Powers and roots — Paper 3 · IGCSE Mathematics 0580

1 (a) Find the value of For Examiner's (i) 50, Use Answer(a)(i) [1] (ii) the square root of 64, Answer(a)(ii) [1] (iii) the cube root of 64, Answer(a)(iii) [1] (iv) the integer closest in value to (1.8)3. Answer(a)(iv) [1] (b) Write down (i) a common factor of 15 and 27, which is greater than 1, Answer(b)(i) [1] (ii) a common multiple of 10 and 12. Answer(b)(ii) [1] (c) (i) Two of the factors of 2007 are square numbers. One of these is 1. Find the other square number. Answer(c)(i) [1] (ii) Write down the two factors of 2007 which are prime. Answer(c)(ii) and [2]

9 marks

Mark scheme: 1 (a) (i) 1 B1 (ii) 8 or −8 or ±8 B1 (iii) 4 B1 Not −4 (iv) 6 B1 (b) (i) 3 B1 (ii) Multiple of 60 B1 (c) (i) 9 B1 (ii) 3 and 223 B1,B1 [9] 2 5

This question in 0580/31 May/June 2007

2 5 (a) –4 –16 0.12 7 144 7 2 For 3 Examiner's Use From this list of numbers, write down (i) the smallest number, Answer(a)(i) [1] (ii) a natural number, Answer(a)(ii) [1] (iii) a square number, Answer(a)(iii) [1] (iv) an irrational number. Answer(a)(iv) [1] (b) Write down 40 as a product of prime numbers. (1 is not a prime number.) Answer(b) 40 = [2] (c) Three pairs of prime numbers have a sum of 40. One pair is 3 and 37. Find the other two pairs. Answer(c) and and [2]

8 marks

Mark scheme: 5 (a) (i) –16 B1 cao (ii) 7 or 144 or both B1 (iii) 144 B1 cao (iv) √7 B1 cao (b) 2 x 2 x 2 x 5 B2 B1 for 8x5, 2x20, 4x10, 2x4x5, or list 2, 2, 2, 5 (c) 11, 29 B1 cao 17, 23 B1 cao [8] IGCSE – October/November 2007 0580 and 0581 3

This question in 0580/31 Oct/Nov 2007

1 (a) (i) 1, 2 and 36 are factors of 36. Examiner's Use Write down all the other factors of 36. Answer(a)(i) [2] (ii) 1 and 2 are common factors of 36 and 90. Write down two more common factors of 36 and 90. Answer(a)(ii) [2] (b) Write down all the square numbers between 20 and 50. Answer(b) [3] (c) p and q are prime numbers. p3 × q = 56 Find p and q. Answer(c) p = q = [2]

9 marks

Mark scheme: Qu. Answers Mark Part Marks 1 (a) (i) 3, 4, 6, 9, 12, 18 2 W1 for 4 or 5 correct and no errors or 6 correct and 1 error. (ii) Any two of 3, 6, 9,18 2 W1 for 1 correct and no errors or 2 correct and one extra, incorrect given. (b) 25, 36, 49 3 −1 each error or omission SC2 for all of 52, 62, 72. SC1 for all of 5, 6, 7 (c) p = 2, q = 7 2 W1 for either correct.

This question in 0580/32 May/June 2010

7 (a) Solve the equation 2(x + 4) = 3(x + 2) + 8 . For Examiner's Use Answer(a) x = [3] (b) Make z the subject of za + b = 3 . Answer(b) z = [2] (c) Find x when 2x3 = 54 . Answer(c) x = [2] (d) A rectangular field has a length of x metres. For The width of the field is (2x – 5) metres. Examiner's Use (i) Show that the perimeter of the field is (6x – 10) metres. Answer (d)(i) [2] (ii) The perimeter of the field is 50 metres. Find the length of the field. Answer(d)(ii) length = m [2]

11 marks

Mark scheme: 7 (a) −6 www 3 M2 for 8 = x + 6 + 8 or better or –x + 8 = 6 + 8 or better M1 for 2x + 8 or 3x + 6 or 3x + 14 3 − b 3 b b 3 (b) or − 2 B1 for 3 – b seen or z + = a a a a a 54 (c) 3 2 B1 for or better 2 SC1 for embedded answer ie 2 × 33 = 54 or 2 × 3 × 3 × 3 = 54 (d) (i) x + x + 2x − 5 + 2x − 5 = 6x – 10 2 M1 accept 2x + 2(2x – 5) or 2(x + 2x – 5) E1 dep (ii) 10 2 M1 for 6x – 10 = 50  0 

This question in 0580/31 Oct/Nov 2011

3 (a) Calculate For Examiner's (i) 33, Use Answer(a)(i) [1] 12 2 (ii) , 81 Answer(a)(ii) [1] (iii) the cube root of 4913. Answer(a)(iii) [1] (b) Find (i) all the square numbers between 6 and 40, Answer(b)(i) [2] (ii) four factors of 76, Answer(b)(ii) [2] (iii) a prime factor of 35, Answer(b)(iii) [1] (iv) the lowest common multiple of 6 and 8, Answer(b)(iv) [2] (v) the highest common factor of 56 and 70. Answer(b)(v) [2]

12 marks

Mark scheme: 3 (a) (i) 27 1 (ii) 16 1 (iii) 17 1 (b) (i) 9, 16, 25, 36 2 B1 for 3 correct or either 3 or 4 correct with other values, or all of 32, 42, 52, 62 IGCSE – May/June 2012 0580 32 (ii) 4 from 1, 2, 4, 19, 38, 76 2 B1 if 3 correct none wrong or 4 correct and 1 wrong or 5 correct and 1 wrong or 6 correct and 1 wrong (iii) 5 or 7 1 (iv) 24 2 B1 for any other multiple of 24 (v) 14 2 B1 for answer of 7 or 2 × 7

This question in 0580/32 May/June 2012

2 (a) Find all the factors of 28 . For Examiner's Use Answer(a) [2] (b) Write down a multiple of 8 that is greater than 20 . Answer(b) [1] (c) Work out 183 . Answer(c) [1] (d) p and q are prime numbers. p3 × q2 = 200 Find the values of p and q. Answer(d) p = q = [2] (e) A town has two bus companies. Buses from Western Travel stop at the Town Hall every 8 minutes. Buses from Eastern Travel stop at the Town Hall every 14 minutes. (i) Write down the lowest common multiple of 8 and 14 . Answer(e)(i) [2] (ii) A bus from each company stops at the Town Hall at 08 00. When is the next time that a bus from each company stop together at the Town Hall? Answer(e)(ii) [1] (iii) The cost of an adult ticket on Western Travel is $a and the cost of a child’s ticket is $c. One day 84 adult tickets and 36 child tickets are sold. Write an expression, in terms of a and c, for the total cost of these tickets. Answer(e)(iii) $ [2]

