Cambridge IGCSE Mathematics 0580 — 2011 May/June Paper 2 · Variant 1

1 A concert hall has 1540 seats. ForFor Examiner'sExaminer's Calculate the number of people in the hall when 55% of the seats are occupied. UseUse Answer [1]

Mark scheme: g g Qu. Answers Mark Part Mark 1 847 1

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2 Shade the required region on each Venn diagram. A B A B A ∪ B' (A ∩ B)' [2]

Mark scheme: 2 correct regions shaded 1, 1

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3 Calculate 810.25 ÷ 4−2 . Answer [2]

Mark scheme: 3 48 2 B1 for 3 and 16 seen

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4 (a) Find m when 4m × 42 = 412 . Answer(a) m = [1] p 5 (b) Find p when 6 ÷ 6 = 6 . Answer(b) p = [1]

Mark scheme: 4 (a) 10 1 (b) 5.5 oe 1

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5 A hummingbird beats its wings 24 times per second. ForFor Examiner'sExaminer's (a) Calculate the number of times the hummingbird beats its wings in one hour. UseUse Answer(a) [1] (b) Write your answer to part (a) in standard form. Answer(b) [1]

Mark scheme: ( ) 5 (a) 86400 1 (b) 8.64 × 104 1ft  3

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6 NOT TO SCALE 6 cm 2 cm A company makes solid chocolate eggs and their shapes are mathematically similar. The diagram shows eggs of height 2 cm and 6 cm. The mass of the small egg is 4 g. Calculate the mass of the large egg. Answer g [2]

Mark scheme: 3 6 108 2 M1 for 33 or 27 or  1  or 1 seen  3  27

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7 Find the length of the straight line from Q (−8, 1) to R (4 , 6). Answer QR = [3]

Mark scheme: 7 13 3 B1 for 12, 5 seen M1 for (their 12)2 + (their 5)2 or M2 √[(–8 – 4)2 + (1 – 6)2] oe or M1 if √ missing 3

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8 Calculate the radius of a sphere with volume 1260 cm3 . ForFor 4 Examiner'sExaminer's [The volume, V, of a sphere with radius r is V = πr3.] UseUse 3 Answer cm [3]

Mark scheme: 3 8 6.70 3 M1 for ( r3 = ) 1260 × oe seen 4 π M1 for 3 of their r3 seen or implied

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9 A B 2x° NOT TO SCALE C D x° 5x° AB is parallel to CD. Calculate the value of x. Answer x = [3]

Mark scheme: 9 22.5 oe 3 B2 180 = 5x +2x + x oe or better B1 for 2x or 6x marked in the correct place on the diagram.

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10 Solve the simultaneous equations. 3x + y = 30 2x – 3y = 53 Answer x = y = [3]

Mark scheme: g 10 x = 13 3 M1 for consistent multiplication and y = –9 addition/subtraction A1 for x = 13 or A1 for y = -9

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11 A rectangular photograph measures 23.3 cm by 19.7 cm, each correct to 1 decimal place. ForFor Calculate the lower bound for Examiner'sExaminer's UseUse (a) the perimeter, Answer(a) cm [2] (b) the area. Answer(b) cm2 [1]

Mark scheme: y 11 (a) 85.8 2 M1 for 23.25 and 19.65 seen (b) 456.8625 cao 1

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12 A train leaves Barcelona at 21 28 and takes 10 hours and 33 minutes to reach Paris. (a) Calculate the time the next day when the train arrives in Paris. Answer(a) [1] (b) The distance from Barcelona to Paris is 827 km. Calculate the average speed of the train in kilometres per hour. Answer(b) km/h [3]

Mark scheme: ( ) 12 (a) (0)8(.)01 (am) 1 Not 8.01pm (b) 78.4 or 78.38 to 78.39 3 M2 for 827 ÷ 10.55 or M1 for figs 827 ÷ their time 2 7 × 20 000

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13 The scale on a map is 1: 20 000. ForFor Examiner'sExaminer's (a) Calculate the actual distance between two points which are 2.7 cm apart on the map. UseUse Give your answer in kilometres. Answer(a) km [2] (b) A field has an area of 64 400 m2 . Calculate the area of the field on the map in cm2 . Answer(b) cm2 [2]

Mark scheme: 7.2 × 20 000 13 (a) 0.54 2 M1 for oe 10 0000 or SC1 for figs 54 in answer (b) 1.61 2 SC1 for figs 161 or M1 2002 or 20 0002 seen IGCSE – May/June 2011 0580 21 ( )( )

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14 Solve the equation 2x2 + 3x – 6 = 0. Show all your working and give your answers correct to 2 decimal places. Answer x = or x = [4]

