CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 0 series 0580 MATHEMATICS 0580/4 Paper 4 (Extended), maximum raw mark 30 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 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 0 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components.
Page Mark Scheme Syllabus Paper IGCSE October/November 0 0580 4 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) 5 M for 3 5 (5 + 3 + ) (ii) 08 M for 60 5 9 oe (b) Correct conversion of money J 0.78 or A 0.78 Correct equalising of weights e.g. [0] 3[0] J or A 3[0] [0] or J 3 and A or J 30 and A 0 97 to 98 or 0[.39 ] and Ann 48.9[4.. ] and 48.[0] and Ann or 68[.6] to 68.[] and 67[.3] and Ann 4.88 to 4.9 and 4.8 and Ann or 6.8[..] to 6.8 and 6.7[ ] and Ann www M M A Correct conversion of money soi by 46.83[] rounded or truncated to 3sf or 34.6[ ] rounded or truncated to 3 sf if done st Correct equalising of weights or money Accept other methods that give a pair of comparable values for method and accuracy marks This mark can be implied by values seen correct to 3 sf or better The underlined values imply M for the money conversion Or A for 97 to 98 or 0[.39 ] or a correct pair of values with wrong/no conclusion (c) 30 Final answer 3 M for 60 60 4 soi by 4400 or figs 6048 or figs 304 and M for (000 0) soi Answer 30.4 implies M (d) 3.6[0] 3 5.3[0] M for oe.5 or M for 5.3[0] associated with.5% (e) Cambridge International Examinations 0
Page 3 Mark Scheme Syllabus Paper IGCSE October/November 0 0580 4 3 + 64 43 (a) (i) [cosa=] 3 64 37.00[...] M A M for correct implicit version 43 = 3 + 64 3 64cosA 37 A for or 0.798 to 0.799 4096 (ii) 66 or 66. to 66.4 M for ½ 3 64 sin 37 oe 64sin 55 (b) [Sin ADC =] soi by 70 48.49 rounded or truncated or x² (73.4 to 73.4) x 804 [= 0] 70sin(5 their 48.5) sin 55 or 64 + 70 64 70cos(5 their 48.5) or solving their 3 term quadratic equation 8 or 8.0 to 8. www M M A M for correct implicit version of sine rule or cosine rule with x M for implicit sine rule or cosine rule or for one error in quadratic solution Ignore negative solutions A for 83.0 to 83. 3 (a) (i) (x + )(x 5) final answer 3 B for (x² 9x 5) and B for (x + ) (x 5) or SC for expansion of brackets gives 3 correct terms e.g. (x + ) (x 0) or (4x + )(x 5) or SC for expansion of brackets gives correct terms e.g. (x )(x + 0) or (4x )(x 4) (ii) /oe, 5 ft Correct or ft their brackets (b) [ ]7 ± ([ ]7) () 4()( 0) B B for ([ ]7) 4()( 0) [= 9 ] If in form p + r q or p r q, B for 7 and () or better.09, 4.59 final answers BB If B0, SC for. and 4.6 as final answers or.089.. and 4.589.. as final answers or.09 and 4.59 seen Cambridge International Examinations 0
Page 4 Mark Scheme Syllabus Paper IGCSE October/November 0 0580 4 (c) 0 (3x )( x ) or 3x 0 7x + 3 M for 6(x ) (3x ) or better. Allow recovery after missing bracket[s] as final answer and B for (3x )(x ) as common denominator seen (may be as two fractions) 4 (a) (i) 48 B for tangent/radius = 90 seen. May be on diagram (ii) 74 ft ft their (a)(i) dep on (a)(i) < 80 (iii) M for 360 90 43 3 their (ii) oe e.g. using quadrilateral AOCD (iv) 0.9 or 0.9 3 M for 6 tan 74 oe or explicit sine rule Or M for implicit version (b) (i) 5 M for ABC = 90. May be on diagram. (ii) 56 M for 39 + 7 or 80 (73 + their 5) or [AXB=] 80 (39 + 7) (iii) Angle at centre twice oe angle at circumference (iv) (v) 68.3 or 68.7 to 68.9 3 36 Allow π as final answer 5 M for 360 34 π 360 34 or π π 360 or π + 80 34 π 360 θ or M for use of π 360 for θ multiples of 90 o Cambridge International Examinations 0
