Indices I — Paper 3 · IGCSE Mathematics 0580

8 (a) A pattern of calculations is shown below. Complete the last four rows. 32 – 12 = 4 × 2 42 – 22 = 4 × 3 52 – 32 = 4 × 4 62 – … = … × 102 – … = … × … – … = 4 × 100 … – … = 4 × n [5] (b) These are the first five terms in a sequence. 3 7 11 15 19 (i) Write down the next term in the sequence. Answer(b)(i) … [1] (ii) Write down an expression for the nth term. Answer(b)(ii) … [2] (iii) Work out the 57th term. Answer(b)(iii) … [1] (iv) Is the number 237 a term in the sequence? Give a reason for your answer. Answer(b)(iv) … because … … [2] __________________________________________________________________________________________ Question 9 is printed on the next page.

11 marks

Mark scheme: 8 (a) 42, 4 × 5 1 82, 4 × 9 1 1012, 992 1 (n + 1)2, (n – 1)2 2 SC1 for (n + 1)2 or (n – 1)2 seen or for n + 12 and n – 12 (b) (i) 23 1 (ii) 4n – 1 oe 2 M1 for 4n seen (iii) 227 1FT FT from (b)(ii) if in form jn + k j,k ≠ 0 (iv) No, oe, with valid reason 2 M1FT for (227), (231), 235 or ft from their their (b)(ii) − k (b)(iii) or 59.5 or ft j A1 for correct deduction and mention of 237 between 235 and 239 or 59.5 is not a whole number oe

This question in 0580/32 Feb/March 2015

5 (a) Write one hundred and twenty thousand and twenty in figures. … [1] (b) Find the value of 3481. … [1] (c) (i) Write down the fraction of the rectangle that is shaded. … [1] (ii) Find the percentage of the rectangle that is not shaded. … % [1] (d) Write these numbers in order, starting with the smallest. 5 7 27% 0.268 17 29 … 1 … 1 … 1 … [2] smallest (e) Write 0.3728 correct to 1 decimal place. … [1] (f) Write down the value of 190. … [1] (g) The height, h metres, of a tower is 128 m, correct to the nearest metre. Complete the statement about the value of h. … G h 1 … [2] (h) Find the highest common factor (HCF) of 126 and 180. … [2] (i) Write down an irrational number with a value between 6 and 7. … [1]

13 marks

Mark scheme: 5(a) 120 020 1 5(b) 59 1 5(c)(i) 5 1 8 5(c)(ii) 37.5 1 5(d) 7 5 2 B1 for 3 in the correct order 0.268 27% or M1 for .27 .29… [.268] .24… 29 17 5(e) 0.4 1 5(f) 1 1 5(g) 127.5 128.5 2 B1 for each or SC1 for both correct but reversed 5(h) 18 2 B1 for an answer of 2 or 3 or 6 or 9 or 2 × 3 × 3 or 2 × 32 as final answer or for [126 =] 2 × 3 × 3 × 7 or 2 × 32 × 7 and [180 =] 2 × 2 × 3 × 3 × 5 or 22 × 32 × 5 or for complete correct list of factors for 126 and 180 5(i) Any irrational number between 6 and 7 1

This question in 0580/31 Oct/Nov 2020

9 (a) Simplify. 4x + 3y + 2x - 8y … [2] (b) A pen costs 60 cents and a ruler costs 29 cents. Write down an expression for the total cost, in cents, of x pens and y rulers. … cents [2] (c) Solve. 5 ( 2x + 4) = 85 x = … [3] (d) (i) 2 8 # 2 m = 2 6 Work out the value of m. m = … [1] (ii) 5 n ' 5 4 = 5 6 Work out the value of n. n = … [1] (e) A plant costs p dollars and a bush costs b dollars. Ana buys 2 plants and 4 bushes for $42. Paola buys 7 plants and 9 bushes for $107. Write down a pair of simultaneous equations and solve them to find the value of p and the value of b. You must show all your working. p = … b = … [6] Question 10 is printed on the next page.