Mark scheme: 14 –2.64, 1.14 cao with working 4 B1 for 32 − 4(2 )(− 6 ) or better seen anywhere B1 for p = –3 and r = 2 × 2 or better as long as in p + q p − q the form or r r After B0B0, SC1 for –2.6 or –2.637(45…) and 1.1 or 1.137(45…)

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15 A teacher asks 36 students which musical instruments they play. ForFor Examiner'sExaminer's P = {students who play the piano} UseUse G = {students who play the guitar} D = {students who play the drums} The Venn diagram shows the results. P G x 2 8 1 5 4 7 5 D (a) Find the value of x. Answer(a) x = [1] (b) A student is chosen at random. Find the probability that this student (i) plays the drums but not the guitar, Answer(b)(i) [1] (ii) plays only 2 different instruments. Answer(b)(ii) [1] (c) A student is chosen at random from those who play the guitar. Find the probability that this student plays no other instrument. Answer(c) [1]

Mark scheme: ( ) 15 (a) 4 1 12 (b) (i) oe 0.333 1 36 11 (ii) , 0.306 or 0.3055 to 1 36 0.3056 1 8 (c) oe 0.533(3…) 15 k 2

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16 ForFor Examiner'sExaminer's UseUse k cm The diagram shows a square of side k cm. The circle inside the square touches all four sides of the square. (a) The shaded area is A cm2. Show that 4A = 4k2 – πk2. Answer (a) [2] (b) Make k the subject of the formula 4A = 4k2 – πk2. Answer(b) k = [3]

Mark scheme: k 16 (a) Answer given 2 M1 (A =)k2 – π    2  πk 2 E1 A = k2 – 4 correctly completed to 4A = 4k2 – πk2 4 A A (b) k = (±) or 2 3 M1 factorising (must contain a π) (4 − π ) (4 − π ) M1 division (by coefficient of k2) M1 square root

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17 ForFor C Examiner'sExaminer's UseUse NOT TO SCALE O B 24° T A A, B and C are points on a circle, centre O. TA is a tangent to the circle at A and OBT is a straight line. AC is a diameter and angle OTA = 24°. Calculate (a) angle AOT, Answer(a) Angle AOT = [2] (b) angle ACB, Answer(b) Angle ACB = [1] (c) angle ABT. Answer(c) Angle ABT = [2]

Mark scheme: q 17 (a) 66° 2 M1 for 90° clearly identified as A (b) 33° 1 (c) 123° 2 B1 for OBA or OAB = 57°

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18 ForFor S Examiner'sExaminer's UseUse Q X NOT TO SCALE P M R In the diagram, PQS, PMR, MXS and QXR are straight lines. PQ = 2 QS. M is the midpoint of PR. QX : XR = 1 : 3. = q and = r. (a) Find, in terms of q and r, (i) , Answer(a)(i) = [1] (ii) . Answer(a)(ii) = [1] (b) By finding , show that X is the midpoint of MS. Answer (b) [3]

Mark scheme: ( ) 18 (a) (i) -r + q or q – r 1 (ii) ½(3q – r) oe 1 Must be simplified (b) correct working 3 M1 for MX = ½ r + ¾ their (–r + q) M1 using a different route for XS or ½ MS E1 dep correct simplification and conclusion

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19 ForFor 15 Examiner'sExaminer's UseUse 10 Speed (metres per second) 5 0 2 4 6 8 10 12 14 Time (minutes) The diagram shows the speed-time graph of a train journey between two stations. The train accelerates for two minutes, travels at a constant maximum speed, then slows to a stop. (a) Write down the number of seconds that the train travels at its constant maximum speed. Answer(a) s [1] (b) Calculate the distance between the two stations in metres. Answer(b) m [3] (c) Find the acceleration of the train in the first two minutes. Give your answer in m/s2. Answer(c) m/s2 [2] Question 20 is printed on the next page.

Mark scheme: p p 19 (a) 480 1 (b) 9900 3 M1 for attempt at area under graph M1 for 0.5 × 15 × (their (a) + 14 × 60) oe or 0.5 × 15 × (8 + 14) oe 1 2 M1 for numerical vertical/horizontal or numerical (c) 0.125 or 8 use of v = u + at but t ≤ 120 or t ≤ 2

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20 f(x) = x3 g(x) = 2x − 3 ForFor Examiner'sExaminer's UseUse (a) Find (i) g(6), Answer(a)(i) [1] (ii) f(2x). Answer(a)(ii) [1] (b) Solve fg(x) = 125. Answer(b) x = [3] (c) Find the inverse function g−1(x) . Answer(c) g –1(x) = [2]

Mark scheme: 20 (a) (i) 9 1 (ii) 8x3 cao 1 (b) 4 www 3 M1 for (2x – 3)3 = 125 M1 2x – 3 = 5 (c) x + 3 2 M1 for x ± 3 = 2y or x = y ± 3 2 2

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