Algebraic fractions — Paper 2 · IGCSE Mathematics 0580

12 Write as a single fraction, in its simplest form. 3 2 − x + 2 x − 1 Answer [3]

3 marks

Mark scheme: x 7 12 3 M1 3(x – 1) – 2(x + 2) seen ( x − 1)( x + 2) B1 denominator correct seen A1 all correct IGCSE – October/November 2010 0580 22 2

This question in 0580/22 Oct/Nov 2010

2 3 For 16 Write + as a single fraction. Examiner's x − 2 x + 2 Use Give your answer in its simplest form. Answer [3]

3 marks

Mark scheme: 16 5 x − 2 3 M1 2(x + 2) + 3(x – 2) seen B1 (x – 2)(x + 2) common denom. seen ( x − 2)( x + 2)

This question in 0580/23 May/June 2011

10 Write as a single fraction in its simplest form. 3 1 O x + 10 x + 4 Answer [3]

3 marks

Mark scheme: 10 2 x + 2 3 B1 common denominator (x + 10)(x + 4) oe seen oe ( x + 10 )( x + 4 ) B1 3(x + 4) – (x + 10) seen oe 3 3

This question in 0580/22 Oct/Nov 2011

18 Write as a single fraction, in its simplest form. For Examiner's Use 1 − x 2 + x − x 1 − 2 x Answer [4]

4 marks

Mark scheme: 18 1 − 5 x + x 2 1 − 5 x + x 2 4 M1 for (1 – x)(1 – 2x) – x(2 + x) seen or 2 B1 for 1 – x – 2x + 2x2 or 1 – 3x + 2x2 seen x (1 − 2 x ) x − 2 x B1 for x(1 – 2x) oe as a common denominator 50 2

This question in 0580/23 Oct/Nov 2011

20 Simplify fully. For x 2 − x − 20 Examiner's Use x 3 − 10 x 2 + 25 x Answer [5] Question 21 is printed on the next page.

5 marks

Mark scheme: 20 5 B2 (x – 5)(x + 4) seen x + 4 oe cao or SC1 (x + a)(x + b) where ab = –20 or a + b = –1 x ( x − 5) B2 x(x – 5)(x – 5) or B1 one of x(x2 –10x + 25), (x – 5)( x – 5), (x – 5)(x2 – 5x) seen 1

This question in 0580/23 May/June 2012

18 (a) Factorise x2 + x – 30. Answer(a) … [2] (x - 5)(x + 4) . (b) Simplify x2 + x - 30 Answer(b) … [1] _____________________________________________________________________________________

3 marks

Mark scheme: 18 (a) (x + 6)(x–5) 2 SC1 for (x + a)(x + b) where ab = –30 or a + b (b) x + 4 1 final answer x + 6 6 k

This question in 0580/21 May/June 2013

13 Write as a single fraction in its simplest form. For Examiner′s Use x + 3 x - 1 - - x 3 x + 1 Answer … [4] _____________________________________________________________________________________

4 marks

Mark scheme: 8 x 13 4 B1 for common denominator (x – 3)(x + 1) ( x − 3)( x + )1 seen B1 for (x + 3)(x + 1) – (x – 1)(x – 3) soi B1 for x2 + 3x + x + 3 or x2 – 3x – x + 3 soi

This question in 0580/23 May/June 2013

8 Write as a single fraction in its simplest form. 2 2 – x x + 1 Answer … [3] __________________________________________________________________________________________

3 marks

Mark scheme: 8 2 3 B1 for common denominator x (x +1) seen M1 for 2 (x + 1) – 2x oe or better x ( x + )1

This question in 0580/21 May/June 2014

21 (a) Write as a single fraction in its simplest form. 3 1 – 2x - 1 x + 2 Answer(a) … [3] (b) Simplify. 4x2 - 16 x 2x2 + 6 x - 56 Answer(b) … [4]

