Cambridge IGCSE Mathematics 0580 — 2009 Oct/Nov Paper 2 · Variant 1

1 For Examiner's Use For the diagram above write down (a) the order of rotational symmetry, Answer(a) [1] (b) the number of lines of symmetry. Answer(b) [1]

Mark scheme: Qu Answers Mark Part Marks 1 (a) 6 1 (b) 0 1

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2 Write down the next two prime numbers after 43. Answer and [2]

Mark scheme: 2 47, 53 2 B1, B1 independent

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(cos30 ) (sin30 ) 3 Use your calculator to find the value of o o . 2(sin120 )(cos120 ) Answer [2]

Mark scheme: 3 − 3 − 1 2 B1 numerator 0.5 –0.577 or or 3 3 or B1 denominator –0.866……or − 3 2

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5 32 1 _ 524 Simplify x ÷ x . 8 2 Answer [2]

Mark scheme: 4 1.25 x4 (or 1 1 x4) 2 B1 1.25 B1 x4 4

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5 In 1970 the population of China was 8.2 x 108. For In 2007 the population of China was 1.322 x 109. Examiner's Calculate the population in 2007 as a percentage of the population in 1970. Use Answer %[2]

Mark scheme: 5 161 2 M1 1.322 × 109 / 8.2 × 108 (× 100)

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6 0 1 7 1 A = B = − 8 − 4 0 − 5 Calculate the value of 5 |A| + |B|, where |A| and |B| are the determinants of A and B. Answer [2]

Mark scheme: 6 5 2 M1 |A| = 0 × –4 – 1 × –8 or better or |B| = 7 × –5 – 0 × 1 or better det symbol can be implied by the working

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7 Shade the region required in each Venn Diagram. A B A B C C A' ∩ (B ∩ C ) A' ∩ (B ∪ C ) [2]

Mark scheme: 7 2 B1, B1

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8 Find the length of the line joining the points A(− 4, 8) and B(−1, 4). For Examiner's Use Answer AB = [2]

Mark scheme: 8 5 www 2 M1 (–4 – –1)2 + (8 – 4)2 or better

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9 Solve the simultaneous equations 6x + 18y = 57, 2x – 3y = −8. Answer x = y = [3]

Mark scheme: 9 x = 0.5 y = 3 www 3 M1 consistent × and – for y or consistent × and + for x A1 one correct provided M1 scored

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10 The braking distance, d, of a car is directly proportional to the square of its speed, v. When d = 5, v = 10. Find d when v = 70. Answer d = [3]

Mark scheme: 10 245 3 M1 d = kv2 A1 k = 1/20 or M1 v2 = kd A1 k = 20

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11 For Examiner's Use 24 cm NOT TO 18 cm 18 cm SCALE 24 cm 24 cm 18 cm 18 cm 24 cm 24 cm 18 cm 18 cm 24 cm Each of the lengths 24 cm and 18 cm is measured correct to the nearest centimetre. Calculate the upper bound for the perimeter of the shape. Answer cm [3]

Mark scheme: 11 258 cao 3 M1 18.5 or 24.5 seen M1 6 × sum of their two upper bounds

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12 Simplify 16 – 4(3x – 2)2. Answer [3]

Mark scheme: 12 –36x2 + 48x or 12x(4 – 3x) oe 3 M1 squaring to “9x2 –12x + 4” algebraic or other partly factorised versions M1 multiplying by –4 terms M1 adding 16 only

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13 Solve the inequality 6(2 − 3x) − 4(1 − 2x)=Y= 0. For Examiner's Use Answer [3]

Mark scheme: 13 x [ 0.8 or x [ 4 cao 3 B1 12 – 18x B1 –4 + 8x these terms may be 5 reversed if moved to the other side of the inequality allow >=

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14 Zainab borrows $198 from a bank to pay for a new bed. The bank charges compound interest at 1.9 % per month. Calculate how much interest she owes at the end of 3 months. Give your answer correct to 2 decimal places. Answer $ [3]

Mark scheme: 14 $11.50 3 M1 198 × r3 r can be anything dep M1 r = 1.019 and subtracting 198 SC2 209.50 on answer line IGCSE – October/November 2009 0580 21

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15 P Q M p O r R O is the origin and OPQR is a parallelogram whose diagonals intersect at M. The vector OP is represented by p and the vector is represented by r. (a) Write down a single vector which is represented by (i) p + r, Answer(a)(i) [1] (ii) 1 p – 1 r. 2 2 Answer(a)(ii) [1] (b) On the diagram, mark with a cross (x) and label with the letter S the point with position vector 1 3 p + r. [2] 2 4