11 marks

Mark scheme: 2 (a) 1, 2, 4, 7, 14, 28 2 1 for four or five correct or 1 × 28 and 2 × 14 and 4 × 7 (b) 24 1 (c) 5832 1 (d) (p =) 2 1 (q =) 5 1 (e) (i) 56 2 M1 for a method to achieve this such as prime factors, 8 = 23 and 14 = 2 × 7 or another multiple of 56, or two trials (ii) 08 56 1ft accept 8 56 (am) (iii) 84a + 36c final answer 2 B1 for either 84a or 36c IGCSE – May/June 2012 0580 33

This question in 0580/33 May/June 2012

2 (a) (i) 1 and 120 are factors of 120. For Examiner′s Use Write down another factor of 120. Answer(a)(i) … [1] (ii) Find the highest common factor of 120 and 900. Answer(a)(ii) … [2] (b) 2 5 15 24 49 60 258 512 From the list, write down (i) a multiple of 30, Answer(b)(i) … [1] (ii) a square number, Answer(b)(ii) … [1] (iii) the cube root of 8. Answer(b)(iii) … [1] (c) Give an example to show that the following statements are not true. (i) An odd number multiplied by an even number gives an odd number. Answer(c)(i) … [1] (ii) The cube of a negative number is positive. Answer(c)(ii) … [1] (d) Use < , > , or = to complete the following statements. Each symbol may be used more than once. 3 (i) 0.5 … [1] 8 (ii) 1.5 … 105% [1] 11 (iii) 0.78 … [1] 14 _____________________________________________________________________________________

11 marks

Mark scheme: 2 (a) (i) 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 1 Award mark for any one from list. 40, 60. (ii) 60 2 B1 for any common factor on answer line, 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 (b) (i) 60 1 (ii) 49 1 (iii) 2 1 (c) (i) Any correct example 1 Calculation and correct answer must be seen IGCSE – October/November 2013 0580 31 (ii) Any correct example 1 Calculation and correct answer must be seen (d) (i) > 1 (ii) > 1 (iii) < 1

This question in 0580/31 Oct/Nov 2013

9 (a) (i) Calculate the cube root of 68 921. Answer(a)(i) … [1] (ii) Write 68 921 in standard form. Answer(a)(ii) … [1] (iii) Write 68 921 correct to 2 significant figures. Answer(a)(iii) … [1] (b) 96 550 kg is reduced to 88 826 kg. Calculate the percentage reduction. Answer(b) … % [3] (c) (i) Work out 5–2. Answer(c)(i) … [1] 2 (ii) Simplify ^ 5 h . Answer(c)(ii) … [1] (iii) Explain why 6 is not a prime number. Answer(c)(iii) … … [1] (iv) Explain the term ‘irrational number’. Answer(c)(iv) … … [1]

10 marks

Mark scheme: 9 (a) (i) 41 1 (ii) 6.8921 × 104 1 (iii) 69 000 1 96550 − 88826 (b) 8% 3 M2 for × 100 oe 96550 88826 or M1 for 7724 seen or 96550 1 (c) (i) or 0.04 1 25 (ii) 5 1 (iii) Has more than 2 factors oe 1 (iv) A decimal that is not truncated and it 1 does not recur (or can’t be written as a fraction) oe

This question in 0580/32 Feb/March 2015

1 (a) Write down (i) two factors of 12, Answer(a)(i) … [1] (ii) the next prime number after 19, Answer(a)(ii) … [1] (iii) the cube root of 64, Answer(a)(iii) … [1] (iv) two million five hundred and seven in figures, Answer(a)(iv) … [1] (v) two multiples of 75, Answer(a)(v) … [1] (vi) the value of π correct to 5 significant figures. Answer(a)(vi) … [1] (b) Write as a percentage. (i) 1.63 Answer(b)(i) … % [1] 3 (ii) 40 Answer(b)(ii) … % [1] (c) (i) Write 63 521.769 correct to 1 decimal place. Answer(c)(i) … [1] (ii) Write 63 521.769 correct to the nearest hundred. Answer(c)(ii) … [1] (d) (i) Change 234 mm into metres. Answer(d)(i) … m [1] (ii) Change 876 m2 into square centimetres. Answer(d)(ii) … cm2 [1] __________________________________________________________________________________________

12 marks

Mark scheme: Question Answer Mark Part marks 1 (a) (i) At least two of 1, 2, 3, 4, 6, 12 1 No incorrect factors (ii) 23 1 (iii) 4 1 (iv) 2 000 507 1 (v) e.g. 75, 150 1 Accept any 75k, k > 0 (vi) 3.1416 1 (b) (i) 163 1 (ii) 7.5 1 (c) (i) 63521.8 1 (ii) 63500 cao 1 (d) (i) [0].234 1 (ii) 8 760 000 1

This question in 0580/31 May/June 2015

1 (a) Write down in figures the number twenty one million. Answer(a) … [1] (b) Write down the four factors of 21. Answer(b) … , … , … , … [2] (c) Write 21% as a fraction. Answer(c) … [1] (d) Put brackets in this calculation to make it correct. 210 + 21 ÷ 2.1 + 21 = 10 [1] (e) Write down the first two prime numbers after 21. Answer(e) … and … [2] (f) Fill in the missing number. 21 210 = 210 ff [1] (g) Calculate 21 2 - 21 . Answer(g) … [1] (h) Work out ( 21 ) 2. Answer(h) … [1] (i) Write down the value of 210. Answer(i) … [1] (j) Write 0.0021 in standard form. Answer(j) … [1] (k) Write down the lowest common multiple (LCM) of 21 and 15. Answer(k) … [2]

14 marks

Mark scheme: Question Answer Mark Part marks 1 (a) 21 000 000 1 (b) 1, 3, 7, 21 2 M1 for 3 correct and one incorrect (or missing) or for 4 correct and one extra 21 (c) 1 100 (d) (210 + 21) ÷ (2.1+ 21) 1 (e) 23 1 If zero scored SC1 for any two other prime 29 1 numbers greater than 21 (f) 2100 1 (g) 436 or 436.4... 1 (h) 21 1 (i) 1 1 (j) 1.2 × 10 − 3 1 (k) 105 2 M1 for [1 ×] 3 × 5 × 7 or 105k or for [1], 3, 7 and [1], 3, 5 seen or for [1], 3, 5, 7 (maybe in a table) or for listing multiples of 15 and 21 to at least 105 with not more than one error

