E2.2· 68 questions · 188 marks · 226 min · 2017–2025· Structured questions
Every Cambridge IGCSE Mathematics - International Paper 2 question on algebraic manipulation, laid out as 22 A4 pages with the mark scheme below. Nothing is left out. Free to read, no account.
![Question 1: Expand. x (x 3 - 4x) ..................................................... [2]](https://img.pastlit.com/crops/1c933872-e827-470b-bc0d-56039c723422/q3.webp)
![Question 2: Expand the brackets and simplify. (4x - 3y)(2x - 5y) ..................................................... [3]](https://img.pastlit.com/crops/1c933872-e827-470b-bc0d-56039c723422/q15.webp)
![Question 3: Factorise completely. 6x 2 - 2x ..................................................... [2]](https://img.pastlit.com/crops/cc9b678b-fab2-4799-9c9c-e77e40f48e9b/q3.webp)
1 / 22![Question 5: Expand. x 3 (x 2 + 3) .................................................... [2] 7](https://img.pastlit.com/crops/8bdc0595-e3d1-4f1d-beed-e3743b70aa7a/q5.webp)
2 / 22![Question 7: (a) Factorise x 2 - 3x - 10 . .................................................... [2] (b) Using your answer to part (a), solve x 2 - 3x - …](https://img.pastlit.com/crops/a86729fc-2891-4456-8bec-728ca1f1332e/q11.webp)
![Question 8: Expand the brackets and simplify. (3a - 5b)(2a - 3b) .................................................... [3]](https://img.pastlit.com/crops/a86729fc-2891-4456-8bec-728ca1f1332e/q13.webp)
3 / 22![Question 10: Factorise completely. 5x 2 - 125y 2 .................................................... [3]](https://img.pastlit.com/crops/a003d9cb-8795-48c4-b24e-de0d103766fc/q15.webp)
![Question 11: (a) Expand and simplify. ( 2p - 7q)(p + q) .................................................... [2] (b) Factorise. 2 - t - 2a + at ........…](https://img.pastlit.com/crops/c5388753-932d-4204-90c7-816ba70655c5/q6.webp)
4 / 22![Question 13: Expand the brackets and simplify. 3x - 5y 5x - 3y ` `j j .................................................... [3]](https://img.pastlit.com/crops/22e806aa-376a-4783-8359-9810d2ee51fc/q15.webp)
![Question 14: Factorise. 4x 2 - 4xy - 3y 2 .................................................... [3]](https://img.pastlit.com/crops/22e806aa-376a-4783-8359-9810d2ee51fc/q17.webp)
![Question 15: Expand and simplify 5 (2x + 3y) - 3 (4y - 2x) . .................................................... [2]](https://img.pastlit.com/crops/f0023cf2-b76c-49be-b9fb-9c44c970bb19/q3.webp)
5 / 22![Question 17: Factorise. 4x 2 - 7x - 2 .................................................... [2]](https://img.pastlit.com/crops/a1827d58-ea35-4b17-9061-7a2853448bfa/q15.webp)
![Question 18: Factorise. (a) 64x 2 - 1 .................................................... [1] (b) 2y 2 - y - 6 ........................................…](https://img.pastlit.com/crops/fdc7aca8-9a2f-4c51-9143-8e71317b3f60/q7.webp)
![Question 19: Expand and simplify. 4(3x + y) - 3(x - 2y) .................................................... [2]](https://img.pastlit.com/crops/325869c6-b11f-45fc-a871-01b4f1c12fb8/q4.webp)
6 / 22![Question 21: Rearrange this formula to make b the subject. (a + b) A = h 2 b = .................................................... [3] Question 14 is p…](https://img.pastlit.com/crops/325869c6-b11f-45fc-a871-01b4f1c12fb8/q13.webp)
![Question 22: Factorise completely. ab - a - b + 1 .................................................... [2]](https://img.pastlit.com/crops/395e00f2-840e-4048-800d-5b6a9ef37bef/q12.webp)
![Question 23: Expand the brackets and simplify. 2 ( 3x - 1) + 3 ( 1 - 2x) .................................................... [2]](https://img.pastlit.com/crops/b0b32256-7c12-404d-a17c-c71fba6beeed/q4.webp)
7 / 22![Question 25: Factorise completely. 8x 2 - 18 .................................................... [2]](https://img.pastlit.com/crops/e81664e8-9e49-4aed-b661-6e110d29fceb/q11.webp)
![Question 26: Rearrange the formula to make x the subject. x y = 1 - 3x - 5 x = ................................................... [4]](https://img.pastlit.com/crops/e81664e8-9e49-4aed-b661-6e110d29fceb/q14.webp)
8 / 22![Question 28: Rearrange this formula to make a the subject. 3a - 2 y = a - 1 .................................................... [3]](https://img.pastlit.com/crops/b8031618-d416-418e-a9d4-a257c353eb3f/q10.webp)
![Question 29: A = 2 r rh + 3rr 2 Rearrange the formula to write h in terms of r, r and A. h = .................................................... [2]](https://img.pastlit.com/crops/1b7ed509-d9be-4bc9-b1dc-d1f40451fb4b/q8.webp)
9 / 22![Question 31: Factorise. 2x 2 - 3x - 5 ................................................. [2]](https://img.pastlit.com/crops/6b34ccf8-1367-4dc1-a744-11d64afcd374/q12.webp)
