GCSE MATHEMATICS PROVISIONAL MARK SCHEME – NOVEMBER 2002 – PAPER 5

No Working Answer Mark Notes1 … 40 0.2 200 2 B2 for integer answer 185 ans. 200

(B1 for 40 or 0.2 seen or for oe or oe)

2 3B1 for two relevant semicircles radii 3cm (±2mm, within guidelines)B2cao for pair of parallel lines equal in length to AB and 3cm from AB (±2mm, within guidelines)

(B1 if 2 parallel lines 3cm (±2mm) from AB but not equal in length to AB ( 3.5 cm ) OR 1 parallel line equal in length to AB and 3cm from AB, within guidelines)

3 a 16 - 20 2 M1 for sight of 32÷2 or 33÷2 oe (possibly implied)A1cao for 16 - 20

b Draw frequency polygon through the points (3,1) (8,3) (13,9) (18,8) and (23,11)

2 B2 for plotting correct points ½ sq and joining with line segments(B1 for joining points either plotted at correct midpoints of intervals, but incorrect heights or plotted at correct heights and positioned consistently within intervals or four correct points joined by line segments or five correct points not joined or joined with a curve)Ignore end closures

4 a Volume = 130 20 2 600 cm3 3 M1 for 130 20 = 2 600 cm3

A1cao for 2600B1(indep) for cm3.

bScale factor oe

x = 10 1.5 = 15

15 2M1 for or 1.5 oe OR for oe if consistent

OR for oe

A1cao for 15.5 a

2

M1 for oe

GCSE MATHEMATICS PROVISIONAL MARK SCHEME – NOVEMBER 2002 – PAPER 5

No Working Answer Mark Notes

= miles A1 for oe (accept decimal)

b OR

3M2 for either oe or oe

= OR (M1 for oe)

= miles A1 for oe (accept decimal)

c oe

2 M1 for valid use of a common denominator that is a multiple of 10 or for 7.5 − 3.8

=

A1 for oe (accept decimal)

621 + x = 6x OR = x

4.2 3 M1 for a correct first step

21 = 6x x M1ft (indep) for a valid second step method

x = A1cao for oe

7 a (i) 6 109 2 B1cao(ii) 1.5 102 B1cao.

8 i 11 < x + 3 + x + 3 < 20 printed ans. 5 M1 for forming 11< p <20 oe or for two correct separate inequalities in terms of x

5< 2x < 14 A1cao [Convincingly shown]ii

x is integer so x = 3, 4, 5, 6

3, 4, 5, 6M1 for (Can be implied by at least A1

below; condone extras within the interval)

GCSE MATHEMATICS PROVISIONAL MARK SCHEME – NOVEMBER 2002 – PAPER 5

No Working Answer Mark NotesA2 cao(A1 for either 3, 4, 5, 6, 7 or for any three of 3, 4, 5, 6 only)

9 Blank; A ; Blank; V; A; V 3 B3 for all 6 correct (mark each box)(B2 for 5 or 4 correct else B1 for 3 correct)

10

a ; Gradient =

0.5 oe 2 B2 cao for 0.5 oe

(B1 for oe seen)

b Sub coords of given pt in y = "0.5"x+c y = 0.5x + 7 2 B2ft for y="0.5"x + 7 oe(B1 for an equation of the form either y ="0.5"x +c where c > 0 or 2y = x + k where 7 ≤ k < 14 or y=px+7 where p > 0)

11

2x ( 4x + 5y) 2 B2cao[B1 for x (8x +10y) only ]

12

a {Reading top to bottom} LHS {0.7}; 0.3 oe RHS 0.6; 0.4; 0.6; 0.4 oe

2 B1 for LHS oeB1 for RHS oe

b(i) P(F & G) = 0.7 “ 0.6” 0.42 oe 2 M1 ft diagram for 0.7 “ 0.6” oe = 0.42 A1cao

(ii) P({F & G} or {S & H}) = (0.7 0.6) + (0.3 0.4)

0.54 oe 3 M1 ft diagram for (" 0.7 0.6") and ("0.3" "0.4") oeM1(dep) for adding

= 0.42 + 0.12 = 0.54 A1caoc P( not [F&G] ) = 1 "0.42" (= 0.58) 116 3 M1ft answer in (b)(i) for 1 "0.42" or 0.58 seen

Estimate for not [F&G] = "0.58" 200 M1(dep) ft for "0.58" 200

GCSE MATHEMATICS PROVISIONAL MARK SCHEME – NOVEMBER 2002 – PAPER 5

No Working Answer Mark Notes = 116 A1caoOR OREstimate for [F&G] =”0.42” 200 (=84) M1ft answer in (b)(i) for ”0.42” 200Estimate for not [F&G] = 200 “84” M1(dep) ft for 200 “84” = 116 A1cao

13

Enlargementcentre (1,1) scale factor 1.5

2 B1cao for ‘Enlargement’ as the only transformation

B1cao (dep on previous B1)14

a mean = 0.3st. dev = 0.2

2 B1cao for mean = 0.3 oeB1cao for st.dev. = 0.2 oe

b 60x + y40x

2 B1cao for mean = 60x + y oeB1cao for standard deviation = 40x oe

15

"200" [40 students] oe= 8000 1160 oe

7 3 M1 for "200" [40 students] oeA2 cao for 7

(A1 for 8000 1160 OR oe dec to 2sf)

