Cambridge IGCSE Mathematics 0580 — 2014 Oct/Nov Paper 3 · Variant 2

1 A building company buys 4 square kilometres of land. On the land the company builds houses, shops and a school. (a) Show that 4 square kilometres is equivalent to 4 000 000 square metres. Answer(a) [1] (b) The company uses 5% of the land for roads and paths. Show that the remaining area of land is 3 800 000 m2. Answer(b) [1] (c) The 3 800 000 m2 of land is divided in the ratio houses : shops : school = 11 : 5 : 3. (i) Show that the area for the school is 600 000 m2. Answer(c)(i) [2] (ii) Calculate the area for houses. Answer(c)(ii) ........................................... m2 [1] (iii) 140 m2 is needed for each house. Calculate, correct to the nearest 10, the number of houses that can be built. Answer(c)(iii) ................................................ [2] 3 1 (d) of the school area is for classrooms and is for other rooms. 5 8 The remainder is for sporting facilities. (i) Without using a calculator, and showing all your working, fi nd the fraction of the school area for sporting facilities. Answer(d)(i) ................................................ [3] (ii) The school has an area of 600 000 m2. Work out the area for sporting facilities. Answer(d)(ii) ........................................... m2 [1] (e) To pay for materials, the building company borrows $250 000 from a bank for 3 years. The bank charges compound interest at a rate of 4% per year. Calculate the total amount the company must pay back at the end of 3 years. Answer(e) $ ................................................ [3] __________________________________________________________________________________________

Mark scheme: 1 (a) 4 × 1000 × 1000 or 4 × 10002 1 (b) 0.95 × 4 000 000 oe 1 (c) (i) 3 ÷ 19 × 3 800 000 2 M1 for 3 ÷ (11 + 5 + 3) or 3 800 000 ÷ (11 + 5 + 3) (ii) 2 200 000 1 (iii) 15 710 2FT M1FT for their 2 200 000 ÷ 140  24 5  24 5 3 × 8 1 × 5 (d) (i) 1 −  +  M2 M1 for or or or  40 40  40 40 5 × 8 8 × 5 11 11 k or final answer A1 If zero scored, 40 40 k 275 SC3 for 1 – (0.6 + 0.125) = 0.275 = = 1000 11 11 k [ or ] 40 40 k or 275 SC2 for 1 – (0.6 + 0.125) = 0.275 = 1000 followed by incorrect fraction 11 11 k SC1 for or final answer 40 40 k (ii) 165 000 1FT FT their (d)(i) × 600 000 (e) 281 216 cao 3 M2 for 250 000 × 1.043 oe or M1 for 250 000 × 1.042 oe If zero scored, SC1 for 31 216

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2 (a) Write down the mathematical name of a polygon with 8 sides. Answer(a) ................................................ [1] (b) Calculate the interior angle of a regular 8-sided polygon. Answer(b) ................................................ [3] (c) Diagram 1 Diagram 2 Diagram 3 The pattern of diagrams above forms a sequence. (i) Complete the table. Diagram 1 2 3 4 5 Number of dots 8 15 [2] (ii) Find an expression, in terms of n, for the number of dots in Diagram n. Answer(c)(ii) ................................................ [2] (iii) Find the number of dots in Diagram 10. Answer(c)(iii) ................................................ [1] (iv) Find the value of n for a diagram with 92 dots. Answer(c)(iv) ................................................ [2] __________________________________________________________________________________________

Mark scheme: 2 (a) Octagon 1 (b) 135 3 M2 for 180 – (360 ÷ 8) or M2 for (8 − 2) × 180 8 or M1 for (360 ÷ 8) or M1 for (8 – 2) × 180 (c) (i) 22 29 36 2 B1 for two terms in correct places or 2 terms with a difference of 7. (ii) 7n + 1 oe 2 B1 for 7n + j or kn + 1 (k ≠ 0) (iii) 71 1FT FT for their (c)(ii) if linear (iv) 13 nfww 2 M1FT for their (c)(ii) = 92 or M1 for (92 – 1) ÷ 7 or 91 ÷ 7 or M1 for 7 × 13 + 1 = 92

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3 O A B (a) Describe fully two single transformations that each map the shaded triangle onto the unshaded triangle. Answer(a) Transformation 1 ............................................................................................................... ........................................................................................................................................... Transformation 2 ............................................................................................................... ........................................................................................................................................... [5] (b) On the grid, draw the image of -2 (i) the shaded triangle after a translation by the vector [2] e 7 o, (ii) the shaded triangle after an enlargement with scale factor 3 and centre O. [2] (c) Draw the line of symmetry of the enlarged triangle in part (b)(ii). [1] __________________________________________________________________________________________

Mark scheme: 3 (a) Reflection 1 [in] AB 1 Rotation 1 180o oe 1 Midpoint of AB oe 1 (b) (i) Translation 2 left and 7 up 2 SC1 for one of 7 up or 2 left (ii) Correct Enlargement 2 SC1 for enlargement scale factor 3 but incorrectly placed (c) Correct line of symmetry 1FT FT is their (b)(ii)