Page 5 Mark Scheme Syllabus Paper IGCSE October/November 0 0580 4 5 (a) 0, 60, 00, 40, 80, 0 (6 0 + 0 60 + 8 00 + 76 40 + 80 + 6 0) (= 640) M M At least 5 correct mid - values soi fm where m is in the correct interval, allow either end of interval as m allow one further slip 58 or f M Depend on second method 37 or 36.9 to 37.0 A SC for 37 or better ww (b) (i) 6, 6, (ii) rectangular bar of height 0. rectangular bar of height.05 correct widths of 80 and 0 with no gaps ft ft Strict ft from their 6 Strict ft from their 6 (c) 35 3 5 36 + 3 30 M for 5 + 3 or M for 5 36 and 3 30 [040] and [390] 6 (a) 5.83 or 5.830 to 5.83 Allow 34 as final answer M for (3² + ([ ]5)²) (b) (i) Vector drawn from P to Q at (4, 3) Must have arrow in correct direction (ii) Points at (8, ) and (3, 4), SC for points at (8, 5) and (3, ) (c) 3a b M for a 3b + a + b or CD + DE Allow mixtures of vector notation. oe (d) 7 6 (e) (i) b c oe Allow unsimplified Cambridge International Examinations 0
Page 6 Mark Scheme Syllabus Paper IGCSE October/November 0 0580 4 (ii) MX = MB + BX ± ¼ or ± ¾ used M M Any order for the M marks For a correct route ¾ c ¼b or ¼ (3c b) or 3c b 4 4 A A for ½ b + ¾ (c b) oe Any correct unsimplified After 0 scored SC for /3c /6b 7 (a) (i) x 5 B for each correct inequality y 8 Penalise the first occurrence only when strict inequalities used x + y 4 y ½ x oe 4 (ii) x = 5 ruled y = 8 ruled x + y = 4 ruled y = ½ x ruled region indicated dep Each line long enough to be boundary of region Check at intercepts Check at (0, 5) Dependent on 4 lines correct (b) (i) 480 M for 0 x + 45 y where x and y are integers and (x, y) is in their quadrilateral (ii) 6, 8 In correct order 8 (a) (i) Tangent drawn at x =.5 reasonable tangent at correct point, no daylight, or chord, crossing x-axis between.7,.0 when extended if necessary (ii).55 to. dep Dependent on correct tangent or close attempt at tangent at x =.5 Mdep attempts y step / x step with correct scales (b).4 to.45 and.8 to.8, (c) (i) 4.4,.5,.5 B for correct values Cambridge International Examinations 0
Page 7 Mark Scheme Syllabus Paper IGCSE October/November 0 0580 4 (ii) 6 correct points plotted curve through all 6 points and correct shape (iii) 0.75 to 0.9 Pft C Pft for 4 or 5 correct plots Smooth curve but last 3 points may be ruled. In absence of plot[s], allow curve to imply plot[s] Solutions may be in any order.6 to.7.6 to.7 9 (a) (i) B for outside of circles in diagram F S or all three of 5,, 7 correctly placed 5 7 (ii) 9 ft ft their + their 7 (iii) 4 ft ft their from diagram / 5 (iv) 5 4 7 (v) 600 oe = 00 ft isw incorrect cancelling ft their 7 from diagram for numerator M for their 7 (7 ) their 5 4 their7 their(7) After 0 scored, SC for 5 5 Cambridge International Examinations 0
Page 8 Mark Scheme Syllabus Paper IGCSE October/November 0 0580 4 (b) (i) F S G 5 4 7 F 5 7 4 G 4 B for any correct diagram with blanks or zeros where needed and labelled unambiguously B for 4 in correct place B for in correct place B for 5 and 7 in correct place S S F G 7 5 4 (ii) 8 ft Correct or ft from their diagram 0 (a) (i) 0 (ii) n 4 oe n + 4 oe n + 6 oe Accept unsimplified B for two correct (iii) (n 4)(n + 4) (n 6)(n + 6) n 4n + 4n 6 (n 6n + 6n 36) or better 0 M E ft from their algebraic expressions can be implied by n 4n + 4n 6 (n 6n + 6n 36) or n 6 (n 36) Must have a line of algebra With no errors or omission of brackets (b) (i) 4 Cambridge International Examinations 0
Page 9 Mark Scheme Syllabus Paper IGCSE October/November 0 0580 4 (ii) (n 5)(n + 5) (n 7)(n + 7) isw or n 5 (n 49) isw or n 5 n + 49 isw M for n 5, n + 5, n 7, n + 7 seen (c) ( 3) (9 5) 53 5 [= 8] E Allow algebraic solution from (n 6)(n + 6) (n 8)(n + 8) (d) 4t oe Accept unsimplified e.g. n (t ) [n (t + ) ] (e) c = 8 and d = 30 5 Cambridge International Examinations 0