![Question 32: Factorise completely. (a) 4x 2 y - 6xy 2 .................................................... [2] (b) 9x 2 - 1 ............................…](https://img.pastlit.com/crops/deff0393-d955-49b0-bd37-dbcb5148bace/q12.webp)
10 / 22![Question 34: J = m ( k 2 + h 2 ) Rearrange the formula to make h the subject. h = ................................................. [3]](https://img.pastlit.com/crops/72464299-f473-4fd3-bf7d-952603ddfebd/q10.webp)
![Question 35: Expand and simplify. 4 ( 2a + 5b) - 3 ( 6b - 3a) ................................................. [2]](https://img.pastlit.com/crops/cddb140a-2a4e-4206-8b26-4cf9dd36b5e6/q8.webp)
11 / 22![Question 37: (a) Factorise a 2 - b 2 . ................................................. [1] (b) Work out 5.37 2 - 4. 63 2 . ...........................…](https://img.pastlit.com/crops/0f67aa1e-a3d9-47a9-a4ec-f88a6cc7a664/q6.webp)
![Question 38: y = x + 3 Rearrange the formula to make x the subject. x = ................................................. [3]](https://img.pastlit.com/crops/0f67aa1e-a3d9-47a9-a4ec-f88a6cc7a664/q11.webp)
12 / 22![Question 40: A = P ( 1 + x) 3 Rearrange the formula to write x in terms of A and P. x = ................................................. [3]](https://img.pastlit.com/crops/fc536852-ae13-450b-9eea-389f83f3f1e8/q14.webp)
![Question 41: Factorise. 3x + 6 - 2xy - 4y ................................................. [2]](https://img.pastlit.com/crops/f762bb7c-3665-42c3-b4d8-45b2802d959e/q16.webp)
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14 / 22![Question 45: Factorise fully. 2cx 2 - 2dx - cx + d ................................................. [2]](https://img.pastlit.com/crops/ee9d8776-92d9-4913-ad66-74da6b18b9aa/q8.webp)
![Question 46: Rearrange the formula to write x in terms of a and y. y = x 2 + 2a 2 x = ................................................ [3]](https://img.pastlit.com/crops/f21449e1-efd1-4c68-a7a7-7429f94d83b5/q10.webp)
15 / 22![Question 48: Expand the brackets and simplify. ( 4x - 3y)( 4x + 3y) ................................................. [2]](https://img.pastlit.com/crops/277e0ace-9d7c-4061-8c57-9912cdd69ebf/q12.webp)
![Question 49: Make x the subject of A = . x x = ................................................ [3]](https://img.pastlit.com/crops/277e0ace-9d7c-4061-8c57-9912cdd69ebf/q14.webp)
![Question 50: Factorise. 5x 2 - xy - 4y 2 ................................................. [2]](https://img.pastlit.com/crops/277e0ace-9d7c-4061-8c57-9912cdd69ebf/q15.webp)
16 / 22![Question 52: Factorise fully. (a) ( 3)y 2 - 16 ................................................. [1] (b) 15ab - 1 - 3a + 5b ............................…](https://img.pastlit.com/crops/76fe45e5-52fe-4985-81a2-022bd41aca69/q12.webp)
![Question 53: J = h 3 + k 3 (a) Find the value of J when h = 3 and k = 4 . J = ................................................ [2] (b) Rearrange the for…](https://img.pastlit.com/crops/fa5b85bf-8762-43bf-9515-0cad06725b62/q7.webp)
17 / 22![Question 55: Rearrange this formula to make R the subject. 2 ( Q + 3 R ) P = 5 R = ................................................. [3]](https://img.pastlit.com/crops/fd9da03e-7e7b-4c8c-863d-a3146ed9ad6f/q12.webp)
![Question 56: Factorise completely. x 3 y 2 - xy ................................................. [2]](https://img.pastlit.com/crops/9fdaa2c8-92d2-491e-bf80-83b9a91170b5/q12.webp)
18 / 22![Question 58: Factorise completely. 48x 2 - 75y 2 ................................................. [3]](https://img.pastlit.com/crops/dda12d37-1cda-46b9-af22-1b8ceed07be6/q11.webp)
![Question 59: Expand x 3 `8 x - x 2j . ................................................. [2] .05](https://img.pastlit.com/crops/7b8fbcbc-cd41-4959-8bab-0bd0249f89b7/q4.webp)
![Question 60: Factorise. (a) 6ax - 8by - 3 ay + 16bx ................................................. [2] (b) 5x 2 - 7 x - 6 ...........................…](https://img.pastlit.com/crops/7b8fbcbc-cd41-4959-8bab-0bd0249f89b7/q8.webp)
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20 / 22![Question 64: (a) Rearrange the formula to make a the subject. a - b = 2c a = ................................................ [2] (b) Rearrange the form…](https://img.pastlit.com/crops/5096a0d7-47d6-4b7e-a449-e26493ab5f77/q18.webp)
21 / 22![Question 66: Expand and simplify. ( x - 4)( 2x + 1)( x + 2) ..................................................................... [3]](https://img.pastlit.com/crops/5d804338-309b-49c6-808a-d157cca34a37/q19.webp)
![Question 67: Rearrange 4y - 3x = 1 - py to write y in terms of p and x. y = ................................................ [2]](https://img.pastlit.com/crops/0c2c9bb6-82d0-4260-8a6c-43dfe5ea6509/q16.webp)
22 / 22Answers below. Sit the paper first if you are practising.
