Cambridge IGCSE Mathematics 0580 — 2012 Oct/Nov Paper 4 · Variant 3

1 (a) The Martinez family travels by car to Seatown. For The distance is 92 km and the journey takes 1 hour 25 minutes. Examiner's Use (i) The family leaves home at 07 50. Write down the time they arrive at Seatown. Answer(a)(i) [1] (ii) Calculate the average speed for the journey. Answer(a)(ii) km/h [2] (iii) During the journey, the family stops for 10 minutes. Calculate 10 minutes as a percentage of 1 hour 25 minutes. Answer(a)(iii) % [1] (iv) 92 km is 15% more than the distance from Seatown to Deecity. Calculate the distance from Seatown to Deecity. Answer(a)(iv) km [3] (b) The Martinez family spends $150 in the ratio For Examiner's fuel : meals : gifts = 11 : 16 : 3 . Use (i) Show that $15 is spent on gifts. Answer (b)(i) [2] (ii) The family buys two gifts. The first gift costs $8.25. Find the ratio cost of first gift : cost of second gift. Give your answer in its simplest form. Answer(b)(ii) : [2]

Mark scheme: Qu. Answers Mark Part Marks 1 (a) (i) [0]9 15 [am] 1 Any acceptable form of time (ii) 64.9 or 65.[0] or 64.92 to 2 M1 for 92 ÷ (1 and 25 mins) or 92/85 × 60 oe 64.98 or 92 ÷ (1.41 to 1.42) (iii) 11.76…or 11.8 1 (iv) 80 3 M2 for 92 ÷ 1.15 oe or M1 for 115% associated with 92 (b) (i) 150 ÷ (11 + 16 + 3) or M1 Correct first step 150 × 3 oe then × 3 or ÷ 30 E1 Correct conclusion (ii) 11 : 9 final answer 2 M1 for 8.25 : (15 – 8.25) oe For M1 e.g. allow 1 : 0.818 [0.8181 to 0.8182] or 1.22 : 1 [1.222…] After M0, SC1 for 9 : 11 as final answer −11  k 

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2 (a) For y Examiner's Use 8 7 X 6 5 4 3 2 1 x –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 –1 –2 Y –3 –4 –5 –6 –7 –8 –9  − 11  (i) Draw the translation of triangle X by the vector   . [2]  − 1 1 (ii) Draw the enlargement of triangle Y with centre (–6, – 4) and scale factor . [2] 2 (b) For y Examiner's Use 8 7 W X 6 5 4 3 2 1 x –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 –1 –2 Y –3 –4 Z –5 –6 –7 –8 –9 Describe fully the single transformation that maps (i) triangle X onto triangle Z, Answer(b)(i) [2] (ii) triangle X onto triangle Y, Answer(b)(ii) [3] (iii) triangle X onto triangle W. Answer(b)(iii) [3] (c) Find the matrix that represents the transformation in part (b)(iii).     Answer(c)   [2]    

Mark scheme: − 11   k  2 (a) (i) Image at (– 3, 1), (– 7, 7), 2 SC1 for translation or k −1     (– 3, 7) (ii) Image at (– 4, – 1), (– 4, – 4), 2 SC1 for enlargement factor 0.5 and correct (– 2, – 4) orientation In each part of (b) must be one transformation only – if more then lose all marks for that part (b) (i) Reflection, y = 1 2 B1 B1 independent (ii) Rotation, (3, 2), 180 oe 3 B1 B1 B1 independent or enlargement, (3, 2), (factor) – 1 (iii) Stretch, (factor) 0.5, 3 B1 B1 B1 independent – must be clear on invariant Invariant line y-axis or x = 0 line IGCSE – October/November 2012 0580 43  5.0 0  (c)   2 ft ft their factor in (b)(iii) only if stretch not 0 or 1 0 1    k 0  SC1 for   [k ≠ 0 or 1] or 0 1    1 0    ft their factor only if stretch in (b)(iii) 0 5.0  

