Surds — Paper 2 · IGCSE Mathematics - International 0607

13 By rationalising the denominator, simplify 12 . 6 - 2 … [3]

3 marks

Mark scheme: 13 6 6 + 12 or 6( 6 + 2) 3 12( 6 + 2) M2 for or better 6 − 4 6 + 2 or M1 for oe seen 6 + 2

This question in 0607/21 May/June 2017

13 p = 5 + 2 3 q = 5 - 2 3 Find p 2 - q 2 , writing your answer in its simplest form. … [3]

3 marks

Mark scheme: 13 40 3 3 M1 for (p + q)(p – q) soi A1 for p + q = 10 or p – q = 4 3 OR B1 for 37 + 20 3 oe B1 for 37 – 20 3 OR B2 for 10 3 +10 3 +10 3 +10 3

This question in 0607/22 May/June 2017

13 (a) Simplify. (4 - 3)(4 + 3 ) … [2] (b) Rationalise the denominator. 5 7 … [1]

3 marks

Mark scheme: 13(a) 13 2 2 2 M1 for 4 − 3 or better ( ) or for three of the terms of 16 + 4 3 − 4 3 − 3 correct 13(b) 5 7 1 7

This question in 0607/23 May/June 2017

12 Rationalise the denominator and simplify. 14 2 3 + 2 … [3]

3 marks

Mark scheme: 12 6 2 – 4 or 2(3 2 – 2) 3 3 - 2 M1 for × final answer 3 - 2 B1 for (3 + 2 )(3 – 2 ) = 7

This question in 0607/21 Oct/Nov 2017

13 (a) Simplify 18 + 72 . … [2] (b) Rationalise the denominator. 1 5 + 2 … [2]

4 marks

Mark scheme: 13(a) 9 2 2 B1 for 3 2 or 6 2 13(b) 5 − 2 2 5 − 2 M1 for × 5 − 2

This question in 0607/22 Oct/Nov 2017

11 Rationalise the denominator and simplify your answer. 32 8 … [2]

2 marks

Mark scheme: 11 8 2 2 8 M1 for × oe 8

This question in 0607/23 Oct/Nov 2017

12 (a) Simplify fully. 700 … [1] (b) Rationalise the denominator. 1 7 - 2 … [2]

3 marks

Mark scheme: 12(a) 10 7 1 12(b) 7 + 2 2 7 + 2 M1 for 47 7 + 2

This question in 0607/21 May/June 2018

12 Simplify. 75 - 12 + 27 … [2]

2 marks

Mark scheme: 12 6 3 2 M1 for 5 3 or 2 3 or 3 3

This question in 0607/22 May/June 2018

11 Simplify. 32 - 72 + 50 … [2]

2 marks

Mark scheme: 11 3 2 2 M1 for 4 2 or 6 2 or 5 2

This question in 0607/21 Oct/Nov 2018

17 Rationalise the denominator, giving your answer in its simplest form. 5 + 3 5 - 3 … [3]

3 marks

Mark scheme: 17 (5 + 3) 2 14 + 5 3 3 (5 + 3) × (5 + 3) or M2 for 22 11 25 + 5 3 − 5 3 − ( 3) 2 5 + 3 5 + 3 as final answer or M1 for × 5 − 3 5 + 3

This question in 0607/21 Oct/Nov 2018

13 Simplify. (5 + 2 3) 2 … [3]

3 marks

Mark scheme: 13 37 + 20 3 3 B2 for 37 + a 3 or b + 20 3 , a, b ≠ 0 or M1 for 52 + 10 3 + 10 3 + 2 3 × 2 3

This question in 0607/22 Oct/Nov 2018

6 (a) Simplify 98 . … [1] (b) Rationalise the denominator. 1 3 - 5 … [2]

3 marks

Mark scheme: 6(a) 7 2 1 6(b) 3 + 5 2 3 + 5 M1 for × 4 3 + 5

This question in 0607/23 Oct/Nov 2018

16 Rationalise the denominator. 1 5 - 1 … [2]

2 marks

Mark scheme: 16 5 + 1 2 5 + 1 M1 for × 4 5 + 1

This question in 0607/22 May/June 2019

13 (a) Simplify 20 + 125 . … [2] (b) Rationalise the denominator and simplify your answer. 18 7 - 1 … [2]

