Cambridge IGCSE Mathematics 0580 — 2012 May/June Paper 2 · Variant 3

1 For A Examiner's Use 73° B NOT TO 120° SCALE 82° x° C D E The diagram shows a quadrilateral ABCD. CDE is a straight line. Calculate the value of x. Answer x = [2]

Mark scheme: Qu Answers Mark Part marks 1 95 2 B1 for 85 seen or M1 x = 180 – their angle ADC, if it is clearly seen 750 × 2 × 8 M1 for oe seen or SC1 870 as final

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2 Hans invests $750 for 8 years at a rate of 2% per year simple interest. Calculate the interest Hans receives. Answer $ [2]

Mark scheme: M1 for oe seen or SC1 870 as final 2 120 2 100 answer

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3 1.5 0.9 3 (a) Calculate 7 + 22 and write down your full calculator display. Answer(a) [1] (b) Write your answer to part (a) correct to 4 significant figures. Answer(b) [1]

Mark scheme: 3 (a) 3.26077… 1 seen (b) 3.261 1ft their (a) to 4 significant figures

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4 Solve the inequality. For 3y + 7 Y 2 – y Examiner's Use Answer [2]

Mark scheme: 4 y ø –1.25 2 M1 inequality with y’s and constants correctly collected

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5 9 cm 5 cm 5 cm NOT TO SCALE 12 cm The diagram shows a quadrilateral. The lengths of the sides are given to the nearest centimetre. Calculate the upper bound of the perimeter of the quadrilateral. Answer cm [2]

Mark scheme: 5 33 cao www 2 M1 any two of 5.5, 9.5, 12.5 seen

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6 C NOT TO 9 cm SCALE B 28° 15 cm A Calculate the area of triangle ABC. Answer cm2 [2]

Mark scheme: 6 31.7 2 M1 0.5 × 9 × 15 × sin 28

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7 For Examiner's Height (h cm) 0 < h Y 10 10 < h Y 15 15 < h Y 30 Use Frequency 25 u 9 Frequency density 2.5 4.8 v The table shows information about the heights of some flowers. Calculate the values of u and v. Answer u = v = [2]

Mark scheme: 7 u = 24(.0), v = 0.6 2 B1 each

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8 During her holiday, Hannah rents a bike. She pays a fixed cost of $8 and then a cost of $4.50 per day. Hannah pays with a $50 note and receives $10.50 change. Calculate for how many days Hannah rents the bike. Answer days [3]

Mark scheme: 8 7 cao 3 B1 for 39.5(0) or 31.5(0) or 42 M1 for (their 39.5 – 8) ÷ 4.5 or (their 42 – 10.5) ÷ 4.5

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9 Make w the subject of the formula. 3 w t = 2 – a Answer w = [3]

Mark scheme: 9 a (2 − t ) 3 M1 correct re-arrangement to isolate the term in w cao oe M1 correct multiplication by a 3 M1 correct division by their 3 An incorrect answer scores a maximum of M2

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10 The periodic time, T, of a pendulum varies directly as the square root of its length, l. For T = 6 when l = 9. Examiner's Use Find T when l = 25. Answer T = [3]

Mark scheme: 10 10 3 M1 T = k√l A1 for k = 2

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11 Boris invests $280 for 2 years at a rate of 3% per year compound interest. Calculate the interest Boris receives at the end of the 2 years. Give your answer correct to 2 decimal places. Answer $ [4]

Mark scheme: 11 17.05 cao www 4 3 M1 for 280 × (1 + )2 oe 100 k M1 subtracting 280 from 280(1 + )2 any k 100 A1 for 17.052 or SC2 297.05 on answer line IGCSE – May/June 2012 0580 23

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12 Without using your calculator, work out the following. For Show all the steps of your working and give each answer as a fraction in its simplest form. Examiner's Use 11 1 (a) − 12 3 Answer(a) [2] 1 11 (b) ÷ 4 13 Answer(b) [2]

Mark scheme: 12 (a) M1 correct use of a common denominator 11 4 − oe 2 12 12 7 cao ww 0 A1 12 13 (b) 1 2 M1 inversion and operation change × oe 4 11 13 cao ww 0 A1 44

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13 (a) Find the value of 7p – 3q when p = 8 and q = O5 . Answer(a) [2] (b) Factorise completely. 3uv + 9vw Answer(b) [2]

