MARK SCHEME for the May/June 2012 question paper for the guidance of teachers 0580 MATHEMATICS

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1 UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 0 question paper for the guidance of teachers 0580 MATHEMATICS 0580/4 Paper 4 (Extended), maximum raw mark 0 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 must be read in conjunction with the question papers and the report on the examination. Cambridge will not enter into discussions or correspondence in connection with these mark schemes. Cambridge is publishing the mark schemes for the May/June 0 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.

2 Page Mark Scheme: Teachers version Syllabus Paper IGCSE May/June Abbreviations cao correct answer only cso correct solution only dep dependent ft follow through after error isw ignore subsequent working oe or equivalent SC Special Case www without wrong working art anything rounding to soi seen or implied Qu. Answers Mark Part Marks (a) (i) 6 correct plots P for 4 or 5 correct plots. (ii) Positive (iii) Line of best fit Ruled line at least from x = 5 to x = 48, with at least points on each side and cuts axes between (5, 0) and (0, 0) (iv) English (integer) value on line at M = (b) ( ) 0 ft Strict ft from their single ruled line 5 ø x ø 48. M M for , condone one slip or SC, for at least values eg ( ) 0 (c) 46 cao www M for ( 8 0) soi by 9 or M for soi by 7 or 9 (a) 445 final answer www M for 5.55 ( 0.) oe or M for 5.55 = (00 ) (%) (b) 640 or or 6.5 Alex by 7.5(0) cao final answer www 6 M for oe M for 4000 (.075) oe or (= 00) and ( their 00) and total interest = the sum of their interests. M for S I amount C I amount or reverse or simple interest compound interest or reverse University of Cambridge International Examinations 0

3 Page Mark Scheme: Teachers version Syllabus Paper IGCSE May/June (a) (i) x > 4 (ii) y > 9 (iii) x + y < 0 (b) 5x + 0y < 70 seen (c) (i) x = 4 ruled y = 9 ruled x + y = 0 ruled x + y = 4 ruled Correct region indicated cao Each line long enough to enclose their region Condone good freehand or dotted y = 9 must be between 8.8 and 9. B for gradient = or y intercept = 0 or x intercept = 0. Exclude lines parallel to either axis. B for y intercept = 7 or x intercept = 4. Exclude lines parallel to either axis. Dependent on all 6 marks for the 4 lines. (ii) 45 cao (from, 9) www M for using 5x + 0y when x + y = 0 and integers (x, y) is in their region University of Cambridge International Examinations 0

4 Page 4 Mark Scheme: Teachers version Syllabus Paper IGCSE May/June In all parts of (a) candidates may refer to angles marked in diagram. Allow if clear even if reason is more complicated as long as it is full. Reasons dependent on correct answers (a) (i) 4 Alternate oe (ii) 90 semicircle oe (iii) 4 same segment oe (iv) 8 cyclic quad oe Not alternate segment Allow diameter same arc key words must not be spoiled (b) 0.9 (0.90 to 0.9) www M for 5 oe i.e explicit or M for = 5 + PQ oe i.e implicit Allow full marks for 9 as final answer Use of trig method must be complete to explicit expression for possible M (c) (i) AD = CD and DE = DG (Angle) CDG = (angle)ade (Sides of) square or 90 + angle ADG oe R Extra pair of sides loses this mark. Extra pair of angles loses this mark As in (a), for all marks allow references to diagram if completely clear. R mark dep on at least one pair of sides stated or pair of angles stated (ii) Congruent University of Cambridge International Examinations 0

5 Page 5 Mark Scheme: Teachers version Syllabus Paper IGCSE May/June (a) ( ).7 or.7 to.7 www M for implied by 9.87 or 9.9 or SC for.77 or.78 or.775 (b) 54 days 4 hours cao M for (. 0 9 ) implied by figs 7 or 54. (.) A for 700 seen or.7 0 seen or 54 6 oe or 54 rem 4 (c) (i) 9.5 (ii) Lower = 5.75 final answer Upper = final answer After 0 scored SC for answers reversed or 9.5 and 5.65 seen or 5.75 and seen 6 (a) (x =) 64 www B for x + x + x = or better or M for x + x x 0 = 60 (b) (i) n oe 5n oe n + 5n oe (ii) 0 M for their n + 5n = 500 or 0 and 5 seen (c) Final answer x 4 x cao www 4 4 B for (x 4)(x + 4) B for (x )(x + 4) or SC for (x + a)(x + b) where either a + b = 7 or ab = 4 7 (a) (5, ) (b) (i) a + c (ii) a + c or (6a + c ) M for OM oe e.g OA+AM or correct unsimplified answer (iii) a + c (iv) a + c or (a + c) M for c + their (iii) or a + their (iii) or correct unsimplified answer or any correct route e.g. CE + ED (c) (CD) parallel (to OB) oe cao dep Part (c) dependent on simplified (i) and (iv) Dep on (i) = k (iv) CD = OB oe cao dep Dep on (i) = (iv) must be scalars University of Cambridge International Examinations 0

