MARK SCHEME for the November 2003 question papers 0580/0581 MATHEMATICS

Transcription

1 CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the November 003 question papers 0580/058 MATHEMATICS 0580/0, 058/0 Paper (Core), maximum raw mark /0, 058/0 Paper (Extended), maximum raw mark /03, 058/03 Paper 3 (Core), maximum raw mark /04, 058/04 Paper 4 (Extended), maximum raw mark 30 These mark schemes are published as an aid to teachers and students, to indicate the requirements of the examination. They show the basis on which Examiners were initially instructed to award marks. They do not indicate the details of the discussions that took place at an Examiners meeting before marking began. Any substantial changes to the mark scheme that arose from these discussions will be recorded in the published Report on the Examination. 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 November 003 question papers for most IGCSE and GCE Advanced Level syllabuses.

2 Grade thresholds taken for Syllabus 0580/058 (Mathematics) in the November 003 examination. maximum mark available minimum mark required for grade: A C E F Component Component Component Component The threshold (minimum mark) for B is set halfway between those for Grades A and C. The threshold (minimum mark) for D is set halfway between those for Grades C and E. The threshold (minimum mark) for G is set as many marks below the F threshold as the E threshold is above it. Grade A* does not exist at the level of an individual component.

3 Notes Mark Scheme Syllabus IGCSE EXAMINATIONS NOVEMBER /058 TYPES OF MARK Most of the marks (those without prefixes, and B marks) are given for accurate results, drawings or statements. M marks are given for a correct method. B marks are given for a correct statement or step. A marks are given for an accurate answer following a correct method. ABBREVIATIONS a.r.t. Anything rounding to b.o.d. Benefit of the doubt has been given to the candidate c.a.o. Correct answer only (i.e. no follow through ) e.e.o. Each error or omission o.e. Or equivalent SC Special case s.o.i. Seen or implied ww Without working www Without wrong working > Work followed through after an error: no further error made= > = Work followed through and another error found

4 November 003 INTERNATIONAL GCSE MARK SCHEME MAXIMUM MARK: 56 SYLLABUS/COMPONENT: 0580/0, 058/0 MATHEMATICS Paper (Core)

5 Page Mark Scheme Syllabus Paper IGCSE EXAMINATIONS NOVEMBER /058 Question Part Mark Scheme Details Number Mark 400 (grams) 5 ($).7(0) M for 8 o.e SC for 8 = (a) Accept equivalent fractions, 5 decimals, percentages (with sign) (b) accept, do not accept, 5 k none, not but condone it with 0 4 (a) 6 o 44 (b) 40(%) M for 00 o.e (0 ) M for 5 sin 0 o or 5 cos70 or M for,, 6 or o ( 0 4) 90 3 M for or 0 ( 0 ) 80 or After 0, SC for answer 36 o 8 50 r.l. < SC if reversed 9 (a) 0x 5xy B for one term correct (b) 6x (x + ) M for 6(x + x) or x(6x + ) or (3x + 6x) or x(3x + 6) or 3(x + 4x) or 3x(x + 4) 0 (a) 87 o 4 (b) 8 o (c) 6 o f.t. is (90 y) 3 University of Cambridge Local Examinations Syndicate 003

6 Page Mark Scheme Syllabus Paper IGCSE EXAMINATIONS NOVEMBER /058 Lines may be freehand but must go completely through the shape Any line through the centre 3 x = 4, y = 3 M for attempting to eliminate one unknown by a correct method A for one correct value (x or y) 3 3 (a) (i).4096 (ii).4 f.t. from (i) (b) 9.3 or 9.3(6 ) B for.68 seen or implied by (a) Monday, Tuesday and Saturday All three and no extras (b) -o 3 B for 4 seen + M for (their 4) (a) (i) 0.8 (ii) 0.75 (iii) or (b) , 8%, 7 or equivalent f.t. from (a) 6 (a) 4.58(m) M for (b) 66.4o or 66.3o 66.45o 5 s.o.i. e.g. 4 M for cos - o.e. incl 5 University of Cambridge Local Examinations Syndicate 003