7 marks

Mark scheme: x + 7 21 (a) 3 B1 for (3 x + )2 − (12 x − )1 seen or better ( 2 x − 1)( x + 2) Final answer B1 for denominator (2x – 1)(x + 2) oe seen x + 5 SC2 for final answer ( 2 x − 1)( x + 2) 2 x (b) 4 M1 for 4x(x – 4) or partial x + 7 factorisation of numerator Final answer and M2 for [2](x + 7)(x – 4) oe or M1 for [2](x2 + 3x – 28) or [2](x + a)(x + b) where ab = –28 or a + b = 3 4 x SC3 for answer oe 2 x + 14

This question in 0580/22 Oct/Nov 2014

13 Write as a single fraction, in its simplest form. 3 2x + + 3 + 2x 2x 3 Answer … [4] __________________________________________________________________________________________

4 marks

Mark scheme: 16 x + 18 x + 9 13 final answer 4 M2 for 9 [+] 4x2 [+] 18x [+] 12x2 or better 6 x or M1 for 2 of these and M1FT for adding their four ‘numerators’ together correctly and B1 for denominator 6x to a maximum of 3 marks 1 1

This question in 0580/23 Oct/Nov 2014

15 Write as a single fraction in its simplest form. 3 4 – x + 2 2x - 5 Answer … [3] __________________________________________________________________________________________

3 marks

Mark scheme: 15 3 B1 for a common denominator of 2 x − 23 final answer (x + 2)(2x – 5) ( x + 2 )( 2 x − 5) B1 for 3(2x – 5) – 4(x + 2) or better 2 x − 7 or SC2 for final answer ( x + 2 )( 2 x − 5 ) or SC1 for numerator of 2x – 7 in final answer

This question in 0580/22 May/June 2015

11 Write the following as single fractions. x (a) x + 2 Answer(a) … [1] 2 (b) x + x Answer(b) … [1] __________________________________________________________________________________________

2 marks

Mark scheme: 3x 11 (a) oe final answer 1 2 x 2 + 2 (b) 1 oe final answer x

This question in 0580/23 Oct/Nov 2015

22 Write as a single fraction in its simplest form. x 2 - 3x (a) 2 x - 9 … [3] 3 2 (b) + x - 4 2x + 5 … [3]

6 marks

Mark scheme: 22(a) x 3 B1 for x(x – 3) final answer B1 for (x – 3)(x + 3) x + 3 22(b) 8 x + 7 3 B1 for common denominator of (x − 4)(2x + 5) final answer oe ( x − 4)(2 x + 5) M1 for 3(2x + 5) + 2(x − 4) oe with an attempt to expand the brackets

This question in 0580/21 May/June 2017

20 Write as a single fraction in its simplest form. 2x - 1 2 - 3 x + 1 … [3]

3 marks

Mark scheme: 20 2 x 2 + x − 7 2 x 2 + x − 7 3 M1 for (2x – 1)(x + 1) – 2 × 3 oe with an or attempt to expand the brackets 3 ( x +1) 3 x + 3 final answer B1 for 3(x + 1) or 3x + 3 for denominator

This question in 0580/22 May/June 2017

19 Write as a single fraction in its simplest form. 5 3 1 + + x - 3 x + 7 2 … [4]

4 marks

Mark scheme: 19 x 2 + 20 x + 31 4 B1 for a common denominator of final answer [2](x ‒ 3)(x + 7) seen isw 2 ( x − 3 )( x + 7 ) M1 for 2×5×(x + 7) + 2×3×(x ‒ 3) + (x ‒ 3)(x + 7) oe and must have attempted to expand all the brackets in the numerator M1 for 10x + 70 + 6x ‒ 18 or x2 − 3x + 7x ‒ 21 or [2](5x + 35 + 3x −9) or better

This question in 0580/21 Oct/Nov 2017

15 Write as a single fraction in its simplest form. x + 1 y - 1 - x y … [3]

3 marks

Mark scheme: 15 y + x 3 B1 for y ( x + 1) − x ( y − 1) final answer xy B1 for common denominator xy y − x or SC2 for final answer xy