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3 A metal cuboid has a volume of 1080 cm3 and a mass of 8 kg. For Examiner's (a) Calculate the mass of one cubic centimetre of the metal. Use Give your answer in grams. Answer(a) g [1] (b) The base of the cuboid measures 12 cm by 10 cm. Calculate the height of the cuboid. Answer(b) cm [2] (c) The cuboid is melted down and made into a sphere with radius r cm. (i) Calculate the value of r. 4 [The volume, V, of a sphere with radius r is V = πr 3.] 3 Answer(c)(i) r = [3] (ii) Calculate the surface area of the sphere. For Examiner's [The surface area, A, of a sphere with radius r is A = 4πr 2.] Use Answer(c)(ii) cm2 [2] (d) A larger sphere has a radius R cm. The surface area of this sphere is double the surface area of the sphere with radius r cm in part (c). R Find the value of . r Answer(d) [2]

Mark scheme: 3 (a) 7.407….. or 7.41 1 (b) 9 2 M1 for 1080 ÷ (12 × 10) oe 1080 (c) (i) 6.36 to 6.37 www 3 M2 for 3 4 oe π 3 1080 or M1 for 4 oe [ 257.7 to 258.7] π 3 Accept 4.18 to 4.19 for 4/3 π (ii) 508 to 510 2 M1 for 4 × π × (their (c)(i))2 (d) 2 or 1.41 [1.414…] www 2 Allow over 1 or 2 : 1 etc M1 for (R / r)2 = 2 oe or [R2 =] (2 × their (c)(ii))/4π or [R2 =] 2 × (their (c)(i))²

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2 For 4 f(x) = O 3x, x ≠ 0 Examiner's 2 x Use (a) Complete the table. x O3 O2.5 O2 O1.5 O1 O0.5 0.5 1 1.5 2 2.5 3 f(x) 9.2 7.8 6.5 5.4 9.5 6.5 O3.6 O5.5 O7.2 O8.8 [2] (b) On the grid, draw the graph of y = f(x), for O3 Y x Y O0.5 and 0.5 Y x Y 3 . y 10 9 8 7 6 5 4 3 2 1 x –3 –2 –1 0 1 2 3 –1 –2 –3 –4 –5 –6 –7 –8 –9 [5] (c) Use your graph to solve the equations. For Examiner's (i) f(x) = 4 Use Answer(c)(i) x = [1] (ii) f(x) = 3x Answer(c)(ii) x = [2] (d) The equation f(x) = 3x can be written as x3 = k. Find the value of k. Answer(d) k = [2] (e) (i) Draw the straight line through the points (–1, 5) and (3, –9). [1] (ii) Find the equation of this line. Answer(e)(ii) [3] (iii) Complete the statement. The straight line in part (e)(ii) is a to the graph of y = f(x). [1]

Mark scheme: 4 (a) 5, – 1 2 B1 B1 (b) 12 points plotted ft P3ft P2ft for 10 or 11, P1ft for 8 or 9 Smooth curve through at least 12 C1 In absence of plot[s], allow curve to imply plot[s]. points No ruled sections Two separate branches B1 Not touching y-axis (c) (i) 0.55 to 0.65 1 (ii) 0.65 to 0.75 2 M1 for y = 3x drawn (ruled) to cross curve 1 0.3& (d) 2 Accept 0.333[3….] or 3 2 M1 for 2 − 3 x = 3 x or better x IGCSE – October/November 2012 0580 43 (e) (i) Ruled line through (– 1, 5) 1 and (3, – 9) (ii) y = −5.3 x + 5.1 oe final 3 B2 for y = kx + 5.1 [k ≠ 0] oe or y = −5.3 x + d oe answer B1 for gradient = – 3.5 oe accept integer/integer or y = kx + 4.1[ to 6.1] oe SC2 for answer − 5.3 x + 5.1 [no ‘y =’ ] (iii) Tangent 1 5 (a) 0.57 B4 Condone use of other variables M1 for 2 w + 3l = 6.3 oe and M1 for l = w + .0 25 oe A1 for correct aw = b or cl = d or M2 for 2 w + (3 w + .025) = 6.3 oe or 2(l − .025) + 3l = 6.3 oe or M1 for w + 0.25 or l – 0.25 seen A1 for 2 w + 3w = 6.3 − .075 or better or 2l + 3l = 6.3 + 5.0 or better l = 0.82 implies M2A1 trial & error scores B4 or zero accept answer 57 if written 57 cents after M0, SC3 if answer 57

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