June 2018 Paper 2 Q18
18
(a) Work out all the integer values of \(x\) for which
\(-5 \lt 4x + 3 \leqslant 13\) [3 marks]
(b) Work out the range of values of \(x\) for which
\(x^2 - 11x + 28 \gt 0\)
You must show your working. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(-5 - 3 \lt 4x \leqslant 13 - 3\) or \(-8 \lt 4x \leqslant 10\) or \(-1.25 \lt x + 0.75 \leqslant 3.25\) or \(x \leqslant 2.5\) or \(x \gt -2\) or \(x \leqslant 2\) or \(x \geqslant -1\) | M1 | could be embedded eg \(-2 \leqslant x \leqslant 2.5\) |
| \(\dfrac{\text{their } -8}{4} \lt x \leqslant \dfrac{\text{their } 10}{4}\) or their \(-1.25 - 0.75 \lt x \leqslant\) their \(3.25 - 0.75\) or \(-2 \lt x \leqslant 2.5\) or \(-2 \lt x \leqslant 2\) or \(-1 \leqslant x \leqslant 2.5\) or \(-1 \leqslant x \leqslant 2\) or \(x \leqslant 2.5\) and \(x \gt -2\) or \(x \leqslant 2\) and \(x \gt -2\) or \(x \leqslant 2.5\) and \(x \geqslant -1\) or \(x \leqslant 2\) and \(x \geqslant -1\) | M1dep | oe eg \((-2,\ 2.5]\) or \([-1,\ 2.5]\) |
| \(-1 \quad 0 \quad 1 \quad 2\) with no incorrect working | A1 |
Additional guidance
| Answer only \(\ -1 \quad 0 \quad 1 \quad 2\) | M2A1 |
| Answer only \(\ -1 \quad 1 \quad 2\) | Zero |
| \(x = 2.5\) and \(x = -2\) (from solving equations) followed by \(\ -1 \quad 0 \quad 1 \quad 2\) | M2A1 |
| \(x = 2.5\) and \(x = -2\) (from solving equations) | Zero |
| \(-1 \quad 0 \quad 1 \quad 2\ \) with no incorrect working and a correct inequality on answer line | M2A1 |
| \(-1 \quad 0 \quad 1 \quad 2\ \) in working but \(\ -1 \quad 1 \quad 2\ \) on answer line | M2A0 |
| Ignore repeated integers eg Answer only \(\ -1 \quad 0 \quad 1 \quad 1 \quad 2 \quad 2\) | M2A1 |
| Answer | Mark | Comments |
|---|---|---|
| \((x - 4)(x - 7)\) or \(\dfrac{--11 \pm \sqrt{(-11)^2 - 4 \times 1 \times 28}}{2 \times 1}\) or \(\dfrac{11}{2} \pm \sqrt{\dfrac{9}{4}}\) | M1 | oe |
| Identifies 4 and 7 | A1 | may be on a graph or implied by an inequality using 4 and 7 |
| \(x \lt 4 \quad x \gt 7\) | A1 | do not allow incorrect notation eg \(4 \gt x \gt 7\) |
Additional guidance
| \(x \lt 4\) with M1 not scored | Zero |
| \(x \gt 7\) with M1 not scored | Zero |
| Both \(x \lt 4\) and \(x \gt 7\) in working but only one on answer line | M1A1A0 |
| \(x \lt 4\) and \(x \gt 7\) | M1A2 |
| \(x \lt 4\) or \(x \gt 7\) | M1A2 |