Pastlit
Mathematics - International 0607 · Algebraic manipulation — Paper 2
IGCSE · topical answer key — answer key (teacher use)
Question
Answer
Marks
2
3
2
3
2
4
4
3
3
3
4
2
3
3
2
3
2
3
2
2
3
2
2
2
2
4
3
3
2
2
2
3
4
3
2
3
3
3
2
3
2
1
3
5
2
3
3
2
3
2
2
3
4
2
3
2
3
3
2
4
2
3
2
5
4
3
2
5| Question | Answer | Marks | From |
|---|---|---|---|
| 1 | see sheet | 2 | 0607/21 May/June 2017 |
| 2 | see sheet | 3 | 0607/21 May/June 2017 |
| 3 | see sheet | 2 | 0607/22 May/June 2017 |
| 4 | see sheet | 3 | 0607/22 May/June 2017 |
| 5 | see sheet | 2 | 0607/23 May/June 2017 |
| 6 | see sheet | 4 | 0607/23 May/June 2017 |
| 7 | see sheet | 4 | 0607/21 Oct/Nov 2017 |
| 8 | see sheet | 3 | 0607/21 Oct/Nov 2017 |
| 9 | see sheet | 3 | 0607/22 Oct/Nov 2017 |
| 10 | see sheet | 3 | 0607/23 Oct/Nov 2017 |
| 11 | see sheet | 4 | 0607/21 May/June 2018 |
| 12 | see sheet | 2 | 0607/21 May/June 2018 |
| 13 | see sheet | 3 | 0607/22 May/June 2018 |
| 14 | see sheet | 3 | 0607/22 May/June 2018 |
| 15 | see sheet | 2 | 0607/23 May/June 2018 |
| 16 | see sheet | 3 | 0607/23 May/June 2018 |
| 17 | see sheet | 2 | 0607/21 Oct/Nov 2018 |
| 18 | see sheet | 3 | 0607/22 Oct/Nov 2018 |
| 19 | see sheet | 2 | 0607/21 May/June 2019 |
| 20 | see sheet | 2 | 0607/21 May/June 2019 |
| 21 | see sheet | 3 | 0607/21 May/June 2019 |
| 22 | see sheet | 2 | 0607/22 May/June 2019 |
| 23 | see sheet | 2 | 0607/23 May/June 2019 |
| 24 | see sheet | 2 | 0607/23 May/June 2019 |
| 25 | see sheet | 2 | 0607/22 Oct/Nov 2019 |
| 26 | see sheet | 4 | 0607/22 Oct/Nov 2019 |
| 27 | see sheet | 3 | 0607/23 Oct/Nov 2019 |
| 28 | see sheet | 3 | 0607/23 Oct/Nov 2019 |
| 29 | see sheet | 2 | 0607/21 May/June 2020 |
| 30 | see sheet | 2 | 0607/22 May/June 2020 |
| 31 | see sheet | 2 | 0607/22 May/June 2020 |
| 32 | see sheet | 3 | 0607/23 May/June 2020 |
| 33 | see sheet | 4 | 0607/22 Oct/Nov 2020 |
| 34 | see sheet | 3 | 0607/23 Oct/Nov 2020 |
| 35 | see sheet | 2 | 0607/22 Feb/March 2021 |
| 36 | see sheet | 3 | 0607/22 Feb/March 2021 |
| 37 | see sheet | 3 | 0607/22 May/June 2021 |
| 38 | see sheet | 3 | 0607/22 May/June 2021 |
| 39 | see sheet | 2 | 0607/23 May/June 2021 |
| 40 | see sheet | 3 | 0607/23 May/June 2021 |
| 41 | see sheet | 2 | 0607/22 Oct/Nov 2021 |
| 42 | see sheet | 1 | 0607/21 May/June 2022 |
| 43 | see sheet | 3 | 0607/21 May/June 2022 |