This question in 0580/32 Oct/Nov 2015

4 (a) Solve. (i) 29 – x = 18 Answer(a)(i) x = … [1] (ii) 4(2y + 7) = 164 Answer(a)(ii) y = … [3] (b) Simplify. 6x4 × 8x Answer(b) … [2] (c) Find (i) 81, Answer(c)(i) … [1] (ii) 73, Answer(c)(ii) … [1] (iii) 80. Answer(c)(iii) … [1] (d) (i) Write 6751 correct to the nearest hundred. Answer(d)(i) … [1] (ii) Write 0.25 as a fraction. Answer(d)(ii) … [1] (iii) Write 0.06 as a percentage. Answer(d)(iii) … % [1] (iv) Write 687 000 000 in standard form. Answer(d)(iv) … [1] __________________________________________________________________________________________

13 marks

Mark scheme: 4 (a) (i) 11 1 (ii) 17 3 M1 for 8y + 28 = 164 or 2y + 7 = 41 M1 FT for a correct further step (b) 48x5 2 M1 for 48xk or jx5 (c) (i) 9 1 Accept ± 9 (ii) 343 1 (iii) 1 1 (d) (i) 6800 1 1 (ii) 1 Accept equivalent fraction 4 (iii) 6 1 (iv) 6.87 ×108 1

This question in 0580/33 Oct/Nov 2015

6 (a) For the integers from 40 to 70, write down (i) a multiple of 19, … [1] (ii) a common multiple of 6 and 8, … [1] (iii) the square root of 2500, … [1] (iv) a factor of 106, … [1] (v) an odd number where the tens digit is double the units digit, … [1] (vi) a number that is both a square number and a cube number, … [1] (vii) a number that has exactly 3 factors, … [1] (viii) three prime numbers. … , … , … [2] (b) Write 234 as a product of its prime factors. … [2] (c) Write the answer to 34 × 37 (i) in the form 3x, … [1] (ii) as an integer, … [1] (iii) in standard form. … [1] (d) (i) Write 3−2 as a fraction. … [1] (ii) Find the value of 3x0 when x = 5. … [1]

16 marks

Mark scheme: 6 (a) (i) 57 1 (ii) 48 1 (iii) 50 1 (iv) 53 1 (v) 63 1 (vi) 64 1 (vii) 49 1 (viii) Any three from 2 B1 for 2 correct and at most one error 41 43 47 53 59 61 67 (b) 2 × 32 × 13 or 2 × 3 × 3 × 13 2 B1 for 2, 3 and 13 only identified as factors or for a correct product eg 2 × 9 × 13 , 18 × 13 (c) (i) 311 1 (ii) 177 147 1 (iii) 1.77[147] × 105 1FT follow through their (c)(ii) (d) (i) 1 1 9 (ii) 3 1

This question in 0580/33 May/June 2016

6 (a) Write down a factor of 24 that is a square number. … [2] (b) Write down the cube number between 100 and 200. … [1] (c) Find (i) 12.25, … [1] (ii) 173, … [1] (iii) 4–2. … [1] 1 2 (d) s = 2 at Find the value of s when a = 0.7 and t = 4.2 . s = … [2] (e) Simplify. (i) a0 … [1] (ii) b 3 # b 2 … [1] c 4 (iii) 8 c … [1]

11 marks

Mark scheme: 6 (a) 4 or 1 2 B1 for 2 or 3 or 6 or 8 or 12 or 24 or 22 or 12 (b) 125 1 1 1 (c) (i) 3.5 or 32 (ii) 4913 1 1 1 (iii) 0.0625 or 16 1 2 (d) 6.174 2 M1 for × 0.7 × 4.2 soi by 6.17 2 (e) (i) 1 1 (ii) b5 1 1 1 (iii) c–4 or 4 c

This question in 0580/32 Oct/Nov 2016

1 (a) Write down (i) a square number between 30 and 40, … [1] (ii) the number three million, three hundred and thirty in figures, … [1] (iii) the next cube number after 64, … [1] (iv) the five factors of 16, … , … , … , … , … [2] (v) a common multiple of 6 and 8, … [1] (vi) a prime number between 20 and 30. … [1] (b) Write 567.4892 correct to (i) the nearest ten, … [1] (ii) 2 decimal places. … [1] (c) Complete these calculations. (i) 4 + 6 ' 2 = … [1] (ii) (8 - 20) ' … = 4 [1] (iii) 6432 # … = 64.32 [1]

12 marks

Mark scheme: Question Answer Marks Part marks 1 (a) (i) 36 1 (ii) 3 000 330 cao 1 (iii) 125 1 (iv) 1, 2, 4, 8, 16 2 M1 for 3 or 4 correct factors and no extras or for 5 correct factors and one extra (v) Any multiple of 24 1 (vi) 23 or 29 1 (b) (i) 570 cao 1 (ii) 567.49 cao 1 (c) (i) 7 1 (ii) –3 1 (iii) [0].01 oe 1

This question in 0580/32 Feb/March 2017

6 (a) Find (i) all the factors of 18, … [2] (ii) a multiple of 30, … [1] (iii) 2134.44, … [1] (iv) 2.53, … [1] (v) (0.2) −1. … [1] (b) Write 72 as a product of its prime factors. … [2] (c) Find the lowest common multiple (LCM) of 16 and 30. … [2] (d) Clock A chimes every 6 hours. Clock B chimes every 9 hours. Both clocks chime at 2 am. At what time will the two clocks next chime together? … [3]

13 marks

Mark scheme: 6(a)(i) 1, 2, 3, 6, 9, 18 only 2 B1 for 4 or 5 correct factors and no extras or 6 correct with one extra 6(a)(ii) Any multiple of 30 1 6(a)(iii) 46.2 1 6(a)(iv) 15.625 1 6(a)(v) 5 1 6(b) 23 × 32 2 M1 for a complete factor tree or 2, 2, 2, 3, 3 clearly identified as factors 6(c) 240 2 M1 for [16=] 24 or 2 × 2 × 2 × 2(×1) or [30=] 2 × 3 × 5(×1) or lists of multiples of both at least up to 240, or any product that equals 240 or B1 for 240n 6(d) 20 00 or 8 pm 3 M1 for [LCM of 6 and 9 =] 18(00) or M1 for lists of multiples B1FT for “2 am” + their 18 correctly worked out soi OR B2 for [clock A = 2] 8, 14, 20… and [clock B = 2] 11, 20…. or B1 for [clock A = 2] 8, 14, 20…or [clock B = 2] 11, 20…

