Higher June 2024 Paper 2 Q19
19
(a) Show that \(\quad 4x(3x + 2) - 2x^2\left(6 - \dfrac{5}{x}\right) - 6x\left(3 + \dfrac{7}{x}\right) \quad\) simplifies to an integer. [3 marks]
(b) Factorise \(\quad 8x^2 - 18x - 35\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| One of \(12x^2 + 8x\) \(-12x^2 + 10x\) \(-18x - 42\) | M1 | may be seen in a grid |
| Two of \(12x^2 + 8x\) \(-12x^2 + 10x\) \(-18x - 42\) | M1dep | may be seen in a grid |
| \(12x^2 + 8x\) and \(-12x^2 + 10x\) and \(-18x - 42\) and \(-42\) | A1 | must see 6 correct terms and a final simplification to \(-42\) |
Additional guidance
| For terms seen in a grid accept eg \(8x\) for \(+8x\) | |
| Accept multiplication signs between coefficients and algebra eg \(12 \times x^2 + 8 \times x\) | 1st M1 |
| Accept eg \(+-12x^2\) for \(-12x^2\) | |
| Do not accept unprocessed brackets eg do not accept \(-(18x + 42)\) | |
| Crossed out terms are likely to be their working rather than deleted work |
| Answer | Mark | Comments |
|---|---|---|
| \((4x + 5)(2x - 7)\) | B2 | oe factorisation eg \((-2x + 7)(-4x - 5)\) B1 \((ax + b)(cx + d)\) where \(ac = 8\) and \(bd = -35\) or \((ax + b)(cx + d)\) where \(ac = 8\) and \(ad + bc = -18\) allow multiplication signs for B2 or B1 |
Additional guidance
| B1 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| \((8x - 7)(x + 5)\) (\(ac = 8\) and \(bd = -35\)) | B1 |
| \((2x - 3)(4x - 3)\) (\(ac = 8\) and \(ad + bc = -18\)) | B1 |
| For B1 allow use of fractions or decimals eg \((4x + 10)(2x - 3.5)\) | B1 |
| For B1 allow eg \(8(x + 1.25)(x - 3.5)\) | B1 |
| Condone missing final bracket for B2 or B1 | |
| Ignore any attempt to ‘solve’ eg \((4x + 5)(2x - 7)\) in working lines with \(-1.25\) and 3.5 on answer lines | B2 |