Cambridge Assessment International Education Cambridge Ordinary Level. Published

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1 Cambridge Assessment International Education Cambridge Ordinary Level ADDITIONAL MATHEMATICS 07/ Paper May/June 08 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 does 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 May/June 08 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 0 printed pages. UCLES 08 [Turn over

2 07/ Cambridge O Level Mark Scheme May/June 08 Generic Marking Principles These general marking principles must be applied by all eaminers when marking candidate answers. They should be applied alongside the specific content of the mark scheme or generic level descriptors for a question. Each question paper and mark scheme will also comply with these marking principles. GENERIC MARKING PRINCIPLE : Marks must be awarded in line with: the specific content of the mark scheme or the generic level descriptors for the question the specific skills defined in the mark scheme or in the generic level descriptors for the question the standard of response required by a candidate as eemplified by the standardisation scripts. GENERIC MARKING PRINCIPLE : Marks awarded are always whole marks (not half marks, or other fractions). GENERIC MARKING PRINCIPLE : Marks must be awarded positively: marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beyond the scope of the syllabus and mark scheme, referring to your Team Leader as appropriate marks are awarded when candidates clearly demonstrate what they know and can do marks are not deducted for errors marks are not deducted for omissions answers should only be judged on the quality of spelling, punctuation and grammar when these features are specifically assessed by the question as indicated by the mark scheme. The meaning, however, should be unambiguous. GENERIC MARKING PRINCIPLE : Rules must be applied consistently e.g. in situations where candidates have not followed instructions or in the application of generic level descriptors. GENERIC MARKING PRINCIPLE 5: Marks should be awarded using the full range of marks defined in the mark scheme for the question (however; the use of the full mark range may be limited according to the quality of the candidate responses seen). GENERIC MARKING PRINCIPLE 6: Marks awarded are based solely on the requirements as defined in the mark scheme. Marks should not be awarded with grade thresholds or grade descriptors in mind. UCLES 08 Page of 0

3 07/ Cambridge O Level Mark Scheme May/June 08 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 08 Page of 0

4 07/ Cambridge O Level Mark Scheme May/June 08 Question Answer Marks Partial Marks (i) cosθ Uses cotθ = sinθ cos θ + sin θ sinθ B cosθ B for using cot θ = oe or sinθ oe at some stage B for use of cos θ + sin θ = oe sin θ tan θ = cos θ Uses cos θ + sin θ = Completes to sinθ = cos ecθ B for common denominator of s sinθ oe either in a compound fraction or in two partial fractions sin θ sin θ or for writing as oe sinθθ sin θ sinθ Maimum of marks if not fully correct or does nott complete too cosecθ (ii) sinθ =.5 or.7[75 ] rot to or more figuress isw A Not fromm wrong working (a) B A B (b) P 8 5 Q B B for 8 correctly placed and all the empty regions correct c B for,, 5 correctly placed B for correctly placed [0] [0] R [0] maimum of marks if fully correct but other values such ass 0, and/ /or 5 present within the diagram their B STRICTT FT their Venn diagram UCLES 08 Page of 0

5 07/ Cambridge O Level Mark Scheme May/June 08 p( ) = 0 or p() = 5 stated or implied 5 + 9a b = 0 or better A finds one correct equation; implies 6 + a 8 + b = 5 or better A finds another correct equation; implies Solves a pair of simultaneous equations in a and b a = 7, b = 5 A dep on first condone one sign or arithmetic error in their solution; as far as finding one unknown 60 cao A Eliminates one of the unknowns Simplifies to a correct -term quadratic: + 6 [= 0] oe or y 6y 6 [= 0] oe Factorises or solves + = 0oe or ( )( ) ( )( ) y+ y 6 = 0oe A FT their -term quadratic in or y; (, 6) and (, ) oe A Not from wrong working A for either (, 6) or (, ) or A for = and = or y = 6 and y = 5(a) 7 P or oe 80 A 5(b)(i) 0 B 5(b)(ii) 5 C C C or 5 oe 0 A 5(b)(iii) 5 C + C oe A UCLES 08 Page 5 of 0

