6} ft cc rrect crect ft their (i), provided nott thet empty set (i) [ P = ] and [Q =] (0 00) B o [P =] andd o [Q =] (00 0) o if onee err mam ay be writtenn as an unevaluated product; B0 B if choice of o P and Q offered (ii) ( ) 760) B mum ust have brackets and must not have comc mmas; must be a by matm trix; must be b fromf crect prodp duct; wking may be sees en in (i) o f any two elements crect ( iii) The total (amount of revenue) frof om alll (three) flights. do d not accept, e.g. The totall amount fromm eace ch flight; ; mustm be a comment not just a figuf ure; must not n contain a contradictionn Cambridge International Examinations 05

3 Page Mark Scheme Syllabus Paper Cambridge IGCSE May/June (i) D cao allow a = and b = Alternative method: ( 6a + 5b) + ( a + 6 ) = b 6a + 5b = 6 a + 6b =5 D a = and b = + 5 (ii) AC = = their their crect crect ft expansions, using Pythagas with and their BC cao igne attempts to square root after crect answer seen (i) cos ( x ) = soi Alternatively sin ( y ) = soi rad.(00 ) rad isw with reference axis indicated by comment, e.g. to the bank upstream, etc. clearly marked on a diagram (0) rad isw with reference axis indicated by comment, e.g. to the perpendicular with the bank, etc. clearly marked on a diagram If M0 then SC f an unsuppted answer of.9. rad rad with reference axis indicated by comment, e.g. on a bearing of from Nth clearly marked on a diagram Cambridge International Examinations 05

4 Page Mark Scheme Syllabus Paper Cambridge IGCSE May/June (ii) Speed = 9 = 5 sin. tan. cos. tan. + cos..6(0 ) rot to me figs. [m/s] soi 0 time = their to 5. (seconds) 5 Substitution of either x y into equation of curve and brackets expanded 6 (a) ( x ) x 5x + [= 0] y y + [= 0] Solve their -term quadratic x = and isw y = and isw log 6 = log log6 = x..6(9 ) rot to me figures isw 0 Or Distance = = 07.(K) sin. soi time their 07.K = igne subsequent rounding attempted conversion to, e.g. minutes but A0 if answer spoiled by continuation of method if no wking, so B0 M0, then allow B f an answer 5.66 to 5. condone one sign err slip in either equation of curve expansion of brackets; condone omission of = 0, BUT x y must be crect dep on a valid substitution attempt x = y = not from wrong wking x = y = not from wrong wking if no wking, allow full marks f fully crect answer only. 6 x log 6 = log x log 6 log 6 = log log crect answer. implies Cambridge International Examinations 05

5 Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE May/June (b) Method y 6y log = log 6y y = B B if at most one err omitted step if at most two errs omitted steps not from wrong wking Method log + log y + log + log + log y 6 log log y = log B,,,0 LHS terms log y = log + log y; log = log ; log 6 y = log + log y; log 6 y = 6 log log y; RHS term log = log y = not from wrong wking 7 n ( n )( n )( n )( ) n( n )( ) = 0 better M condone omitting the fact of n and/ n ; must have dealt with factials M if one slip/omission if two slips/omissions n 5n [= 0] (n + ) (n ) [= 0] n = only f and f n n ( n ) ( ) [ x ] seen ( n )( n )( n ) ( ) [ x ] seen equivalent must be -terms, e.g. n 5n = any valid method of solution f their -term quadratic A0 if also given as a final solution, i.e. not discarded If zero sced, allow SC f n = unsuppted without crect method Cambridge International Examinations 05

