Mark Scheme Summer 2007

Transcription

1 Mark Scheme Summer 007 IGCSE IGCSE Mathematics (4400) Edexcel Limited. Registered in England and Wales No Registered Office: One90 High Holborn, London WCV 7BH

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3 Contents. 4400/F /F /3H /4H

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5 4400 IGCSE Mathematics Summer 007 Paper F Q Working Answer Mark Notes. (ai) 8 rectangles B shaded (ii) 60 B cao (b) 0.4(0) B (c) for, etc cao 5 Total 5 marks. (ai) (,5) B (ii) (5,0) B (b) 64 B Allow + mm (c) 8(0) + 64 Also award for inc ft from 64 (d) isosceles B (ei) 77 B Allow + (ii) acute B Total 8 marks 5

6 3. (ai) 4 5 B cao (ii) 5 B cao (iii) 7 B cao (iv) 5 B cao (v) 3 B cao (bi) 4 B cao (ii) 39 B cao Total 7 marks 4. (a) 3 B cao (b) eg Add 6 B (c) 6 B cao (d) 89 B ft from (b) (e) eg Sum of two odd numbers is always even B Accept if odd used correctly Total 5 marks 5. (ai) B 4 B cao (ii) F B cao (iii) I B cao (iv) D B cao (b) H B cao Total 5 marks 6

7 6. (a) 300 B cao (b) B (c) Bengali B (d) 00 < bar < 50 B (e) 300 : 5 for 300:5, 60:5 also for 5 : 300, 5 : 60, 5 : : 5 (f) (g) for or for 43. or better Total 0 marks 7. (a) x = 9 4 oe (b) 5y y = 7 + 4, 3 oe 3 3 Also accept or more d.p. rounded or truncated e.g. 3.66, 3.67 Total 4 marks 7

8 8. (ai) pm B Accept 300 (ii) 0 pm B Accept 00 (b) 6 B cao (c) Rio de Janeiro B Accept Rio (di) 7 B Accept 7 (ii) 5 B Accept 5 9. (a) 0 B cao (b) (5 ) B cao (c) B (d).6 B cao 0. (i) y = 3 3 B (ii) x = 5 B (iii) y = x B Total 6 marks Total 4 marks Total 3 marks. (a) for 8.3, 68.89, 9. or Accept if first 5 figures correct (b) 7.57 B ft from (a) if non-trivial Total 3 marks 8

9 . (a) (b) (c) B Accept 0., 0% 5 3 for fraction with denominator for 5 Accept 0.6, 60% ft from Do not accept 5 50 Total 5 marks 3. (a) cao (b) rectangle 5.6 cm long and 3 cm wide B B for each Allow + mm Total 4 marks 4. (a) ( 3) 5 3 for substn or 9 or 5 see 4 cao (b) x ( x 5) B B for factors which, when expanded and simplified, give two terms, one of which is correct SC B for x(5 x), x(x 5x) Total 4 marks 9

10 5. (a) 47 B cao (b) 5 46 for 5 46, 46-5 etc 5 cao (c) or or for finding at least 4 products and adding (dep) for division by 0 48 cao Total 6 marks 6. (a) translation 3 squares to the right and square down B B for translation Accept translate, translated etc Accept across instead of to the right B for 3 right and 3 down or but not (3, ) (b) rotation of 90 clockwise about (, ) 3 B3 B for rotation Accept rotate, rotated etc B for 90 clockwise or 90 or 70 B for (, ) These marks are independent but award no marks if answer is not a single transformation Total 5 marks 0

11 7. (ai) 7 8 B cao (ii) 5 6 B cao (b) n = 8 or 3 n = 8 Also award for n = 5, n = 3 or 5 on answer line 5 cao Total 4 marks 8. (a) x 5 8x 4 for at least 3 terms correct 4x 9 (b) y + 3y + 8y + 4 for 3 terms correct or y + y seen y + y + 4 (c) 5p 3 + 4p 3 B cao B for either 5p or for +4p Total 6 marks

12 9. (a) or = ; for or 38.5 or or or or for or 0 3 cao (b) π M for π (no with digits 49) no with digits for or for answer which rounds to (π ) Total 6 marks 38.5 '0.35' PAPER TOTAL 00 MARKS

13 4400 IGCSE Mathematics Summer 007 Paper F Q Working Answer Mark Notes. (ai) (ii) (iii) (bi) (ii) 99, 999, 009, 999, or B B B B B B for begin with or end with or 3 Total 6 marks. A at mm B at C at < d < 5mm from 0 B B B ie between 0 & 0.5 exclusive Total 3 marks 3. (ai) (ii) (iii) (b) Radius Chord Tangent Sector 3 B allow misspellings B allow misspellings B allow misspellings B allow misspellings Total 4 marks 3

14 4 (a) (b) (c) (d) (e) (fi) (ii) B B or 3000 or 3 thousand or 000 or thousands B B B B B ft Total 7 marks 5 (a) (b) / 4 7 Allow 30/4 or ans 7.5 oe Total 4 marks 6 (a) (b) / 36 3 / 0 B B (c) Attempt convert all to dec or % or c. d. All correctly converted 9 / 5 3 Total 5 marks 4

15 or dep Total 3 marks 8 (a) (b) (c) (di) (ii) Σx attempted or 56 / 8 Arrange in order oe Same Middle unchanged 3 B eg 48.5 dep or answer 5 or 6 Bindep B or still 5.5 (e) /8 oe B num B denom Ratio subtr B Total 3 marks 9 (a) (b) 6 cm Triangle correct + mm 3 B B for 5 to 7 incl B B B for a vertex correct + mm or correct size & orientation + mm Total 5 marks 5

16 0 (a) (b) (c) (d) (6 + 3) 5 oe 70/5 or 4 (-3) -85/5 3 or (x + 3) or (x + 3) 5 or 5x + 5 oe Bracket essential unless ans correct Allow 67/5 or 3. B B for answer x B0 for answer 5x + 3 or x + 5 x = subtract B Total 8 marks (a) (bi) (ii) < sum of triangle B B B Not 80 ( ) Total 3 marks (a) (b) / 5 x x 90 oe Allow 0.85 x 4800 oe Total 4 marks 3 (i) (ii) 6x = or 6x = 0 etc x + x + + 3x - 5 = 7 x = 3.5 oe eg / 6 B oe eg 6x 4 = 7 ISW not = p ft (i) if 6x = c Total 3 marks 6

17 4 9 seen 7/9 7 or 7 7/9 oe 3 B dep B seen, & ans = 3 BA0 Total 3 marks 5 5x 0 = 35 5x = 55 3 dep or M for x 4 = 7 Total 3 marks 6 (a) (b) 7 50 or 7, 50, or 00 B for 7 and B for 50 (6or7) (48or50) correctly eval d or 3 eg 68 f If no wking: ft (a) (c) Num incr or 6.8 & 47.6 incr denom decr or.09 decr (b) rnded up (not rnd to sf) or 75 rnded to 00 B any two of these B any one of these Ignore other Total 6 marks 7 (a) (b) ( + 3)/ 6 or 6 + / 6 oe 5 / / / or 0.75 ft 5 for only Total 5 marks 7

18 8 x + 3 = 7x 7y = 7x + (6x = 3 oe) (6y = ) x = /, y = 3 / 3 y = 7(y-3) y = 7y- 0 = 6x -3 Total 3 marks 9 (a) (b) tan used tan x = 5./4. or tanx =.. oe sin 9 = AB/5 or c/sin9 = 5/sin90 AB = 5sin 9 x = 50.5 AB =.4... cm 3 3 (sin or cos) & ( ( ) or 6.6) used sinx = 5./ ( ) or cosx = 4./ ( ) BC = 5cos9 AB = (5 +(5cos9) ) or 5cos9 tan9 Total 6 marks 0 (a) ( ) or 0.4 oe 0.6 or 0.6 in table allow in table if not contrad on line (b) or ( ) 0.3 or 0.3 seen (c) (Poss) overlap or mut excl or doesn t wk for B or Y } No or poss or poss yes } Can t tell & (No or poss) B B Correct reason only: B Incorrect reason: B0 Unqualified Yes: B0 Total 6 marks 8