15 marks

Mark scheme: 9(a) 6x – 5y final answer 2 B1 for 6x or – 5y in final answer or for 6x – 5y seen then spoilt 9(b) 60x + 29y final answer 2 B1 for 60x or 29y in final answer or for 60x + 29y seen then spoilt or for 60p + 29r or [$] 0.60x + 0.29y 9(c) 6.5 oe 3 M1 for a first correct step e.g. 10x + 20 = 85 or 2x + 4 = 17 M1FT for a second correct step e.g. 10x = 65 or 2x = 13 9(d)(i) –2 cao 1 9(d)(ii) 10 cao 1 9(e) 2p + 4b = 42 and 7p + 9b = 107 B2 B1 for each Correctly equating one set of M1 FT coefficients Correct method to eliminate one M1 FT variable Dependent on the coefficients being the same for one of the variables. Correct consistent use of addition or subtraction using their equations [p =] 5 A1 [b =] 8 A1 If M0 scored, SC1 for 2 values satisfying one of the/their original equations or SC1 if no working, but 2 correct answers

This question in 0580/32 Oct/Nov 2020

7 (a) W = 3a + 5c Find the value of W when a = 6 and c = 2 . W = … [2] (b) Factorise completely. 12b + 8b 2 … [2] (c) Make m the subject of the formula y = 4m - p . m = … [2] (d) Find the value of x when 5 x # 5 3 = 5 12 . x = … [1] (e) Find the value of (i) 30, … [1] (ii) 5 - 2 . … [1] (f) In this part, all measurements are in centimetres. (3x – 1) NOT TO SCALE (2x + 5) The diagram shows a kite with sides ( 2x + 5) and ( 3x - 1) . The perimeter of the kite is 33 cm. Work out the length of a shorter side. … cm [5]

14 marks

Mark scheme: 7(a) 28 2 M1 for 3 × 6 + 5 × 2 or better or B1 for 18 or 10 seen 7(b) 4b(3 + 2b) final answer 2 B1 for final answer 2(6b + 4b2) or 4(3b + 2b2) or 2b(6 + 4b) or b(12 + 8b) or 4b(3 + 2b) seen then spoilt 7(c) y + p y p 2 y p or + oe final answer M1 for 4m = y + p or = m − 4 4 4 4 4 7(d) 9 cao 1 7(e)(i) 1 1 7(e)(ii) 1 1 or 0.04 25 7(f) 6.5 5 M2 for 10x + 8 = 33 or 5x + 4 = 16.5 or M1 for [2 ×] (3x – 1 + 2x + 5) oe M1 for a correct first step in solving their linear equation A1 for [x =] 2.5 If A0 scored, SC1 for 3x – 1 correctly evaluated FT their positive x

This question in 0580/33 Oct/Nov 2020

5 (a) Find. (i) 320.41 … [1] (ii) 6.4 2 + 1. 2 3 … [1] (iii) the reciprocal of 2 … [1] (iv) 90 … [1] 3 (v) of $42 7 $ … [1] (vi) 12% of $62 $ … [1] (b) Insert one pair of brackets in each statement to make it correct. (i) 20 - 5 ' 5 - 3 = 0 [1] (ii) 20 - 5 ' 5 - 3 = 17.5 [1] (c) Write one of the symbols 1 , 2 or = in each statement to make it correct. 7 … 0.07 10 1 … 20% 5 3 … 0.38 8 [2] (d) (i) Write 90 as the product of its prime factors. … [2] (ii) Find the lowest common multiple (LCM) of 35 and 90. … [1] (iii) Find the highest common factor (HCF) of 35 and 90. … [1]