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4 600 550 500 Seville 450 400 Distance from Madrid (km) 350 Cordoba 300 250 200 150 100 50 Madrid 0 07 00 07 30 08 00 08 30 09 00 09 30 10 00 10 30 11 00 Time (a) A train leaves Madrid at 07 00. It arrives at Cordoba at 08 40 and stays at the station for 10 minutes. It then continues to Seville arriving at 09 40. (i) Show this journey on the grid opposite. [3] (ii) Write down, in hours and minutes, the total time for this journey. Answer(a)(ii) .................. h .................. min [1] (iii) Calculate, in kilometres per hour, the average speed for the whole journey. Answer(a)(iii) ....................................... km/h [2] (b) Another train leaves Seville at 07 45. It travels to Madrid without stopping at an average speed of 200 km/h. (i) Calculate, in hours and minutes, the time taken for this journey. Answer(b)(i) .................. h .................. min [2] (ii) Show this journey on the grid. [2] (c) How far from Madrid were the trains when they passed each other? Answer(c) .......................................... km [1] __________________________________________________________________________________________

Mark scheme: 4 (a) (i) Line (0700, 0) to (08 40, 310) 1 Lines need not be ruled and could be curves Horizontal line 2 squares 1FT with positive gradients throughout. Line their (08 50, 310) to (09 40, 470) 1FT (ii) 2[h]40[min] 1 (iii) 176.25 2 M1FT for 470 ÷ their (a)(ii) (b) (i) 2[h]21[min] 2 M1 for 470 ÷ 200 soi (ii) Line from (07 45, 470) to (their 10 06, 2FT B1 for (07 45, 470) correctly plotted 0) or B1FT for (their 10 06, 0) correctly plotted (c) 290 to 300 1FT (Correct or follow through) FT from intersection on their graph.

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5 A 9 cm F G 50 cm B NOT TO SCALE E 52 cm D 12 cm H 70 cm C The diagram shows a rectangle ABCD divided into three sections by the lines EF and HG. AF = 9 cm, GB = 50 cm, DH = 12 cm, HC = 70 cm and HG = 52 cm. (a) Write down the mathematical name of (i) quadrilateral BCHG, Answer(a)(i) ................................................ [1] (ii) the shaded polygon. Answer(a)(ii) ................................................ [1] (b) (i) Show by calculation that BC = 48 cm. Answer(b)(i) [2] (ii) Calculate the area of rectangle ABCD. Answer(b)(ii) ......................................... cm2 [2] (c) Calculate (i) the perimeter of BCHG, Answer(c)(i) .......................................... cm [1] (ii) the area of BCHG. Answer(c)(ii) ......................................... cm2 [2] (d) E is the midpoint of AD. Find the area of triangle AEF. Answer(d) ......................................... cm2 [3] (e) Work out the area of the shaded polygon. Answer(e) ......................................... cm2 [1] __________________________________________________________________________________________

Mark scheme: 5 (a) (i) Trapezium 1 (ii) Pentagon 1 (b) (i) [BC =] 52 2 − 20 2 [= 48] B2 B1 for 522 = BC2 + (70 – 50)2 or 522 = BC2 + 202 or BC2 = 522 – 202 (ii) 3936 or 3940 2 M1 for (70 + 12) × 48 oe (c) (i) 220 1 (ii) 2880 2 M1 for 0.5(50 + 70) × 48 oe (d) 108 3 B1 for [AE=] 24 M1 for 0.5 × their AE × 9 (e) 948 1FT FT their (b)(ii) – (their (c)(ii) + their (d))

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20 .6 (a) (i) Complete the table of values for y = x x –8 –5 –4 –2.5 2.5 4 5 8 y –2.5 –4 8 4 [2] 20 (ii) On the grid, draw the graph of y = for –8 Y x Y –2.5 and 2.5 Y x Y 8. x y 9 8 7 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 –3 –4 –5 –6 –7 –8 –9 [4] 20 (iii) By drawing a suitable line on your graph solve the equation = 6. x Answer(a)(iii) x = ................................................ [2] (b) x –8 0 8 y 1 (i) Complete the table for y = x – 1. [2] 2 1 (ii) On the grid, draw the graph of y = x – 1 for –8 Y x Y 8. [1] 2 1 (iii) Write down the gradient of y = x – 1. 2 Answer(b)(iii) ................................................ [1] 20 1 (c) Write down the values of x at the points of intersection of the graphs of y = and y = x – 1. x 2 Answer(c) x = ...................... and x = ...................... [2] __________________________________________________________________________________________