6 Page 6 Mark Scheme Syllabus Paper Cambridge IGCSE May/June Method (Separate areas subtracted) [ x = x = ] 7 soi B C x 6x ( ) x 6x+ 0 dx= + 0x Crect crect ft substitution of limits 0 x 6x and their 7 into their + 0x ( 0 + 7) 7 ( 7) 7 7 x ( x+ 0) dx= 0x 0( 7) 0 + = their to sf 57.6(6...) 6 6 rot to figs isw Method (Subtracting and using integration once) M D B f at least one term crect dep on at least being earned; evidence of substitution must be seen in their integral which must be at least two terms; condone omission of lower limit; f (their 0 + their 7) their 7 f ( ) x x + 0 dx = + 0x dep on a genuine attempt to integrate the equation of the curve; must be their area trapezium/under the line their attempt at area under curve from full and crect wking with no omitted steps [ x x ] 7 B = = soi c ( x 7x) + x 7x + dx x 7x M condone omission of dx M f px qx ( px + qx) dx = + either with p = ± q = ±7 Crect crect ft substitution of limits 0 and their 7 x 7x into their to sf 57.6(6...) 6 6 rot to figs isw M px qx dx = + with non-zero constants p and q, with p ± and q ±7 dep on a valid integration attempt; evidence of substitution must be seen; condone omission of lower limit; f ( px + qx) from full and crect wking with no omitted steps Cambridge International Examinations 05

7 Page 7 Mark Scheme Syllabus Paper Cambridge IGCSE May/June (i) 0 = m + soi m = 0 m soi 0 [ = ] (ii) 0 y (iii) R = soi (, 7) x = and y = 7 y = x + if y = 7 only stated, provided that x = is soi in wking allow both marks if M0 then f y = 7 only with no wking (iv) Use of m m = with their m from (i) y 0 = their ( x ) may be implied by perpendicular gradient seen in equation their x + c and y + x = isw 0 = their + c allow f crect equation with integer cfficients in any simplified fm (v), their isw,ft ft their y Q f 0+, seen (vi).5 cao B not from wrong wking f any crect method with crect codinates 0 (a) y B,,0 crect sinusoidal/reflected sinusoidal shape, all above x-axis with intent to have all maximum points of equal height; O x maximum points of intended equal height only over 0 to 60; all max points clearly at y = ; cusp at 0 Cambridge International Examinations 05

8 Page Mark Scheme Syllabus Paper Cambridge IGCSE May/June (b)(i) [ hg( x) = ] ln( x ) e + fully crect and completion to [ hg( x ) = ] x Alternative method y = ln(x ) and change of subject to x, fully crect and comment that h( x) = g ( x) (ii) y = h(x) y = g(x B,,0 crect shape; marked on the y-axis (0, ) stated close by; curve with positive gradient in first quadrant only O (iii) x 0 [ 0, ) not domain 0 (iv) y [, ) ( ) h x, h etc. (i) h : h soi h h : soi h h h h h must be in the numerat of the expression f this mark; expand and simplify to h h + 6h AG (ii) h h + 6 their h h + 6 = 0 and attempt to solve only A must be a -term quadratic; must be an attempt at a derivative f h = and allow.67.66(6...) rot to me figs f Cambridge International Examinations 05

9 Page 9 Mark Scheme Syllabus Paper Cambridge IGCSE May/June (i) = 0 crect unsimplified fm, e.g. 5( ) + 6( ) ( ) 6 = 0 5( ) + 6() ( ) 6 = 0 crect synthetic division (ii) Substituting x = into 5x + 6x x crect synthetic division crect answer implies ; must be explicitly identified as answer if using synthetic/long division methods by e.g. circling (iii) (x )(5x + 6x x 6) soi by inspection division; may be implied by e.g. (ax + b)(5x + 6x x 6) and a =, b = seen in later wk comparing cfficients Multiply out (x ± )(5x + 6x x 6) and compare cfficients of x x to quartic multiply out, e.g. (ax + b)(5x + 6x x 6) and compare cfficients of x x to quartic p = q = 5 crect p q implies ; crect p and q www implies Cambridge International Examinations 05

Cambridge International Examinations Cambridge International General Certificate of Secondary Education. Published

Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 07/6 Paper 6 (Extended) May/June 07 MARK SCHEME Maximum Mark: