Cambridge IGCSE Mathematics 0580 — 2013 May/June Paper 2 · Variant 3

1 Sheila can pay her hotel bill in Euros (€) or Pounds (£). For Examiner′s The bill was €425 or £365 when the exchange rate was £1 = €1.14 . Use In which currency was the bill cheaper? Show all your working. Answer ............................................... [2] _____________________________________________________________________________________

Mark scheme: Qu Answers Mark Part Marks 1 £ or pound[s] 2 M1 for 425 ÷ 1.14 or 365 × 1.14 Correct working must be shown 30 k

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2 The Ocean View Hotel has 300 rooms numbered from 100 to 399. A room is chosen at random. Find the probability that the room number ends in zero. Answer ............................................... [2] _____________________________________________________________________________________

Mark scheme: 30 k 2 oe www 2 M1 for 30 seen or seen 300 300

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3 The time in Lisbon is the same as the time in Funchal. A plane left Lisbon at 08 30 and arrived in Funchal at 10 20. It then left Funchal at 12 55 and returned to Lisbon. The return journey took 15 minutes more. What time did the plane arrive in Lisbon? Answer ............................................... [2] _____________________________________________________________________________________

Mark scheme: 3 1500 or 3 pm 2 B1 for 1h50 or 2h[0]5 or SC1 for 1255 + their 1h50 + 15mins correctly evaluated

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4 Use a calculator to fi nd For Examiner′s Use 5 (a) 5 24 , Answer(a) ............................................... [1] cos 40° (b) . 7 Answer(b) ............................................... [1] _____________________________________________________________________________________

Mark scheme: 4 (a) [ ± ] 2.28 or 2.282 to 2.2822 1 (b) 0.109 or 0.1094[3…] 1 2 5.1 − 5.1  2   2  3 2  2 

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5 Write the following in order of size, smallest fi rst. 2 2 3 2 2 1.5 2 -1.5 3 - .15 ^ h c 3 m c 3 m c 3 m Answer ................... < ................... < ................... < ................... [2] _____________________________________________________________________________________

Mark scheme:  2  2  3 2  2  2 M1 for at least 2 correct decimals seen5   −  (5.1 )3    3   3   3  1.3[1..] 0.5[4..] 1.8[3..] or 1.84 0.7[6..] 288π

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6 The volumes of two similar cones are 36π cm3 and 288π cm3. The base radius of the smaller cone is 3 cm. Calculate the base radius of the larger cone. Answer ......................................... cm [3] _____________________________________________________________________________________

Mark scheme: 288π 6 6 3 M2 for 3 × 3 36π 288π 36π or M1 for 3 × 3 or 3 × 3 36π 288π

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7 For Examiner′s 4 cm Use NOT TO SCALE 5 cm 10 cm 18 cm The shaded shape has rotational symmetry of order 2. Work out the shaded area. Answer ........................................ cm2 [3] _____________________________________________________________________________________

Mark scheme: 7 260 3 M2 for [2 × ](4 × 10 + 18 × 5) oe or M1 for a correct area statement 2500 3

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8 The mass, m, of a sphere varies directly with the cube of its radius, r. m = 160 when r = 2. Find m when r = 5. Answer m = ............................................... [3] _____________________________________________________________________________________

Mark scheme: 8 2500 3 M1 for m = kr 3 A1 for k = 20 5 4

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9 Calculate, giving your answers in standard form, For Examiner′s Use (a) 2 × (5.5 × 104) , Answer(a) ............................................... [2] (b) (5.5 × 104) – (5 × 104) . Answer(b) ............................................... [2] _____________________________________________________________________________________

Mark scheme: 9 (a) 1.1 × 105 2 B1 for 110 000 oe e.g.11 × 104 (b) 5 × 103 2 B1 for 5000 oe e.g. 0.5 × 104 IGCSE – May/June 2013 0580 23

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10 Find the value of 2x + y for the simultaneous equations. 3x + 5y = 48 2x – y = 19 Answer 2x + y = ............................................... [4] _____________________________________________________________________________________

Mark scheme: 10 25 4 M1 for correct method to eliminate one variable A1 for x = 11 A1 for y = 3 B1 FT for 2 × their x + their y correctly evaluated

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11 The sum of the prime numbers less than 8 is equal to 17. For Examiner′s Use (a) Find the sum of the prime numbers less than 21. Answer(a) ............................................... [2] (b) The sum of the prime numbers less than x is 58. Find an integer value for x. Answer(b) x = ............................................... [2] _____________________________________________________________________________________

Mark scheme: 11 (a) 77 2 M1 for 11,13,17,19 clearly identified, ignore numbers less than 8 with no other numbers greater than or equal to 8 besides possibly an extra 17 (b) either 18 or 19 or both 2FT M1 for 11,13,17 clearly identified, ignore numbers less than 8 with no other numbers greater than or equal to 8 besides possibly an extra 17 or for their (a) − 58

