The four operations — Paper 4 · IGCSE Mathematics 0580

10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 x b c A 3 by 3 square can be chosen from the 6 by 6 grid above. d e f g h i 8 9 10 (a) One of these squares is . 14 15 16 20 21 22 In this square, x = 8, c = 10, g = 20 and i = 22. For this square, calculate the value of (i) (i − x) − (g − c), [1] (ii) cg − xi. [1] (b) x b c d e f g h i (i) c = x + 2. Write down g and i in terms of x. [2] (ii) Use your answers to part(b)(i) to show that (i − x) − (g − c) is constant. [1] (iii) Use your answers to part(b)(i) to show that cg − xi is constant. [2] (c) The 6 by 6 grid is replaced by a 5 by 5 grid as shown. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 x b c A 3 by 3 square can be chosen from the 5 by 5 grid. d e f g h i For any 3 by 3 square chosen from this 5 by 5 grid, calculate the value of (i) (i − x) − (g − c), [1] (ii) cg − xi. [1] (d) A 3 by 3 square is chosen from an n by n grid. (i) Write down the value of (i − x) − (g − c). [1] (ii) Find g and i in terms of x and n. [2] (iii) Find cg − xi in its simplest form. [1]

13 marks

Mark scheme: 10(a) (i) 4 B1 (ii) 24 B1 (b) (i) x + 12, x + 14 o.e. B1,B1 Any order ignore ref to g and i (ii) (x + 14 – x) and (x + 12 – (x + 2)) x + 12 and x + 14 must be seen to be used 14 – 10 or 14 – 12 + 2 or 4 E1 No errors seen (iii) (x + 2)(x + 12) – x(x + 14) B1 Subtraction can be implied later 24 E1 Dep on B1 and no errors anywhere for the E mark (c) (i) 4 B1 (ii) 20 B1 (d) (i) 4 B1 (ii) x + 2n o.e., x + 2+ 2n o.e. B1,B1 (iii) 4n B1 Allow 4×n, n×4, n4 [13]

This question in 0580/41 May/June 2008

7 For Examiner's NOT TO Use SCALE x cm x cm 250 cm A solid metal bar is in the shape of a cuboid of length of 250 cm. The cross-section is a square of side x cm. The volume of the cuboid is 4840 cm3. (a) Show that x = 4.4. Answer (a) [2] (b) The mass of 1 cm3 of the metal is 8.8 grams. Calculate the mass of the whole metal bar in kilograms. Answer(b) kg [2] (c) A box, in the shape of a cuboid measures 250 cm by 88 cm by h cm. 120 of the metal bars fit exactly in the box. Calculate the value of h. Answer(c) h = [2] (d) One metal bar, of volume 4840 cm3, is melted down to make 4200 identical small spheres. For Examiner's All the metal is used. Use (i) Calculate the radius of each sphere. Show that your answer rounds to 0.65 cm, correct to 2 decimal places. 4 3 [The volume, V, of a sphere, radius r, is given by V = πr .] 3 Answer(d)(i) [4] (ii) Calculate the surface area of each sphere, using 0.65 cm for the radius. [The surface area, A, of a sphere, radius r, is given by A = 4 πr 2 .] Answer(d)(ii) cm2 [1] (iii) Calculate the total surface area of all 4200 spheres as a percentage of the surface area of the metal bar. Answer(d)(iii) % [4]