4 marks

Mark scheme: 13(a) 7 5 2 B1 for 2 5 or 5 5 13(b) 3( 7 + 1) or 3 7 + 3 2 7 + 1 M1 for × 7 + 1

This question in 0607/21 Oct/Nov 2019

12 (a) Simplify. 300 - 27 … [2] (b) Rationalise the denominator and simplify your answer. 14 3 - 2 … [3]

5 marks

Mark scheme: 12(a) 7 3 2 M1 for 10 3 or 3 3 12(b) 2(3 + 2) or 6 + 2 2 3 14(3 + 2) M2 for (3 − 2)(3 + 2) (3 + 2) or M1 for × (3 + 2)

This question in 0607/22 Oct/Nov 2019

11 Expand and simplify. (3 2 + 7 ) 2 … [3]

3 marks

Mark scheme: 11 67 + 42 2 3 B2 for a + 42 2 or 67 + b 2 or M1 for 3 2 × 3 2 + 21 2 + 21 2 + 49 or better

This question in 0607/23 Oct/Nov 2019

13 Rationalise the denominator. 9 7 - 2 … [2]

2 marks

Mark scheme: 13 3( 7 + 2) or 3 7 + 6 2 7 + 2 M1 for × 7 + 2

This question in 0607/21 May/June 2020

13 2 3 - 3 2 = p + q 6 Find the value of p and the value of q. p = … q = … [3]

3 marks

Mark scheme: 13 [p =] 30 3 B2 for either [q =] –12 or B1 for 12 − 6 6 − 6 6 + 18 oe

This question in 0607/23 Oct/Nov 2020

13 Rationalise the denominator and simplify. 2 5 + 1 … [3]

3 marks

Mark scheme: 113 5 − 1 5 5 − 1 1 3 2 ( ) 2 5 − 2 − or 5 − 1 or ( ) 2 2 2 2 M2 for or 5 − 1 5 − 1 5 − 1 or M1 for × 5 − 1

This question in 0607/22 Feb/March 2021

8 Expand and simplify ( 2 3 - 5 )( 4 + 3 ) . … [2]

2 marks

Mark scheme: 8 3 3 – 14 or–14 + 3 3 2 B1 for 3 3 + k or k 3 – 14 or 8 3 + 6 – 20 – 5 3

This question in 0607/22 May/June 2021

17 Simplify by rationalising the denominator. 3 2 2 - 1 … [2] Questions 18, 19 and 20 are printed on the next page.

2 marks

Mark scheme: 17 3(2 2 + 1) 6 2 + 3 2 2 2 + 1 or M1 for × 7 7 2 2 + 1

This question in 0607/23 May/June 2021

14 Simplify. 125 + 80 … [2]

2 marks

Mark scheme: 14 9 5 2 B1 for 5 5 or 4 5

This question in 0607/22 Oct/Nov 2021

6 (a) Simplify. 75 - 27 … [2] (b) Rationalise the denominator and simplify your answer. 10 5 - 5 … [3]

5 marks

Mark scheme: 6(a) 2 3 2 M1 for 5 3 or 3 3 6(b) 5 + 5 1 3 10(5 + 5) or (5 + 5) M2 for 2 2 25 − 5 (5 + 5) or M1 for × (5 + 5)

This question in 0607/23 Oct/Nov 2021

13 (a) Simplify fully. 75 - 48 + 12 … [2] (b) Rationalise the denominator, giving your answer in its simplest form. 1 3 + 5 … [2]

4 marks

Mark scheme: 13(a) 3 3 final answer 2 M1 for either 5 3 or 2 3 13(b) 3  5 5  3 2 3  5  or oe final answer M1 for  oe 22 22 3  5 Must be convinced that 3  5 and NOT 3  5

This question in 0607/21 May/June 2022

17 (a) Expand the brackets and simplify. a + b a - b ` `j j … [2] (b) Rationalise the denominator. 1 7 + 6 … [1] (c) Work out the value of 1 1 1 1 1 + + + + . 9 + 8 8 + 7 7 + 6 6 + 5 5 + 4 … [2]

5 marks

Mark scheme: 17(a) a  b 2 M1 for a a  a b  b a  b b oe 17(b) 7  6 1 17(c) 1 2 M1 for at least 3 of 9  8  8  7 …  4 or B1 for 9  4