Mark scheme: 15 (a) (i) OQ 1 (ii) RM or MP 1 Allow ½RP (b) 2 B1, B1 correct position wrt each direction of the vector ± 1 mm S ×

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16 For Examiner's B Use North NOT TO 3 km North SCALE 3 km C 85° A A, B and C are three places in a desert. Tom leaves A at 06 40 and takes 30 minutes to walk directly to B, a distance of 3 kilometres. He then takes an hour to walk directly from B to C, also a distance of 3 kilometres. (a) At what time did Tom arrive at C? Answer (a) [1] (b) Calculate his average speed for the whole journey. Answer (b) km/h [2] (c) The bearing of C from A is 085°. Find the bearing of A from C. Answer (c) [1]

Mark scheme: 16 (a) (0)810 or 8:10 etc. 1 (b) 4 2 M1 (3 + 3)/(1 + 0.5) (c) 265 1

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17 (a) In 2007, a tourist changed 4000 Chinese Yuan into pounds (£) when the exchange rate was £1 = 15.2978 Chinese Yuan. Calculate the amount he received, giving your answer correct to 2 decimal places. Answer(a) £ [2] (b) In 2006, the exchange rate was £1 = 15.9128 Chinese Yuan. Calculate the percentage decrease in the number of Chinese Yuan for each £1 from 2006 to 2007. Answer(b) % [2]

Mark scheme: 17 (a) 261.48 cao 2 M1 4000 / 15.2978 (b) (±)3.86(48…) or 3.865 2 M1 (15.9128 – 15.2978)/15.9128 (× 100) or (“261.48 – 4000/15.9128) / “261.48”

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18 For Examiner's Use NOT TO y SCALE y = 2x + 4 y = mx + c A B x 6 units The line y = mx + c is parallel to the line y = 2x + 4. The distance AB is 6 units. Find the value of m and the value of c. Answer m = and c = [4]

Mark scheme: 18 m = 2 c = –8 4 B1 B(4, 0) or A(–2, 0) seen or used B1 m = 2 0 −c M1 substituting (4, 0) into y = 2x + c or = 2 4 − 0

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19 For Examiner's Use NOT TO B D 68° SCALE O C E F A Points A, B and C lie on a circle, centre O, with diameter AB. BD, OCE and AF are parallel lines. Angle CBD = 68°. Calculate (a) angle BOC, Answer(a) Angle BOC = [2] (b) angle ACE. Answer(b) Angle ACE = [2]

Mark scheme: 19 (a) 44 2 M1 OCB = 68 (b) 158 2

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20 The number of hours that a group of 80 students spent using a computer in a week was recorded. For The results are shown by the cumulative frequency curve. Examiner's Use 80 60 Cumulative 40 frequency 20 0 10 20 30 40 50 60 70 Number of hours Use the cumulative frequency curve to find (a) the median, Answer(a) h [1] (b) the upper quartile, Answer(b) h [1] (c) the interquartile range, Answer(c) h [1] (d) the number of students who spent more than 50 hours using a computer in a week. Answer(d) [2]

Mark scheme: 20 (a) 38 1 (b) 45 to 46 1 (c) 15 to 16 1 (d) 10 or 11 2 SC1 70 on answer line

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21 For Examiner's Use 40 30 Speed car 20 (m / s) truck 10 0 10 20 30 40 50 60 Time (seconds) The graph shows the speed of a truck and a car over 60 seconds. (a) Calculate the acceleration of the car over the first 45 seconds. Answer(a) m/s2 [2] (b) Calculate the distance travelled by the car while it was travelling faster than the truck. Answer(b) m [3] Question 22 is printed on the next page.

Mark scheme: 21 (a) 0.8 or 4/5 cao 2 M1 speed/time (b) 960 www 3 M1 30 × (12 + 36)/2 M1 12 × 40 M1 10 × (12 + 36)/2 M1 ½ × 40 × 24 IGCSE – October/November 2009 0580 21

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2 x + 1 For 22 f(x) = 4x + 1 g(x) = x3 + 1 h(x) = Examiner's 3 Use (a) Find the value of gf(0). Answer(a) [2] (b) Find fg(x). Simplify your answer. Answer(b) [2] (c) Find h -1(x). Answer(c) [2]

Mark scheme: 22 (a) 2 2 M1 f(0) = 1 (b) 4x3 + 5 2 M1 4(x3 + 1) + 1 (3 x − )1 (c) 2 M1 rearranging y = (2x + 1)/3 to make x the subject 2 and interchanging x and y. Allow any one error in the working 70

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