15 marks

Mark scheme: 7 (a) 250x2 = 4840 o.e. M1 Allow M1 for 250 × 4.42 = 4840 x² = 19.36 or (x =) 4840 ÷ 250 (= 4.4) E1 Then E1 for 250 × 19.36 = 4840 (b) 42.6 (kg) cao (42.592 or 42.59) B2 SC1 for figures 426 or 4259… (c) 26.4 (cm) c.a.o. B2 If B0, M1 for any of following 88 ÷ 4.4 = 20 and 120 ÷ 20 = 6 (accept 6 bars high o.e.) or 88h = 4.42 × 120 or 250 × 88 × h = 120 × 4840 (d) (i) 4840 ÷ 4200 (implied by 1.15(2)) M1 4200 × 4 3 π r3 = 4840 ÷ 4 3 π (implied by 0.274 to 0.276) M1 (r3 =) 4840 ÷ (4200 × 4 3 π ) 3 (seen or implied by correct answer to M1 3 Third M dependent on M1M1 dep more than 2 dp) 0.649 – 0.651 A1 Must be 3dp or better (ii) 5.31 (5.306 – 5.31) (cm2) B1 4200 × their (ii) (iii) × 100 M3 If M0, M1 for 4200 × their (ii) (22299) 2 × 4.4 2 + 4 × 4.4 × 250 and M1 (independent) for correct method for surface area of solid cuboid (4438.72) 501.9 – 503 (%) c.a.o. www4 A1 [15]

This question in 0580/41 May/June 2009

10 For Examiner's Total Use Row 1 1 = 1 Row 2 3 + 5 = 8 Row 3 7 + 9 + 11 = 27 Row 4 13 + 15 + 17 + 19 = 64 Row 5 Row 6 The rows above show sets of consecutive odd numbers and their totals. (a) Complete Row 5 and Row 6. [2] (b) What is the special name given to the numbers 1, 8, 27, 64…? Answer(b) [1] (c) Write down in terms of n, (i) how many consecutive odd numbers there are in Row n, Answer(c)(i) [1] (ii) the total of these numbers. Answer(c)(ii) [1] (d) The first number in Row n is given by n2 − n + 1. Show that this formula is true for Row 4. Answer(d) [1] (e) The total of Row 3 is 27. This can be calculated by (3 × 7) + 2 + 4. For Examiner's The total of Row 4 is 64. This can be calculated by (4 × 13) + 2 + 4 + 6. Use The total of Row 7 is 343. Show how this can be calculated in the same way. Answer(e) [1] (f) The total of the first n even numbers is n(n + 1). Write down a formula for the total of the first (n – 1) even numbers. Answer(f) [1] (g) Use the results of parts (d), (e) and (f) to show clearly that the total of the numbers in Row n gives your answer to part (c)(ii). Answer(g) [2]

10 marks

Mark scheme: 10 (a) 21 + 23 + 25 + 27 + 29 = 125 B1 31 + 33 + 35 + 37 + 39 + 41 = 216 B1 (b) Cubes B1 (c) (i) n oe B1 (ii) n3 oe B1 (d) 42 – 4 + 1 = 13 www E1 Allow 16 for 42, otherwise all must be seen (e) 7 × 43 + 2 + 4 + 6 + 8 + 10 + 12 B1 All must be seen (f) n(n – 1) final answer oe B1 (g) n(n2 – n + 1) + their (f) M1 n3 – n2 + n + n2 – n = n3 E1 All must be seen, no errors or omissions [10]

This question in 0580/41 Oct/Nov 2009

8 Examiner's Use NOT TO SCALE 3 cm 6 cm 10 cm A solid metal cuboid measures 10 cm by 6 cm by 3 cm. (a) Show that 16 of these solid metal cuboids will fit exactly into a box which has internal measurements 40 cm by 12 cm by 6 cm. Answer(a) [2] (b) Calculate the volume of one metal cuboid. Answer(b) cm3 [1] (c) One cubic centimetre of the metal has a mass of 8 grams. The box has a mass of 600 grams. Calculate the total mass of the 16 cuboids and the box in (i) grams, Answer(c)(i) g [2] (ii) kilograms. Answer(c)(ii) kg [1] For (d) (i) Calculate the surface area of one of the solid metal cuboids. Examiner's Use Answer(d)(i) cm2 [2] (ii) The surface of each cuboid is painted. The cost of the paint is $25 per square metre. Calculate the cost of painting all 16 cuboids. Answer(d)(ii) $ [3] (e) One of the solid metal cuboids is melted down. Some of the metal is used to make 200 identical solid spheres of radius 0.5 cm. Calculate the volume of metal from this cuboid which is not used. 4 [The volume, V, of a sphere of radius r is V = π r 3.] 3 Answer(e) cm3 [3] (f) 50 cm3 of metal is used to make 20 identical solid spheres of radius r. Calculate the radius r. Answer(f) r = cm [3]

