Cambridge Assessment International Education Cambridge Ordinary Level. Published

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1 Cambridge Assessment International Education Cambridge Ordinary Level ADDITIONAL MATHEMATICS 407/ Paper 07 MARK SCHEME Maimum Mark: 80 Published This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the eamination. It shows the basis on which Eaminers were instructed to award marks. It ds not indicate the details of the discussions that took place at an Eaminers 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 Eaminer Report for Teachers. Cambridge International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the 07 series for most Cambridge IGCSE, Cambridge International A and AS Level components and some Cambridge O Level components. IGCSE is a registered trademark. This document consists of printed pages. UCLES 07 [Turn over

2 407/ Cambridge O Level Mark Scheme 07 MARK SCHEME NOTES The following notes are intended to aid interpretation of mark schemes in general, but individual mark schemes may include marks awarded for specific reasons outside the scope of these notes. Types of mark M A B Method marks, awarded for a valid method applied to the problem. Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. For accuracy marks to be given, the associated Method mark must be earned or implied. 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 in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. The notation dep is used to indicate that a particular M or B mark is dependent on an earlier mark in the scheme. Abbreviations awrt cao dep FT isw nfww rot SC soi answers which round to correct answer only dependent follow through after error ignore subsequent working not from wrong working or equivalent rounded or truncated Special Case seen or implied UCLES 07 Page of

3 407/ Cambridge O Level Mark Scheme 07 (a) B for each (b) n ( ') n n P 8 (( ) ) Q R P ( ) Q P not earned if incorrect equation(s) present p( + ) + ( ) ( )( + ) ( ) ( ) q + on LHS take common denominator or rationalise each term or multiply throughout p + + q + correct eqn with no surds in denominators of LHS equate surd/non surd parts equate and solve for p or q ( 0) p 5 and q 4 log or log9 implied by one correct equation + y y 9 solve correct equations for or y 4 and y 5 5(i) OX λ (.5b+ a) 5(ii) AB b a or BA a b OX a+ µ ( b a) 5(iii).5 λ µ or λ µ OX OX and equate for a or b µ λ A 9 for each UCLES 07 Page of

4 407/ Cambridge O Level Mark Scheme 07 5(iv) AX XB 5(v) OX XD 7 Accept : but not : Accept : 7 but not : 7 (i) f f(f) used algebraic ([( + ) + ] + ) + numerical or algebraic 7 (ii) y y change and y ( ) y y y dep multiply, collect y terms, factorise y g( ) correct completion (iii) gf ( ) ( ) + + ( ) + + ( ) ( ) ( ) their gf and simplify to 9 quadratic equation solve quadratic dep Must be of equivalent form 5 7(i) v 0 cost set v 0 and solve for cost t 0.5 or 0. 7(ii) s sin t t + c ( ) for sin t and ± t π π t s ( ) 7(iii) a sint for sint t 0.5 a 5. or 5.5 or 8 condone substitution of degrees UCLES 07 Page 4 of

5 407/ Cambridge O Level Mark Scheme 07 8(i) cos α α (ii) speed Pythagoras/trig ratio/cosine rule 8 or or.8 m s 8(iii) time 50 their 8 5 or 7.7s 8(iv) their 8( iii) seen BC 0 or 4. m or 4. m 9(i) d (ln ) and d d d d or d 4 seen Substitution of their derivatives into quotient rule ln d ln d correct completion 9(ii) dy 0 ln 0 d ln dy equate given d for ln or to zero and solve e seen y e seen 9(iii) ln ln 4 d use given statement in (i) d 4 seen anywhere ln ln d (+C) 4 9 A for each term UCLES 07 Page 5 of

6 407/ Cambridge O Level Mark Scheme 07 0(a) LHS ( ) ( + ) sin + + cos sin cos correct addition of fractions + cos + sin cos ( + ) epansion and use of identity ( + ) ( + ) cos cosec sin cos factorisation and completion 0(b)(i) cosec y + cosecy 5 0 cosec y + cosecy 0 use of identity for cot y to obtain quadratic in cosecy ( )( ) cosecy cosecy + 0 solve term quadratic for cosecy sin y, siny obtain values for siny y 0, 50, 99.5, 40.5 A for values 0(b)(ii) π 5π z + or 4 7π (.,. ) M 5π equate to 7π equate to 7π π z or (0.9,.44) 4 4 A for value (i) Other root 4 f ( ) ( )( )( 4) 0 + multiply out ( )( )( ± p) a 0 b A for each Can be implied by correct cubic (ii),,,, 9,, B4 for each of first two sets B for third set UCLES 07 Page of

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

Cambridge International Examinations Cambridge International General Certificate of Secondary Education. Published Cambridge International Eaminations Cambridge International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/ Paper May/June 07 MARK SCHEME Maimum Mark: 80 Published This mark scheme