This question in 0580/31 Oct/Nov 2017

2 (a) Fill in the missing number in each calculation. (i) 6 + 2 # … = 24 [1] (ii) (10 – … ) ÷ 3 = 2 [1] (b) Find the value of (i) .196, … [1] (ii) 163. … [1] 7. 82 - 4. 15 (c) Work out . 5.25 # 16. 4 Give your answer correct to 2 significant figures. … [2] 1 2 (d) V = a h 3 Calculate V when a = 4.5 and h = 9.6 . V = … [2] (e) Put a ring around the irrational number in the list below. 2 5 4 5 - 36 1 [1] 3 7 5 (f) Written as a product of its prime factors, T = 22 # 3 # 52 . (i) Work out the value of T. T = … [1] (ii) Write 80 as a product of its prime factors. … [2] (iii) Find the highest common factor (HCF) of T and 80. … [2]

14 marks

Mark scheme: 2(a)(i) 9 1 2(a)(ii) 4 1 2(b)(i) 1.4 1 2(b)(ii) 4096 1 2(c) [0].043 cao 2 367 M1 for 0.0426… or 8610 2(d) 64.8 2 1 2 324 M1 for × 4.5 × 9.6 or 3 5 2(e) 5 indicated 1 2(f)(i) 300 1 2(f)(ii) 24 × 5 or 2 × 2 × 2 × 2 × 5 2 M1 for 2, 2, 2, 2, 5 or 24,5 or 1 × 2 × 2 × 2 × 2 × 5 or 1 × 24 × 5 2(f)(iii) 20 2 B1 for 2 or 4 or 5 or 10 as answer or 22 × 5 as answer

This question in 0580/33 Oct/Nov 2017

1 (a) Find the value of (i) the square root of 19 044, … [1] (ii) 27. … [1] (b) n is an integer and 120 < n < 140. Find the value of n when it is (i) a multiple of 45, n = … [1] (ii) a square number, n = … [1] (iii) a factor of 402, n = … [1] (iv) a cube number. n = … [1] 21 - 15 # 3 (c) Work out the value of . 18 ' 6 - 4 … [2] 19.2 # 8.64 (d) Estimate the value of by rounding each number in the calculation to 1 significant 31.6 ' 6.32 figure. Show all your working by filling in the calculation below. … # … = … [2] … ' …

10 marks

Mark scheme: Question Answer Marks Partial Marks 1(a)(i) 138 1 1(a)(ii) 128 1 1(b)(i) 135 1 1(b)(ii) 121 1 1(b)(iii) 134 1 1(b)(iv) 125 1 1(c) 24 2 B1 for numerator of −24 or denominator of −1 or answer of –24 1(d) 20 × 9 M1 M1 for all correct roundings 30 ÷ 6 12 A1 If 0 scored SC1 for 3 correct roundings or 20.[0] and 9.0[0] and 30.[0] and 6.0[0]

This question in 0580/32 May/June 2018

2 (a) Write down all the factors of 18. … [2] (b) Write down a prime number between 40 and 50. … [1] 7. 85 . (c) Calculate 1.09 + 6.21 - 4.37 Give your answer correct to 1 decimal place. … [2] (d) Find the value of (i) .2 89, … [1] (ii) 143, … [1] (iii) 4–2. … [1] (e) (i) 126 = 2 # 32 # k Find the value of k. k = … [1] (ii) Write 90 as the product of its prime factors. … [2] (iii) Find the lowest common multiple (LCM) of 90 and 126. … [2]

13 marks

Mark scheme: 2(a) 1, 2, 3, 6, 9, 18 2 B1 for four or more correct and no extras or six correct and one extra 2(b) 41 or 43 or 47 1 2(c) 5.4 2 B1 for 5.35[6…] or 5.36 2(d)(i) 1.7 1 2(d)(ii) 2744 1 2(d)(iii) 1 1 0.0625 or 16 2(e)(i) 7 1 2(e)(ii) 2 × 32 × 5 or 2 × 3 × 3 × 5 2 M1 for a complete factor tree or 2, 3, 3, 5 clearly identified as factors or B1 for a correct product that equals 90 2(e)(iii) 630 2 B1 for 630k, where k ⩾ 2 or for list of multiples of 90 and 126 to at least 630

This question in 0580/31 Oct/Nov 2018

6 (a) Write the number 602 047 in words. … [1] (b) Find (i) a multiple of 14, … [1] (ii) 562, … [1] (iii) 3 103823 , … [1] (iv) 120. … [1] (c) Find the lowest common multiple (LCM) of 12 and 78. … [2] (d) Find the highest common factor (HCF) of 12 and 78. … [2] (e) Write 432 as a product of its prime factors. … [2]

11 marks

Mark scheme: 6(a) Six hundred (and) two thousand (and) 1 forty seven 6(b)(i) Any multiple of 14 1 6(b)(ii) 3136 1 6(b)(iii) 47 1 6(b)(iv) 1 1 6(c) 156 2 M1 for (12 =) 2, 2, 3 and (78 =) 2, 3, 13 or for 2, 2, 3, 13 or for list of multiples of 12 and 78 to at least 156 If 0 scored SC1 for 156k 6(d) 6 2 B1 for 2 or 3 as final answer 6(e) 24 × 33 or 2 × 2 × 2 × 2 × 3 × 3 × 3 2 M1 for a complete factor tree with at most one error or 2, 2, 2, 2, 3, 3, 3 clearly identified as factors or B1 for a correct product that equals 432

This question in 0580/33 Oct/Nov 2018

2 (a) Work out 48 ' 3 - 5 # 2 . … [1] (b) Insert one pair of brackets to make this statement correct. 3 + 2 # 12 - 4 = 19 [1] (c) Write the following in order, starting with the smallest. 3 11 0.749 76% 4 15 … 1 … 1 … 1 … [2] smallest (d) Find the value of (i) 265.69, … [1] (ii) 83. … [1] (e) Write down the smallest prime number. … [1] (f) Write down all the factors of 18. … [2] (g) Write down a common factor of 16 and 72 that is greater than 2. … [1] 28(h) Write as a fraction in its simplest form. 140 … [1] (i) Jeff and his friends win a prize. 5 Jeff’s share is $160 which is of the prize. 11 Work out the value of the prize. $ … [2]