7 Page 3 Mark Scheme Syllabus Paper IGCSE EXAMINATIONS NOVEMBER /058 7 (a) etc. penalise once only (b) -4 accept 04 (c) 0 4 (d) - 8 (a) 0.4 or.6 B for one correct SC if (0.4,0) (.6,0) (b) (i) 0 (ii) Correct line from x = - to x = 4 Must be ruled 6 (c) (0,), (4,5) B for one correct f.t. from (b) (ii) University of Cambridge Local Examinations Syndicate 003

8 November 003 INTERNATIONAL GCSE MARK SCHEME MAXIMUM MARK: 70 SYLLABUS/COMPONENT: 0580/0, 058/0 MATHEMATICS Paper (Extended)

9 Page Mark Scheme Syllabus Paper IGCSE EXAMINATIONS NOVEMBER /058 * indicates that it is necessary to look in the working following a wrong answer 0.5 or c.a.o. (-)4504 Allow (-) (a) (b) (n + ) Allow 49, 64, 8, 00, n + n + 4 3/500, /8, * M for all 3 evaluated as decimals (or fractions or percentages or stand. form) SC reversed order 5 (a) -, 36 (b), 30 Allow, S6 SC (a) and (b) 36,, 30 6 I = mr/5 * M for 40 r m o.e. 00 ( ) M for 00 o.e (a) - (b) 5k 9 (a) 3000 (b) , SC both correct and reversed 0.5() 3* M F = kv M k = 8/40 or better (a) 30 (b) 3 * M for or 936 x.6 Allow , 307.5, 308 or SC (a) (a) 45, 5 (b) (a) 5.5 or 5½ (b).5, Allow 58 * M (a) x + 3 x( x + ) 3* M 3(x + ) - x M denominator x(x + ) (b) -3 University of Cambridge Local Examinations Syndicate 003

10 Page Mark Scheme Syllabus Paper IGCSE EXAMINATIONS NOVEMBER /058 5 (a) angle bisector of angle P (b) radius from T or U 6 (a) A(,0) B(0,-6) (b) 6.3 (c) (,-3) * *, * M correct construction method AS o SC for accurate line but no arcs M radius drawn, meets (a) and O labelled. AS o SC correct and reversed M (AB ) = (0 ) + (-6-0) from (a) 7 (a) 0 (b) 98 (c) 6 (d) 4 (e) 36 (b) (c) 8 (a) 5.8 x 0 8 (b) 98 (c) (a) -6 (b) (i) 0.4 (ii) (0.4, 0.) 0 (a) (i) - / 3 p + q (ii) - 3 / 4 q + p (b) / 3 p / q (a) 60x + seen (b) x A y (c) line y = x line through (0,0) and (0,5) shading out or R labelled (d) 0 c.a.o. * * * * * * M figs 58 figs 59 or figs M figs 59 figs 58 x 0 n or x 0 n (b) n = 3 or 6 M (7/) M 5x o.e., - 4x = x or better M use of AQ = S / 3 p S q or AO + OQ M use of BQ = S 3 / 4 q S p or BO + OP M - / 4 q + / 3 BP Allow 0.6x + M intention A accurate Dep. on both lines Allow 0, 0 or Total 70 TOTAL MARKS 70 University of Cambridge Local Examinations Syndicate 003

11 November 003 INTERNATIONAL GCSE MARK SCHEME MAXIMUM MARK: 04 SYLLABUS/COMPONENT: 0580/03, 058/03 MATHEMATICS Paper 3 (Core)