| 44 | see sheet | 5 | 0607/21 May/June 2022 |
| 45 | see sheet | 2 | 0607/23 May/June 2022 |
| 46 | see sheet | 3 | 0607/23 Oct/Nov 2022 |
| 47 | see sheet | 3 | 0607/23 Oct/Nov 2022 |
| 48 | see sheet | 2 | 0607/22 Feb/March 2023 |
| 49 | see sheet | 3 | 0607/22 Feb/March 2023 |
| 50 | see sheet | 2 | 0607/22 Feb/March 2023 |
| 51 | see sheet | 2 | 0607/23 May/June 2023 |
| 52 | see sheet | 3 | 0607/23 May/June 2023 |
| 53 | see sheet | 4 | 0607/21 Oct/Nov 2023 |
| 54 | see sheet | 2 | 0607/22 Oct/Nov 2023 |
| 55 | see sheet | 3 | 0607/22 Feb/March 2024 |
| 56 | see sheet | 2 | 0607/21 May/June 2024 |
| 57 | see sheet | 3 | 0607/21 May/June 2024 |
| 58 | see sheet | 3 | 0607/23 May/June 2024 |
| 59 | see sheet | 2 | 0607/21 Oct/Nov 2024 |
| 60 | see sheet | 4 | 0607/21 Oct/Nov 2024 |
| 61 | see sheet | 2 | 0607/22 Oct/Nov 2024 |
| 62 | see sheet | 3 | 0607/22 Oct/Nov 2024 |
| 63 | see sheet | 2 | 0607/22 Feb/March 2025 |
| 64 | see sheet | 5 | 0607/22 Feb/March 2025 |
| 65 | see sheet | 4 | 0607/21 May/June 2025 |
| 66 | see sheet | 3 | 0607/21 May/June 2025 |
| 67 | see sheet | 2 | 0607/23 May/June 2025 |
| 68 | see sheet | 5 | 0607/21 Oct/Nov 2025 |
3 Expand. x (x 3 - 4x) … [2]
2 marks
Mark scheme: 3 x 4 − 4 x 2 2 M1 for either term correct
15 Expand the brackets and simplify. (4x - 3y)(2x - 5y) … [3]
3 marks
Mark scheme: 15 8 x 2 − 26 xy + 15 y 2 final answer 3 M2 for 8 x 2 − 6 xy − 20 xy + 15 y 2 or M1 for 3 terms correct
3 Factorise completely. 6x 2 - 2x … [2]
2 marks
Mark scheme: 3 2x(3x – 1) final answer 2 B1 for 2(3x² – x) or x(6x – 2)
10 Expand the brackets and simplify. (2x - 3y)(3x - 4y) … [3]
3 marks
Mark scheme: 10 6x² – 17xy + 12y² final answer 3 B2 for 6x² – 8xy – 9xy + 12y² or B1 for 3 terms correct
5 Expand. x 3 (x 2 + 3) … [2] 7
2 marks
Mark scheme: 5 x 5 + 3 x 3 final answer 2 B1 for x 5 + kx n or kx n + 3 x 3 , k ≠ 0
14 Factorise. (a) p 2 - p - 30 … [2] (b) x (u - v) - y (v - u) … [2]
4 marks
Mark scheme: 14(a) ( p − 6)( p + 5) 2 B1 for ( p + a )( p + b ) where ab = – 30 or a + b = – 1 or p ( p + 5) − 6( p + 5) or p ( p − 6) + 5( p − 6) 14(b) (u − v )( x + y ) 2 M1 for x (u − v ) + y (u − v ) or u ( x + y ) − v ( x + y )
11 (a) Factorise x 2 - 3x - 10 . … [2] (b) Using your answer to part (a), solve x 2 - 3x - 10 2 0 . … [2] Questions 12 and 13 are printed on the next page.