17 marks

Mark scheme: (iv) 8 ft 1ft ft a positive root –4 if positive answer − ( −)1 ± ( −)1 2 − 41()( −4) 2 (c) 2 B1 for ( −)1 − 41()( −4) or better 2 )1( p + q p − q If in form or r r then B1 for –(–1) and 2(1) or better Brackets and full line may be implied later –1.56, 2.56 2 B1 B1 If B0, SC1 for –1.6 or –1.562 to –1.561 and 2.6 or 2.561 to 2.562 10 (a) Dots all correctly placed in Diagram 4 1 (b) Column 4 16, 25, 16, 41 7 B2 or B1 for three correct Column 5 25, 41, 20, 61 B2 or B1 for three correct Column n: n2, 4n, n2 + (n + 1)2 oe B1 B1 B1 oe likely to be (n –1)2 + n2 + 4n or 2n2 + 2n + 1 After any correct answer for column n, apply isw (c)(i) 79 601 cao 1 (ii) 800 ft 1ft ft their 4n linear expression only

This question in 0580/43 May/June 2010

10 Consecutive integers are set out in rows in a grid. For Examiner's Use (a) This grid has 5 columns. 1 2 3 4 5 6 7 8 9 10 a b 11 12 13 14 15 n 16 17 18 19 20 c d 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 The shape drawn encloses five numbers 7, 9, 13, 17 and 19. This is the n = 13 shape. In this shape, a = 7, b = 9, c = 17 and d = 19. (i) Calculate bc O ad for the n = 13 shape. Answer(a)(i) [1] (ii) For the 5 column grid, a = n O 6. Write down b, c and d in terms of n for this grid. Answer(a)(ii) b = c = d = [2] (iii) Write down bc O ad in terms of n. Show clearly that it simplifies to 20. Answer(a)(iii) [2] (b) This grid has 6 columns. The shape is drawn for n = 10. For Examiner's Use 1 2 3 4 5 6 a b 7 8 9 10 11 12 n 13 14 15 16 17 18 c d 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 (i) Calculate the value of bc O ad for n = 10. Answer(b)(i) [1] (ii) Without simplifying, write down bc O ad in terms of n for this grid. Answer(b)(ii) [2] (c) This grid has 7 columns. 1 2 3 4 5 6 7 a b 8 9 10 11 12 13 14 n 15 16 17 18 19 20 21 c d 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Show clearly that bc O ad = 28 for n = 17. Answer(c) [1] Question 10 continues on the next page. (d) Write down the value of bc O ad when there are t columns in the grid. For Examiner's Use Answer(d) [1] (e) Find the values of c, d and bc O ad for this shape. 2 3 4 16 c d Answer (e) c = d = bc O ad = [2]

12 marks

Mark scheme: 10 (a) (i) 20 1 (ii) n – 4 oe Accept unsimplified n + 4 oe n + 6 oe 2 B1 for two correct (iii) (n – 4)(n + 4) – (n – 6)(n + 6) M1 ft from their algebraic expressions can be implied by n2 – 4n + 4n – 16 – (n2 – 6n + 6n – 36) or n2 – 16 – (n2 – 36) n2 – 4n + 4n – 16 – (n2 – 6n + 6n Must have a line of algebra – 36) or better 20 E1 With no errors or omission of brackets (b) (i) 24 1 IGCSE – October/November 2012 0580 42 (ii) (n – 5)(n + 5) – (n – 7)(n + 7) 2 M1 for n – 5, n + 5, n – 7, n + 7 seen isw or n2 – 25 – (n2 – 49) isw or n2 – 25 – n2 + 49 isw (c) (11 × 23) – (9 × 25) Allow algebraic solution from 253 – 225 (n – 6)(n + 6) – (n – 8)(n + 8) [= 28] E1 (d) 4t oe 1 Accept unsimplified e.g. n2 – (t – 1)2 – [n2 – (t + 1)2] (e) c = 28 and d = 30 1 52 1