19 4 + 6 (= 5) (4 + 6 ) or 5 or 3 h = dep Total 3 marks PAPER TOTAL 00 MARKS 9

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21 4400 IGCSE Mathematics Summer 007 Paper 3H Q Working Answer Mark Notes. (a) for 8.3, 68.89, 9. or Accept if first 5 figures correct Also accept (b) 7.57 B ft from (a) if non-trivial ie (a) must have more than d.p. Total 3 marks. (a) ( 3) 5 3 for substn or 9 or 5 seen 4 cao (b) x ( x 5) B B for factors which, when expanded and simplified, give two terms, one of which is correct SC B for x(5 x) and for x(x 5x) Total 4 marks or or for finding at least 4 products and adding (dep) for division by 0 48 cao Total 3 marks

22 4. (a) translation 3 squares to the right and square down B B for translation Accept translate, translated etc B for 3 right and down (accept across instead of to the right ) or but not (3, ) (b) rotation of 90 clockwise about (, ) 3 B3 B for rotation Accept rotate, rotated etc B for 90 clockwise or 90 or 70 B for (, ) 5. (ai) 7 8 B cao (ii) 5 6 B cao (b) n = 8 or 3 n = 8 Also award for n = 5 or 5 on answer line 5 cao 3 These marks are independent but award no marks if answer is not a single transformation Total 5 marks Total 4 marks

23 6. (a) x 5 8x 4 for at least 3 terms correct inc signs 4x 9 cao (b) y + 3y + 8y + 4 for 3 terms correct or y + y seen y + y + 4 (c) 5p p B cao B for either 3 5p or for + 4p Total 6 marks 7. (a) or = ; for or.83 or better 38.5 or or 83.3 or better 0. or or for or 0 3 cao (b) π M for π (no with digits 49) no with digits for or for answer which rounds to Total 6 marks 38.5 '0.35' 3

24 8. (a) for or 0.06 or or cao (b) (c) or for or 6 seen cao 3 M for or for, 04% = 338 or 3 seen cao Total 7 marks 4

25 9. (a) 5x x = for correct rearrangement (b) 7 y 4 or 7 y 4 = 4(y + 3), oe Also accept or more d.p. rounded or truncated e.g. 3.66, for clear intention to multiply both sides by 4 or a multiple of 4 For example, award for 7 y 4 or 7 y 4 = 4 y + 3 or 8y + 3 or y or y + 7 y = 8y + or simpler for correct expansion of brackets (usually 8 y + ) or for correct rearrangement of correct terms e.g. 8y + y = 7 0y = 5 for reduction to correct equation of form ay = b oe Total 6 marks 5

26 0. Accept decimals in parts (a) and (b) (a) B for seen for cao Do not accept 50 3 (bi) (ii) for or 5 oe for 4 3 seen or SC for (dep) for adding both above products 8 or oe 8 for or oe or SC (dep) for adding both above products Total 8 marks 6

27 . (a) tangent at any point of a circle and the radius at that point are perpendicular B for mention of tangent and radius or line from centre (b) or for squaring and subtracting or (dep) for square root for 3.89 or better " " for only 9. for 9. or answer which rounds to 9. ( ) Total 6 marks. (a) 0, 6, 4, 50, 56, 60 B cao (b) Points correct B + ½ sq ft from sensible table Curve or line segments B ft if 4 or 5 points correct or if points are plotted consistently within each interval (inc end points) at the correct height (c) Use of w = 430 on graph may be shown on graph or implied by 43, 44 or 45 stated Approx 6 If scored, ft from cumulative frequency graph If no method shown, ft only from correct curve Total 5 marks 7

28 3. lines 4 B3 B for each correct line (full or broken) Ignore additional lines region B for correct region shaded in or out or for correct region labelled R Total 4 marks 4. (a) (bi) (ii) A r = π 3.5 π 4.5 π or.4836 for r = A Ignore ± π A π or r = A π for 3.5 seen for answer which rounds to for or value which rounds to π.48 or.49 cao dep on previous 3 marks in (b) Total 6 marks 8

29 5. (ai) k f = 4 w f = w (ii) f w B k May be implied by 500 = 00 k Also award if answer is f = but w k is evaluated as in (a) or (b) B for graph with negative gradient (increasing or constant) even if it touches or crosses one or both axes e.g. f (b) O w f = for substitution in ft from k f = k w Total 6 marks 9