14 marks

Mark scheme: 5(a)(i) 17.9 1 5(a)(ii) 42.688 cao 1 5(a)(iii) 1 2 or 0.5 1 5(a)(iv) 1 1 5(a)(v) 18 1 5(a)(vi) 7.44 1 5(b)(i) (20 – 5) ÷ 5 – 3 = 0 1 5(b)(ii) 20 – 5 ÷ (5 – 3) = 17.5 1 5(c) 7 2 B1 for 2 correct > 0.07 10 1 = 20% 5 3 < 0.38 8 5(d)(i) 2 × 3 × 3 × 5 or 2 × 32 × 5 2 B1 for 2, 3, 3, 5 or M1 for correct factor tree / diagram / table 5(d)(ii) 630 1 5(d)(iii) 5 1

This question in 0580/33 May/June 2021

9 (a) Simplify. 3g + 7g - 4g … [1] (b) Solve. 4x + 5 = 27 x = … [2] (c) 6 p # 6 3 = 6 17 Work out the value of p. p = … [1] (d) Mia buys 4 calculators and 2 pens for $20.60 . Heidi buys 5 calculators and 3 pens for $26.90 . Write down a pair of simultaneous equations and solve them to find the cost of a calculator and the cost of a pen. Calculator $ … Pen $ … [6]

10 marks

Mark scheme: 9(a) 6g 1 9(b) 5.5 2 4 x 5 27 M1 for 4x = 27– 5 or + = 4 4 4 or better 9(c) 14 1 9(d) 4c + 2p = 20.60 B1 5c + 3p = 26.90 B1 correctly equating one set of coefficients M1 correct method to eliminating one variable M1 Dependent on the coefficients being the same for one of the variables Correct consistent use of addition or subtraction using their equations [c =] 4 A1 [p =] 2.30 A1 If M0 scored, SC1 for two values that satisfy one of the original or FT equations SC1 if no working shown, but 2 correct answers given If A0A0 working in cents SC1 for final answers of 400 and 230

This question in 0580/33 Oct/Nov 2021

7 (a) Simplify. 5g - 3h - 7g + 6h … [2] (b) j = 4k + 7m Find the value of j when k =- 5 and m = 6 . j = … [2] (c) Factorise completely. 14x 3 + 49x … [2] (d) Solve. 8 ( 3t - 9) = 108 t = … [3] (e) (i) 9 24 ' 9 w = 9 5 Find the value of w. w = … [1] (ii) 4x 2 = 256 Find the value of x. x = … [1] (f) Ranjit’s age is x years. Suzi’s age is 3 times Ranjit’s age. Juan’s age is 4 years more than Suzi’s age. The total of their ages is 46 years. Use this information to write down an equation and solve it to find the value of x. x = … [4]

15 marks

Mark scheme: 7(a) –2g +3h final answer 2 B1 for –2g or 3h in final answer or –2g+ 3h seen then spoilt 7(b) 22 2 M1 for 4  –5 + 7  6 or B1 for –20 or [+]42 7(c) 7x(2x2 + 7) final answer 2 B1 for 7(2x3 +7x) or x(14x2 + 49) or correct answer seen then spoilt 7(d) 7.5 3 M1 for a first correct step 24t – 72 = 108 or 3t –9 =13.5 M1FT for a second correct step e.g. 24t =180 or 3t =22.5 7(e)(i) 19 1 7(e)(ii) 8 1 7(f) x + 3x + 3x + 4 = 46 4 M2 for a correct equation which would or 7x + 4 = 46 lead to 7x + 4 = 46 leading to x = 6 or B1 for 3x or 3x + 4 seen M1 for 7x = 42 or for rearranging their equation to ax = b B1 for [x =] 6

This question in 0580/31 Oct/Nov 2022

1 (a) List all the factors of 68. … [2] (b) Put one pair of brackets into each calculation to make it correct. (i) 7 + 3 # 5 - 1 = 19 [1] (ii) 12 + 16 ' 2 + 5 = 19 [1] (c) Find (i) the reciprocal of 2, 7 … [1] (ii) the value of 100. … [1] (d) Calculate. (i) 3 2 + 3 4 … [1] (ii) 3 # 12 … [1] (iii) 5 -3 … [1] (e) Write these numbers in order of size, starting with the smallest. 22 10 .3142 .182 r 7 … 1 … 1 … 1 … 1 … [2] smallest (f) By writing each number in the calculation correct to 1 significant figure, work out an estimate for the value of 136 + 47.2 . 62.9 ' 18.1 You must show all your working. … [2] (g) Write .473 # 106 as an ordinary number. … [1] (h) Write down a prime number between 30 and 40. … [1]

