Cambridge International Examinations Cambridge International General Certificate of Secondary Education

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1 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 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 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 Method marks, awarded for a valid method applied to the problem. A 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. B 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 0606/ Cambridge IGCSE Mark Scheme May/June 07 + = or + = = and = 0. only mark final answer Alternative method A nfww for = ignoring etras implies if no etras seen If M0 then SC for any correct value with at most one etra value ( ) ( ) + = soi = 0 = 0., = only; mark final answer Without using a calculator Sufficient evidence must be seen to be convinced that a calculator has not been used. Withhold the mark for any step that is unsupported. deals with the negative inde soi e.g. + rationalises allow for + multiplies out correctly + squares correct binomial ( ) ( ) + = allow for 9 allow for ( + ) = ( + + ) 9 cao dep on all previous marks awarded UCLES 07 Page of

4 0606/ Cambridge IGCSE Mark Scheme May/June 07 Alternative method : dealing with the negative inde soi correctly squaring with at least terms in the numerator and denominator = rationalising 6 6 their 6+ 6 multiplying out correctly; at least terms in the numerator but condone a single value for the denominator cao Alternative method dealing with the negative inde soi ( a b )( ) = = 6a+ 0b 6= a+ 6b a = 9 cao b = cao Alternative method for dealing with the negative inde soi [ ( c d )( ) = + + leading to] c+ d = c+ d = c = and d = ( + ) = + 9 cao UCLES 07 Page of

5 0606/ Cambridge IGCSE Mark Scheme May/June 07 Correctly finding a correct linear factor or root from a valid method, e.g. factor theorem used or long division or synthetic division: f( ) = 0( ) ( ) + = 0 0 or or correct linear factor stated or implied by, e.g. ( )( 0 ) ( ) or( ) or( + ) do not allow or + Correct quadratic factor ( 0 ) or ( 8 ) or ( + ) B found using any valid method; for any terms correct ( )( )( ) + mark final answer must be written as a correct product of all linear factors; only award the final if all previous marks have been awarded If quadratic factor is not found but correct remaining linear factors are found using e.g. the factor theorem or long division or synthetic division etc. with correct, sufficient, complete working to justify that no calculator has been used allow: for correctly finding a correct linear factor or root for a correct linear factor stated or implied SC for the full, complete and correct working to find the remaining two linear factors and arrive at the correct product of linear factors UCLES 07 Page of

6 0606/ Cambridge IGCSE Mark Scheme May/June 07 dy 6 7 d = soi m normal = soi finds or uses correct gradient of normal m = soi or ( ) tangent 6 7 = uses mm = with numerical gradients 6 7= = y = 9 k = 7 Alternative method m normal = m tangent = + c = 0 solving + = 0 to find = y = 9 k = 7 (i) ( ) their (0. ln ) + 0. their or ( ) their0. their ln + 0. their clearly applies correct form of product rule ln isw nfww (ii) ln or ln + ln + ln UCLES 07 Page 6 of

7 0606/ Cambridge IGCSE Mark Scheme May/June 07 (iii) ( ln) d FT k = from (i) allow for ( ln) d or, when k =, for ( ) = ln d 0. (0. ln ) or ( ln) d = 0. (0. ln ) or, when FT k =, for ( ) = ln d 0. (0. ln ) or ( ) = ln d 0. (0. ln ) ( 0. (0. ln ) )[ + c] isw cao nfww; implies An answer of 0.6 (0. ln ) following k = from (i) implies A0 6 Uses b ac ( p q) ( p)( q) implies p + pq + q correctly simplifies ( p q) 0 + cao isw Alternative method ( p q)( ) [ 0] + = or q =, = p ( p q) ± ( p+ q) M or for ( p + q)( ) [ = 0] p ± or p ( p q) ( p q) ( p)( q) for conclusion, e.g. p and q are real therefore q p is real [and is real] 7(a)(i) 7 7(a)(ii) 7 or their7 FT their 7 must not be if following through UCLES 07 Page 7 of

8 0606/ Cambridge IGCSE Mark Scheme May/June 07 7(b) y = 8 or y = or y = 9 Anti-logs y = only or 0.[.] only nfww; implies the ; y =. must be seen at least once If M0 then SC for e.g. 8 = as final answer 7(c) ( ) ( ) or ( ) or or log log = log6 converts the terms given left hand side to powers of or ; may have crossmultiplied or separates the power in the numerator correctly or applies a correct log law = 6 = or = 6 = 8 or = 6 log8 = log or ( )log log = log6 [ = ] ± isw cao or ± rot to or more figs. isw combines powers and takes logs or equates powers; or brings down all powers for an equation already in logs condone omission of necessary brackets for ; condone one slip 8(i) 8 y 8 = ( ( 8 )) isw 8 or y[ 0] = ( ) isw or y = + 8 isw B 8 for m AB = or for (ii) ( ) ( ) 8 + 8[ 0] any valid method 08 isw or isw or.0 rot to or more sf implies provided nfww UCLES 07 Page 8 of