4 marks
Mark scheme: 11(a) (x – 5)(x + 2) 2 B1 for (x + a)(x + b) where ab = –10 or a + b = –3 or for x(x – 5) + 2(x – 5) or x(x + 2) – 5(x + 2) 11(b) x < –2, x > 5 2 B1FT for correct 'inequalities' from (a)
13 Expand the brackets and simplify. (3a - 5b)(2a - 3b) … [3]
3 marks
Mark scheme: 13 6a2 – 19ab + 15b2 3 B2 for 6a2 – 9ab – 10ab + 15b2 final answer or B1 for 3 correct terms above
6 y = 2x2 - 1 Rearrange the formula to write x in terms of y. x = … [3]
3 marks
Mark scheme: 6 y + 1 3 M1 for correct rearrangement [ ± ] oe M1 for correct division by 2 2 M1 for correct square root
15 Factorise completely. 5x 2 - 125y 2 … [3]
3 marks
Mark scheme: 15 5( x − 5 y )( x + 5 y ) 3 M2 for (5 x − 25 y )( x + 5 y ) or (5 x + 25 y )( x − 5 y ) final answer or M1 for one correct factor identified
6 (a) Expand and simplify. ( 2p - 7q)(p + q) … [2] (b) Factorise. 2 - t - 2a + at … [2]
4 marks
Mark scheme: 6(a) 2 p 2 − 5 pq − 7 q 2 final answer 2 B1for three terms of 2 p 2 − 7 pq + 2 pq − 7 q 2 correct 6(b) (1 − a )( 2 − t ) oe 2 M1 for 2 – t – a(2 – t) or 2(1 – a) – t(1 – a)
10 v 2 = u 2 - 2as Find s in terms of a, u and v. s = … [2]
2 marks
Mark scheme: 10 u 2 − v 2 2 M1 for correct rearrangement to isolate the s oe term 2 a or M1 for correct division by 2a or –2a
15 Expand the brackets and simplify. 3x - 5y 5x - 3y ` `j j … [3]
3 marks
Mark scheme: 15 15 x 2 − 34 xy + 15 y 2 final answer 3 M2 for 15 x 2 − 9 xy − 25 xy + 15 y 2 or M1 for 3 correct terms
17 Factorise. 4x 2 - 4xy - 3y 2 … [3]
3 marks
Mark scheme: 17 (2 x − 3 y )(2 x + y ) final answer 3 M2 for (ax + by)(cx + dy) where ac = 4 and bd = –3 or M1 for ac = 4 or bd = –3 OR M2 for 2 x (2 x − 3 y ) + y (2 x − 3 y ) or 2 x (2 x + y ) − 3 y (2 x + y ) or M1 for 4 x 2 − 6 xy + 2 xy − 3 y 2 oe
3 Expand and simplify 5 (2x + 3y) - 3 (4y - 2x) . … [2]
2 marks
Mark scheme: 3 16x + 3y final answer 2 M1 for 10x + 15y or –12y + 6x
14 Rearrange this formula to make x the subject. ax y = bx + c x = … [3]
3 marks
Mark scheme: 14 cy 3 M1 for bxy + cy = ax oe x = oe final answer M1FT for cy = ax – bxy oe a − by M1FT for completion from their isolated x terms
15 Factorise. 4x 2 - 7x - 2 … [2]
2 marks
Mark scheme: 15 (4 x + 1)( x − 2) 2 M1 for ( ax ± b )( cx ± d ) where two of ac = 4, bd = –2, ad + bc = –7 are correct. or M1 for 4x(x – 2) + x – 2 or x(4x + 1) – 2(4x + 1)
7 Factorise. (a) 64x 2 - 1 … [1] (b) 2y 2 - y - 6 … [2]
3 marks
Mark scheme: 7(a) (8x + 1)(8x – 1) 1 7(b) (2y + 3)(y – 2) 2 B1 for (2y + a)(y + b) where ab = –6 or a + 2b = –1 or 2y(y – 2) + 3(y – 2) or y(2y + 3) – 2(y + 3)
4 Expand and simplify. 4(3x + y) - 3(x - 2y) … [2]
2 marks
Mark scheme: 4 9x + 10y final answer 2 B1 for 12x + 4y – 3x + 6y or ax + 10y, a ≠ 0 or 9x + by, b ≠ 0
9 Factorise completely. 2x2 - 18 … [2]
2 marks
Mark scheme: 9 2(x + 3)(x – 3) final answer 2 B1 for 2(x2 – 9) or (2x + 6)(x – 3) or (x + 3)(2x – 6) in working or answer space