6 Page 6 Mark Scheme: Teachers version Syllabus Paper IGCSE May/June Throughout question, penalise non-reduced fraction only once; isw any conversion and allow decimals in working and on branches but not final answers if fractions not seen. (a) (i) (ii) 5,,,,, correctly placed B for and and 5 or 6 5 correctly placed For method marks in (b) and (c), ft tree with each probability 0< p < (b) 4 cao www 9 5 M for or or M for or two of 5, 5, 6 added (c) 4 4 cao www M for + or their M for one of 5 or 6 from a 9 Accurate ruled perp. bisector with correct intersecting arcs Accurate ruled angle bisector with correct intersecting arcs Compass drawn arc centre F radius 5.5 cm long enough to enclose region Correct region indicated cao maximum of two products added. B for accurate with no/wrong arcs or M for correct intersecting arcs Ignore one extra perp. bisector B for accurate with no/wrong arcs or M for correct intersecting arcs Ignore one extra angle bisector M for compass drawn arc centre F Accept dotty lines but not freehand for all three University of Cambridge International Examinations 0

7 Page 7 Mark Scheme: Teachers version Syllabus Paper IGCSE May/June (a) (i) 8x 6 y 9 final answer B for any two of 8, x 6, y 9 in a single term in answer (ii) x oe but not x oe final answer B for or x 6 x or B for 7 x or x oe as answer or as answer seen 7 6 x or SC for or x or x seen in answer (b) 6x + xy 0y final answer B for of 6x 4xy + 5xy 0y (xy implies terms) or B for of 6x 4xy + 5xy 0y (c) (i) V πr V r or oe but not triple πr πr fractions final answer M for correct subtraction or correct division by πr seen (ii) V final answer B for V = h or V = h or h = V (d) 5x 6x 0x x final answer B for of,, 0x oe implied by 4 ie with common denominator = at least 6 (a) 45 or 45. to 45.4 M for π Allow full marks for 44π as final answer (b) 59.9 or to 59.9 cao www 5 5 M for π 4 7 (soi by 57 to 58) oe or π 4 oe (soi by 4.60 to 4.6) 60 and M dep for 60 π 4 7 (soi by. to.) and M for their (a) oe may restart (soi by to 7.7) and M dep on M for adding two areas (c) cos50 soi by 7.( ) oe (their AC) + their AC cos00 art 7.9 cao www 6 M M A M for cos 50 = oe i.e. implicit AC M for implicit cos rule A for 4 to 44 University of Cambridge International Examinations 0

8 Page 8 Mark Scheme: Teachers version Syllabus Paper IGCSE May/June (a) 0x + 4y = 0.7 oe 8x + 6y = 0. oe Multiplying or dividing equation(s) by number(s) suitable for elimination Elimination of one variable x = 0.85 cao y = 0.55 cao M M A A Allow one arithmetic error. If substitution, correctly making one variable the subject of one equation. Allow one arithmetic error. If substitution method then M is for the actual substitution. SC for correct fractions After M0, SC for both correct answers If working in cents, likely mark is 0 for equations, M for method, A if answers converted to dollars, A if left in cents (b) 5 ± ( 5) B B for ( 5) p + B for or r and r = or 4 ( 89 ) p r with p = 5 or 5 5 Completing the square B for x 4 B for or. final answer BB After B0 B0 for answers, SC for.6 or.608 and. or.08 or.6 and. seen Correct answers without working score max and University of Cambridge International Examinations 0

MARK SCHEME for the May/June 2012 question paper for the guidance of teachers 0580 MATHEMATICS

MARK SCHEME for the May/June 2012 question paper for the guidance of teachers 0580 MATHEMATICS UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 0 question paper for the guidance of teachers 0580 MATHEMATICS 0580/4