16

4 x 8 = 3 y 5x y4

M1 expanding or splitting into four correct terms

4x + 5x y = 3 y + 8 M1(indep) rearranging correctly to isolate x-terms from non x terms

x (4+5 y) = 3 y + 8 M1(indep) factorising x with other factor a function of y

A1cao for oe

17

(a) = + oe b a2 M1 for = + oe

= b a A1 for b a oe

GCSE MATHEMATICS PROVISIONAL MARK SCHEME – NOVEMBER 2002 – PAPER 5

No Working Answer Mark Notes(b) = 2 b Printed result 3 B1 (possibly implied by correct if clear)

= + = a + 2b M1 valid method to find in terms of andor in terms of a and b

= 2 (b a ) AC is parallel to PB A1cao (Printed result convincingly shown)

(c) 16 1 B1cao for 16

18

(a) x31

B1cao

(b) Raise all to power 6 gives 83 , 44 , 25 , 72 , 36, 2 M1 for raising all to power 6

, , , A1cao correct order[SC if M0 award B1 for either a list with four in correct order or a correct list of five but reversed.]

(c) = 24 (3)4 . 144 2 M1 for either 24 (3)4 or A1cao for 144

19

(a)Area = (x + 3 + x + 5) (2x + 1)

2x2 +9x + 4 3 M1 for any correct unsimplified form for the areaA2 for 2x2 +9x + 4

= (x + 4) (2x + 1) = 2x2 + 8x + x + 4

[else B1 for either (2x+1)(x+3) = 2x2 + 7x + 3 or (2x+8)(2x+1) = 4x2 + 18x + 8or 3 terms correct in expn (x+4) (2x+1) =2x2+8x +x+4 ]

(b) Area of square = (2x+5)2 = 4x2 +20x +25 Printed result 3 B1 for 4x2 +20x +25Shaded area = "4x2+20x+25" ("2x2+9x+4") M1 must both be in form ax2+bx+c = 2x2 + 11x + 21 A1cao (Printed result obtained convincingly)

(c) 2x2 + 11x + 21 42 = 0 x = 1.5 3 M1 (putting sh. area = 42 and rearranging appropriately)(2x 3)(x +7) = 0

A1 for either (2x 3)(x +7) or

Length cannot be negative so A1cao for x = 1.5[Must have formed and solved a quadratic algebraically]

2 (a) (i) Draws horizontal line y = 0.2 13 and 167 2 The M1…may be implied by one ‘correct’ solution

GCSE MATHEMATICS PROVISIONAL MARK SCHEME – NOVEMBER 2002 – PAPER 5

No Working Answer Mark Notes0

A1cao for both 13 3 and 167 3 (ii) EITHER completes the graph, as a curve, correctlyOR uses 180+p (or 360p) where sinp =0.6

218 and 322 2 M1 for completed curve, scale correct, through (270, 1) and beyond y = 0.6 OR uses 180+p (or 360p) where sinp =0.6 (M1…may be implied by one ‘correct’ solution)A1cao for both 218 3 and 322 3

(b) Use of x = [a (i) 90] or x = [ 90 a (i) ] oe eg x = 77 , eg x = 283

2 M1 for use of x = [a (i) 90] or x = [ 90 a (i) ] oe(oe eg x = [a (i) 90 + 360n] or x = [ 90 a (i) + 360n]where n is an integer)A1 ft for two correct ft answers

ALT Draw y = cos x graph M1 for y = cos x graph drawn as a curve through (0 , 1)(90 , 0) (180 , 1) (270 , 0) and beyond y = 0.2 and also intersecting the y = sin x graph within 1 sq of horizontal 45A1 for both 77 3 and 283 3 oe alternatives

21

(a) (i) ( 4 , 25) 1 B1cao for ( 4 , 25)

(ii) ( 2 , 3) 1 B1cao for ( 2 , 3)(iii) ( 1 , 25) 1 B1cao for ( 1 , 25)

(b) a = 1b = 9

2 B1caoB1cao[SC if 0/2 award M1 if either (i) indicates curve crossing +'ve x-axis at 4.5 or(ii) substitutes to get 25 = (4+a)(4b) oe or(iii) substitutes to get 0 = (9+a)(9b) oe]

22

(a) Printed result 2B1 for

= which is always rational B1cao (Printed result convincingly explained)

(b) Explanation eg (x y) = (x + y) 2y 2 M1 (accept equiv)

GCSE MATHEMATICS PROVISIONAL MARK SCHEME – NOVEMBER 2002 – PAPER 5

No Working Answer Mark Notes = rational irrational = irrational A1cao (convincing valid argument)

(c) eg x = 3 + 2 ; y = 3 2 2 B2 cao[B1 if both x and y are different irrational numbers AND satisfy two of the three conditions ie(I) x and y are both positive(II) xy is rational(III) x + y is rational ]