13 Rearrange this formula to make b the subject. (a + b) A = h 2 b = … [3] Question 14 is printed on the next page.
3 marks
Mark scheme: 13 2A 2 A − ah 3 2A [b =] – a or [b =] oe M2 for 2A – ah = bh or = a + b h h h Final answer A a + b or M1 for 2A = (a + b)h or = h 2
12 Factorise completely. ab - a - b + 1 … [2]
2 marks
Mark scheme: 12 (b − 1)( a − 1) 2 M1 for a (b − 1) − (b − 1) or for b ( a − 1) − ( a − 1) or for either factor correct
4 Expand the brackets and simplify. 2 ( 3x - 1) + 3 ( 1 - 2x) … [2]
2 marks
Mark scheme: 4 1 2 M1 for 6 x − 2 + 3 − 6 x
14 Factorise completely. 6ac - 9bc - 8ad + 12bd … [2]
2 marks
Mark scheme: 14 (3c − 4 d )(2 a − 3b ) 2 M1 for either factor correct or M1 for 3c (2 a − 3b ) − 4 d (2 a − 3b ) or 2 a (3c − 4 d ) − 3b (3c − 4 d )
11 Factorise completely. 8x 2 - 18 … [2]
2 marks
Mark scheme: 11 2(2 x − 3)(2 x + 3) 2 M1 for 2(4 x 2 − 9) or (4 x − 6)(2 x + 3) or (2 x − 3)(4 x + 6)
14 Rearrange the formula to make x the subject. x y = 1 - 3x - 5 x = … [4]
4 marks
Mark scheme: 14 5 y − 5 5 − 5 y 4 M1 for correctly eliminating fractions x = or x = M1 for correctly collecting their x terms 3 y − 2 2 − 3 y M1 for correct final division of their terms
5 Factorise. (a) x 2 - 1 … [1] (b) 3x 2 - 6ax - axy + 2a 2 y … [2]
3 marks
Mark scheme: 5(a) (x + 1)(x – 1) 1 5(b) (3x – ay)(x – 2a) oe 2 B1 for 3x( x – 2a) or –ay(x –2a) or ay(2a – x) or x(3x – ay) or 2a(–3x + ay) or –2a(3x – ay)
10 Rearrange this formula to make a the subject. 3a - 2 y = a - 1 … [3]
3 marks
Mark scheme: 10 y − 2 3 M1 for y(a – 1) = 3a – 2 oe oe M1 FT for ay – 3a = y – 2 y − 3 M1 FT for completion
8 A = 2 r rh + 3rr 2 Rearrange the formula to write h in terms of r, r and A. h = … [2]
2 marks
Mark scheme: 8 A − 3πr 2 2 M1 for correct rearrangement for term oe final answer in h 2πr M1 for correct division by 2πr
10 Rearrange the formula to make x the subject. y ( x + 4) = 2 x = … [2]
2 marks
Mark scheme: 10 2 − 4y 2 2 2 x = oe or x = − 4 oe B1 for yx + 4 y = 2 or x + 4 = y y y
12 Factorise. 2x 2 - 3x - 5 … [2]
2 marks
Mark scheme: 12 (2 x − 5)( x + 1) 2 M1 for (2 x + a )( x + b ) where ab = −5 or a + 2b = −3
12 Factorise completely. (a) 4x 2 y - 6xy 2 … [2] (b) 9x 2 - 1 … [1]
3 marks
Mark scheme: 12(a) 2xy(2x – 3y) 2 B1 for any correct partially factorised expression 12(b) (3x + 1)(3x – 1) 1
10 Factorise (a) x 2 - x - 6 , … [2] (b) 3ax + 2bx - 4by - 6ay . … [2]
4 marks
Mark scheme: 10(a) (x – 3)(x + 2) 2 B1 for (x + a)(x + b) where ab = –6 or a + b = –1 or B1 for x(x + 2) – 3(x + 2) or x(x – 3) + 2(x – 3) 10(b) (x – 2y)(3a + 2b) 2 B1 for x(3a + 2b) – 2y(3a + 2b) oe or 3a(x – 2y) + 2b(x – 2y) oe
10 J = m ( k 2 + h 2 ) Rearrange the formula to make h the subject. h = … [3]
3 marks
Mark scheme: 10 2 3 M1 for correct division by m J − mk J 2 [ ± ] or [ ± ] − k M1 for correct rearrangement for h or h2 term m m M1 for correct square root final answer
8 Expand and simplify. 4 ( 2a + 5b) - 3 ( 6b - 3a) … [2]
2 marks
Mark scheme: 8 17 a + 2 b final answer 2 B1 for answer 17a + kb or ka + 2b k ≠ 0 or M1 for 8 a + 20 b or − 18b + 9 a
12 Rearrange this formula to make x the subject. a - x y = 3x x = … [3] Questions 13 and 14 are printed on the next page.
3 marks
Mark scheme: 12 a 3 M1 for correct removal of fraction(s) [ x = final answer M1 for isolating terms in x ]3 y + 1 M1 for correct division by a linear expression maximum 2 marks if final answer incorrect
6 (a) Factorise a 2 - b 2 . … [1] (b) Work out 5.37 2 - 4. 63 2 . … [2]
3 marks
Mark scheme: 6(a) (a + b)(a – b) final answer 1 6(b) 7.4 2 M1 for (5.37 + 4.63)( 5.37 – 4.63) or better
11 y = x + 3 Rearrange the formula to make x the subject. x = … [3]
3 marks
Mark scheme: 11 2 − 3y 2 3 M1 for y(x + 3) = 2 oe or − 3 oe final answer M1FT for xy = 2 – 3y oe y y OR 2 M2 for = x + 3 y
12 Factorise. 1+ a - c - ac … [2]
2 marks
Mark scheme: 12 (1 + a )(1 − c ) 2 M1 for 1 + a − c (1 + a ) or 1 − c + a (1 − c )
14 A = P ( 1 + x) 3 Rearrange the formula to write x in terms of A and P. x = … [3]
3 marks
Mark scheme: 14 A 3 M1 for correct division by P 3 − 1 final answer M1 for correct cube root P M1 for correct subtraction of 1 Maximum mark of M2 if answer incorrect
16 Factorise. 3x + 6 - 2xy - 4y … [2]
2 marks
Mark scheme: 16 ( x + 2 )( 3 − 2 y ) final answer 2 B1 for 3( x + 2) − 2 y ( x + 2) or x (3 − 2 y ) + 2(3 − 2 y )
4 Factorise x 3 - 2x . … [1]
1 marks
Mark scheme: 4 x ( x 2 2) final answer 1
14 x 2 - 14 x + c = ( x + d ) 2 Find the value of c and the value of d. c = … d = … [3] Questions 15 and 16 are printed on the next page.