This question in 0607/22 May/June 2022

14 Rationalise the denominator. 5 3 - 2 … [2]

2 marks

Mark scheme: 14 5 3 + 2 2 3 + 2 ( ) M1 for  3 + 2

This question in 0607/23 Oct/Nov 2022

13 Rationalise the denominator and simplify. 2 3 - 5 … [3]

3 marks

Mark scheme: 13 3  5 3 2(3  5) 6  2 5 cao final answer B2 for or 2 4 4 3  5 or M1 for  3  5

This question in 0607/22 May/June 2023

15 Rationalise the denominator. 5 5 - 1 … [2]

2 marks

Mark scheme: 15 2 5 + 1 5 5 + 1 5 + 5 ( ) M1 for  or 5 + 1 4 4

This question in 0607/21 Oct/Nov 2023

11 Expand and simplify. ( 4 + 2 3)( 5 - 3) … [2]

2 marks

Mark scheme: 11 14 + 6 3 oe 2 B1 for 14 + k 3 or k + 6 3 or M1 for 20 + 10 3 – 4 3 – 2 3 3

This question in 0607/23 Oct/Nov 2023

13 Write in the form a + b 3 where a and b are integers. (a) ( 5 + 2 3) 2 … [2] 5 (b) 2 + 3 … [2] Questions 14 and 15 are printed on the next page.

4 marks

Mark scheme: 13(a) 37 + 20 3 2 M1 for 25 + 5  2 3 + 5  2 3 + 2 3  2 3 , 3 terms correct 13(b) 10 − 5 3 2 5(2 − 3) M1 for (2 + 3)(2 − 3)

This question in 0607/22 Feb/March 2024

11 (a) Simplify. 50 - 8 … [2] (b) By rationalising the denominator, simplify 12 . 7 - 3 … [3]

5 marks

Mark scheme: 11(a) 3 2 cao 2 B1 for 5 2 or 2 2 11(b) 3 3 7  3 or 3 7  3 3 12 7  3     M2 for oe 7 7  7 3  7 3  3 3 7  3 or M1 for  7  3

This question in 0607/21 May/June 2024

11 (a) Simplify. 2 `5 8 - 7 2j … [2] (b) Rationalise the denominator. 21 3 - 2 … [2]

4 marks

Mark scheme: 11(a) 6 2 B1 for 5 16 or 10 2 or 7 4 or better 11(b) 3(3 + 2) or 9 + 3 2 final answer 2 3 + 2 B1 for  3 + 2

This question in 0607/21 Oct/Nov 2024

13 Simplify fully. 20 + 45 + 8 0 … [2]

2 marks

Mark scheme: 13 9 5 final answer 2 B1 for 2 5 or 3 5 or 4 5 seen

This question in 0607/22 Oct/Nov 2024

21 (a) Simplify. 3 12 - 48 + 75 … [3] (b) Rationalise the denominator and simplify. 6 5 - 2 … [3]

6 marks

Mark scheme: 21(a) 7 3 3 B1 for 3  2 3 B1 for 4 3 or 5 3 21(b) 3 2 5 + 2 2 or 2 5 + 2 6 5 + 2 ( ) ( ) M2 for oe 5 − 2 6 5 + 2 or M1 for  5 − 2 5 + 2

This question in 0607/22 Feb/March 2025

22 Simplify. (a) 120 # 27 … [2] 1 (b) 5 - 2 3 … [2]

4 marks

Mark scheme: 22(a) 18 10 2 B1 for 2 10 3 or 2 30 or 3 3 or 6 90 or 9 40 22(b) 5 + 2 3 2 1 (5 + 2 3) M1 for  13 (5 − 2 3) (5 + 2 3)

This question in 0607/21 May/June 2025

11 (a) Simplify. 27 - 2 75 … [2] (b) Rationalise the denominator and simplify. 6 5 - 2 … [3]

5 marks

Mark scheme: 11(a) − 7 3 2 M1 for 3 3 or 2  5 3 311(b) 2 5 + 2 2 or 2 5 + 2 6 5 + 2 ( ) ( ) M2 for 2 2 or better 5 − 2 ( ) ( ) 5 + 2 or M1 for × 5 + 2

This question in 0607/23 May/June 2025

17 Write ( 5 - 2 3 ) 2 in the form a + b 3 . … [2]

2 marks

Mark scheme: 17 2 B1 for 3 terms correct 37 – 20 3 52 – 10 3 – 10 3 + 2 3 ×2 3 oe

This question in 0607/21 Oct/Nov 2025