Cambridge IGCSE Mathematics 0580 — 2015 Oct/Nov Paper 4 · Variant 1

1 (a) Luc is painting the doors in his house. He uses 34 of a tin of paint for each door. Work out the least number of tins of paint Luc needs to paint 7 doors. Answer(a) ................................................ [3] (b) Jan buys tins of paint for $17.16 each. He sells the paint at a profit of 25%. For how much does Jan sell each tin of paint? Answer(b) $ ................................................ [2] (c) The cost of $17.16 for each tin of paint is 4% more than the cost in the previous year. Work out the cost of each tin of paint in the previous year. Answer(c) $ ................................................ [3] (d) In America a tin of paint costs $17.16 . In Italy the same tin of paint costs €13.32 . The exchange rate is $1 = €0.72 . Calculate, in dollars, the difference in the cost of the tin of paint. Answer(d) $ ................................................ [2] (e) Paint is sold in cylindrical tins of height 11 cm. Each tin holds 750 ml of paint. (i) Write 750 ml in cm3. Answer(e)(i) ......................................... cm3 [1] (ii) Calculate the radius of the tin. Give your answer correct to 1 decimal place. Answer(e)(ii) .......................................... cm [3] (iii) A mathematically similar tin has a height of 22 cm. How many litres of paint does this tin hold? Answer(e)(iii) ....................................... litres [2] (f) The mass of a tin of paint is 890 grams, correct to the nearest 10 grams. Work out the upper bound of the total mass of 10 tins of paint. Answer(f) ............................................. g [1] (g) The probability that a tin of paint is dented is 0.07 . Out of 3000 tins of paint, how many would you expect to be dented? Answer(g) ................................................ [2] (h) Tins of paint are filled at the rate of 2 m3 per minute. How many 750 ml tins of paint can be filled in 1 hour? Answer(h) ................................................ [3] __________________________________________________________________________________________

Mark scheme: Question Answer Mark Part marks 1 (a) 6 3 1 B2 for 5 or 5.25 shown in working isw 4 3 or M1 for × 7 soi by answer 5 4 (b) 21.45 cao final answer 2 M1 for 17.16 × 0.25 or 17.16 × 1.25 (c) 16.5[0] nfww 3 M2 for 17.16 ÷ 1.04 oe or M1 for 17.16 associated with 104[%] oe isw (d) 1.34 cao final answer 2 M1 for 13.32 ÷ 0.72 soi by 18.5[0] or for any correct complete longer method If zero scored, SC1 for 0.96 [euros] seen (e) (i) 750 1 (ii) 4.7 cao 3 B2 for 4.658 to 4.66 or M2 for their (e)(i) ÷ 11π or M1 for 11πr2 = their (e)(i) 1 (iii) 6 2 M1 for 23 or 3 oe seen 2 or for π × (2 × their (e)(ii))2 × 22 If zero scored, SC1 for answer 6 000 (f) 8 950 1 (g) 210 2 M1 for 0.07 × 3 000 (h) 160 000 3 M2 for 2 × 60 × 1003 ÷ 750 oe or M1 for figs 16 as answer or 1003 seen

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2 (a) Calculate 20.7. Answer(a) ................................................ [1] (b) Find the value of x in each of the following. (i) 2x = 128 Answer(b)(i) x = ................................................ [1] (ii) 2x × 29 = 213 Answer(b)(ii) x = ................................................ [1] (iii) 29 ÷ 2x = 4 Answer(b)(iii) x = ................................................ [1] (iv) 2x = 3 2 Answer(b)(iv) x = ................................................ [1] (c) (i) Complete this table of values for y = 2x. x –3 –2 –1 0 1 2 3 y 0.125 0.5 2 4 8 [2] (ii) On the grid, draw the graph of y = 2x for –3 G x G 3. y 8 7 6 5 4 3 2 1 x –3 –2 –1 0 1 2 3 –1 [4] (iii) Use your graph to solve 2x = 5. Answer(c)(iii) x = ................................................ [1] (iv) Find the equation of the line joining the points (1, 2) and (3, 8). Answer(c)(iv) ................................................ [3] (v) By drawing a suitable line on your graph, solve 2x – 2 – x = 0. Answer(c)(v) x = .......................... or x = .......................... [2] __________________________________________________________________________________________