This question in 0580/22 Oct/Nov 2017

21 Write as a single fraction in its simplest form. 1 1 - y - 1 y … [3]

3 marks

Mark scheme: 21 1 1 3 B1 for common denominator of y ( y − 1) or y2 – y or final y ( y − 1) y 2 − y B1 for y − ( y − 1) or y – y + 1 answer

This question in 0580/23 May/June 2018

15 Write as a single fraction in its simplest form. x - 5 6 + 3 x + 2 … [3]

3 marks

Mark scheme: 15 x 2 − 3 x + 8 x 2 − 3 x + 8 3 B1 for common denominator 3(x + 2) or x − 5 )( x + 2 ) + 3 × 6 3 ( x + 2 ) 3 x + 6 M1 for ( final answer

This question in 0580/21 Oct/Nov 2018

18 Write as a single fraction in its simplest form. 1 1 - x x + 1 … [3]

3 marks

Mark scheme: 18 1 3 B1 for common denominator x ( x + 1) oe oe final answer nfww x ( x + 1) M1 for x + 1 – x

This question in 0580/23 Oct/Nov 2018

20 Write as a single fraction in its simplest form. 1 2 - x + 2 3x - 1 … [3]

3 marks

Mark scheme: 20 x − 5 3 B1 for common denominator isw expansion final answer ( x + 2)(3 x − 1) M1 for 3 x −−1 2( x + 2) or better

This question in 0580/21 May/June 2019

17 Write as a single fraction in its simplest form. 2 x x + 3 + x + 3 x - 5 … [3]

3 marks

Mark scheme: 17 3 x 2 − 4 x + 9 3 B1 for common denominator final answer ( x + 3 )( x − 5 ) oe isw ( x + 3 )( x − 5 ) M1 for 2 x ( x − 5 ) + ( x + 3 )( x + 3 ) or better

This question in 0580/22 May/June 2019

x 3 + 5x 217 Simplify 2 , giving your answer as a single fraction. x - 25 … [3]

3 marks

Mark scheme: 17 2 3 B1 for x2(x + 5) x final answer B1 for (x – 5)(x + 5) x − 5

This question in 0580/23 May/June 2019

x 2x + 418 Write - as a single fraction, in its simplest form. 2 x + 1 … [3]

3 marks

Mark scheme: 18 x 2 − 3 x − 8 x 2 − 3 x − 8 3 B1 for common denominator 2( x + )1 or or final answer 2( x + )1 2 x + 2 2x + 2 M1 for x ( x + )1 − 2 ( 2 x + 4 ) or better

This question in 0580/21 Oct/Nov 2019

22 Simplify. x 2 - 5x 2x 2 - 50 … [4] Question 23 is printed on the next page.

4 marks

Mark scheme: 22 x x 4 B1 for x(x – 5) or final answer B2 for 2(x – 5)(x + 5) or (x – 5)(2x + 10) 2 ( x + 5 ) 2 x + 10 or (2x – 10)(x + 5) or B1 for 2(x2 – 25) or (x – 5)(x + 5)

This question in 0580/21 Oct/Nov 2020

23 Write as a single fraction in its simplest form. 2x - 1 2 - x + 1 … [3]

3 marks

Mark scheme: 23 3 3 B1 for 2( x + 1) − (2 x − 1) oe final answer x + 1 B1 for common denominator x + 1

This question in 0580/22 Oct/Nov 2020

22 Simplify. 2x 2 - 5x - 12 3x 2 - 12x … [4]

4 marks

Mark scheme: 22 2 x + 3 4 B2 for ( x − 4 )( 2 x + 3 ) final answer 3 x or B1 for (x + a) (2x + b) where ab = –12 or 2a + b = – 5 or x ( 2 x + 3 ) − 4 ( 2 x + 3 ) or 2 x ( x − 4 ) + 3 ( x − 4 ) B1 for 3 x ( x − 4 )

This question in 0580/22 May/June 2021

19 Write as a single fraction in its simplest form. 2 x + 2 - x + 3 7 … [3]

3 marks

Mark scheme: 19 8 − 5 x − x 2 8 − 5 x − x 2 3 B1 for 7 × 2 – (x + 2)(x + 3) or better seen or 7( x + 3) 7 x + 21 B1 for common denominator 7(x + 3) oe isw final answer

