0580 and 0581 MATHEMATICS

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1 UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 007 question paper 0580 and 0581 MATHEMATICS 0580/04 and 0581/04 Paper 4 (Extended), maximum raw mark 10 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 ds not indicate the details of the discussions that took place at an Examiners meeting before marking began. All Examiners are instructed that alternative correct answers and unexpected approaches in candidates scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes must be read in conjunction with the question papers and the report on the examination. CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 007 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.

2 Page Mark Scheme Syllabus Paper 1 (a) 400 SC1 for figures SC1 for figures 5 1 : or n = SC1 for seen in final answer or n = figs 5 in final answer Time = hrs 8 mins or 18 (mins) =.1(..) (hours) soi 1580 their time cso Implies previous Accept soi is by correct answer www 4 (1. seen earns ) [10] (a) Axes to correct scale S1 Accept mm accuracy throughout Correct triangle A(,1)B(,)C(5,1) Condone absence of labels (c) A 1 (1,), C 1 (1,5), B 1 (,) ft their ABC (d) A (,1), C ( 5,1), B (,) ft their A 1 B 1 C 1 for correct points Condone absence of labels and sides but not incorrect suffices for correct points Condone absence of labels and sides but not incorrect suffices SC1 for rotation of their A 1 B 1 C 1 90 clockwise about the origin If triangle ABC is rotated correctly treat as mis-read (e) Reflection y-axis cso Indep (Only possible answer) (f) A (, 1), C (5, 4), B (,0) B for correct points plotted Condone absence of labels and sides If B0, for any set up of matrix multiplication seen for at least one point and for correct result (If correct triangle A B C used treat as MR, and the co-ords are (, ), ( 5, 6), (, 6)) Shear, y-axis invariant, Allow factor of either +1 or 1 if invariant line omitted, but dependent on shear or stretch for the left hand column [15]

3 Page Mark Scheme Syllabus Paper (a) sin9 AB sin 9 ( AB ) sin sin 9 sin or Any other method must be complete (s = ) ww scores zero ( AB ) cos 9 (AB = ) square root of above and a correct combination (dep) Accept if found in ww scores zero 55 Angles in the same segment dep (iv) correct or ft ft 88 their 55, if answer is positive (v) Similar or enlarged (vi) XD (5) XD 0.1 sin their( iii) sin their( iv) 0.(09 ) cao Any other method must be complete ww scores zero y y + 1 y + y 1 y(y 1) = (y + 1)(y + ) y y = y + y + y + y 4y = 0 E1 May be implied by next line Accept correct ratio statement May be implied by next line Implies previous M Dep (no errors in any line) If M0, SC1 for y(y 1) (y + 1)(y + ) = y y y y y = y 4y 4 ± , 4.45 cao,, p + ( or ) q If of form r for 4 and, for 4-4(1)(-) q If of form p + ( or ) r for 4-4(1)(-) but may recover the other from answers SC1 for rounding or truncating to 1 dp or more , ww scores max of 7.9(0) or better ft ft ft a positive root -1 [19]

4 Page 4 Mark Scheme Syllabus Paper 4 (a) 4.5 to 4 1.6,.0, 8.6 to 8.6 for any one correct 9. (c) 9,, 1 each extra incorrect value (d) 0<x<6, ( i.e.0 to 6 only) Accept (0,6), [0,6], (0, ) to (6, 9). SC1 for other inequality errors or answers using 0 and 6 as boundaries (e) 1 x If re-arranged it must be correct equation with y or f(x) in it but exclude f(x) + x 1 = 0 [11]

5 Page 5 Mark Scheme Syllabus Paper 5 (a) Using a right-angled triangle with 5 and (or ) (BD) = 48 (or 4 ) E1 5 and 7 seen is sufficient (or 50, 14) Must be a correct numerical calculation includes trig methods, which can round to 4, then 48 for the E mark Dep on M, correctly established 7 cos 1 5 If scale drawing seen then M0 147 cao www score only air -4 or ft ft 180 their 147 (c) q + p q p (d) OC + CE e.g. their (q p) + their (q + p) p + q cao any correct unsimplified expression q + their (c) www (e) 1 OC + OB 0.5p +.5q cao any correct unsimplified expression q + ½ their (c) www (f) 0 4 Accept any reasonable notation in both parts 7 4 (g) 50 [16]

6 Page 6 Mark Scheme Syllabus Paper 6 (a) 1.5 < x ( ) their , 1.77, 1.78 or 1.7 cso For mid-values (allow two slips) For Σfx (allow two slips) dep on first, or mid-values ± 0.05 for 00 dep on second www 4 (c) 8, 5, 80, 10, 169, 190, 197, 00 If B0, allow for clear attempt to add accumulatively (d) axes correct scale 8 points plotted ft part (c) (0.5, 8), (1, 5), (1.5, 80), (, 10), (.5, 169), (, 190), (.5, 197), (4, 00) S1 Pdep Not reversed and must reach 00 vertically, even if not labelled dep on at least in (c) 8 points from their values For x-values (upper boundary values), points must touch grid line For y-values, even, must touch grid line, odd must be inside square. P for 6 or 7 points ft P1 for 4 or 5 points ft curve (or polygon) either correct or through 8 points and correct shape C1 Allow 1 mm tolerance Ignore any bars drawn if they do not compromise the points and graph (e) integers only If B0 allow SC1 for non-integer in correct range, or seen (may be written on graph) (f) SC1 for figures or 87 9 Accept if written on graph www [18]

7 Page 7 Mark Scheme Syllabus Paper 7 (a) cao dep (1.08) or 60 (180) (0.6) www (= 864 seen) Their 864 their (a) their (a) (%) or better (1. ) cao ind dep Their (a) 8 5 seen or their 864 their (a) 100 (1...) subtract 100 (Dep on second ) www 4 (10 or 1. www M0) (c) πr figs1 = figs ( r ) π or.1.14 cao ind dep (implied by ) Dep on M ( ) www 4 figs 1 imply first (d) = (.8) = area 0.1 Length their perimeter = their area ind Accept.8 seen Accept 40 seen 150 cao www 4 [15] 8 (a) 105 Do not allow x =, but allow other letter x and condone presence of units 105 x + 4 Do not allow x =, but allow other letter and condone presence of units (c) 105 x 105 x (x + 4) 105x = 0.8x(x + 4) 0.8x +.x 40 = 0 x + 4x 55 = 0 M E1 SC1 if ± signs between terms incorrect or SC1 for their (a) their = 0.8 if (a) and are fractions with linear denominators Dep on M or SC1 and allow all over x(x + 4) at this stage Condone any sign error in any expanding done first (this is taken into account in the E mark) Completed without any errors dep on M (d) (x + 5)(x 1) for (x 5)(x + 1) -5, 1 ft - allow 5 and -1 from above only (e) 46 ft ft a positive root + 4 (f) 10 ( their (e)) 4.57 or better (4.565 ) ft ft www, but 4.6 ww scores zero [1]

8 Page 8 Mark Scheme Syllabus Paper 9 (a) Sketch of 4 by 4 diagram 5, 40, n (n + 1) (n + 1) + n 1 or n + n) or n(n + 1) Any one of these isw and if B0 allow SC1 for their (n + 1) + their (n ) 1 or an expression containing n, as the highest order term, soi (c) + f + g 4 + f + g 4 f + g E1 ie for substituting No errors Allow 10, 10., 10.7, f + g 4 f + g 4 0 ( f ),( g ) cao for correctly setting up for elimination of one variable www accept 6 for (iv) 880 cao [14]

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