0580 MATHEMATICS. Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for Teachers.
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1 CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME f the May/June 4 series 58 MATHEMATICS 58/4 Paper 4 (Extended), maximum raw mark 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 befe 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 Rept f Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes f the May/June 4 series f most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components.
2 Page Mark Scheme Syllabus Paper IGCSE May/June Abbreviations cao crect answer only dep dependent FT follow through after err isw igne subsequent wking equivalent SC Special Case nfww not from wrong wking soi seen implied (a) 4 ( 5 + 7) 7 [=4] M M f 4 (5 + 7) 4 7 (b) : final answer B f ratio of fm x : x seen SC f : (c) 44 M f M f 4 5 (d) cao mark final answer B f 89.9[8 ] shown but not spoiled answer 9[...] nfww M f 8 4 (e) 4.8 M f If M spoiled by adding 8 subtracting 8 then SC f answers r =.4 M f.4 [6.[]] r Cambridge International Examinations 4
3 Page Mark Scheme Syllabus Paper IGCSE May/June (a),, B B B (b) Complete crect curve 5 BFT points BFT f 9 points BFT f 7 8 points And Bindep two separate branches not touching crossing y-axis (c).5 to.6 (d) Crect line and.4 to.5 no line and.4 to.5 nfww Must check line - not if wrong line B f y = x + ruled crectly SC f crect freehand line ruled line with either gradient y-intercept but not y = (e) (i) Tangent at x =.5 No daylight at x =.5. Consider point of contact as midpoint between two vertices of daylight, the midpoint must be between x =.7 and. (ii) to Dependent on tangent mark awarded Allow integer/integer if in range Or M f rise/run also dep on any tangent drawn close attempt at tangent at any point Must see crect implied calculation from a drawn tangent Cambridge International Examinations 4
4 Page 4 Mark Scheme Syllabus Paper IGCSE May/June (a) M f 8sin 55 sin 49 M f 8 x = sin 49 sin 55 (b) to M f [cos =].95.9 M f 8 = cos BCD 75 4 A f etc (c) to 67 M f.5 8 their(a) sin( ) [68 7 ] [If AB used then AB=.8 to ] sin(their(b)) [9 ] M f one of these triangle area methods (d) 8 76 to 79 FT FT their (c).5 crectly evaluated to 5 sf better M f their (c) SC FT f figs 8 figs 76 to 79 Cambridge International Examinations 4
5 Page 5 Mark Scheme Syllabus Paper IGCSE May/June (a) Image at (, ), ( 5, ), ( 5, 4), (, ) SC reflection in y = x = k 4 crect points not joined (b) (i) Image at (, 4), ( 6, 4), ( 6, 8), (, 6) SC other enlargement of scale fact -, crect size and crect ientation 4 crect points not joined (ii) k SC f, k may be algebraic k numeric but not (c) (i) Image at (, 4), (, 4), (, 8), (, 6) SC f trapezium with vertices at (, 6) and (, 8) crect stretch with y-axis invariant 4 crect points not joined (ii) SC f k may be algebraic k numeric but not f (iii) isw FT FT inverse of their (c)(ii) (algebraic numeric) a c BFT their (c)(ii) f b d p ie FT their crect fraction their transposed matrix FT f and mark dependent on det (iv) Stretch, [fact ], invariant [line] x-axis B B B each independent cao Cambridge International Examinations 4
6 Page 6 Mark Scheme Syllabus Paper IGCSE May/June (a) (i) 4 to 4. B Must be at least 4 figures shown M f π 8 (ii).4[] B (b) min 4 s 4 B f 8.76 to 8.8[] to.97.4 min.76 to min.8 seen M f π 4 [ 8/π ] M f π 4 [6/π 67.5 to 67.6] (c) 4 M f 6 (a) (i) [x =], [y =] 4 B B 4 π 4 4 their cone vol from part ( b) A f 4. to 4.4. (ii).79.8[].79 to M f = TQ 9.4 sin sin their x TQ = sin7 9.4 (b) 4 4 B f angle between HE and tangent = 5 GFH = 4 EGH = 5 and angle EHG = 5 (accept 9 and 5 at H f 5) B f angle EGH = 5 angle EHG = 5 (accept 9 and 5 at H f 5) B f angle FEG = 5 angle EFG = 65 (c) 8 5 B4 f angle ADC = 4 M4 f x x + 7 =8 better B f angle OBA = and angle OBC = 56 angle CBA = 76 reflex angle AOC = 8 B f angle OAB OBA = and angle ACB = 7 obtuse angle AOC = 5 angle BOC = 68 B f angle OAB OBA = angle ACB = 7 Cambridge International Examinations 4
7 Page 7 Mark Scheme Syllabus Paper IGCSE May/June (a) (i) ( 7).4 [= ] better Accept 9 78 (ii) nfww 5 B f 4 crect extra frequencies, 6,, 8 soi M f at least 4 of mid-interval values 5, 4, 55, 65, 85 soi M f Σ fx where x is any value in each interval allow their frequencies provided integers and they must be shown [ ] [75] M (dependent on second M) f 9 ( ) (b) 6.5 M f M f either (a) (i) 6 x Not 6 x = x (ii) 6 x + Not 6 x = x + (b) (i) 6 6 = x x + MFT FT their (a)(i) their (a)(ii) = If M, SCFT f their(a)(ii) their (a)(i) = 6 ( x + ) 6x = x( x + ) better A May still be over common denominat and can be implied by third line. Allow recovery if bracket omitted + 6x + 6 6x = x x = x + x 6 x + x = A Dep on MA and conclusion reached with at least one of the interim lines and without any errs omissions Cambridge International Examinations 4
8 Page 8 Mark Scheme Syllabus Paper IGCSE May/June (ii) x = 5 B BFT B f ( x + 6)( x 5) [= ] SC f ( x + a)( x + b) where ab = a + b = B f + B f x q FT 6 (their x + ) if x > crectly evaluated 9 (a), 4 9,, B f 4 B f 9 B f and (b) 45 (c) (d) (e) M f M f + only 4 4 their (c) only 4 9 M f 4 4 their (c) M f 4 4 Cambridge International Examinations 4
9 Page 9 Mark Scheme Syllabus Paper IGCSE May/June (a) B f g = soi [fg=] x (b) x Accept equivalents e.g. (x ) (c) x x + M f ( x ) + (d) 6 [ x ] = x x + x B f ( ) better (e) ( ) 4()() better B f x p = ( ) and r = B Must see p + r q p r q both.8,.6 BB f + SC f answers.4 and.6.89 to.8 and.68.8 and.6 seen in wking f.8 and.6 as final ans (f) f(x) and g(x) Accept f and g /x and x Cambridge International Examinations 4
10 Page Mark Scheme Syllabus Paper IGCSE May/June Allow equivalent decimal throughout (sf better where necessary) M f () : : seen 6 π π to π M f [X = 6 ].5 l sin6 [X = 6 ].5 l sin Or recognition that the area of the obtuseangled triangle shaded is equal to the area of one of the 6 equilateral triangles from the centre 4 If fraction given as answer, check if it falls into range B f [sect=] π r 4 B f [triangle =] r their sect their triangle Mdep f dep their sect on BB earned Cambridge International Examinations 4
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