This question in 0580/21 Oct/Nov 2021

23 Simplify. 3 xy + 36 y - 5 x - 60 2 x 2 - 288 … [4]

4 marks

Mark scheme: 23 3 y – 5 3 y – 5 4 3 y – 5 or final answer SC3 for answer 2 ( x –1 2 ) 2 x – 24 x –12 or B3 for (3y – 5)(x + 12) and 2(x – 12)(x + 12) or (2x – 24)(x + 12) or B2 for (3y – 5)(x + 12) or 2(x – 12)(x + 12) or (2x – 24)(x + 12) or (2x + 24)(x – 12) or B1 for 3y(x + 12) – 5(x + 12) or x(3y – 5) + 12(3y – 5) or 2(x2 – 144) or (x – 12)(x + 12)

This question in 0580/22 Oct/Nov 2021

22 Simplify. 5 x - x 2 25 - x 2 … [3]

3 marks

Mark scheme: 22 x 3 B1 for x(5 – x) final answer nfww B1 for (5 – x)(5 + x) or 5  x

This question in 0580/23 May/June 2022

22 (a) Expand and simplify. ( 2x - 1)( x + 4)( x - 3) … [3] (b) Write as a single fraction in its simplest form. 4 2 x 2 + 14 x ' 2x - 3 2x 2 + 11x - 21 … [4]

7 marks

Mark scheme: 22(a) 2x3 + x2 – 25x + 12 final answer 3 B2 for correct unsimplified expanded expression or for simplified four-term expression of correct form with 3 terms correct or B1 for correct expansion of 2 brackets with at least 3 terms out of 4 correct 22(b) 2 4  4 2 x 2 + 11x − 21 final answer M1 for  oe soi   2 x  2 x − 3  2 x + 14 x B1 for (x + 7)(2x – 3) oe factorised B1 for 2x(x + 7) oe factorised

This question in 0580/23 Oct/Nov 2022

23 Simplify. 5x 2 - 19x + 12 x 2 - 9 … [4]

4 marks

Mark scheme: 23 5 x − 4 4 B2 for ( 5 x − 4 )( x − 3 ) final answer x + 3 or B1 for ( 5x + a )( x + b ) with ab = 12 or a + 5b = –19 or for 5 x ( x − 3 ) − 4 ( x − 3 ) or x ( 5 x − 4 ) − 3 ( 5 x − 4 ) B1 for ( x + 3 )( x − 3 )

This question in 0580/22 Feb/March 2023

22 Write as a single fraction in its simplest form. 5 4 + 3x + 2 2x - 1 … [3]

3 marks

Mark scheme: 22 22 x  3 3 B1 for a common denominator (3x + 2)(2x – 1) oe final answer  3 x  2  2 x  1 isw B1 for 5(2x – 1) + 4(3x + 2) oe isw

This question in 0580/22 May/June 2023

21 Simplify. 2 x 2 + 5 x - 12 4x 2 - 9 … [4]

4 marks

Mark scheme: 21 x  4 4 B1 for  2 x  3  2 x  3  final answer 2 x  3 B2 for 2 x  3  x  4  or B1 for (2x + a)(x + b) where ab = −12 or a + 2b = 5 or x  2 x  3   4  2 x  3  or 2 x  x  4   3  x  4 

This question in 0580/23 May/June 2023

20 Write as a single fraction in its simplest form. 10x 2 - 60x (a) 2 x - x - 30 … [3] 7 5 (b) + x + 3 8x - 1 … [3]

6 marks

Mark scheme: 20(a) 10 x 3 B1 for 10x(x – 6) final answer B1 for (x – 6)(x + 5) x + 5 20(b) 61x + 8 3 B1 for common denominator of (x + 3)(8x – 1) final answer ( x + 3 )( 8 x − 1) isw B1 for 7(8x – 1) + 5(x + 3) or better isw