This question in 0580/42 Oct/Nov 2012

10 (a) (3x – 5) cm NOT TO (2x – 3) cm SCALE (15 – 2x) cm (2x + 7) cm (i) Write an expression, in terms of x, for the perimeter of the quadrilateral. Give your answer in its simplest form. Answer(a)(i) … cm [2] (ii) The perimeter of the quadrilateral is 32 cm. Find the length of the longest side of the quadrilateral. Answer(a)(ii) … cm [3] Question 10(b) is printed on the next page. (b) (5a – 2b) m (6b – a) m 14 m (7a – 6b) m NOT TO SCALE a m 13.5 m (3b + a) m The triangle has a perimeter of 32.5 m. The quadrilateral has a perimeter of 39.75 m. Write two equations in terms of a and b and simplify them. Use an algebraic method to fi nd the values of a and b. Show all your working. Answer(b) a = … b = … [6]

11 marks

Mark scheme: 10 (a) (i) 5x + 14 final answer 2 M1 for 5x + k or kx + 14 (ii) 14.2 3 M1 for 5x = 32 – 14 FT their expression in (a)(i) A1FT for x = 3.6 (b) 8a – 3b + 14 = 32.5 or better B1 8a – 3b = 18.5 5a + 4b + 13.5 = 39.75 or better B1 5a + 4b = 26.25 Equates coefficients of either a or b M1 or rearranges one of their equations to make a or b the subject 40a – 15b = 92.5 3b + 185. e.g. a = 40a + 32b = 210 8 or 32a – 12b = 74 15a + 12b = 78.75 Adds or subtracts to eliminate M1 Dep on previous method 47b = 117.5 or correctly substitutes into the second equation 47a = 152.75 5(3b + 185. ) e.g. + 4b = 26.25 8 [a =] 3.25 A1 After M0 scored [b =] 2.5 A1 SC1 for 2 correct values with no working or for two values that satisfy one of their original equations

This question in 0580/43 Oct/Nov 2014

3 The diagram shows a horizontal water trough in the shape of a prism. NOT TO 35 cm SCALE 12 cm 6 cm 120 cm 25 cm The cross section of this prism is a trapezium. The trapezium has parallel sides of lengths 35 cm and 25 cm and a perpendicular height of 12 cm. The length of the prism is 120 cm. (a) Calculate the volume of the trough. Answer(a) … cm3 [3] (b) The trough contains water to a depth of 6 cm. (i) Show that the volume of water is 19 800 cm3. Answer (b)(i) [2] (ii) Calculate the percentage of the trough that contains water. Answer(b)(ii) … % [1] (c) The water is drained from the trough at a rate of 12 litres per hour. Calculate the time it takes to empty the trough. Give your answer in hours and minutes. Answer(c) … h … min [4] (d) The water from the trough just fills a cylinder of radius r cm and height 3r cm. Calculate the value of r. Answer(d) r = … [3] (e) The cylinder has a mass of 1.2 kg. 1 cm3 of water has a mass of 1 g. Calculate the total mass of the cylinder and the water. Give your answer in kilograms. Answer(e) … kg [2] __________________________________________________________________________________________

