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

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1 Cambridge International Examinations Cambridge International General Certificate of Secondary Education ADDITIONAL MATHEMATICS 0606/ Paper May/June 07 MARK SCHEME Maximum Mark: 80 Published 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 07 series for most Cambridge IGCSE, Cambridge International A and AS Level and Cambridge Pre-U components, and some Cambridge O Level components. IGCSE is a registered trademark. This document consists of printed pages. UCLES 07 [Turn over

2 0606/ Cambridge IGCSE Mark Scheme May/June 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 oe 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 0606/ Cambridge IGCSE Mark Scheme May/June 07 B B for each ( A B) C ( ) A B C ( ) A B C attempt at differentiating a quotient, must have minus sign and ( x +) in the denominator for ( ) 5x + 4 for ( )( ) 0 x 5 x + 4 ( x+ ) ( 0x)( 5x + 4) ( 5x + 4) dy dx ( x + ) M B DB A all else correct When dy x, d x A must be exact Alternative ( ) ( ) y 5x + 4 x + for ( ) 5x + 4 for ( )( ) 0 x 5 x + 4 ( ) ( ) ( ) ( ) ( ) dy 0 x 5 x + 4 x+ + 5 x + 4 x+ dx M B DB A attempt to differentiate a product all else correct When dy x, d x A A must be exact UCLES 07 Page of

4 0606/ Cambridge IGCSE Mark Scheme May/June 07 (a) v 5 ( i j ) 5 M attempt to find the magnitude of ( i j ) and use i 6j A for i 6j only (b) w cos0 + o o i sin0 j M attempt to use trigonometry correctly to obtain components i+ j A 4 n n x + ( ) n x n n n 6 6 n 8, so n 4 4 a 6 B M for n x n n, 6 n x C or 6 n n x, with/without their n 6 a 8 A using their n and equating to a to obtain a b M for ( ) x 6 n n n, C or 6 n n x n n x, with/without their n 6 b A using their n and equating to b to obtain b 5(i) v sin t B 5(ii) B FT on their (i) of the form ksin t, must be k 5(iii) a 6cost B allow unsimplified π t,.57 or better π t or B B 5(iv) 4 cao B may be obtained from knowledge of cosine curve UCLES 07 Page 4 of

5 0606/ Cambridge IGCSE Mark Scheme May/June 07 6(i) + M for cot θ, tan θ, cosecθ dealing with the fractions correctly M sin cos sin θ θ + θ M use of identity A correct simplification, with all correct Alternative cosecθ tanθ ( + tan θ ) tanθcosecθ sec θ M M dealing with fractions use of appropriate identity cos θ M for cot θ, tan θ, tanθ sec θ, cosecθ A correct simplification, with all correct Alternative cosecθ cotθ ( cot θ + ) cotθcosecθ cosec θ M M dealing with fractions use of appropriate identity cotθ cosecθ M for cot θ, tan θ, cosecθ A correct simplification, with all correct UCLES 07 Page 5 of

6 0606/ Cambridge IGCSE Mark Scheme May/June 07 6(ii) a a B cos θ dθ sin θ 0 0 sin a 4 M a use of [ k sin θ ]0 to obtain 4 ksin a 4 π a π a, 0.67π or better 6 DM A attempt to solve equation of the form ksin a, with, must 4 4k have a correct order of operations dealing with the double angle 7(i) lg y lg A+ bx B straight line form, may be implied by correct values of both A and b later Gradient b, M equating gradient to b b A Use of substitution into one of the following. lg A+ 0.5b.7 lg A+ b A 0 0.5b A 0 b M or equivalent valid method leads to lg A A 5, 5.0 or 0 A Alternative lg y lg A+ bx B straight line form, may be implied by correct work later. lg A+ 0.5b M one correct equation.7 lg A+ b A both equations correct attempt to solve correct equations M 0.7 leading to b and A 5, 5.0 or 0 A UCLES 07 Page 6 of

7 0606/ Cambridge IGCSE Mark Scheme May/June 07 7(i) Alternative y A bx ( 0 ) A 0 0.5b M one correct equation A 0 b A both correct b M attempt to solve correct equations leading to b A correct b 0.7 Use of substitution leads to A 5, 5.0 or 0 A correct A 7(ii) Substitute A and b correctly into either ( 0 0.6b ) y A, lg y lg A+ 0.6b or lg y lg A + 0.6lg0 or using lg y b M correct statement using their A and b correctly in either equation or using lg y x y 6, 5 or 0 A 7(iii) Substitute A and b correctly into either bx ( ) 600 A 0, lg 600 lg A+ bx or lg 600 lg A+ x lg0 or using lg 600 x b M correct statement using their A and b correctly in either equation or using lg y x x 0.69 A 8(a)(i) 50 B 8(a)(ii) 60 B 8(a)(iii) 080 B 8(a)(iv) 6 or 8 to start with No of ways to start with No of ways B B Total number of ways 40 DB Dependent on both previous B marks UCLES 07 Page 7 of