15 marks

Mark scheme: Question Answer Marks Partial Marks 1(a) 1, 2, 4, 17, 34, 68 2 B1 for 4 or 5 correct and no extras or 6 correct and one extra 1(b)(i) 7 + 3  (5 – 1) = 19 1 1(b)(ii) (12 + 16) ÷ 2 + 5 = 19 1 1(c)(i) 7 1 1 or 3.5 or 3 2 2 1(c)(ii) 1 1 1(d)(i) 90 1 1(d)(ii) 6 1 1(d)(iii) 1 1 or 0.008 125 1(e) 22 2 B1 for 4 in correct order π, 3.142, , 10 , 1.82 7 or M1 for [ 10 =] 3.16…, [1.82=] 3.24, 22 [π =] 3.141…, [ =] 3.1428 to 3.1429 or 7 3.143 1(f) 100 + 50 M1 60  20 50 cao nfww A1 If 0 scored, SC1 for 3 correct roundings or for all correct but with any trailing zeros 1(g) 4 730 000 1 1(h) 31 or 37 1

This question in 0580/33 Oct/Nov 2022

1 (a) Write the number forty thousand and thirty-three in figures. … [1] (b) Find the value of 3 729 . … [1] 7 (c) Find the reciprocal of . 9 Give your answer as a decimal, correct to 3 decimal places. … [2] (d) Find the value of 6 5 ' 3 4 . … [2] (e) Work out ( - 9 ) # ( - 7 ) ' ( - 3) . … [1] (f) Work out. (i) 11 + 9 # 5 - 4 … [1] (ii) ( 11 + 9) # 5 - 4 … [1] 5(g) - 0 .67 123 49 3 .142 9 From this list, write down an irrational number. … [1] (h) (i) Find the lowest common multiple (LCM) of 24 and 104. … [2] (ii) Find the highest common factor (HCF) of 24 and 104. … [2]

14 marks

Mark scheme: Question Answer Marks Partial Marks 1(a) 40 033 1 1(b) 9 1 1(c) 1.286 cao 2 B1 for 97 or 1.29 or 1.285 or 1.285…… 1(d) 96 2 B1 for 7776 or 81 1(e) −21 1 1(f)(i) 52 1 1(f)(ii) 96 1 1(g) √123 1 1(h)(i) 312 2 B1 for 312k as final answer or M1 for [24 =] 2 × 2 × 2 × 3 or 23 × 3 and [104 =] 2 × 2 × 2 × 13 or 23 × 13 or 2 correct factor trees or tables or a list of multiples of both 24 and 104 with at least 3 of each or 2 × 2 × 2 × 3 × 13 oe 1(h)(ii) 8 2 B1 for 2 or 4 or 2 × 2 × 2 or 23 as final answer, or for a complete list of factors of 24 and 104

This question in 0580/31 May/June 2023

7 (a) P = 3a + 5 Find the value of P when a = 2 . P = … [1] (b) Solve these equations. (i) 7x =-42 x = … [1] (ii) 9 ( 8x - 7) = 72 x = … [3] (c) 5 8 # 5 k = 5 -24 Find the value of k. k = … [1] (d) Solve the simultaneous equations. -6x - y = 13 8x + y =- 51 x = … y = … [2] (e) n is an integer where n 2- 3 and n G 1. Write down all the possible values of n. … [2] (f) A boy walks for 35 minutes at x metres per minute. He then runs for t minutes at 160 metres per minute. Write down an expression, in terms of x and t, for the total distance, in metres, the boy travels. … m [2] (g) NOT TO SCALE x – 2 x + 5 Find an expression for the area of this rectangle. Give your answer in the form x 2 + ax + b . … [3] Question 8 is printed on the next page.