Mark scheme: 13 (a) 71 2 M1 for 7×8 – 3×–5 or B1 56 and –15 (b) 3v (u + 3w) final answer 2 B1 for 3(uv + 3vw) or v (3u + 9w) As final answer

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14 Simplify the following. For Examiner's 2 Use (a) (4pq )3 Answer(a) [2] 1 − (b) (16 x 8 ) 4 Answer(b) [2]

Mark scheme: 14 (a) 64p3q6 2 B1 64puqv or kp3q6 1 1 1 (b) 0.5x–2 or 2 oe 2 B1 u oe or 2 oe 2 x 2 x kx

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

Mark scheme: 15 –3.44, 0.44 4 B1 for ( 6) 2 − 4( 2)( −3) or better seen p + ( or − ) q B1 if in form , for p = –6 and r = 2×2 oe correct working must be shown r B1, B1 (SC1 –3.4 or –3.436… and 0.4 or 0.436...)

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16 For Examiner's NOT TO Use SCALE 4 cm 15 cm The diagram shows a solid prism of length 15 cm. The cross-section of the prism is a semi-circle of radius 4 cm. Calculate the total surface area of the prism. Answer cm2 [4]

Mark scheme: 16 359 www 4 M1 π × 42 or 12 π × 4 2 M1 for 0.5 × π × 8 × 15 oe M1 for 8 × 15 + their 2 ends + their curved surface area

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 2 4 17 A =   B = (1 2 )  1 3  (a) Calculate BA. Answer(a) [2] (b) Find A– 1 , the inverse of A. Answer(b) [2]

Mark scheme: 17 (a) (4 10) 2 B1 each element or correct without brackets  3 − 4  1  a c   3 − 4  seen  (b) 2  oe 2 B1 for 12  or k   b d    − 1 2    − 1 2 

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18 For Q R Examiner's Use M NOT TO q X SCALE O p P O is the origin and OPRQ is a parallelogram. The position vectors of P and Q are p and q. X is on PR so that PX = 2XR. Find, in terms of p and q, in their simplest forms (a) , Answer(a) = [2] (b) the position vector of M, the midpoint of QX. Answer(b) [2]

Mark scheme: 18 (a) p – 13 q oe 2 M1 QR + RX oe or –q + p + ( 2 )q oe 3 (b) 1 2 p + 56 q oe 2 ft ft q + 12 their (a) but must be vectors or M1 for OQ + QM oe

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19 For 120 Examiner's Use 100 80 Speed 60(km / h) 40 20 0 0.5 1 1.5 2 2.5 3 3.5 Time (minutes) The diagram shows the speed-time graph for part of a car journey. The speed of the car is shown in kilometres / hour. Calculate the distance travelled by the car during the 3.5 minutes shown in the diagram. Give your answer in kilometres. Answer km [4]

Mark scheme: 19 6(.00) www 4 M1 use of area = distance M1 complete, correct set of area statements, ignoring units M1 changing min to hours or km/h to km/min IGCSE – May/June 2012 0580 23

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20 Simplify fully. For x 2 − x − 20 Examiner's Use x 3 − 10 x 2 + 25 x Answer [5] Question 21 is printed on the next page.

Mark scheme: 20 5 B2 (x – 5)(x + 4) seen x + 4 oe cao or SC1 (x + a)(x + b) where ab = –20 or a + b = –1 x ( x − 5) B2 x(x – 5)(x – 5) or B1 one of x(x2 –10x + 25), (x – 5)( x – 5), (x – 5)(x2 – 5x) seen 1

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21 For P Examiner's Use NOT TO SCALE 5 cm D C 8 cm M A 8 cm B The diagram shows a pyramid on a square base ABCD. The diagonals of the base, AC and BD, intersect at M. The sides of the square are 8 cm and the vertical height of the pyramid, PM, is 5 cm. Calculate (a) the length of the edge PB, Answer(a) PB = cm [3] (b) the angle between PB and the base ABCD. Answer(b) [3]

Mark scheme: 1 21 (a) 7.55 www 3 M2 ( √(82 + 82))2 + 52 or 42 + 52 + 42 seen 2 or M1 82 + 82 or 52 + 42 or 42 + 42 or 52 + (their MB)2 seen 5 5 (b) 41.5 www 3 M2 sin(B) = or tan(B) = or a( ) their MB their MB cos(B) = (a) or M1 recognition of angle PBM

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