This question in 0580/21 Oct/Nov 2023

17 (a) Simplify. 2 64y 27 3 ` j … [2] (b) Simplify. x - 5 x 2 - 25 … [2]

4 marks

Mark scheme: 17(a) 16y18 final answer 2 B1 for 16yk or ky18 as final answer or correct answer spoiled 17(b) 1 2 B1 for (x + 5)(x – 5) final answer x + 5

This question in 0580/23 Oct/Nov 2023

23 Simplify. 2 3 - y + 1 y Give your answer as a single fraction in its simplest form. … [3]

3 marks

Mark scheme: 23 y 3 y 3 y  3 3 B1 for 2 y  3  y  1 oe or or  y  y  1 y 2  y y  y  1 B1 for common denominator y( y + 1) y  3 or  final answer 2 2 or y  y isw y  y

This question in 0580/23 May/June 2024

25 Simplify. pt - p - t + 1 1 - t 2 … [4]

4 marks

Mark scheme: 25 1  p 4 oe final answer 1  t B2 for  p  1 t  1 oe or B1 for p  t  1 t  1 or t  p  1 p  1 B1 for 1  t 1  t  oe

This question in 0580/23 May/June 2024

14 f ( x) = 3x - 4 g ( x) = 4x + 1 (a) Find f ( - 2 ) . … [1] (b) Find f -1 ( x) . f -1 ( x) = … [2] (c) fg ( x) = ax + b Find the value of a, and the value of b. a = … b = … [2] (d) Simplify. 2 5 - f ( x) g ( x) Give your answer as a single fraction in terms of x. … [3]

8 marks

Mark scheme: 14(a) –10 1 14(b) x + 4 2 M1 for correct first step oe y + 4 = 3x or x = 3y – 4 3 y 4 or = x − 3 3 14(c) a = 12, b = –1 2 B1 for either a or b correct or M1 for 3(4x + 1) – 4 14(d) 22 − 7 x 22 − 7 x 3 B1 for 2(4x + 1) – 5(3x – 4) oe or better isw or ( 3 x − 4 )( 4 x + 1) 12 x 2 − 13 x − 4 B1 for common denominator (3x – 4)(4x + 1) final answer oe isw

This question in 0580/21 May/June 2025

19 Write as a single fraction in its simplest form. 5a 3 b (a) # 6 a … [2] p 3t (b) + 2 4 … [2] 2 3 (c) - x - 2 x + 1 … [3]

7 marks

Mark scheme: 19(a) 5b 2 15 ab cao final answer B1 for or better seen 2 6 a 19(b) 2 p + 3t 2 M1 for adding two correct fractions with a cao final answer 4 p 6t 4 common denominator e.g. + 8 8 19(c) 8 − x 8 − x 3 B1 for 2(x + 1) – 3(x – 2) or better isw or 2 −−x 2 ( x − 2 )( x + 1) x B1 for common denominator (x – 2)(x + 1) oe isw cao final answer

This question in 0580/22 May/June 2025

26 Write as a single fraction in its simplest form. mp 15 (a) # 25 y m … [2] 3 4 (b) + 2x - 5 x - 3 … [3]

5 marks

Mark scheme: 26(a) 3 p 2 15 p 3mp final answer cao B1 for or final answer 5 y 25 y 5my or for correct answer spoilt 26(b) 11 x − 29 3 B1 for 3( x − 3) + 4(2 x − 5) or better final answer (2 x − 5)( x − 3) seen B1 for common denominator of or (2 x − 5)( x − 3) oe isw 11x − 29 2 final answer 2 x − 11x + 15

This question in 0580/23 Oct/Nov 2025

28 Simplify. 2 x 2 - 11 x - 21 x 2 - 49 … [4]

4 marks

Mark scheme: 28 2 x + 3 4 B2 for (2x + 3)(x – 7) final answer or B1 for 2x(x – 7) + 3(x – 7) x + 7 or x(2x + 3) – 7(2x + 3) or (2x + a)(x + b) where ab = – 21 or a + 2b = – 11 B1 for (x + 7)(x – 7)

This question in 0580/23 Oct/Nov 2025