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 MATHEMATICS 0580/4 Paper 4 Paper 4 (Extended) October/November 06 MARK SCHEME Maximum Mark: 30 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 ds 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 October/November 06 series for most Cambridge IGCSE, Cambridge International A and AS Level components and some Cambridge O Level components. IGCSE is the registered trademark of Cambridge International Examinations. This syllabus is approved for use in England, Wales and Northern Ireland as a Cambridge International Level /Level Certificate. This document consists of 8 printed pages. UCLES 06 [Turn over

2 Page Mark Scheme Syllabus Paper Cambridge IGCSE October/November Abbreviations cao correct answer only dep dependent FT follow through after error isw ignore subsequent working or equivalent SC Special Case nfww not from wrong working soi seen or implied 5 (a) (i) final answer M for (ii) M for or M for recognition of 4 05 as 85% soi 3 (b) 60 final answer M for 5 P + = (c) (i) M for or or M for 00 r or or soi by If zero scored, SC for 56.8 or 56. as final answer 4.7 (ii) 6 3 M for 00 or or 0.6 soi by.06 or or M for r 00 + = UCLES 06

3 Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE October/November (a) (b) Fully correct graph 4 B3FT for 6 or 7 points plotted or BFT for 4 or 5 points plotted or BFT for or 3 points plotted (c) (i) < ans < < ans <.45.5 < ans <.6 (ii) 0.7 < ans < 0.5 M for evidence of y = x or 3 x 3 x + = x (d) (i) y = to. FT FT only if a clear maximum point y = 0.4 to 0.33 FT FT only if a clear minimum point (ii) 0.4 to 0.33 FT Correct or FT their graph 3 (a) 40sin 85 sin50 M or M for sin50 = sin85 40 AB 3 or 3.. B (b) sin A = 000 M to A A for sin = 4000 or better or or or 0.5 or to (c) 38 or 38.3 to B for [angle A =] seen (d) (i) or 08.7 or 08.8 M for 80 + ( their AB) 80 their AB cos ( theirab) x or M for cos78.75 = 80 ( theirab) A for to (ii) or 88.7 or 88.8 FT FT 80 + their (d)(i) M for 80 + their (d)(i) or 360 (80 their(d)(i)) UCLES 06

4 Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE October/November (a) 5 M for 0 40 [ 60] (b) 49. nfww 4 M for 35, 4.5, 47.5, 5.5, 57.5, 70 soi M for Σ fx M dep for their Σ fx 00 (c) Fully correct histogram 4 B3 for 4 correct blocks or B for or 3 correct blocks or B for correct block (d) (i) 5, 80 If zero scored, SC for correct frequency densities 0.8, 9,,.8, 0.3 soi (ii) Correct diagram 3 BFT their (d)(i) for 6 correct heights within correct square(including boundaries) or touching correct line if should be on a grid line and B for 6 points at upper ends of intervals on correct vertical line and BFT (dep on at least B) for increasing curve or polygon through 6 points (iii) (a) 48 to 49 (b) 55 If zero scored, SCFT for 5 correct points plotted (c) 8 to 4 FT BFT for 86 to 9 seen UCLES 06

5 Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE October/November (a) (i) 3 4, 4 7 8, 8 B for any correct (ii) 3 M for (iii) FT M for or their ((a)(ii)) (b) 75 M for (c) 400 M for 575 their(a)(ii) UCLES 06

6 Page 6 Mark Scheme Syllabus Paper Cambridge IGCSE October/November (a) (i).3 M for (ii) 0.75 or to M for πr 0.8 = their(a)(i) or πr =.5. (iii) 0.53 to 0.58 nfww 5 M for ( ) M for [ ] π ( their (a)(ii)) M for π ( their (a)(ii)) 0.8 or M for π ( their (a)(ii)) (b) (i) x+ y 9 y If zero scored, SC for x + y > 9 and y > (ii) Fully correct diagram with unwanted region shaded 4 B for x + 3y = 4 ruled B for x + y = 9 ruled B for y = ruled (iii) 0 [x = ] 7 [y =] If zero scored, SC for x + 3y evaluated from integers UCLES 06

7 Page 7 Mark Scheme Syllabus Paper Cambridge IGCSE October/November (a) final answer B for to or 54.5 or M for (b) (i) 000 xx+ ( ) final answer 3 M for 000 (x + ) 000x M for denominator x(x + ) (ii) x x + = 4.5[0] 000 or = 4.5 xx+ ( ) 000 = 4.5x (x + ) 4.5x + 4.5x 000 = 0 9x + 9x 000 = 0 M Mdep A Allow their (b)(i) for first M only for a single fraction Correctly multiplying by algebraic denominator Equation reached without any errors or omissions and at least one step after clearing the denominators of the fractions still with brackets included (iii) 9 ± 9 4(9)( 000) (9) B for 9 4(9)( 000) If in form p + q or p r r B for p = 9 and r = (9) q then B B SC for answers 5.4 or 5.4 to 5.4 and 4.4 or 4.4 to 4.4 or for 4.4 and 5.4 or 5.4 and 4.4 seen but not final answer Answers without working only score B, B or SC (iv) to final answer must be dp FT FT 000 their positive x with final answer rounded up or down to dp or M for 000 their positive x UCLES 06

8 Page 8 Mark Scheme Syllabus Paper Cambridge IGCSE October/November (a) [u = ] 80 [v = ] 60 (b) 6.4 or 6.44 to M for 8 5 or M for l + 5 = 8 or B for suitable right angled triangle drawn with 5 on correct side (c) 5.05 or M for =.5 MN 5 (d) 4 nfww 4 M3 for [ x ]( x+ ) = 4 [ x ]( x ), n =, or 3 n n or M for [ x]( x+ ) [ x] = [ x]( x ) [ x] 5 or M for or soi 9 (a) (i).5 (ii) 3 y final answer 3 M for correct removal of fraction M for collection of terms in x and factorises OR M subtracts from both sides M multiplies by x to remove fraction and M for correct division by expression of the form ay + b, a and b 0 (b) (i) 3 (ii) final answer B for h(6) e.g. h( 4 ) (iii) 6 M for x = 3 (iv) 3 0 (a) 7.5 M for 3x + x + 3x + x = 60 (b) 5 3 B for 3x + 4x + 5x [= 60] or better or M for (3 x) + (4 x) (c) 6.8 or M for 90 x + x+ π x [ = 60] 360 or M for 90 π x 360 UCLES 06