13 marks

Mark scheme: 2(a) 6 1 2(b) 3 + 2 × (12 – 4) = 19 1 2(c) 15 11 [0].749 34 76[%] 2 B1 for 3 in the correct order or 0.75, (0.749) , 0.76, 0.73… or 75%, 74.9%, (76%), 73….% 2(d)(i) 16.3 1 2(d)(ii) 512 1 2(e) 2 1 2(f) 1 2 3 6 9 18 2 B1 for 4 or 5 correct factors only or 6 correct factors with one extra or 1 × 18, 2 × 9, 3 × 6 2(g) 4 or 8 1 2(h) 1 5 cao 1 2(i) 352 2 M1 for 160 ÷ 5 [ × 11]

This question in 0580/31 May/June 2019

4 (a) Write the number four hundred and eighteen thousand and seventy two in figures. … [1] (b) Write down all the factors of 16. … [2] (c) Write down a prime number between 30 and 40. … [1] (d) Find the value of (i) 729, … [1] (ii) 183, … [1] (iii) 70. … [1] (e) Saskia has $600. 1 1 She spends of the $600 on a coat and gives of the $600 to her son. 5 3 What fraction of the $600 does she have left? Give your answer in its simplest form. … [3] (f) Find the lowest common multiple (LCM) of 15 and 27. … [2] (g) Write 432 as the product of its prime factors. … [2] (h) Ella invests $4000 for 3 years at a rate of 1.2% per year compound interest. Calculate the value of her investment at the end of the 3 years. $ … [3]

17 marks

Mark scheme: 4(a) 418 072 1 4(b) 1 2 4 8 16 2 B1 for 3 or 4 correct and no extra or all correct and one extra 4(c) 31 or 37 1 4(d)(i) 27 1 4(d)(ii) 5832 1 4(d)(iii) 1 1 4(e) 7 3 5 3 8k cao M2 for + or 15 15 15 15k 320 280 7 k or or or , k must be an 600 600 15k integer 1 1 or M1 for + 5 3 or 120 + 200 or 320 or 280 or 600 – 120 − 200 oe 47 If M0 scored, SC1 for answer of 100 467 4667 1000 10000 4(f) 135 2 M1 for listing at least 3 multiples of 15 and 27 or [15=]3 × 5 and [27=]3 × 3 × 3 or 3³ or B1 for 135k as final answer or B1 for 3 × 3 × 3 × 5 or 33 × 5 4(g) 24 × 33 or 2 × 2 × 2 × 2 × 3 × 3 × 3 2 M1 for a complete correct factor tree or 2,2,2,2,3,3,3 clearly identified as factors or B1 for a correct product that equals 432 4(h) 4145.7[3] or 4145.70 or 4150 or 4146 3  1.2  3 M2 for 4000 ×  1 +  oe  100   1.2  2 or M1 for 4000 ×  1 +  oe  100 

This question in 0580/31 Oct/Nov 2019

5 (a) Using the integers from 60 to 75 only, find (i) a multiple of 17, … [1] (ii) the prime numbers. … [2] (b) Find (i) the square root of 4489, … [1] (ii) 43, … [1] (iii) 3 274 625 , … [1] (iv) 2 -3 # 24 2 . … [1] (c) Write down the reciprocal of 7. … [1] (d) Write 3.72194 correct to 3 decimal places. … [1] (e) Find the lowest common multiple (LCM) of 8 and 14. … [2] (f) The average temperature at the North Pole is -23 °C in January and -11 °C in March. (i) Find the difference between these temperatures. … °C [1] (ii) The average temperature in July is 28 °C higher than the average temperature in March. Find the average temperature in July. … °C [1]

13 marks

Mark scheme: 5(a)(i) 68 1 5(a)(ii) 61, 67, 71, 73 2 B1 for 3 correct and none incorrect or 4 correct and one incorrect 5(b)(i) 67 1 5(b)(ii) 64 1 5(b)(iii) 65 1 5(b)(iv) 72 1 5(c) 1 1 7 5(d) 3.722 1 5(e) 56 2 B1 for 56k or lists of multiples of 8 and 14 (at least 3 of each) 5(f)(i) 12 1 5(f)(ii) 17 1

This question in 0580/31 May/June 2020

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

6 (a) Write 60 025 in words. … [1] (b) Write 849.481 correct to 1 decimal place. … [1] (c) Write down (i) all the factors of 21, … [2] (ii) a prime number between 40 and 50. … [1] 2 (d) Write as a decimal. 5 … [1] (e) Find the value of (i) 3 2744 , … [1] (ii) 70. … [1] (f) Gino invests $6000 for 5 years at a rate of 1.2% per year compound interest. Calculate the value of his investment at the end of the 5 years. Give your answer correct to the nearest dollar. $ … [3]

11 marks

Mark scheme: 6(a) Sixty thousand [and] twenty five 1 6(b) 849.5 cao 1 6(c)(i) 1, 3, 7, 21 2 B1 for 3 factors and no extras or 4 correct and 1 extra 6(c)(ii) 41, 43 or 47 1 6(d) 0.4 cao 1 6(e)(i) 14 1 6(e)(ii) 1 1 6(f) 6369 cao 3  1.2  5 M1 for 6000 ×  1 +  or better  100  A1 for 6368.7 … , or 6368 or 6370 If A0 scored, SC1 for correctly rounding their decimal answer

This question in 0580/32 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

1 (a) 14 17 25 27 30 36 48 From the list, write down (i) the square root of 289, … [1] (ii) a factor of 81, … [1] (iii) a common multiple of 3 and 5. … [1] (b) A, B and C are three consecutive whole numbers. • A is a prime number. • B is a cube number. • C is a square number. • A + B + C is less than 40. Find A, B and C. A = … B = … C = … [2] (c) Put one pair of brackets into each of these calculations to make them correct. (i) 4 # 3 + 7 ' 2 = 20 [1] (ii) 51 - 12 ' 3 + 6 = 19 [1] (d) Write down (i) the reciprocal of 8, … [1] (ii) the value of 140. … [1] (e) Calculate. (i) 54 … [1] (ii) 3 6859 … [1] 1 (iii) 16 - 2 … [1]