12 Page Mark Scheme Syllabus Paper MATHEMATICS NOVEMBER /058 3 Question Number Mark Scheme Part Marks Notes Question Total a) 4 b) 5 or 5 c) 7 or 3 3 d) 3 9 e) 6 condone 6, 6 or 6 x 6 f) 8 cao g) and 7 condone x 7 8 a) i) 300 or pm ii) 030 allow 0.30, 0:30 etc iii) 9 B for either 4 or 33 seen or M for correct horizontal lines drawn or 4 and 33 marked on axis b) i) 4.35, 8.7(0) B for one correct ii) Correct straight line (through (0, 8.6 to 8.8) P for (5, 4. to 4.4) or (0, 8.6 to 8.8) iii) 9.(0) (± 0.) no ft. iv) 575 (± 5) no ft a) 6000 M for 5 x 30 x 8 b) i) art M for π x 0 x 4 or SC for π x 5 x 4 ii) art 0400 ft their a + bi iii) art ft for (their bii) (5 x 30) M for (their bii) (5 x 30) oe or M for (their bi) (5 x 30) 9 4 a) 4, 7, 6, 4, 4,, 3 SC for 5 or 6 correct or 7 correct tallies b) cao c) cao M for attempt at ranking list seen d).5 cao M their f ( ) _ imp by.5 seen e) i) 7 allow 3% 0.3(3...) or 30 ft from their table ii) 3 9 ft from their table 0.3 or or 0 30 f) 40 ft their table x 0. Allow 40/ a) 6 4 b) i) Rotation through 80 ii) about (.5, 6) o.e. Enlargement s.f. 3 centre (,7) M A A B B B Half turn M Al, for "symmetry" allow correct description of point accept scale 3, x3 etc accept'b' for (,7) c) i) 3 cao ignore units ii) : 9 cao SC for 7 seen M for correct answer nlt d) 6 ', 0.66 or better SC for oe or k University of Cambridge Local Examinations Syndicate 003

13 Page Mark Scheme Syllabus Paper MATHEMATICS NOVEMBER / a) i) 7 ii) 6 M for (39-3) 6 iii) P 3 oe M for P 3 seen or = x oe seen b) i) 4x + 3 M for 9x + 4 x (3x + ) oe allow 9x + 4 x 3x + oe for M or SC for 4x or (+)3 in answer space ii) 0, 6 and 3 3 M for 9x + 4 = 49 oe A for x = a) i) 44 SC for 40 to 48 ii) 5 3 B for 6 or 8 or or 9 or or 8 or 3 or seen +M for adding 6 rectangles o.e. iii) cuboid or rectangular prism allow rectangular cuboid but not cube or cubical iv) 5 ft from their aii (not strict ft) v) 4 M for x 3 x 4 b) i) (pq + qr + pr) oe as final answer SC for pq or qr or pr seen or imp. for both parts. Other letters used consistently MR ii) pqr as final answer M for pqr seen 3 8 a).5 3 M for 7.5 x oe or 80/ oe seen NB 40 answer M for x00 (explicit) or working uses 75 and x00 (explicit) after M0 SC for figs 4 to 6 ww or SC for.5 b) 0 minutes 3 3 B for or 80 or x 300 seen 5 5 c) i) Accurate bisector of AB, with arcs ± ±mm complete inside figure Accurate bisector of <C with arcs as above +M for 5 x 300 oe or SC if accurate without arcs or incomplete line. Ignore extra lines SC if accurate without arcs or incomplete line as above ii) correct area shaded Areas marked as diagram ft from clear intention to draw perp. bisector and angle bisector 9 a) i) 50 (km) ii) oe ( ) Ml for their a)i) x 00 x 000 or SC for their a)i) x 0 n when n>0 b) i) 70 to 30 M for their 8.6 x their 50 must have some evidence for their 8.6 ii) (0)45 to (0)48 oe iii) 45 to 48 SC for any answer in the range 80 < x < University of Cambridge Local Examinations Syndicate 003

14 Page 3 Mark Scheme Syllabus Paper MATHEMATICS NOVEMBER / a) Sum Sum 8 SC if 6 or 7 correct b) i) 5 accept 9 SC for 56 ii) n SC for x x seen or description c) The first 6 numbers repeated in reverse order TOTAL 04 University of Cambridge Local Examinations Syndicate 003

15 November 003 INTERNATIONAL GCSE MARK SCHEME MAXIMUM MARK: 30 SYLLABUS/COMPONENT: 0580/04, 058/04 MATHEMATICS Paper 4 (Extended)

