FP2 June 2005 Q6
6.
(a) On the same diagram, sketch the graphs of \(y = |x^2 - 4|\) and \(y = |2x - 1|\), showing the coordinates of the points where the graphs meet the axes. (4)
(b) Solve \(|x^2 - 4| = |2x - 1|\), giving your answers in surd form where appropriate. (5)
(c) Hence, or otherwise, find the set of values of \(x\) for which \(|x^2 - 4| \gt |2x - 1|\). (3)
| Scheme | Marks |
|---|---|
 | B1 |
 | B1 |
| \(-2,\ 2\) | B1 |
| \(\dfrac{1}{2}\) | B1 |
| (4) |
Notes
(corrected from the printed mark scheme: the third B1 line reads “\(-z, z\)”; the graph meets the \(x\)-axis at \(-2\) and \(2\))
| Scheme | Marks |
|---|---|
| \(x^2 - 4 = 2x - 1\) | M1 |
| \(x^2 - 2x - 3 = 0 \Rightarrow \underline{x = 3, -1}\) | A1 |
| \(x^2 - 4 = -(2x - 1)\) | M1 |
| \(x^2 + 2x - 5 = 0, \Rightarrow x = \dfrac{-2 \pm \sqrt{4 + 20}}{2}\) | A1, |
| \(x = \underline{-1 \pm \sqrt{6}}\) correct 3 term quadratic = 0 | A1 |
| (5) |
| Scheme | Marks |
|---|---|
| \(x \lt -1 - \sqrt{6};\ -1 \lt x \lt \sqrt{6} - 1,\ x \gt 3\) (√surds) Accept 3sf. | B1ft; B1ft; B1 |
| (3) | |
| (12 marks) |