15 marks

Mark scheme: 3 (a) 43 200 3 M2 for 0.5 × (35 + 25) × 12 × 120 oe or M1 for 0.5 × (35 + 25) × 12 oe (b) (i) 0.5 × (25 + 30) × 6 ×120 [= 19 800] M2 Dep on a valid method for obtaining the width of 30 cm B1 for 0.5 × (25 + 35) oe 19 800 (ii) 45.8 or 45.83… 1FT FT for × 100 their (a) (c) 1 hr 39 min 4 33 B3 for 1.65 [h] or 99 mins or 20 19 800 or M2 for oe 12 × 1000 19 800 19 800 or M1 for or or 12 × 1000 12 1000 If zero scored then SC1 for figs 165 and B1 for converting their time (in hours) into hours and minutes 19 800 (d) 12.8 or 12.80 to 12.81 3 M2 for 3 3 π or M1 for π r 2 3r = 19 800 19 800 (e) 21[.0] 2 M1 for + 2.1 1000

This question in 0580/43 Oct/Nov 2015

1 The Smith family paid $5635 for a holiday in India. The total cost was divided in the ratio travel : accommodation : entertainment = 10 : 17 : 8. (a) Calculate the percentage of the total cost spent on entertainment. … % [2] (b) Show that the amount spent on accommodation was $2737. [2] (c) The $5635 was the total amount Mr Smith received from an investment he made 5 years ago. Compound interest at a rate of 2.42% per year was paid on this investment. Calculate the amount he invested 5 years ago. $ … [3] (d) Mr Smith, his wife and their three children visit a theme park. The tickets cost 2500 Rupees for an adult and 1650 Rupees for a child. Calculate the total cost of the tickets. … Rupees [2] (e) One day the youngest child spent 130 Rupees on sweets. On this day the exchange rate was 1 Rupee = $0.0152 . Calculate the value of the sweets in dollars, correct to the nearest cent. $ … [2]

11 marks

Mark scheme: Question Answer Marks Part Marks 1 (a) 22.9 or 22.85 to 22.86 2 8 M1 for [× 100] oe 10 + 17 + 8 17 5635 (b) 5635 × or better [= 2737] 2 M1 for 10 + 17 + 8 (10 + 17 + 8)  2.42  5 (c) 5000 3 M2 for 5635 = k 1 + oe    100   2.42  or B1 for  1 +   100  (d) 9950 2 M1 for 2 × 2500 or 3 × 1650 (e) 1.98 final answer 2 B1 for 1.976 or 1.98 not final answer or M1 for 130 × 0.0152

This question in 0580/42 Feb/March 2017

1 Adele, Barbara and Collette share $680 in the ratio 9 : 7 : 4. (a) Show that Adele receives $306. [1] (b) Calculate the amount that Barbara and Collette each receives. Barbara $ … Collette $ … [3] (c) Adele changes her $306 into euros (€) when the exchange rate is €1 = $1.125 . Calculate the number of euros she receives. € … [2] (d) Barbara spends a total of $17.56 on 5 kg of apples and 3 kg of bananas. Apples cost $2.69 per kilogram. Calculate the cost per kilogram of bananas. $ … [3] 1 (e) Collette spends half of her share on clothes and of her share on books. 5 Calculate the amount she has left. $ … [3]

12 marks

Mark scheme: Question Answer Marks Partial Marks 1(a) 9 1 × 680 9 + 7 + 4 1(b) 238 136 3 B2 for 238 or 136 7 or M1 for × 680 oe or 9 + 7 + 4 4 × 680 oe seen 9 + 7 + 4 1(c) 272 2 M1 for 306 ÷ 1.125 1(d) 1.37 3 M2 for (17.56 −×5 2.69 ) ÷ 3 or M1 for 17.56 − 5 × 2.69 or B1 for 13.45 [cost of apples] 1(e) 40.8[0] 3 3FT for 0.3 × their 136 from part (b) 1 1 or M2 for their 136( + ) or better 2 5 1 1 or M1 for their 136 × or their 136 × 2 5 3 or B1 for 68 or 27.2 or or 0.3 seen 10