16 Page Mark Scheme Syllabus Paper IGCSE EXAMINATIONS NOVEMBER /058 4 Marks in brackets are totals for questions or part questions. (a) 44:96 Final answer 3: or.5: or :0.667 B B () (b) (i) 3 (children) B (ii) 54 (adults off) B (iii) 0 (adults on) B (iv) 6 (=x) w.w.w. B (4) (c) 4 M 300 thier ( ) After B0, allow SC for reversed correct final ans. www 80 children A www () (d) (i) Final Ans. 3 or (0)9 3 pm B Condone hrs but hrs and minutes BO (ii) h 0 min (o.e) or 0 0 M 40 min A (3) () Implied by 6 h 40 min or 400 min www (a) (i).8(0..) B Throughout (a)(i)(ii)(iii) NO misreads allowed. (ii).99 80h = 3600 o.e. M Must be h, not h (h =) 78(. ) A ww (Must be correct e.g MO ww) (iii) A hm = 3600 M (First step must be correct from correct formula for first M.) Correctly squares at any stage 3600A = hm 3600A = h m M M (6) Correctly multiplies at any stage Correctly divides at any stage Only a correct answer in this form can get M3. (b) (i) (x + 4) (x 4) B i.s.w. solutions in all (b) (ii) x(x 6) B Condone loss of final bracket in any (b) (iii) (x 8)(x ) B (4) University of Cambridge Local Examinations Syndicate 003

17 Page Mark Scheme Syllabus Paper IGCSE EXAMINATIONS NOVEMBER /058 4 (c) (i) x(3x 9) = x 8 o.e. M x 8 = 3x 9x x 9x + 8 = 0 E (ii) x = B x = 8 B (iii) time = 5 (sec) c.a.o. B distance = 0 (m) c.a.o. B (6) (6) 3 (a) (i) cos40 M No error seen and some working to reach final quoted equation. Must have = 0. (E = established) Allow M for sign error or correct implicit eqn (ii) their M Dep M. NOT for 46 o 5 cos 40 or Answer in range.89 to.9 (m) A www4 o sint sin40 M or 7 = 3 + (their.9).3. (their =.9) cost 7 their.9 sint = o 7 sin40 their.9 (0.499) M cost = 3 + ( their.9) 7.3.(their.9) 9.9 A Accept 9.93 to www3 (7) (b) (i) 5 c.a.o. B All bearings must be 0 Y==θ Y=360 to score ** (ii) 305 B (80 + their 5 ) correct ** (iii) 335 or B (3) (their their T) correct (c) M 30 or FXT tan( Fˆ ) = = tan 3 clearly identified. o.e A ( ) www NOT 43. () () 4 (a) Scale correct S 0 Y t Y 7 (4 cm) and 0 60 ( cm) 8 correct plots (0, 0), (, 5), (, 37.5), (3, 43.8), (4, 46.9), (5, 48.4), (6, 49.), (7, 49.6) P3 Allow P for 6 or 7 correct P for 4 or 5 correct Accuracy better than mm horizontally. In correct square Reasonable curve through 8 points C (5) Not for linear or bad quality University of Cambridge Local Examinations Syndicate 003

18 Page 3 Mark Scheme Syllabus Paper IGCSE EXAMINATIONS NOVEMBER /058 4 (b) (i) 03 B Do not accept improper fractions f(8) = 49.8 or 49 o.e. 8 3 B f(9) = 49.9 or 49 o.e. 56 (ii) f(t large) 50 B (3) (c) (i) Tangent drawn at t = B Not a chord and not daylight Uses vert/horiz using scale M Can be given after B0 if line not too far out ** Answer correct for their tangent A (ii) Acceleration or units B (4) Accept ms, m/s, m/s/s. (d) (i) Straight line through (0, 0) B Straight line gradient 6 B Must be ruled and full length to earn B ** ** (ii) one intersection value for t Second t and range B B (iii) Distance = area (under curve) First particle (f(t)) goes further M A (6) (8) Marking final answers throughout this question 5 (a) (i) 0. o.e. B Accept /0, /5, 0% (ii) 0.4 o.e. B After first B0, condone in 0 type answers. (iii) 0.5 o.e. B Never condone : 0 type (iv) 0. o.e. B (v) 0 B Accept none, nothing, 0/0, nil, zero (5) (b) (i) /0 x /9 M /45 o.e. A Accept /90, 0.0.% www (ii) 3/0 x /9 M /5 o.e. A Accept 6/90 etc, (or 7), 6.66 or 6.67% www (iii) (their) /45 + (their) /5 M 4/45 o.e. A Accept 8/90 etc, (or 9), 8.88 or 8.89% www (iv) Clearly (their) 4.45 o.e. M Alternative method must be complete 4/45 A (8) (3) Accept 8/90 etc, 0.9, 9.% www University of Cambridge Local Examinations Syndicate 003