Mark scheme: 2 (a) 1.62 or 1.62… 1 (b) (i) 7 1 (ii) 4 1 (iii) 7 1 (iv) 1 1 oe 3

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3 (a) T 60 m NOT TO SCALE 50 m B 130° A 70 m C A, B and C are points on horizontal ground. BT is a vertical pole. AT = 60 m, AB = 50 m, BC = 70 m and angle ABC = 130°. (i) Calculate the angle of elevation of T from C. Answer(a)(i) ................................................ [5] (ii) Calculate the length AC. Answer(a)(ii) AC = ............................................ m [4] (iii) Calculate the area of triangle ABC. Answer(a)(iii) ........................................... m2 [2] (b) Y 12 cm NOT TO SCALE 22 cm X 45 cm A cuboid has length 45 cm, width 22 cm and height 12 cm. Calculate the length of the straight line XY. Answer(b) XY = .......................................... cm [4] __________________________________________________________________________________________

Mark scheme: 3 (a) (i) 25.4 or 25.35… nfww 5 M2 for 60 2 − 50 2 oe soi by 33.1 to 33.2 or M1 for TB2 + 502 = 602 oe and theirTB M2 for tan = oe 70 or B1 for recognising angle TCB as required angle (ii) 109 or 109.0 to 109.1 4 M2 for 502 + 702 – 2 × 50 × 70 × cos130 M1 for implicit cos rule A1 for 11 899 to 11 900 (iii) 1 340 or 1 340.0 to 1 341 2 1 M1 for × 50 × 70 × sin130 oe 2 (b) 51.5 or 51.50 to 51.51 4 M3 for [XY] = 45 2 + 22 2 + 12 2 or M2 for [XY 2 =] 452 + 222 + 122 soi by 2 653 or M1 for 452 + 222 oe or 452 + 122 oe or 122 + 222 oe

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4 Ali buys x rose bushes and y lavender bushes. He buys: • at least 5 rose bushes • at most 8 lavender bushes • at most 15 bushes in total • more lavender bushes than rose bushes. (a) (i) Write down four inequalities, in terms of x and/or y, to show this information. Answer(a)(i) ................................................ ................................................ ................................................ ................................................ [4] (ii) On the grid, show the information in part (a)(i) by drawing four straight lines. Label the region R where all four inequalities are true. y 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 [5] (b) Rose bushes cost $6 each and lavender bushes cost $4.50 each. What is the greatest amount of money Ali could spend? Answer(b) $ ................................................ [2] __________________________________________________________________________________________

Mark scheme: 4 (a) (i) x ⩾ 5 oe Condone 5 ⩽ x ⩽ 15 y ⩽ 8 oe Condone 0 < y ⩽ 8 x + y ⩽ 15 oe y > x oe or y ⩾ x + 1 4 B1 for each – 1 for first occurrence of strict inequalities used in first 3 inequalities (ii) x = 5 ruled 1 y = 8 ruled 1 x + y = 15 ruled 1 y = x ruled broken line 1 Allow y = x + 1 ruled only after y ⩾ x + 1 in (a)(i) Correct region indicated 1dep Dependent on all marks for lines earned Accept R written in correct quadrilateral or any other unambiguous indication or accept in triangle if y = x + 1 used and all marks for lines earned (b) 78 2 B1 for (7, 8) chosen or M1 for a calculation shown of the form 6x + 4.5y where (x, y) is clearly in their region and both x and y are integers

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5 B NOT TO C SCALE A 37° O D E A, B, C, D and E are points on the circle, centre O. Angle BAD = 37°. Complete the following statements. (a) Angle BED = .................... because ................................................................................................. ............................................................................................................................................................. [2] (b) Angle BOD = .................... because ................................................................................................ ............................................................................................................................................................. [2] (c) Angle BCD = .................... because ................................................................................................ ............................................................................................................................................................. [2] __________________________________________________________________________________________

Mark scheme: 5 (a) 37 or [angle] BAD 1 [Angles in ] same segment [are 1dep Dependent on 37 or [angle] BAD equal] (b) 74 or 2 [× angle] BAD or 1 2 [× angle] BED Angle at centre is twice angle at 1dep Dependent on 2 × 37 or 2 [× angle] BAD circumference or 2 [× angle] BED Must use the terms circumference, centre and angle (c) 143 or 180 – [angle] BAD 1 or 180 – [angle] BED [Opposite angles of] cyclic quad 1dep Dependent on 180 – 37 [are supplementary] or 180 – [angle] BAD or 180 – [angle] BED