3 marks
Mark scheme: 14 [c =] 49 3 2 M1 for correct expansion of x d [d =]−7 M1 for equating their coefficients (dependent on a three term expression) OR B2 for d = −7 only or B1 for 2d = [±] 14 soi If 0 scored SC1 for c = (their d)2
15 (a) Factorise fully. 6x 2 - 7x - 3 … [2] (b) Solve. 6x 2 - 7x - 3 1 0 … [3]
5 marks
Mark scheme: 15(a) (2 x 3)(3 x 1) final answer 2 M1 for ( ax b )( cx d ) where ac= 6 AND bd = –3 or ad bc 7 or for 2 x (3 x 1) 3(3 x 1) or for 3 x (2 x 3) (2 x 3) 15(b) 1 3 3 FT their factors from (a) x final answer 3 2 or 1 3 ‘ x AND x ’ final answer 1 3 3 2 B2 for ‘ x [or] x ’ 3 2 (must include the word AND) 1 3 or B1 for x or B1 for x 3 2 1 or B1 for x AND x 3 soi 3 2
8 Factorise fully. 2cx 2 - 2dx - cx + d … [2]
2 marks
Mark scheme: 8 (cx – d)(2x – 1) final answer 2 B1 for 2 pairs factorised e.g. cx(2x – 1) – d(2x – 1) or 2x(cx – d) – (cx – d)
10 Rearrange the formula to write x in terms of a and y. y = x 2 + 2a 2 x = … [3]
3 marks
Mark scheme: 10 2 2 3 M1 for correct squaring [+] y − 2 a final answer M1 for correct rearranging for x term M1 for correct square root Max 2 marks if answer incorrect
13 Factorise. (a) 49 - 16u 2 … [1] (b) 1 + 4xy - 2x - 2y … [2]
3 marks
Mark scheme: 13(a) ( 7 + 4u )( 7 − 4u ) final answer 1 13(b) (1 − 2 x )(1 − 2 y ) final answer 2 M1 for 1 − 2 y − 2 x (1 − 2 y ) or 1 − 2 x − 2 y (1 − 2 x)
12 Expand the brackets and simplify. ( 4x - 3y)( 4x + 3y) … [2]
2 marks
Mark scheme: 12 16 x 2 − 9 y 2 final answer 2 M1 for 3 terms correct of 16 x 2 + 12 xy − 12 xy − 9 y 2
14 Make x the subject of A = . x x = … [3]
3 marks
Mark scheme: 14 3 y 3 M1 for correct elimination of x = oe fractions ( A − 3) M1 for correct collection of terms M1 for correct division of equation of form ( p + k ) x = q Only scores 3/3 for a correct answer
15 Factorise. 5x 2 - xy - 4y 2 … [2]
2 marks
Mark scheme: 15 (5 x + 4 y )( x − y ) oe 2 M1 for ( ax + b)(cx + d ) with two of ac = 5, bd = −4, ad + bc = −1 or 5 x ( x − y ) + 4 y ( x − y )
6 Expand. 3x ( 2x 4 - 5) … [2]
2 marks
Mark scheme: 6 6 x 5 15 x final answer 2 M1 for 6x 5 kx or px k 15 x oe
12 Factorise fully. (a) ( 3)y 2 - 16 … [1] (b) 15ab - 1 - 3a + 5b … [2]
3 marks
Mark scheme: 12(a) (3 y 4)(3 y 4) final answer 1 12(b) (5b 1)(3a 1) oe final answer 2 M1 for 5b (3a 1) 1 3a or 3a(5b – 1) + 5b – 1
7 J = h 3 + k 3 (a) Find the value of J when h = 3 and k = 4 . J = … [2] (b) Rearrange the formula to write h in terms of J and k. h = … [2]
4 marks
Mark scheme: 7(a) 91 2 M1 for 27 or 64 or 33 + 43 7(b) 3 3 2 M1 for correct rearrangement for J − k final answer h3 = ... or h = ... M1 for correct cube root Max 1 mark if answer incorrect
9 Expand 3p 2 ( 4 - 3p). … [2]
2 marks
Mark scheme: 9 12p2 – 9p3 final answer 2 2 3 2 3 M1 for either 12 p + kp or kp − 9 p
12 Rearrange this formula to make R the subject. 2 ( Q + 3 R ) P = 5 R = … [3]
3 marks
Mark scheme: 12 5 P − 2Q 3 M1 for eliminating fractions oe final answer M1 for expanding and isolating terms 6 M1 for dividing Incorrect answer scores M2 maximum
12 Factorise completely. x 3 y 2 - xy … [2]
2 marks
Mark scheme: 12 2 2 2 2 2 3 xy x y 1 or xy 1 x y M1 for x ( x y y ) or y x y x final answer