19 Page 4 Mark Scheme Syllabus Paper IGCSE EXAMINATIONS NOVEMBER / (a) π(30) (50) M (cm 3 ) A (b) (i) 8 (cm) B (ii) cos AOB = (their 8)/30 x M () Mdep AOB = 06.6 c.a.o A (4) (4 300 to 4 430) www Allow M or M at similar stages for other methods e.g. sin A = 8/30 then (80 A) Must have decimal places seen. ww (condone = 06.3 afterwards) (c) (i) 06.3 M (their) used 360 π(30) used 834 to (cm ) M A www3 (ii) M.30.30sin (their) 06.3 or to 43 (cm ) A www (iii) Ans. Rounds to 403 cm A (6) (d) (i) 50 x (their) 403 M ** 0 00 to 0 00 (cm 3 ) A correct for their 403 www ** (ii) 0. to 0. (litres) B (3) (e) M k their(a) their(d)(i ) 7 (a) (i) 50.3 to 5 (litres) A () (7) their previous answer 000 k = (cm 3 ) k =.00 (litres) k = other consistent conversion error. Marking final answer www F M A M marks for letters, A marks for 4 descriptions. If no letter given, allow SC for correct description (ii) D x = M A (iii) E (, ) M A (iv) C (s.f.) 3 M A (v) A Shear M A (0) University of Cambridge Local Examinations Syndicate 003

20 Page 5 Mark Scheme Syllabus Paper IGCSE EXAMINATIONS NOVEMBER /058 4 (b) 3 ( ) or QP 5 7 M Penalty for each wrong one thought possible. ( 7) final ans A Allow SC for one correct ( 3) or RS 3 M () A (6) 8 (a) (i) 0 < M Y 5 B Must clearly mean this and not 3 (ii) Midpoints 5,.5, 7.5,.5, M Allow for 3 or 4 correct 3.5 fx ( ) M (70) Needs previous M or only marginally out (their) 70 0 M dep previous M 8.9 () (kg) A www4 () (iii) 36 B (6) (b) Horizontal scale cm 5 units S 0 Y M Y 40. Accuracy < mm. (numbered or used correctly) If S0 (e.g. cm 5 units) can score B5 If S0 (e.g. 0, 0, 5) can only score on correct width bars. Penalty for polygon superimposed. Heights 3k, 6k, 4k, k, 4k cm B5 If not scored, decide on their k and allow SC for each correct bar. (Needs [= bars to decide on value of k if k.) Their k = 9 (a) (i) (Diagram) 5 only B B (ii) (Diagram) 4 only B (iii) (Diagram) only B (6) (7) (3) (3) Brackets essential here. Allow SC for or University of Cambridge Local Examinations Syndicate 003

21 Page 6 Mark Scheme Syllabus Paper IGCSE EXAMINATIONS NOVEMBER /058 4 (b) Diagram 9 (cm ) Diagrams and 3 have same area One of them x 3 x 3 4 (cm ) B B M A 9.00 to 3 s.f. www Diagram 4 4 π3 s.o.i. M (7.07 cm ) M indep. i.e. 8 kπ where k numerical x 6 x 6 their 9π/4 0.9 (cm ) A www3 M Diagram 5 s.o.i (bc = 7 ) 6 tan M (.485) (This is AD or DE) dep.m (6 their.485) x 6 or 8 x 6 x their.485. (o.e.) 0.5 (cm ) A () (4) www4 University of Cambridge Local Examinations Syndicate 003