30 6. (ai) 3b 3 B (ii) 3b a B (iii) a + b or a + (3b a) or 3b (3b a) oe 3 3 B (b) a 3 or 3 PQ or k = 3 or a + 3 (3b a) b 3 or 3 a + b b or (a)(iii) b or b + a + 3 (3b a) B for a or PQ or k = unless clearly obtained by non-vector method or for expression in terms of a and/or b (need not be simplified) for EF either correct or ft from (a) B for correct vector statement with at least 3 terms which includes EF (or FE) in terms of capital letters and/or a, b or b + a + 3 (a)(ii) or b 3 (3b a) or b 3 (a)(ii) oe eg PQ = PE + EF + FQ PF = PE + EF a = b + EF + FQ If an attempt is crossed out and replaced, mark all attempts, including crossed out one, and award best mark. Total 5 marks 30

31 7. dy dx 6 = x x " 6 x ± = 0 x " 4 B B for x 6 for ± x or + 6x 8. (a) π.8 + 4π.8 (b) 3 5 or 5 seen 3 (, ) cao For answer (, ) with no preceding marks scored, award B0 B0 Total 4 marks 3 M for each term Also award for values rounding to 4.6 and to 49. or for 73.9 or for answer which rounds to for 5 (a) or for π (.8 5) + π (.8 5) or for substituting r =.8 5 in the expression used in (a) 850 for 850 or for any value in range ft from 5 (a) Total 6 marks 3

32 9. x + (3x ) = 5 6 for correct substitution x + 9x 3x 3x + 5 B (indep) for correct expansion of or x + 9x 6x + (3x ) even if unsimplified 5 0x 6x 4 = 0 B for correct simplification ( 5x + )(x ) = 0 or ( 5x + )( x ) = 0 or ( 0x + 4)( x ) = 0 B for correct factorisation or 6 ± 96 0 or 3 ± 49 0 or for correct substitution into the quadratic formula and correct evaluation of b 4ac or 3 ± or for using square completion correctly as far as indicated x = or x = 5 for both values of x x =, y = 5 5 x =, y = for complete, correct solutions Total 6 marks PAPER TOTAL 00 MARKS 3

33 4400 IGCSE Mathematics Summer 007 Paper 4H Q Working Answer Mark Notes. (i) x + x + + 3x - 5 = 7 B B oe eg 6x 4 = 7 ISW not = p (ii) 6x = or 6x = 0 etc ft (i) if 6x = c x = 3.5 oe eg / 6 Total 3 marks. 9 seen B 7/9 7 or 7 7/9 oe dep B 3 seen, & ans = 3 BA0 Total 3 marks 3. 5x 0 = 35 5x = 55 or M for x 4 = 7 3 Total 3 marks 33

34 4. (a) 7 50 or 7, 50, B for 7 and B for 50 (b) 75 (6or7) (48or50) correctly eval d or 3 eg 68 (c) 00 or 00 f If no wking: ft (a) Num incr or 6.8 & 47.6 incr denom decr or.09 decr B any two of these (b) rnded up (not rnd to sf) or 75 rnded to 00 B B B any two of these B any one of these Ignore other Total 6 marks 5. (a) ( + 3)/ 6 or 6 + / 6 oe 5 (b) 5 / / / 0 or 0.75 ft 5 for only Total 5 marks 34

35 6. x + 3 = 7x 7y = 7x + (6x = 3 oe) (6y = ) y = 7(y-3) y = 7y- 0 = 6x -3 x = /, y = 3 / 3 Total 3 marks 7. (a) tan used (sin or cos) & ( ( ) or (6.6) used tan x = 5./4. or tanx =.. oe sinx = 5./ ( ) or cosx = 4./ ( ) x = (b) sin 9 = AB/5 or c/sin9 = 5/sin90 BC = 5cos9 AB = 5sin 9 AB = (5+(5cos9)) or 5cos9 tan9 AB =.4... cm 3 Total 6 marks 8. (a) ( ) or 0.4 oe or 0.6 in table 0.6 allow in table if not contrad on line (b) or ( ) or 0.3 seen 0.3 (c) (Poss) overlap or mut excl or doesn t wk for B or Y } No or poss or poss yes } B Can t tell & (No or poss) B Correct reason only: B Incorrect reason: B0 Unqualified Yes: B0 Total 6 marks 35