This question in 0580/41 May/June 2018

1 (a) Campsite fees (per day) Tent … $15.00 Caravan … $25.00 The sign shows the fees charged at a campsite. Today there are 54 tents and 18 caravans on the site. Calculate the fees charged today. $ … [2] (b) In September the total income at the campsite was $37 054. This was a decrease of 4.5% on the total income in August. Calculate the total income in August. $ … [2] (c) The visitors to the campsite today are in the ratio men : women = 5 : 4 and women : children = 3 : 7. (i) Calculate the ratio men : women : children in its simplest form. … : … : … [2] (ii) Today there are 224 children at the campsite. Calculate the total number of men and women. … [3] (d) The space allowed for each tent is a rectangle measuring 8 m by 6 m, each correct to the nearest metre. Calculate the upper bound for the area of the space allowed for each tent. … m2 [2] (e) The value of the campsite has increased exponentially by 1.5% every year since it opened 30 years ago. Calculate the value of the campsite now as a percentage of its value 30 years ago. … % [2]

13 marks

Mark scheme: Question Answer Marks Partial Marks 1(a) 1260 2 M1 for 15 × 54 + 25 × 18 1(b) 38 800 2  4.5  M1 for 37054 ÷  1 −  oe  100  1(c)(i) 15 : 12 : 28 2 M1 for correct attempt to find a common multiple for the women oe 1(c)(ii) 216 3 M2 for 224 ÷ their 28 × their (15 + 12) or M1 for 224 ÷ their 28 1(d) 55.25 2 M1 for 8 + 0.5 or 6 + 0.5 seen 1(e) 156 or 156.3… 2 30  1.5  M1 for  1 +   100 

This question in 0580/43 May/June 2020

3 2 6 (a) Work out 2 e o - e o. - 5 - 7 f p [2] - 6 (b) MN = e o. 4 (i) M is the point (2, -5). Find the coordinates of N. ( … , … ) [1] (ii) Find MN . … [2] (c) A Q C NOT TO SCALE a P O B 2c OACB is a trapezium with OB = 2AC. OA = a and OB = 2c . 4 AP : PB = 4 : 1 and AQ = AC . 5 (i) Write each of the following in terms of a and c. Give each answer in its simplest form. (a) AB … [1] (b) CB … [1] (c) OP … [2] (d) QP … [2] (ii) Use your answers to make two statements about the relationship between lines QP and CB. … … [2]

13 marks

Mark scheme: 6(a)  4  2  6  4  k    B1 for   or answer  or    −3   −10  k  −3  6(b)(i) (–4, –1) 1 6(b)(ii) 7.21 or 7.211… 2 M1 for (–6)2 + 42 6(c)(i)(a) 2c – a 1 6(c)(i)(b) c – a 1 6(c)(i)(c) 1 2 4 (a + 8c) final answer M1 for [ AP =]  their(2c – a) 5 5 1 or [ BP = ]  – their (2c – a) 5 or for a correct vector route using the lines on the diagram 6(c)(i)(d) 4 2 4 4 (– a + c) final answer M1 for [QP = ] – c +  their(2c – a) 5 5 5 or for a correct vector route 6(c)(ii) [QP is] parallel [to CB ] 2 Dep both statements consistent with 4 their (c)(i)(b) and their (c)(i)(d) and both vectors QP = CB oe in terms of a and c 5 B1 for each dep on statement consistent with their (c)(i)(b) and their (c)(i)(d) and both vectors in terms of a and c

This question in 0580/42 Oct/Nov 2024

20.24 - 3 30 4 Calculate . 6.5 Give your answer correct to 1 decimal place. … [2]

2 marks

Mark scheme: 4 2.6 cao 2 B1 for 2.64 or 2.635 to 2.636 If 0 scored, SC1 for their more accurate value seen rounded correctly to 1 decimal place

This question in 0580/42 Feb/March 2025