0580 MATHEMATICS. Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for Teachers.

Transcription

1 CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME f the May/June 04 series 080 MATHEMATICS 080/4 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 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 04 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 oe equivalent SC Special Case nfww not from wrong wking soi seen implied (a) (i) 6 4 (ii) Not possible (iii) 6 4 B f one row column crect (iv) 3 oe isw B f a b c d seen k seen 3 (b) column in C and rows in D Any clear indication (c) Enlargement [Fact] [Centre] (0, 0) oe (a) 8 M f. oe (b) [Distance =] 36 their36 3 [= ] oe B M (c) 00 M f oe e.g (d) Hizontal line at 36 to 3 4 (their 3 4, 36) joined to (6 4, 0) FT 3 (a) 60 M f 46 6 soi by 4 46 (b) ( ) M f 00 oe 3890 M f crect fraction soi by soi by Cambridge International Examinations 04

3 Page 3 Mark Scheme Syllabus Paper IGCSE May/June (c) M f oe M f 3890 associated with 0% (d) (i) 9 to to nfww 3 M f π (30 ) oe Or M f π 30 π (ii) (a) 0.0 (b) nfww 4 M f 9. their 0.0 [=.96.8[0]] and M f their (d)(i) ( 9. their 0.0) A f 80 to 83. (non integer) If 0 then SC f their (d)(i) ( 9. 0.) 4 (a) 4 x crectly placed x crectly placed crectly placed (b) (6 x) + x (4 x) + ( x) + = 40 oe 46 x = 40 nfww x = 3 M A B SC3 f, and all crectly placed instead of expressions in x FT from their Venn diagram, condone omission of one subset Must be in the fm a + bx = c, ie each side simplified, better (c) (i) 9 ISW cancelling conversion after crect 0..% 40 answer seen (ii) FT FT from their Venn diagram and their x provided n(b P T ) (iii) FT FT from their Venn diagram (iv) FT FT from their Venn diagram (v) 4 (d) Crect region shaded. T B P Cambridge International Examinations 04

4 Page 4 Mark Scheme Syllabus Paper IGCSE May/June (a) [0]44 to [0]48 (b).6 to 3. B f 8.4 to 8.8 seen (c) 340 (d) : 0000 M f soi (e) Arcs f perp bisect of SL Two pairs of crect arcs Ruled perp bisect of SL Within tolerance of overlay Arcs f bisect of angle PSL Marks on PS and SL plus one pair of crect arcs Ruled bisect of angle PSL Within tolerance of overlay B marked within accuracy Within tolerance of overlay Dep on two crect bisects drawn (f) 3.3 M f.. (/3) seen 6 (a) (i) 0.6 oe M f (ii) 00 (iii) 0.03 oe M f (b) 8 3 oe = 0.848[4 ] 3 33 M f M f addition of any two of 6,, sum of 3 products with an err in the numerat of one product 4 f identified Cambridge International Examinations 04

5 Page Mark Scheme Syllabus Paper IGCSE May/June k (a) (i) Image: ( 4, 3), ( 4, ), ( 3, ) SC f translation k 4 (ii) Image: (, ), (3, ), (3, ) SC f rotation about the igin but 90º anticlockwise (b) (i) Image: (, ), (, 3), (4, 3) 3 B f crect vertices plotted SC f 3 vertices shown in wking SC f vertices shown in wking 0 M (ii) Stretch [fact] Invariant line y-axis oe Accept x = 0, stays the same 8 (a). and.3 B f one crect value (b) Crect curve B4 B3FT f crect plots BFT f 9 0 crect plots BFT f 8 crect plots (c) Ruled tangent at x = Gradient from.8 to 0. B No daylight at x =. Consider point of contact as midpoint between two vertices of daylight, this must be between x =.8 and. Dep on B awarded Allow integer/integer a mixed number if within range M dep f (change in y) (change in x) Dependent on any tangent drawn close attempt at a tangent at any point Must see crect implied calculation from a drawn tangent (d) 0 and. to.6 and.6 to. B f two crect values (e). to 0.8 < k <.8 to 3. B f each crect SC f reversed answers Cambridge International Examinations 04

6 Page 6 Mark Scheme Syllabus Paper IGCSE May/June (a) (i) 3. to 38. (ii) 9. to 0. nfww B f [LQ =] 3. to 4 [UQ =] 43. to 44 (iii) 43 B f 6 seen hizontal line drawn at cf = 6 (b) (i) 3.8[4 ] nfww 4 M f midpoints soi (condone err omission) and M f use of ft with t in crect interval including both boundaries (condone further err omission) and M (dep on nd M) f ft 80 (4. 80) (ii) Crect histogram 4 B f each crect block with crect width and height If B0 then SC f four crect f.d.s four crect widths 0 (a) (i) (ii) 3 oe B f [h( ) =] 3 soi M f (3 x ) 3 (iii) x + 3 x +. as final ans M f y + 3 = x x = y 3 y = x. better crect reverse flowchart (iv) 4x 9 as final answer nfww M f (x 3) 3 (v) (x 3)(x + ) = + (x + ) x 3x + x 3 better seen x 3x 6 = 0 M B A (x )(x + ) = (eliminate fractions) x x + x better seen No errs omissions seen Cambridge International Examinations 04

7 Page Mark Scheme Syllabus Paper IGCSE May/June (vi) ( 3) ± ( 3) 4 6 B B f ( 3) 4 6 better [ ] p + q p q and if in fm r r B f p = ( 3) and r = better.64 and.4 cao BB SC f.64 and.4 seen in wking.6 and. as final ans.63. and.3.. as final ans.64 and.4 as final ans (b) x x + as final answer nfww 4 B3 f (x )(x ) and (x + )(x ) B f (x )(x ) (x + )(x ) SC f (x + a)(x + b) where a + b = 3 3 ab = 0 (a) (i) (, ) (ii) M f ( 3) + 4 better (b) (i) (a) 3 a + b (3a + b) (a) final answer M f any crect vect path f ON (b) a M f any crect vect path f NY (ii) NY = BC oe dep dep on (b)(i)(b) crect [NY] parallel to [BC] dep dep on NY = ka, k Cambridge International Examinations 04