36 (= 5) (4 + 6 ) or 5 or 3 dep h = Total 3 marks 0. (a) V/H in any correct triangle attempted eg 3 not 3 0 Grad =, may be embedded or implied Bf Bf for grad. B for y-int (lin y = x + 4 eqn) or Bf for just x + No wking, ans x + : B (b) y = -x + c B y = -x ± any no. (not 5) or letter or y = -x (c) (0, -4) B Total 6 marks. (a) 56 B (b) x/0 = 6/ or 4/8 oe or x/sin30 = 0/sin( ) 0 or 0.0. (c) y/0 = 4/6 or 8/ oe or y = ( cos 56 ) or y/sin56 = 8/sin( ) 6.6 to 6.7 incl oe (a)(b): ft (a) M-mks only Total 5 marks 36

37 . (a) a 7 / a or a a 4 or a 3 a a 5 (b) x 3 B (c) Correctly cancel numbers or (x + ) eg / or 0.5 or denom = 3 ( x + ) 3 x + 3 or or or k(x + ) (k ) 6 6 / (x + ) or 0.5(x + ) x + x or or + or equiv Not ISW Total 5 marks 3. (a) Attempt arrange one set in order State or indicate correct 5 & 4 or4 & 6 A: B: 8 4 (b) A more spread or gter dispersion or less consistent than B NB: IQR for B = 8, check wking B Bf Consistent with (a). Ignore other. Not: gter range or difference or more constant or gter IQR or gter variance Total 5 marks 37

38 4. 5x 7 = x or 5x 7 = (x )(x + ) condone 5x 7 = x x + x 5x + 6 = 0 allow different order with = 0 (x.5) 5 ± ( 5) (x )(x 3) (= 0) or x = or 3 4 T & I or no wking: 4 mks or 0 mks Total 4 marks 5. overlapping circles, in overlap 6 in H only or 6 play H only M in T only or 0 6, 6++x=0, 0-8, 35-33: M3 4 4 ans : M3A0 Total 4 marks cosx = 6 90cosx = 70 or -90cosx = -70 (cosx = 70 / 90 ) x = 38.9 or better 3 or cosx = M Total 3 marks 38

39 7. (ai) - B or x - or x = - (ii) x < B B for x < or 0, -, -, (b) \/9 or (0 ) _ or her \ /9 + x + / 5 or 0. 3 ignore ans = - (c) y = \ / (x ) -, \ / y = x - Reverse order y = \ /(x ) dep x = \ /(y ) condone x- if next step correct x = y + squ, + (g (x) =) x + oe 4 x = y - y + M3 y = x + M3 x = x + M3 SC (g - (x) =) (x + ) : B Total 0 marks 8. (a) / 6 / 6 / 6 alone or or better. Not k / 6 or (b),,4 or,,6 or,,6 seen or implied ie one route,, 4 and,, 6 (or,, 6) seen or implied ie two routes incl,, 4 ( / 6 ) 3 3 ie three routes and correct exp n / 7 or 3 / 6 or 0.04 or 4 better ( / 6 ) 3 or / 08, no wking: M0A0 Total 6 marks 39

40 9. (a) / 5 5 sin / 5 (5 ( 5 / ) ) or / (5 3)/4 A0 (b) sect = / 6 π x 5 or ( / 6 π ) or or / 6 π cm 3 + (sect - ) or sect - Allow eg = / 5 5 Total 5 marks 0 (i) (ii) 0 30 B B or sq reps freq of 5 seen anywhere: B Total 3 marks. (a) (bi) 6 0 seen or implied (4 0 )(4 0 + ) or 39 4 (4x )(4x + ) B 3 or 39 or 4 or 3 as factor factors 3, 3, 4 or 39, 4 or 3, 3 Ans M0A0 (ii) or oe f or tree including 000 or 0 and 00 ft her (i) Total 3 marks 40

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