9 0606/ Cambridge IGCSE Mark Scheme May/June 07 8(iii) [coordinates of D =] (, ) soi If coordinates of D not stated then a calculation for m CD or a relevant length with the coordinates clearly embedded must be shown to imply Gradient methods: or Length of sides methods: m CD 7 their = = their 0 their( ) y A 6 8 C 6 D B finds or states AC or ( 8 0) ( 8 7) = 6 or AC = 6 AC = + or ( 8 0) ( 8 7) and AC = + CD = their CD = their or or CD ( 0 their ( ) ) ( 7 their) = + or CD ( 0 their ( ) ) ( 7 their) and = + AD = their AD = their or or AD = ( 8 their ( ) ) + ( 8 their) or AD = ( 8 their ( ) ) + ( 8 their) or uses a valid method with their coordinates of D to find the eact area of the triangle and equates to ( AD )( CD )sin( ADC ) states 8 = or is the negative reciprocal of or finds the equation of the perpendicular bisector of AB as y = + 7 independently of C and states that C lies on this line. applies Pythagoras to confirm, using integer values, that 6 = + or finds e.g. AC = or 6 using solves ( )( ) ( ) + ( ) sin ADC = or ( ) 6 = ( ) + ( ) ( )( ) cos ADC to show ADC is a right angle 8(iv) or i+ j condone coordinates UCLES 07 Page 9 of

10 0606/ Cambridge IGCSE Mark Scheme May/June 07 8(v) Full valid method e.g. 0 for showing that e.g. CB = = or showing that e.g AC = = and EB = = 0 B for incomplete method e.g. for stating that CB = 7 or 8 AC = = EB or just showing that one pair of opposite sides is parallel or has the same length or comparing gradients of both pairs of opposite sides and showing they are pairwise the same or comparing the lengths of both pairs of opposite sides and showing that they are pairwise the same or showing that length AC = length AE or that the length BC = length BE or comparing the gradients and lengths of a pair of opposite sides or showing that D is the midpoint of CE or showing that length DC = length DE and that C, D and E are collinear or just showing that length DC = length DE or just showing that C, D and E are collinear A(-8, 8) m AE m AC 6 7 = E(-, ) = 8 D(-, ) 6 6 m C(0, 7) EB m BC = 8 7 = 6 B (, 0) 9(i) (.) + 0. isw B or B for a = and b =. and c = 0. provided not from wrong format isw or B for (.) or a = and b =. + c where c 0. or SC for (.) + 0. or. ( ) + seen or for (.) seen or for b =. or for c = 0. or SC for correct values seen in or incorrect format e.g. ( ) ( ) UCLES 07 Page 0 of

11 0606/ Cambridge IGCSE Mark Scheme May/June 07 9(ii) y B for correct graph for f over correct domain or correct graph for f over correct domain for verte marked for f or f and intercept marked for f or f. for idea of symmetry either symmetrical by eye, ignoring any scale or line y = drawn and labelled Maimum of marks if not fully correct 9(iii) 0. ( y.) = FT their a,b,c, provided their a and a,b,c are all non-zero constants y 0. or = (.) and reverses variables at some point f ( ). 0. = must have selected negative square root only; condone y =... etc.; must be in terms of If M0 then SC for f ( ) = 6 8 or SC for ( 6) ± 6 ( )( ) f ( ) = () 0(i) sin soi implied by 0.88[06 ] 0.88[06 ] rot to or more figs or.9[ ] rot to or more figs implied by a correct answer of acceptable accuracy rot to or more figs isw.0 or rot to or more figs isw Maimum marks if etra angles in range; no penalty for etra values outside π range 0 UCLES 07 Page of

12 0606/ Cambridge IGCSE Mark Scheme May/June 07 0(ii) Correctly uses tan y sec y sin y cos y and sin y= cos y = and/or for using correct relationship(s) to find an equation in terms of a single trigonometric ratio sec y sec y = 0 ( sec y )( sec y ) + or cos y+ cos y = 0 (cos y )(cos y+ ) D for factorising or solving their -term quadratic dependent on the first being awarded [ cos y = ] cos y = 78. or rot to or more decimal places isw 8. or rot to or more decimal places isw Maimum marks if etra angles in range; no penalty for etra values outside range 0 60 (i) + + [ + c] isw (ii) + + = and rearrange to ( ) + = 0 soi B for any correct terms y B A(, ) C(, ) E (, 0) O D(, 0) Solves their + [ = 0] soi =, = soi OEAB =, OBCD = UCLES 07 Page of

13 0606/ Cambridge IGCSE Mark Scheme May/June 07 (iii) Correct or correct FT substitution of 0, their seen in + + their 0 dependent on at least in (i) Correct or correct FT substitution of their, 0 seen in + + their 0 dependent on at least in (i) 7 9 their theiroeab + theirobcd their for the strategy needed to combine the areas; may be in steps; 97.96& +.08& 886 or 7 or 7.8 & rot to or more sig 6 figs all method steps must be seen; not from wrong working If M0 then allow SC for 0 ( + )d ( + )d 0 0 = = = 6 or SC for 0 their ( + )d ( + )d their ( ) 0 0 their = + + their( ) 0 [ F(0) F( their( )) ] [ F( their) F(0) ] = (i) + 6 6( ) or ( + ) isw Allow + ( + ) ( ) or Denominator or ( + ) is positive [and numerator negative therefore g( ) is always negative] k FT their g ( ) of the form ( ) + where k > 0; Allow ( + ) is always positive (ii) g > 0 (iii) k + + isw UCLES 07 Page of

14 0606/ Cambridge IGCSE Mark Scheme May/June 07 (iv) k + = (0) + k = isw (v) > implies the first 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