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6 120 students take a mathematics examination. (a) The time taken, m minutes, for each student to answer question 1 is shown in this table. Time (m minutes) 0 < m G 1 1 < m G 2 2 < m G 3 3 < m G 4 4 < m G 5 5 < m G 6 Frequency 72 21 9 11 5 2 Calculate an estimate of the mean time taken. Answer(a) ..........................................min [4] (b) (i) Using the table in part (a), complete this cumulative frequency table. Time (m minutes) m G 1 m G 2 m G 3 m G 4 m G 5 m G 6 Cumulative frequency 72 120 [2] (ii) Draw a cumulative frequency diagram to show the time taken. 120 110 100 90 80 70 Cumulative frequency 60 50 40 30 20 10 0 m 1 2 3 4 5 6 Time (minutes) [3] (iii) Use your cumulative frequency diagram to find (a) the median, Answer(b)(iii)(a) ..........................................min [1] (b) the inter-quartile range, Answer(b)(iii)(b) ..........................................min [2] (c) the 35th percentile. Answer(b)(iii)(c) ..........................................min [2] (c) A new frequency table is made from the table shown in part (a). Time (m minutes) 0 < m G 1 1 < m G 3 3 < m G 6 Frequency 72 (i) Complete the table above. [2] (ii) A histogram was drawn and the height of the first block representing the time 0 < m G 1 was 3.6 cm. Calculate the heights of the other two blocks. Answer(c)(ii) ................. cm and ................. cm [3] __________________________________________________________________________________________

Mark scheme: 6 (a) 1.35 nfww 4 M1 for 0.5, 1.5, 2.5, 3.5, 4.5, 5.5 soi, M1 for Σfm soi by 162 where m is in correct interval including boundaries M1dep for Σfm ÷ 120 or Σfm ÷ Σf dependent on second M1 earned (b) (i) 93, 102, 113, 118 2 SC1FT for 1 error (ii) Correct diagram 3 B1FT for correct vertical plots and B1 for correct horizontal plots and B1FT dep on at least B1 for reasonable increasing curve or polygon through their 6 points If zero scored, SC1FT for 5 out of 6 correct plots (iii) (a) 0.6 to 0.85 1 (b) 1.3 to 1.7 2 B1 for UQ = 1.7 to 1.9 or LQ = 0.2 to 0.4 (c) 0.3 to 0.6 2FT Allow in correct range provided there is no evidence of reading at 35 or FT their reading at 42 B1 for 42 soi (c) (i) 30 and 18 2 B1 for each (ii) 0.75 and 0.3 3FT FT (their 30) ÷ 40 and (their 18) ÷ 60 B2FT for either 0.75 or 0.3 or M1 for their 30 ÷ 2 or ÷ 20 or for their 18 ÷ 3 or ÷ 20

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7 The scale drawing shows the positions of three towns A, B and C on a map. The scale of the map is 1 centimetre represents 10 kilometres. C North A North Scale: 1 cm to 10 km B (a) Find the actual distance AB. Answer(a) .......................................... km [1] (b) Measure the bearing of A from B. Answer(b) ................................................ [1] (c) Write the scale 1 cm to 10 km in the form 1 : n. Answer(c) 1 : ................................................ [1] (d) A national park lies inside the triangle ABC. The four boundaries of the national park are • equidistant from C and B • equidistant from AC and CB • 15 km from CB • along AB. On the scale drawing, shade the region which represents the national park. Leave in your construction arcs. [7] (e) On the scale drawing, a lake inside the national park has area 0.4 cm2. Calculate the actual area of the lake. Answer(e) ......................................... km2 [2] __________________________________________________________________________________________

Mark scheme: 7 (a) 123 to 127 1 (b) 288 to 292 1 (c) [1:] 1 000 000 1

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8 (a) Factorise x2 – 3x – 10. Answer(a) ................................................ [2] x + 2 3 (b) (i) Show that + = 3 simplifies to 2x2 – 2x – 3 = 0. x + 1 x Answer(b)(i) [3] (ii) Solve 2x2 – 2x – 3 = 0. Give your answers correct to 3 decimal places. Show all your working. Answer(b)(ii) x = ........................... or x = ........................... [4] 2x + 3 x (c) Simplify – . x + 2 x + 1 Answer(c) ................................................ [4] __________________________________________________________________________________________

