FP2 June 2008 Q6
6.
(a) Find, in the simplest surd form where appropriate, the exact values of \(x\) for which \[\frac{x}{2} + 3 = \left|\frac{4}{x}\right|.\] (5)
(b) Sketch, on the same axes, the line with equation \(y = \dfrac{x}{2} + 3\) and the graph of \(y = \left|\dfrac{4}{x}\right|,\ x \neq 0.\) (3)
(c) Find the set of values of \(x\) for which \(\dfrac{x}{2} + 3 > \left|\dfrac{4}{x}\right|.\) (2)
| Scheme | Marks |
|---|---|
| \(\dfrac{4}{x} = \dfrac{x}{2} + 3\) \(x^2 + 6x - 8 = 0\) \(x = \ldots,\ \left(\dfrac{-6 \pm \sqrt{68}}{2}\right)\) \(-3 \pm \sqrt{17}\) – root not needed | M1, A1 |
| \(-\dfrac{4}{x} = \dfrac{x}{2} + 3,\ x^2 + 6x + 8 = 0\) \(x = -4\) and \(-2\) | M1, A1 |
| Three correct solutions (and no extras): \(-4,\ -2,\ -3 + \sqrt{17}\) | A1 |
| (5) |
Alternative using squaring method
| Scheme | Marks |
|---|---|
| Square both sides and attempt to find roots | M1 |
| \(x^4 + 12x^3 + 36x^2 - 64 = 0\) gives \(x = -2\) and \(x = -4\) | A1 |
| Obtain quadratic factor, divide find solutions of quadratic and obtain \((-3 \pm \sqrt{17})\) | M1 A1 |
| Last mark as before |
| Scheme | Marks |
|---|---|
| Line through point on −ve \(x\) axis and \(+\ y\) axis | B1 |
| Curve | B1 |
| 3 Intersections in correct quadrants | B1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(-4 < x < -2,\ x > -3 + \sqrt{17}\) o.e. | B1, B1 |
| (2) | |
| (10 marks) |
Notes
Use of \(\leqslant\) instead of \(<\) lose last B1 Extra inequalities lose last B1