June 2025 Paper 2 Q11
11
(a) Simplify fully \(\quad \dfrac{3}{4w} + \dfrac{1}{6w}\) [2 marks]
(b) Factorise fully \(\quad (x - 1)^9 + (x - 1)^8(x + 4)\) [2 marks]
(c) Factorise fully \(\quad 8y^2 - 50\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Valid common denominator with at least one numerator correct | M1 | eg \(\dfrac{9}{12w}\) and \(\dfrac{2}{12w}\) or \(\dfrac{18w}{24w^2}\) and \(\dfrac{4w}{24w^2}\) implied by a correct single fraction not in simplest form eg \(\dfrac{9 + 2}{12w}\) or \(\dfrac{22}{24w}\) or \(\dfrac{22w}{24w^2}\) for common denominator \(4w\) or \(6w\) both numerators must be correct |
| \(\dfrac{11}{12w}\) | A1 | accept \(\dfrac{11}{12}w^{-1}\) |
Additional guidance
| \(\dfrac{11}{12w}\) followed by incorrect further work | M1A0 |
| \(\dfrac{4.5}{6w} + \dfrac{1}{6w}\) \(\dfrac{5.5}{6w}\) | M1 A0 |
| For M1 accept numerator(s) and/or denominator(s) seen as products eg \(\dfrac{6w \times 3}{4w \times 6w}\) and \(\dfrac{4w}{4w \times 6w}\) | M1 |
| \(\dfrac{1}{w}\left(\dfrac{3}{4} + \dfrac{1}{6}\right)\) (does not score M1 at this stage) \(\dfrac{1}{w}\left(\dfrac{9}{12} + \dfrac{2}{12}\right)\) | M1 |
| Omitting \(w\) in working with recovery to answer \(\dfrac{11}{12w}\) | M1A1 |
| Omitting \(w\) in working with no or incorrect answer | M0A0 |
| Answer | Mark | Comments |
|---|---|---|
| \((x - 1)^8(2x + 3)\) | B2 | B1 factorised expression with \((x - 1)^8\) as one of the factors allow multiplication signs for B2 or B1 |
Additional guidance
| Condone missing final bracket for B2 or B1 unless final factor is \((x - 1)^8\) eg1 \((x - 1)^8(2x + 3\) eg2 \((2x + 3)(x - 1^8\) | B2 B0 |
| Ignore expansion of a B2 or B1 response | |
| \((x - 1)^8(3 + 2x)\) etc | B2 |
| B2 response followed by further, incorrect work is B1 eg \((x - 1)^8(2x + 3) = 5x(x - 1)^8\) | B1 |
| B1 response followed by further, incorrect work is B1 eg \((x - 1)^8(3x + 4) = 7x(x - 1)^8\) | B1 |
| Ignore any attempt to ‘solve’ | |
| \((x - 1)^8(x - 1) + (x - 1)^8(x + 4)\) | B0 |
| \((x - 1)^8(x - 1) + (x + 4)\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(2(2y + 5)(2y - 5)\) | B2 | B1 \(2(4y^2 - 25)\) or \((4y + 10)(2y - 5)\) or \((4y - 10)(2y + 5)\) or correct use of difference of two squares eg \(8(y + 2.5)(y - 2.5)\) or \(\dfrac{1}{2}(4y + 10)(4y - 10)\) or \(\left(\sqrt{8}\,y + \sqrt{50}\right)\left(\sqrt{8}\,y - \sqrt{50}\right)\) or \(\left(2\sqrt{2}\,y + 5\sqrt{2}\right)\left(2\sqrt{2}\,y - 5\sqrt{2}\right)\) allow multiplication signs for B2 or B1 |
Additional guidance
| All B1 responses must be equivalent to \(8y^2 - 50\) | |
| Condone missing final bracket for B2 or B1 | |
| Ignore expansion of a B2 or B1 response | |
| \(2(5 + 2y)(2y - 5)\) etc | B2 |
| \((10 + 4y)(2y - 5)\) etc | B1 |
| B2 response followed by further, incorrect work is B1 eg \(2(2y + 5)(2y - 5) = (4y + 10)(4y - 10)\) | B1 |
| B1 response followed by further, incorrect work is B1 eg \((4y + 10)(2y - 5) = 14y(2y - 5)\) | B1 |
| Ignore any attempt to ‘solve’ |