0606 ADDITIONAL MATHEMATICS

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1 CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the May/June 5 series 66 ADDITIONAL MATHEMATICS 66/ Paper, maximum raw mark 8 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 5 series for 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 y x Page Mark Scheme Syllabus Paper Cambridge IGCSE May/June 5 66 Abbreviations awrt cao dep FT isw oe rot SC soi www answers which round to correct answer only dependent follow through after error ignore subsequent working or equivalent rounded or truncated Special Case seen or implied without wrong working (i) 8 or π radians or. radians ( or better) B (ii) B (iii) (a) B y sin x all correct (b) B B for either starting at their highest value and ending at their lowest value Or a curve with highest value at y and lowest value at y completely correct graph (iv) B ( )( ) (i) tanθ ( + )( ) M cao A attempt to obtain tan θ and rationalise. Must be convinced that no calculators are being used Cambridge International Examinations 5

3 Page Mark Scheme Syllabus Paper Cambridge IGCSE May/June 5 66 (ii) sec θ + tan θ ( ) Alternative solution: AC ( + ) + ( ) sec θ ( + ) 8 + ( + ) M DM A M DM attempt to use sec θ + tan θ, with their answer to (i) attempt to simplify, must be convinced no calculators are being used. Need to expand ( + ) as terms A (i) 6 + 9x + x + 6 B,,, each error 6 x (ii) ( 6 9x + x ) + x x B expansion of x Terms needed 6 (9 6) + ( 9) 7 M A attempt to obtain or terms using their (i) Cambridge International Examinations 5

4 Page Mark Scheme Syllabus Paper Cambridge IGCSE May/June 5 66 (a) X k 8 k k B,, each incorrect element (b) Use of AA I 5 a 6 6 b 5 Any equations will give a, b M A,A use of AA - I and an attempt to obtain at least one equation. Alternative method : a b b a Compare any terms to give a, b M A,A correct attempt to obtain A - and comparison of at least one term. Alternative method : Inverse of M A,A reasoning and attempt at inverse 5 x x(x ) + x or y + y + y y + x x or y y 5 ( x )( x + ) or ( y 7)( y + 5) leading to x, x and 7 5 y, y Midpoint, Perpendicular gradient Perp bisector: y x ( y + x ) M DM A A B M M A equate and attempt to obtain an equation in variable forming a term quadratic equation and attempt to solve x values y values for midpoint, allow anywhere correct attempt to obtain the gradient of the perpendicular, using AB straight line equation through the midpoint; must be convinced it is a perpendicular gradient. allow unsimplified Cambridge International Examinations 5

5 Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE May/June (i) a 5 b f + 8 leading to a + b 6 f () a 5 + b 5 leading to a + b M A correct use of either f or f() paired correctly both equations correct (allow unsimplified) giving b 8 (AG), a (ii) ( )( 7x x + ) M,A M for solution of equations A for both a and b. AG for b. x M,A M for valid attempt to obtain g(x), by either observation or by algebraic long division. (iii) 7x x + has no real solutions as b < ac 6 < 56 8x ( x ) dy ( x + ) 7 (i) ( x ) ln When x, y ln oe ( x + ) M A M B A B use of b ac correct conclusion; must be from a correct g(x) or g(x) www differentiation of a quotient (or product) ln x all else correct correct differentiation of ( + ) for y value dy ln so gradient of normal is ln (allow numerical equivalent) M valid attempt to obtain gradient of the normal normal equation y + ln or y.9x., or (Allow y.9x. ) (ii) when x, y ln when y, ( ) ln x y x cao x ln Area ±.66 or ±.67 or awrt these or ( ln) M A M A attempt at normal equation must be using a perpendicular valid attempt at area Cambridge International Examinations 5

6 Page 6 Mark Scheme Syllabus Paper Cambridge IGCSE May/June (i) Range for f: y Range for g: y 9 B B (ii) x + y 5 g ( x ) + x 5 Domain of g : x 9 Alternative method: y + y + 9 x + 6 (9 x) y x (iii) Need g ( e ) x ( e + ) + 5 x x or 9e + e x x ( e )( e + 8) leading to e x + ± 6 so or e x so x ln x ln Alternative method: Using f ( x ) g (), g () leading to e x, so x ln M A B M A M DM M A M DM M A attempt to obtain the inverse function Must be correct form for domain attempt to use quadratic formula and find inverse must have + not ± correct order correct attempt to solve the equation dealing with the exponential correctly in order to reach a solution for x Allow equivalent logarithmic forms correct use of dealing with g () to obtain an x equation in terms of e dealing with the exponential correctly in order to reach a solution for x Allow equivalent logarithmic forms g (iv) x g ( x) 6e g (ln ) 96 B B B for each Cambridge International Examinations 5

7 Page 7 Mark Scheme Syllabus Paper Cambridge IGCSE May/June (i) d y x x + d y When x, for curve, gradient of line also so line is a tangent. Alternate method: x + x 5x + x + leading to x, so tangent at x d y (ii) When, ( x )( x ) x, x M A M A M A,A for differentiation comparing both gradients attempt to deal with simultaneous equations obtaining x equating gradient to zero and valid attempt to solve A for each (iii) Area ( + 9) d x x x x 87 x 5x x + + x or.8 B M A DM A area of the trapezium attempt to obtain the area enclosed by the curve and the coordinate axes, by integration integration all correct correct application of limits (must be using their from (ii) and ) Alternative method: ( x+ ) x 5 x + x+ d x Area ( ) + x 5x x 5x + 99 B M A DM A correct use of Y y attempt to integrate integration all correct correct application of limits (a) sin x sin x ( ± ) x,5,, M A,A using cosecx and obtaining sin x sin x A for one correct pair, A for another correct pair with no extra solutions Cambridge International Examinations 5

8 Page 8 Mark Scheme Syllabus Paper Cambridge IGCSE May/June 5 66 (b) ( sec y ) secy sec y secy (sec y + )(secy ) leading to cosy, cos y y 8, 5 y 7.5, 89.5,.5 y 6,8,.5, 96.5,. 5 M M M A,A A use of the correct identity attempt to obtain a term quadratic equation in sec y and attempt to solve dealing with sec and y correctly A for a correct pair, A for a second correct pair, A for correct 5 th solution and no other within the range Alternative : sec y secy leading to cos y + cosy ( cos y )(cos y + ) Alternative : sin y cos y cos y ( cos x ) cos x cos x M M M M use of the correct identity attempt to obtain a quadratic equation in cos y and attempt to solve dealing with y correctly A marks as above use of the correct identity, sin y tan y and sec y, then cos y cos y as before (c) π π π z, π 5π z, or.9 or., 5. M A,A correct order of operations A for a correct solution A for a second correct solution and no other within the range Cambridge International Examinations 5

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

Cambridge International Examinations Cambridge International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/ Paper October/November 06 MARK SCHEME Maximum Mark: 80 Published This