8 0606/ Cambridge IGCSE Mark Scheme May/June 07 8(a)(iv) Alternative All numbers > 6000 all odd numbers > 6000 B plan and attempt to use, must be using B for 80 and 480 Total number of ways 40 DB Dependent on both previous B marks Alternative Even numbers > : Odd numbers > : B correct ratio Total number of ways B 40 DB Dependent on both previous B marks 8(b)(i) B 8(b)(ii) 6460 B 8(b)(iii) With brother and sister C B for C 5 or C5 k C k Without brother and sister C B for C 7 or C7 k C k Total number of ways B for C + C and evaluation 9(a)(i) B 9(a)(ii) correct attempt to multiply the matrices M 5 7 C A for each incorrect element 9(b)(i) 7 X 4 5 B B for correct use of determinant B for correct matrix 9(b)(ii) x 7 6 y attempt to evaluate using inverse from (i) together with pre-multiplication to obtain a matrix B M x 4, y A A for each 0(i) 0.5 B for 0.5 from correct work only UCLES 07 Page 8 of

9 0606/ Cambridge IGCSE Mark Scheme May/June 07 0(ii) ( 8 8 cos AOB ) AOB.4075 rads M use of cosine rule (or equivalent) to obtain angle AOB. ( ) DOC AOB their AOD M use of angle AOD and symmetry DOC.4 to dp A Answer Given: need to have seen either.4 or better, or.4 or better in previous calculations Alternative + DOC M use of basic trigonometry 5 8 sin use of DOC M may be implied DOC.4 to dp A Answer Given: need to have seen either.4 or better, or.4 or better or.5 or better in previous calculations Alternative ( ) cos AOB AOB.4075 rads AOB 8 arc AB arcab 8 DOC 8 M M use of cosine rule (or equivalent) to obtain angle AOB. attempt at DOC, must be a complete method with AOB found DOC.4 to dp A Answer Given: need to have seen either.4 or better, or.4 or better or.5 or better in previous calculations Alternative Equating different forms for the area of triangle AOB 5 8 sin AOB, AOB.4075 rads 4 M using both different forms of the area of triangle AOB DOC AOB ( their AOD ) M use of angle AOD and symmetry DOC.4 to dp A Answer Given: need to have seen either.4 or better, or.4 or better in previous calculations UCLES 07 Page 9 of

10 0606/ Cambridge IGCSE Mark Scheme May/June 07 0(iii) DC.4 sin 8 DC or ( ) cos.4 M use of cosine rule or basic trigomoetry to obtain DC DC 0.49 A awrt 0.5, may be implied Perimeter A awrt.5 0(iv) 8.4 sin sin.4 ( ) ( ) B area of one appropriate sector; allow unsimplified; may be implied by a correct segment area of one appropriate triangle, allow unsimplified B an appropriate segment, allow unsimplified B 4.8 (allow awrt 4.8) B final answer Alternative Area of a trapezium + small segments B one appropriate small sector, allow unsimpified (could be doubled) Each small segment 8 ( 0.5 sin 0.5 ) B an appropriate triangle, allow unsimplfied (could be doubled) Area of trapezium ( + ) ( ) B attempt at trapezium, must have a correct attempt at finding the distance between the parallel sides allow unsimplified Total area 4.8 (allow awrt 4.8) B final answer Alternative Area of small sectors + area of triangle ODC the area of triangle OAB Area of a small sector 8 B area of small sector, allow unsimplified, (could be doubled) Area of triangle ODC Area of triangle OAB 8 sin.4 B area of triangle ODC, allow unsimplified 8 sin.4 B area of triangle OAB, allow unsimplified Total area 4.8 (allow awrt 4.8) B final answer UCLES 07 Page 0 of

11 0606/ Cambridge IGCSE Mark Scheme May/June 07 0(iv) Alternative Area of rectangle + small triangles + small segments Each small segment 8 ( 0.5 sin 0.5 ) ( ) ( ) B B area of a small segment, allow unsimplified, could be doubled area of a small triangle, allow unsimplified, could be doubled Area of rectangle 0.49 ( ) B allow unsimplified, could be doubled Total area 4.8 (allow awrt 4.8) B final answer Alternative 4 Sector AOB sector AOD sector COB triangle DOC sin.4 Area sector AOB segment DC triangle AOB 8.4 (their segment) 8 sin.4 B B B area of one appropriate sector; allow unsimplified; may be implied by a correct segment area of one appropriate triangle, allow unsimplified an appropriate segment, allow unsimplified Total area 4.8 (allow awrt 4.8) B final answer (i) m e x where m is numeric constant M f e x ( x) ( + c ) A condone omission of +c 7 +c x f( x ) e + DM A correct attempt to find arbitrary constant must be an equation (ii) k e x where k is a numeric constant M x ( ) f'' x e A 4 x ln k x + ln DM A attempt to equate to 4 and use logarithms UCLES 07 Page of