6 07/ Cambridge O Level Mark Scheme May/June 08 6(i) (Arc length = ).5 5 oe soi implied by 7.5 (DE =) 0sin(0.75) oe soi implied by awrt 6.8. or answer in range. to. A 6(ii) (Area sector = ) 5.5 oe implied by 8.75 (Area triangle =) 5 sin(.5) oe implied by awrt.7. or answer in range. to. A 7(i) their(a + c) = their (5 + ) A mark final answer 7(ii) [( + m)i + ( 5m)j therefore] their ( + m ) = 0 for attempting to form a + mb and equate the scalar of the i component to 0 m = only A implies 7(iii) [(n )i + (n + 5)j = i + j or n(i + j) = (i + j) + (i 5j) oe leading to] n = or n + 5 = oe, soi n = only A implies UCLES 08 Page 6 of 0

7 07/ Cambridge O Level Mark Scheme May/June 08 8(a) 6 B B for a by matri with or correct elements 6 their oe isw 0 B FT their non-singular BA B FT for either their( 0) 6... their or If their BA is singular, B0 then SC for 6... their OR Alternative method A B : B for A = isw 5 or B = isw or B for a multiplier of 5 or for or for a multipler of 6 or for 5 0 B FT for A B = 6 their their 0 or B FT for a by matri with or correct elements Maimum of marks if not fully correct 8(b)(i) B 8(b)(ii) oe isw B B for each correct element; must be in a by matri or for a full method as far as finding values for the two elements UCLES 08 Page 7 of 0

8 07/ Cambridge O Level Mark Scheme May/June 08 9(i) d sin = (sin ) cos d ( ) ( ) oe B B for (sin )... or for (sin ) or for (...) cos their or for (sin ) their cos ( ) their sin Applies correct form of product rule + their ( sin ) ( cos ) oe sin + ( sin) ( cos ) oe isw A Not from wrong working 9(ii) ( ) sin d + ( sin ) ( cos ) d = sin oe or cos d+ + sin d sin FT their (i) oe sin [ ] + + c A A for d + sin 0(a)(i) y B B for correct shape B for roots marked on the graph or seen nearby provided graph drawn and one root is negative and one is positive - 5 0(a)(ii) Any correct domain B 0(b)(i) B mark final answer UCLES 08 Page 8 of 0

9 07/ Cambridge O Level Mark Scheme May/June 08 0(b)(ii) Correct method for finding inverse function e.g. swopping variables and changing subject or vice versa; or indicates (hg) ( ) = g h ( ) and + finds g ( ) = and h ( ) = FT only if theirhg( ) of the form a, b and c are integers a b + c where (hg) ( ) = + oe isw or (hg) ( ) + = oe isw A FT their (hg) ( ) = a c oe b If M0 then SC for their hg() of the form a y= + b oe leading to their (hg) - () of the a form y = b isw 0(c) a cao B (a) ( ) [ + c ] oe isw B B for ( ) + k where k 0 + (b)(i) (b)(ii) kcos [ + c ] where k < 0 or cos + [ c ] k = Sight of correct substitution of limits: π π cos cos 8 oe A FT their kcos from (b)(i) dep on awarded in (b)(i) A does not imply UCLES 08 Page 9 of 0

10 07/ Cambridge O Level Mark Scheme May/June 08 (c) e d = ke [ + c ] k any non-zero constant k = Sight of correct substitution of limits: ln8 0 their ke their k e oe Shows how to deal with the power of the first term e.g. ln8 ln8 = or ln8 ln seen = or ( 8 ) A B dep on first 6 = A Not from wrong working (i) π tan = r h oe r = h ( ) or π r = h tan oe A ( ) [ V = ] π h h oe ( ) π + h [ V = ] correctly leading to ( ) oe π 7 h [ V = ] AG A Correctly uses their epression for r in terms of h in formula for volume of a cone dependent on finding an epression connecting r and h (ii) Correct derivative of V e.g. B π( 7 ) h oe isw dh dh dv = dt dv dt soi B dv their d h h = 5 0 if correct implies B B; if incorrect, a correct FT statement implies the second B 5. A UCLES 08 Page 0 of 0

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 March 07 MARK SCHEME Maximum Mark: 80 Published This mark scheme