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12 Two spinners have sections numbered from 1 to 5. Each is spun once and each number is equally likely. The possibility diagram is shown below. 5 4 1 Second 2 3 5 spinner 1 2 3 4 2 3 5 4 1 1 2 3 4 5 First spinner Find the probability that (a) both spinners show the same number, Answer(a) ............................................... [2] (b) the sum of the numbers shown on the two spinners is 7. Answer(b) ............................................... [2] _____________________________________________________________________________________

Mark scheme: 12 (a) 5 5 k oe 2 B1 for answer or 25 k 25 4 4 k (b) oe 2 B1 for answer or 25 k 25 8x

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13 Write as a single fraction in its simplest form. For Examiner′s Use x + 3 x - 1 - - x 3 x + 1 Answer ............................................... [4] _____________________________________________________________________________________

Mark scheme: 8 x 13 4 B1 for common denominator (x – 3)(x + 1) ( x − 3)( x + )1 seen B1 for (x + 3)(x + 1) – (x – 1)(x – 3) soi B1 for x2 + 3x + x + 3 or x2 – 3x – x + 3 soi

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14 (a) Solve 3n + 23 < n + 41. Answer(a) ............................................... [2] (b) Factorise completely ab + bc + ad + cd. Answer(b) ............................................... [2] _____________________________________________________________________________________

Mark scheme: 14 (a) n < 9 2 M1 for 2n < 18 or 2n – 18 < 0 oe If 0 scored SC1 for 9 with incorrect inequality. (b) (b + d)(a + c) 2 B1 for b(a + c) + d(a + c) or a(b + d) + c (b + d)

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15 For Examiner′s Use A B 12 – x 13 14 x 20 – x 15 – x 8 y C The Venn diagram shows the number of elements in sets A, B and C. (a) n(A ∪ B ∪ C ) = 74 Find x. Answer(a) x = ............................................... [2] (b) n( ) = 100 Find y. Answer(b) y = ............................................... [1] (c) Find the value of n((A ∪ B )' ∩ C ). Answer(c) ............................................... [1] _____________________________________________________________________________________

Mark scheme: 15 (a) 4 2 M1 for attempt at sum of all numeric and x terms equated to 74 (b) 26 1FT =18 + 2 × their (a) (c) 8 1

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x 2 Examiner′s – 3, x ¸ 0 g(x) = – 516 f(x) = x + x 2 Use Find (a) fg(18), Answer(a) ............................................... [2] (b) g–1(x). Answer(b) g–1(x) = ............................................... [2] _____________________________________________________________________________________

Mark scheme: 16 (a) 1.5 2 B1 for [g(18) =] 4 y (b) 2(x + 5) or 2x + 10 2 M1 for correct first step e.g. x = − 5 or 5 x = + 5 or 2y = x – 10 y 2 IGCSE – May/June 2013 0580 23

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2 3 2 1 5 17 M = N = e 3 6 o e 1 7 2 o (a) Work out MN. Answer(a) [2] (b) Find M–1, the inverse of M. Answer(b) [2] _____________________________________________________________________________________

Mark scheme: 17 (a)  7 23 16  2 B1 for any one row or column correct, must   be in a 2 by 3 matrix 12 45 27   1  6 − 3  (b)  6 − 3  1  a b    or 3   B1 for k    − 3 2  2 c d 3    − 3 2 

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18 For Examiner′s Use B A NOT TO SCALE 5 cm 120° O A and B lie on a circle centre O, radius 5 cm. Angle AOB = 120°. Find the area of the shaded segment. Answer ........................................ cm2 [4] _____________________________________________________________________________________

Mark scheme: 18 15.4 or 15.35 to 15.36 4 120 2 M1 for ×π × 5 oe 360 1 2 M1 for × 5 × sin 120 oe 2 120 2 1 2 M1 for ×π × 5 − × 5 × sin 120 oe 360 2

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19 For Examiner′s C Use D B c b E A O OABCDE is a regular polygon. (a) Write down the geometrical name for this polygon. Answer(a) ............................................... [1] (b) O is the origin. = b and = c. Find, in terms of b and c, in their simplest form, (i) , Answer(b)(i) = ............................................... [1] (ii) , Answer(b)(ii) = ............................................... [2] (iii) the position vector of E. Answer(b)(iii) ............................................... [1] _____________________________________________________________________________________ Question 20 is printed on the next page.

Mark scheme: 19 (a) hexagon 1 (b) (i) −b + c 1 (ii) b − 12 c 2 B1 for OB + BA or any correct route (iii) −b + c 1FT = their (b)(i) √

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20 (a) For Examiner′s 4 Use y = 8 + x Find y when x = 2. Give your answer correct to 4 decimal places. Answer(a) y = ............................................... [2] 4 (b) Rearrange y = 8 + to make x the subject. x Answer(b) x = ............................................... [4]

Mark scheme: 20 (a) [ ± ] 3.1623 cao 2 M1 for √10 seen 4 (b) 2 oe final answer 4 M1 first move completed correctly − 8 y M1 second move completed correctly M1 third move completed correctly M1 final move completed correctly on answer line

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