15 marks

Mark scheme: 7(a) 11 1 7(b)(i) −6 1 7(b)(ii) 1.875 oe 3 M1 for a correct first step e.g. 8 x − 7 = 8 or 72 x− 63 = 72 M1FT for a correct second step e.g. 8 x = 15 or 72 x = 135 7(c) −32 1 7(d)  x =  − 19  y = 101 2 B1 for  x =  − 19 B1 for  y = 101 7(e) −2, −1, 0, 1 2 B1 for 3 correct and no extras or 4 correct and one extra 7(f) 35x + 160t final answer 2 B1 for 35x or 160t seen in final answer or 35 x + 160t seen and spoilt 7(g) x 2 + 3 x − 10 final answer 3 B2 for x 2 + 5 x − 2 x − 10 with at least 3 terms correct or B1 for ( x + 5 )( x − 2 ) oe

This question in 0580/32 Feb/March 2024

2 (a) Write the number 845 024 in words. … [1] (b) Write these numbers in order, starting with the smallest. 7 15 39% 0.388 18 40 … 1 … 1 … 1 … [2] smallest (c) Find the value of (i) 45 … [1] (ii) 60 … [1] (iii) 16 # 49 . … [1] (d) Solve. 6x + 5 = 29 x = … [2] (e) By writing each number in the calculation correct to 1 significant figure, find an estimate for the value of 2.7 # 42.4 . 8.6 - 4.3 You must show all your working. … [2] (f) 3 5 # 3 x = 3 20 Find the value of x. x = … [1] (g) Simplify ( x4 ) 3. … [1] (h) A boat to town A leaves a port every 16 minutes. A boat to town B leaves the same port every 34 minutes. Both boats leave the port at 07 30. Work out the next time both boats leave the port together. … [3]

15 marks

Mark scheme: 2(a) Eight hundred and forty five thousand 1 and twenty four 2(b) 15 7 2 0.388 39% 40 18 B1 for 3 in the correct order Or M1 for 0.389 or 0.3888 or 0.3889 and 0.39 and 0.375 2(c)(i) 1024 1 2(c)(ii) 1 1 2(c)(iii) 28 1 2(d) 4 2 5 29 M1 for 6x = 29 – 5 or x + = oe 6 6 2(e) 3  40 M1 9 – 4 24 A1 If 0 scored SC1 for 3 correct from 3, 40, 9 and 4 or for all correct but with trailing zeros 2(f) 15 1 2(g) x12 1 2(h) 12 02 3 B2 for 272 or 4 h 32 mins or M1 for 272k or 2  2  2  2  17 oe or [16 = ] 2  2  2  2 and [34 = ] 2  17 or 2 correct factor trees/tables of both 16 and 34 OR M2 for listing times/multiples of both 16 and 34 to at least 1202 or 272 or M1 for listing at least the next 2 of each or 1 full list

This question in 0580/33 Oct/Nov 2024

17 (a) Solve. x = 18 3 x = … [1] 6 9 11 (b) y = 6 6 Find the value of y. y = … [1] (c) Simplify. a 6 b -2 a 4 b 3 … [2]

4 marks

Mark scheme: 17(a) 54 1 17(b) −2 1 17(c) a 2 2 −5 2 B1 for a 2 b k or a k b − 5 as final answer 5 or a b final answer or for correct answer spoilt b

This question in 0580/32 Feb/March 2025

6 0 1 8 27 39 51 59 81 From the list of numbers, write down (a) a square number … [1] (b) the value of 390 … [1] (c) a prime number. … [1]

3 marks

Mark scheme: 6(a) 1 or 81 accept 0 1 6(b) 1 1 6(c) 59 1

This question in 0580/32 Oct/Nov 2025

18 Calculate. 8 .4 2 + 9 .3 (a) 26.5 … [1] (b) 4 -3 … [1]

2 marks

Mark scheme: 18(a) 1.74 or 1.735 to 1.736 1 18(b) 1 1 0.015625 or 64

This question in 0580/33 Oct/Nov 2025