14 Make x the subject of the formula. p q = x x - 2 x = … [3]
3 marks
Mark scheme: 14 2 p 2 p 3 M1 for correctly eliminating terms in x from [ x ] or [ x ] p q q p denominators final answer M1 for correctly rearranging their equation into the form x ( a b ) cp
11 Factorise completely. 48x 2 - 75y 2 … [3]
3 marks
Mark scheme: 11 3(4x + 5y)(4x – 5y) Final answer 3 B2 for (12x + 15y)(4x – 5y) or (4x + 5y)(12x – 15y) or B1 for 3(16x2 – 25y2) If 0 scored, SC1 for 48 x 75 y )( 48 x 75 y
4 Expand x 3 `8 x - x 2j . … [2] .05
2 marks
Mark scheme: 4 8 x 4 − x 5 final answer 2 B1 for 8 x 4 − kx 5 or kx 4 − x 5
8 Factorise. (a) 6ax - 8by - 3 ay + 16bx … [2] (b) 5x 2 - 7 x - 6 … [2]
4 marks
Mark scheme: 8(a) (2 x − y )(3a + 8b) oe 2 M1 for 3a (2 x − y ) + 8b( −−−y ( 2 x )) oe or 2 x (3a + 8b) − y (8b + 3a ) oe 8(b) (5 x + 3)( x − 2) oe 2 M1 for (5 x + a )( x + b) where ab = –6 or 5b + a = –7 or 5 x ( x − 2) + 3( x − 2) or x (5 x + 3) − 2(5 x + 3)
3 Expand. x 3 ( x - 2) … [2]
2 marks
Mark scheme: 3 x 4 − 2 x 3 final answer 2 B1 for x 4 + kx c or kx − 2 x 3
15 x 2 + 18x + 2 = ( x + p) 2 + t Find the value of p and the value of t. p = … t = … [3]
3 marks
Mark scheme: 15 [ p = ] 9 3 B2 for one correct [ t = ] –79 or B1 for ( x + 9) 2 seen or for x2 + 2px + p2 + t oe
15 Expand and simplify. ( 3a + 4b)( a - 2 b) … [2]
2 marks
Mark scheme: 15 3a 2 − 2ab − 8b 2 final answer 2 B1 for 3 out of 4 terms correct in 3a 2 + 4 ab − 6ab − 8b 2
18 (a) Rearrange the formula to make a the subject. a - b = 2c a = … [2] (b) Rearrange the formula to make p the subject. p - 5 = q 3p - 2 p = … [3]
5 marks
Mark scheme: 18(a) 4 c 2 + b final answer 2 B1 for a − b = ( 2c ) 2 oe 18(b) 5 − 2 q 3 M1 for removing fraction e.g. p −=5 (3 p − 2) q final answer 1 − 3q M1 for isolating terms in p e.g. p − 3 pq = 5 − 2q a ( q ) M1 for factorising and dividing e.g. p = b ( q ) maximum M2 if final answer incorrect
9 Factorise. (a) 4x 2 y 3 - 6xy 4 … [2] (b) 18p 2 - 2 … [2]
4 marks
Mark scheme: 9(a) 2 xy 3 ( 2 x − 3 y ) final answer 2 B1 for correct factorisation with at least two factors outside. 2 x 2 xy 3 − 3 y 4 ( ) or xy 3 ( 4 x − 6 y ) or 2 y 3 2 x 2 − 3 xy or better ( ) 9 p 2 − 19(b) 2 ( 3 p + 1)( 3 p − 1) final answer 2 B1 for 2 ( ) or ( 6 p + 2 )( 3 p − 1) or ( 3 p + 1)( 6 p − 2 )
19 Expand and simplify. ( x - 4)( 2x + 1)( x + 2) … [3]
3 marks
Mark scheme: 19 3 2 3 B2 for correct unsimplified expression or 2 x − 3 x − 18 x − 8 final answer for simplified 4-term expression with 3 terms correct in final answer. or B1 for one pair of brackets expanded with at least 3 out of 4 terms correct
16 Rearrange 4y - 3x = 1 - py to write y in terms of p and x. y = … [2]
2 marks
Mark scheme: 16 1 + 3 x 2 M1 for isolating terms in y oe final answer 4 + p M1 for factorising 2 terms in y and dividing to final answer Max M1 if answer incorrect
8 Rearrange each formula to make b the subject. (a) c = 5 ab b = … [2] 3a + 2b (b) P = 5a - b b = … [3]
5 marks
Mark scheme: 8(a) c 2 2 c oe final answer B1 for c2 = 5ab or b = 5 a 5a 8(b) 5aP − 3a 3 M1 for clearing fraction oe final answer M1 for expanding and rearranging 2 + P M1 for factorising and dividing Incorrect answer scores M2 maximum