FP2 June 2006 Q3
3.
(a) Use algebra to find the exact solutions of the equation \[|2x^2 + x - 6| = 6 - 3x.\] (6)
(b) On the same diagram, sketch the curve with equation \(y = |2x^2 + x - 6|\) and the line with equation \(y = 6 - 3x\). (3)
(c) Find the set of values of \(x\) for which \[|2x^2 + x - 6| \gt 6 - 3x.\] (3)
| Scheme | Marks |
|---|---|
| \(2x^2 + x - 6 = 6 - 3x\) | M1 |
| Leading to \(x^2 + 2x - 6 = 0\) \((x + 1)^2 = 7 \Rightarrow x = -1 \pm \sqrt{7}\) surds required | M1 A1 |
| \(-2x^2 - x + 6 = 6 - 3x\) | M1 |
| Leading to \(2x^2 - 2x = 0, \Rightarrow x = 0, 1\) | A1 A1 |
| (6) |
Accept if parts (a) and (b) done in reverse order
| Scheme | Marks |
|---|---|
| Curved shape | B1 |
| Line | B1 |
| At least 3 intersections | B1 |
| (3) |
| Scheme | Marks |
|---|---|
| Using all 4 CVs and getting all into inequalities | M1 |
| \(x \gt \sqrt{7} - 1,\ x \lt -\sqrt{7} - 1\) both ft their greatest positive and their least negative CVs | A1ft |
| \(0 \lt x \lt 1\) | A1 |
| (3) | |
| (12 marks) |