Mark scheme: 6 (a) (i) –5 –8 5 2.5 2 B1 for 3 correct (ii) 8 points correctly plotted B3FT B2FT for 6 or 7 correct points Correct curve 1 B1FT for 4 or 5 correct points (iii) Ruled line y = 6 drawn 1 Independent marks 3.1 to 3.6 1 (b) (i) –5 –1 3 2 B1 for 2 correct (ii) Ruled correct line 1 (iii) 1 1 oe 2 (c) 7.2 to 7.6 1FT –5.2 to –5.6 1FT

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7 (a) 21 11 7 29 3 20 24 8 18 14 For these numbers (i) calculate the mean, Answer(a)(i) ................................................ [2] (ii) fi nd the median, Answer(a)(ii) ................................................ [2] (iii) fi nd the range. Answer(a)(iii) ................................................ [1] (b) The table shows the number of births for each month of 2013 in a hospital. Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 319 299 336 309 334 336 348 363 351 347 331 335 (i) On the grid opposite, complete the bar chart. The fi rst 6 months have been drawn for you. [2] (ii) Write down the modal month. Answer(b)(ii) ................................................ [1] (iii) A month is chosen at random. Find the probability that the number of births in that month is greater than 340. Answer(b)(iii) ................................................ [1] 370 360 350 340 Number of births 330 320 310 300 290 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month __________________________________________________________________________________________

Mark scheme: 7 (a) (i) 15.5 2 M1 Sum of the 10 items of data ÷ 10 (ii) 16 2 M1 for ordering at least first or last 6 items or for 14 and 18 indicated (iii) 26 1 (b) (i) 6 correct bars 2 B1 for 4 or 5 correct bars or 6 correct heights (ii) Aug[ust] 1 (iii) 1 4 oe 12

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8 North Q North 48 km P (a) The scale drawing shows a ship’s voyage from port P to port Q. The straight line distance from P to Q is 48 km. (i) Measure the bearing of Q from P. Answer(a)(i) ................................................ [1] (ii) Complete the following statement. The scale of the drawing is 1 centimetre represents ........................ kilometres. [2] (b) From port Q, the ship sails on a bearing of 125° for 76 km to port R. Show this part of the voyage on the scale drawing. [3] (c) L North NOT TO 8.5 km SCALE P W 297° Another ship leaves port P and sails on a bearing of 297° to a lighthouse, L. PL = 8.5 km. (i) Show that angle LPW = 27°. Answer(c)(i) [1] (ii) Using trigonometry, calculate PW. Give your answer correct to 2 signifi cant fi gures. Answer(c)(ii) PW = .......................................... km [3] (d) The diagram shows the positions of two beacons, A and B. A ship sails on a course that is the perpendicular bisector of the line AB. Using a straight edge and compasses only, construct the ship’s course. B A [2] __________________________________________________________________________________________

Mark scheme: 8 (a) (i) [0]63 to [0]67 1 (ii) 8 2 B1 for 6 ± 0.2 [cm] seen in working (b) QR on bearing 123o to 127o 1 B1 for bearing of 123o to 127o 9.3 cm to 9.7 cm continuous ruled line 2FT M1FT for 76 ÷ their (a)(ii) soi by calculation or distance on diagram (c) (i) 297 – 270 1 or 90 – (360 – 297) PW PW (ii) 7.6 cao nfww 3 M1 for cos27° = or sin63° = or 5.8 5.8 better A1 for 7.57(...) B1ind for correctly rounding their 7.57(...) to 2 sig figs if their 7.57(…) is to 3 sig figs or more (d) Correct continuous perpendicular 2 B1 for correct continuous bisector without arc bisector of AB with two pairs of correct or with incorrect arcs arcs

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9 Adriano hires a car. The cost of hiring the car is $36 per day plus 24 cents for each kilometre travelled. He hires the car for 5 days and travels a total of 660 km. (a) (i) Calculate the cost to hire the car. Answer(a)(i) $ ................................................ [3] (ii) 15% tax is then added to this cost. Calculate the total cost of hiring the car including tax. Answer(a)(ii) $ ................................................ [2] (b) The car uses one litre of fuel to travel 11 km. Fuel costs $1.80 per litre. (i) Work out the number of litres used to travel the 660 km. Answer(b)(i) ....................................... litres [1] (ii) Work out the cost of this fuel. Answer(b)(ii) $ ................................................ [1] (iii) Find the total cost of hiring the car including tax and the fuel used. Answer(b)(iii) $ ................................................ [1] (c) During the 5 days Adriano earns $1600. What percentage of his earnings is your answer to part (b)(iii)? Give your answer correct to the nearest whole number. Answer(c) .............................................% [2]

Mark scheme: 9 (a) (i) 338.4[0] 3 M2 for 5 × 36 + 660 × 0.24 or better or M1 for 5 × 36 or 660 × 0.24 or better (ii) 389.16 2FT M1FT for 1.15 × their (a)(i) oe (b) (i) 60 1 (ii) 108 1FT 1.8 × their (b)(i) (iii) 497.16 1FT FT their (a)(ii) + their (b)(ii) their (b)(iii) (c) 31 nfww 2FT M1FT for × 100 1600

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