12 marks

Mark scheme: Question Answer Marks Partial Marks 1(a)(i) 17 1 1(a)(ii) 27 1 1(a)(iii) 30 1 1(b) 7, 8, 9 2 M1 for any 2 conditions in final answer from: A prime or B cube or C square or consecutive A + B + C < 40 1(c)(i) 4 × (3 + 7) ÷ 2 = 20 1 1(c)(ii) (51 – 12) ÷ 3 + 6 = 19 1 1(d)(i) 1 1 or 0.125 8 1(d)(ii) 1 1 1(e)(i) 625 1 1(e)(ii) 19 1 1(e)(iii) 1 1 or 0.25 4

This question in 0580/31 Oct/Nov 2021

7 (a) Write the number six hundred and three thousand eight hundred and twenty-one in figures. … [1] (b) Pens cost 47 cents each. Aroha buys 8 pens. How much change does she receive from $5? $ … [2] (c) Find the value of (i) 81, … [1] (ii) 63, … [1] (iii) 30. … [1] (d) Write 130 as a product of its prime factors. … [2] (e) A tower has two bells, A and B. Bell A rings every 12 minutes. Bell B rings every 14 minutes. Both bells ring at 09 30. Find the next time both bells ring together. … [3]

11 marks

Mark scheme: 7(a) 603 821 1 7(b) 1.24 2 M1 for 5 – (0.47 × 8) oe 7(c)(i) 9 1 7(c)(ii) 216 1 7(c)(iii) 1 1 7(d) 2 × 5 × 13 2 B1 for 2, 5, 13 or M1 for correct factor tree/diagram/list/ table 7(e) 1054 3 B2 for 84 or 1 hr 24 mins or M1 for 84k or 2 × 2 × 3 × 7 or [12 =] 2 × 2 × 3 and [14 =] 2 × 7 or 2 correct factor trees / tables of both 12 and 14 OR M2 for listing times/multiples of both 12 and 14 to at least 1054 or 84 or M1 for listing at least 3 of each or one full list

This question in 0580/33 Oct/Nov 2021

1 (a) Write the number six and a half million in figures. … [1] (b) Write 6538 correct to the nearest ten. … [1] (c) Work out 6 # 5 + 12 ' 3 . … [1] (d) 9 16 18 29 57 64 87 96 From this list of numbers, write down (i) a factor of 48, … [1] (ii) a cube number, … [1] (iii) a prime number. … [1] (e) Find the value of .0001225 . … [1] (f) Find the reciprocal of 8. … [1] (g) Find the value of 80. … [1] (h) (i) Write 180 as a product of its prime factors. … [2] (ii) Find the lowest common multiple (LCM) of 160 and 180. … [2] (i) The mass of an aircraft, m tonnes, is 473 tonnes, correct to the nearest tonne. Complete this statement about the value of m. … G m 1 … [2]

15 marks

Mark scheme: Question Answer Marks Partial Marks 1(a) 6 500 000 1 1(b) 6540 1 1(c) 34 1 1(d)(i) 16 1 1(d)(ii) 64 1 1(d)(iii) 29 1 1(e) 0.035 1 1(f) 1 1 8 or 0.125 1(g) 1 1 1(h)(i) 2  2  3  3  5 2 B1 for 2, 2, 3, 3, 5 or M1 for correct factor tree or table 1(h)(ii) 1440 2 B1 for 1440k as final answer or M1 for [160 =] 2  2  2  2  2  5 and [180 =] 2  2  3  3  5 or a list of multiples of 160 and 180 with at least the first three correct or two correct factor trees or tables or 2, 2, 2, 2, 2, 5, 3, 3 or 2  2  2  2  2  5  3  3 oe 1(i) 472.5 473.5 2 B1 for each If zero scored, SC1 for both correct but reversed

This question in 0580/31 May/June 2022

5 (a) Work out the number of days in seven weeks. … days [1] (b) The summit of Mount Everest is 8848 metres above sea level. Ayding Lake is 154 metres below sea level. Work out the difference in height between these places. … m [1] (c) Find two integers that have a sum of - 12 and a product of 32. … and … [1] 3 (d) Write as 8 (i) a decimal, … [1] (ii) a percentage. … % [1] (e) Write down the reciprocal of 1. 9 … [1] (f) Find the value of (i) 45, … [1] (ii) 3 512 . … [1] (g) (i) Write 587 000 in standard form. … [1] (ii) Calculate 4.9 # 10 - 3 + 8.1 # 10 - 4 . Give your answer in standard form. … [1] (h) The height, h metres, of a fence post is 2.43 m, correct to the nearest centimetre. Complete the statement about the value of h. … G h 1 … [2]

12 marks

Mark scheme: 5(a) 49 1 5(b) 9002 1 5(c) −8 and −4 1 5(d)(i) [0].375 1 5(d)(ii) 37.5 1 5(e) 9 1 5(f)(i) 1024 1 5(f)(ii) 8 1 5(g)(i) 5.87 × 105 cao 1 5(g)(ii) 5.71 × 10−3 cao 1 5(h) 2.425 2.435 2 B1 for each If zero scored, SC1 for 242.5 ⩽ h < 243.5 or for both correct but reversed

This question in 0580/33 May/June 2022

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

9 (a) Write down the reciprocal of 1. 3 … [1] (b) Write down the value of 30. … [1] 3 4 (c) Find a fraction between and . 25 25 … [1] (d) Find the difference in temperature between −5 °C and 9 °C. … °C [1] (e) Write in standard form. (i) 5 600 000 … [1] (ii) 0.000 072 … [1] (f) Calculate ( 5.2 # 10 6 ) # (3 .8 # 10 -2 ) . Give your answer in standard form. … [1]

7 marks

Mark scheme: 9(a) 3 1 9(b) 1 1 9(c) Correct fraction e.g. 507 1 9(d) 14 1 9(e)(i) 5.6 × 106 1 9(e)(ii) 7.2 × 10-5 1 9(f) 1.976 × 105 1

This question in 0580/32 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

1 (a) Write the number three hundred thousand and three in figures. … [1] (b) Write 15 896 correct to (i) the nearest thousand … [1] (ii) the nearest ten. … [1] (c) By writing each number in the calculation correct to 1 significant figure, work out an estimate for the value of 28.9 # 5.49 . 0.472 + 0.97 You must show all your working. … [2] (d) Find the value of (i) 1849 … [1] (ii) 5 0 - 5 -1 … [1] 5 sin 30 - 8 (iii) . 11 … [1] (e) A cyclist travels at a constant speed of 8.5 metres per second. (i) Work out how long the cyclist takes to travel a distance of 5.27 kilometres. Give your answer in minutes and seconds. … min … s [4] (ii) The cyclist increases speed from 8.5 m/s to 10.2 m/s. Work out the percentage increase in speed. … % [2]