Mark scheme: 8 (a) (x – 5)(x + 2) final answer 2 B1 for (x – 5)(x + 2) seen and then spoiled or M1 for (x + a)(x + b) where a + b = – 3 or ab = –10 [a, b integers] (b) (i) x(x + 2) + 3(x + 1) = 3x(x + 1) or M2 M1 for x(x + 2) + 3(x + 1) or better seen x2 + 2x + 3x + 3 = 3x2 + 3x Allow recovery of omitted brackets for M marks but not A mark 0 = 2x2 – 2x – 3 A1 Brackets expanded correctly and/or no errors or omission of brackets seen (ii) [ −− ]2 ± ([ − ] 2) 2 − 4( 2)( −3) B2 B1 for ([ − ]2 ) 2 − 4 ( 2 )( −3) or 28 2( 2) or .175 oe in completion of square p + q p − q or 0.5 ± .175 and B1 for in form or , r r p = – –2 and r = 2(2) or better or (x – 0.5)2 oe in completion of square – 0.823 and 1.823 final answer B1 B1 If B0B0 for answers, SC1 for – 0.82 or – 0.822… and 1.82 or 1.822.. as final answers or – 0.823 and 1.823 seen or –1.823 and 0.823 as final answers

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9 The first three diagrams in a sequence are shown below. The diagrams are made by drawing lines of length 1 cm. Diagram 1 Diagram 2 Diagram 3 (a) The areas of each of the first three diagrams are shown in this table. Diagram 1 2 3 Area (cm2) 1 4 9 (i) Find the area of Diagram 4. Answer(a)(i) ......................................... cm2 [1] (ii) Find, in terms of n, the area of Diagram n. Answer(a)(ii) ......................................... cm2 [1] (b) The numbers of 1 cm lines needed to draw each of the first three diagrams are shown in this table. Diagram 1 2 3 Number of 1 cm lines 4 13 26 (i) Find the number of 1 cm lines needed to draw Diagram 4. Answer(b)(i) ................................................ [1] (ii) In which diagram are 118 lines of length 1 cm needed? Answer(b)(ii) ................................................ [1] (c) The total number of 1 cm lines needed to draw both Diagram 1 and Diagram 2 is 17. The total number of 1 cm lines needed to draw all of the first n diagrams is 2 n3 + an2 + bn. 3 Find the value of a and the value of b. Show all your working. Answer(c) a = ................................................ b = ................................................ [6] __________________________________________________________________________________________ Question 10 is printed on the next page.

Mark scheme: 9 (a) (i) 16 1 (ii) n2 1 (b) (i) 43 1 (ii) 7 1 5 5 (c) a = oe, b = oe with 6 M1 for any correct substitution 2 6 2 supporting working eg (2)3 + 22a + 2b 3 A1 for one of 2 eg + a + b = 4 or better 3 16 eg + 4a + 2b = 17 or better 3 54 eg + 9a + 3b = 43 or better 3 A1 for another of 2 eg + a + b = 4 or better 3 16 eg + 4a + 2b = 17 or better 3 54 eg + 9a + 3b = 43 or better 3 M1 for correctly eliminating one variable from two of their equations in a and b 5 A1 for a = oe 2 5 A1 for b = oe 6 After zero scored, SC2 for 2 correct answers without supporting working or SC1 for 2 of 17, 43, 86, 150, 239 seen

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10 C NOT TO b SCALE M a A X B BC = a and AC = b. (a) Find AB in terms of a and b. Answer(a) AB = ................................................ [1] (b) M is the midpoint of BC. X divides AB in the ratio 1 : 4. Find XM in terms of a and b. Show all your working and write your answer in its simplest form. Answer(b) XM = ................................................ [4]

Mark scheme: 10 (a) b – a or – a + b 1 4 3 1 (b) b – a or (8b – 3a) 4 B3 for correct unsimplified expression 5 10 10 in a and b or M1 for XA + AC + CM or XB + BM 1 1 or −5 (their (a)) + b – 2 a 4 1 or (their (a)) + a 5 2 and M1 indep 1 4 for ± oe or ± oe used 5 5 After zero scored, SC2 for answer 1 3 1 (3b – a) or b – a 4 4 4

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