14 marks

Mark scheme: Question Answer Marks Partial Marks 1(a) 300 003 1 1(b)(i) 16 000 1 1(b)(ii) 15 900 1 1(c) 30  5 M1 0.5  1 100 A1 If 0 scored, SC1 for three correct from 30, 5, [0].5 and 1 or if all correct but with trailing zeros 1(d)(i) 43 1 1(d)(ii) 0.8 1 1(d)(iii) −0.5 1 1(e)(i) 10 (min) 20 (s) 4 B3 for 620 or 10.3… or 0.172… OR B1 for 5270 or 0.0085 or 510 or 30600 or 0.51 or 30.6 M1 for figs 527÷ figs 85 (imp by figs 62) or figs 527 ÷ figs 510 (imp by figs 103…) or figs 527 ÷ figs 306 (imp by figs172 … ) B1 for their time seen (assume time is in seconds unless units stated) and converted correctly to minutes and seconds (seconds must be correct to 3sf or better) 1(e)(ii) 20 2 10.2 8.5 M1 for [×100] oe 8.5 or (10.2  100 ) [–100 ] oe 8.5 or (10.2 )1 [×100] oe 8.5

This question in 0580/32 May/June 2023

3 (a) The diagram shows a shape on a 1 cm 2 grid. Work out the area of the shape. … cm2 [1] (b) 7 cm NOT TO SCALE 12 cm Work out the perimeter of the rectangle. … cm [1] (c) A square has an area of 841 cm 2. Work out the length of one side of the square. … cm [1] (d) The diagram shows a cuboid made from 1 cm 3 cubes. NOT TO SCALE (i) Work out the volume of the cuboid. … cm3 [1] (ii) Write down the dimensions of a different cuboid that can be made using all of the cubes. … cm by … cm by … cm [1] (e) NOT TO SCALE The diagram shows three small circles and one large circle. The large circle has radius 20 cm. The small circles each have radius 4 cm. Work out the shaded area. Give your answer in terms of r. … cm2 [3] (f) The exterior angle of a 9-sided regular polygon is 40°. (i) Work out the size of the interior angle of this polygon. … [1] (ii) NOT TO SCALE x° x° The diagram shows a regular pentagon inside part of a regular 9-sided polygon. Work out the value of x. x = … [4]

13 marks

Mark scheme: 3(a) 9.5 1 3(b) 38 1 3(c) 29 1 3(d)(i) 48 1 3(d)(ii) Correct dimensions 1 3(e) 352π cao nfww 3 M2 for π20 2 3 π4 2 oe or M1 for π20 2 or π4 2 oe 3(f)(i) 140 1 3(f)(ii) 16 4 B2 for 108 360 or M1 for 180  oe or 5  5  2   180 oe 5 AND 140  108 M1 for or 2 their (f)(i)  their108 2

This question in 0580/33 May/June 2023

3 (a) Write the number fourteen thousand and ninety-seven in figures. … [1] (b) Write down a common multiple of 17 and 5. … [1] (c) Write 0.25 as a percentage. … % [1] (d) Find the value of (i) 75 … [1] (ii) 80. … [1] 5 (e) Ranjit buys some plants and sells of them. 11 He sells 190 plants. Work out how many plants he buys. … [2] (f) Factorise completely. 15x 3 y - 3x … [2] (g) Make n the subject of the formula V = 3n + t . n = … [2] (h) 7 15 ' 7 x = 7 9 Find the value of x. x = … [1]

12 marks

Mark scheme: 3(a) 14 097 1 3(b) Any correct multiple i.e. 85k 1 3(c) 25 1 3(d)(i) 16 807 1 3(d)(ii) 1 1 3(e) 418 2 M1 for 190 ÷ 5 soi by 38 3(f) 3x(5x2y – 1) final answer 2 B1 for 3(5x3y – x) or for x(15x2y – 3) or correct answer spoilt 3(g) V − t 2 V t oe final answer M1 for V – t =3n or = n + 3 3 3 3(h) 6 1

This question in 0580/32 Oct/Nov 2023

1 (a) 6 7 10 12 18 32 49 63 From this list of numbers, write down (i) a factor of 21 … [1] (ii) a square number … [1] (iii) a prime number. … [1] (b) Find the value of (i) the cube root of 1728 … [1] (ii) 25 … [1] (iii) 50 … [1] 2 (iv) 36 1. … [1] (c) Put one pair of brackets into this calculation to make it correct. 3 # 2 - 6 - 2 ' 2 = 4 [1] (d) Find the lowest common multiple (LCM) of 30 and 68. … [2]

10 marks

Mark scheme: Question Answer Marks Partial Marks 1(a)(i) 7 1 1(a)(ii) 49 1 1(a)(iii) 7 1 1(b)(i) 12 1 1(b)(ii) 32 1 1(b)(iii) 1 1 1(b)(iv) 6 1 1(c) 3 × 2 – (6 – 2) ÷ 2 = 4 1 1(d) 1020 2 B1 for 1020k as final answer or M1 for [30 =] 2 × 3 × 5 and [68 =] 2 × 2 × 17 or for [30 =] 2 × 15 and [68 =] 2 × 34 or 2 correct factor trees or correct tables or correct Venn diagram or a list of multiples of both 30 and 68 with at least their first 3 correct or 2 × 2 × 3 × 5 × 17 oe

This question in 0580/32 May/June 2024

2 (a) Simplify. (i) 5a - 6a + 3a … [1] (ii) 6x 2 - 6x - 4x 2 - x … [2] (b) Find the value of c 2 + d 2 when c = 7 and d =- 5 . … [2] (c) The time, T minutes, to cook a chicken with a mass of m kg is T = 35m + 20 . (i) Make m the subject of the formula. m = … [2] (ii) Find the mass of a chicken that takes 83 minutes to cook. … kg [2] (d) Solve these simultaneous equations. You must show all your working. 5x - 6y = 24 15x + 8y = 33 x = … y = … [3]

12 marks

Mark scheme: 2(a)(i) 2a final answer 1 2(a)(ii) 2x2 – 7x final answer 2 B1 for 2x2 or −7x in final answer or for 2x2 – 7x seen then spoilt. 2(b) 74 2 B1 for 49 or 25 2(c)(i) T  20 2 T 20 [m =] final answer M1 for T – 20 = 35m or  m  35 35 35 2(c)(ii) 1.8 2 FT their (c)(i) for 2 marks or 1 mark M1 for (83 – 20) ÷ 35 2(d) Correctly eliminates one variable M1 Making the coefficients the same for one of the variables and correct consistent use of addition or subtraction using their equations alternative substitution method. M1 for correct rearrangement of one equation to make either x or y the subject and correct substitution of their rearrangement into 2nd equation. [x =] 3 A1 If A0 scored SC1 for 2 values satisfying one of the original equations. [y =] −1.5 A1

This question in 0580/32 May/June 2024

4 (a) Write down the value of the 8 in the number 39 829. … [1] (b) Write down all the factors of 18. … [2] (c) Show that 57 is not a prime number. [1] (d) x = 64 Find the value of x. x = … [1] (e) Find the first multiple of 40 that is greater than 620. … [1] (f) Find the reciprocal of 2. 3 … [1] 1(g) Find a fraction between and 1. 5 4 … [1] (h) Write down an irrational number with a value between 9 and 10. … [1] (i) Find the highest common factor (HCF) of 72 and 180. … [2]

11 marks

Mark scheme: 4(a) 800 1 4(b) 1 2 3 6 9 18 2 B1 for 4 or 5 correct no extras or 6 correct and 1 extra or for 1×18 and 2×9 and 3×6 4(c) [57 =] 3× 19 oe 1 4(d) 4096 1 4(e) 640 1 4(f) 1 12 or 1.5 1 4(g) Any correct fraction 1 4(h) Any correct irrational number 1 4(i) 36 2 B1 for any of 2, 3, 4, 6, 9, 12, 18 as answer or M1 for 2×2×3×3 oe as answer or [72 = ] 2×2×2 ×3×3 and [180 = ] 2×2 ×5× 3×3 or a complete list of factors of 72 and 180 or 2 correct factor trees or tables or Venn diagram

This question in 0580/33 May/June 2024

2 (a) 3.142 87 14 41 56 117 121 From this list, write down (i) an odd number … [1] (ii) a factor of 28 … [1] (iii) a square number … [1] (iv) a prime number … [1] (v) a multiple of 9. … [1] (b) Write down the value of 50. … [1] (c) Write 60 as a product of its prime factors. … [2] (d) Calculate the value of 3 157 464 . … [1] (e) By rounding each number in the calculation correct to 1 significant figure, find an estimate for the value of 67.8 # 2.38 . 4.803 + 29.87 You must show all your working. … [2] (f) (i) .978 # 108 2 .04 # 10 9 Which of these two numbers is larger? Give a reason for your answer. … is larger because … [1] (ii) Calculate 1.732 # 10 3 ' 5 .73 # 10 -1 . Give your answer in standard form. … [2] (g) Two cars go round a track. One car completes each lap of the track in 96 seconds. The other car completes each lap in 120 seconds. Both cars start a lap together at 08 37. Find the next time when both cars start a lap together. … [3]

17 marks

Mark scheme: 2(a)(i) 41 or 117 or 121 1 2(a)(ii) 14 1 2(a)(iii) 121 1 2(a)(iv) 41 1 2(a)(v) 117 1 2(b) 1 1 2(c) 2 × 2 × 3 × 5 or 2 2 × 3 × 5 2 B1 for 2, 2, 3, 5 OR M1 for correct factor tree/table/list 2(d) 54 1 2(e) 70  2 M1 5 + 30 4 A1 If 0 scored SC1 for three correct from 70, 2, 5 and 30 or all correct but with trailing zeros 2(f)(i) 2.04 × 109 and it has the larger power of 1 10 oe 2(f)(ii) 3.02[2...] × 103 or 3.023 × 103 2 B1 for 3020 or 3023 or 3022[. ...] or or correct value but not in correct form 3.02[2...] × 101 or 3.023 × 101 or 30.2 or 30.23 or 30.2[2 ...] or correct value but not in correct form or for their value seen and correctly converted to standard form to at least 3 sf 2(g) 08 45 3 B2 for 480 or 8 mins or M1 for 480k or 2×2×2×2×2×3×5 or [96=] 25 × 3 and [120=] 23 × 3 ×5 or two correct factor trees/tables of both 96 and 120 OR M2 for listing the times/multiples of both 96 and 120 up to 480 or 08 45 or M1 for listing at least the next 2 of each or one full list

This question in 0580/32 Oct/Nov 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

19 (a) Write 2025 as the product of its prime factors. … [2] (b) Write 2025 as a product of two square numbers that are both greater than one. … [1]

3 marks

Mark scheme: 19(a) 4 2 2 B1 for 3,3,3,3,5,5 3  5 or M1 for correct factor tree/diagram/list/table 19(b) 25×81 or 225×9 1

This question in 0580/32 Feb/March 2025

3 Find the value of .196 . … [1]

1 marks

Mark scheme: 3 1.4 1

This question in 0580/31 May/June 2025

5 Find the value of (a) (i) .196 … [1] (ii) 173 … [1] (iii) -0.8 # - 1 .2 . … [1] (b) Write these numbers in order, starting with the smallest. 11 27 58% 0.574 19 47 … 1 … 1 … 1 … [2] smallest

5 marks

Mark scheme: 5(a)(i) 1.4 1 5(a)(ii) 4913 1 5(a)(iii) 0.96 1 5(b) 27 11 2 B1 for 3 in correct order 0.574 58[%] or M1 for all decimal or % equivalents to 47 19 enable comparison. 0.578… or 0.579 , 0.58 , 0.5744… or 0.5745 , [0.574]

This question in 0580/31 Oct/Nov 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

7 Calculate 361 + 1. … [1]

1 marks

Mark scheme: 7 20 1

This question in 0580/32 Oct/Nov 2025

13 Find the value of 85. Give your answer correct to 3 significant figures. … [2]

2 marks

Mark scheme: 13 32 800 cao 2 B1 for 32 768 or their answer of more than 3 figures correctly rounded to 3 sf

This question in 0580/32 Oct/Nov 2025

6 (a) Write down the reciprocal of 8. … [1] (b) Work out 193. … [1]

2 marks

Mark scheme: 6(a) 1 1 or 0.125 8 6(b) 6859 1

This question in 0580/33 Oct/Nov 2025