FP2 June 2009 Q7
7.
(a) Sketch the graph of \(y = |x^2 - a^2|\), where \(a > 1\), showing the coordinates of the points where the graph meets the axes. (2)
(b) Solve \(|x^2 - a^2| = a^2 - x,\ a > 1\). (6)
(c) Find the set of values of \(x\) for which \(|x^2 - a^2| > a^2 - x,\ a > 1\). (4)
\(y = |x^2 - a^2|,\ a > 1\)
| Scheme | Marks |
|---|---|
| Correct Shape. Ignore cusps. | B1 |
| Correct coordinates. | B1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(|x^2 - a^2| = a^2 - x,\ a > 1\) | |
| \(\{|x| > a\},\quad x^2 - a^2 = a^2 - x\) \(x^2 - a^2 = a^2 - x\) | M1 aef |
| \(\Rightarrow x^2 + x - 2a^2 = 0\) | |
| \(\Rightarrow x = \dfrac{-1 \pm \sqrt{1 - 4(1)(-2a^2)}}{2}\) Applies the quadratic formula or completes the square in order to find the roots. | M1 |
| \(\Rightarrow x = \dfrac{-1 \pm \sqrt{1 + 8a^2}}{2}\) Both correct “simplified down” solutions. | A1 |
| \(\{|x| < a\},\quad -x^2 + a^2 = a^2 - x\) \(-x^2 + a^2 = a^2 - x\) or \(x^2 - a^2 = x - a^2\) | M1 aef |
| \(\{\Rightarrow x^2 - x = 0 \Rightarrow x(x - 1) = 0\}\) | |
| \(\Rightarrow x = 0, 1\) \(x = 0\) \(x = 1\) | B1 A1 |
| (6) |
| Scheme | Marks |
|---|---|
| \(|x^2 - a^2| > a^2 - x,\ a > 1\) | |
| \(x < \dfrac{-1 - \sqrt{1 + 8a^2}}{2}\) {or} \(x > \dfrac{-1 + \sqrt{1 + 8a^2}}{2}\) \(x\) is less than their least value \(x\) is greater than their maximum value | B1 ft B1 ft |
| {or} \(0 < x < 1\) For \(\{|x| < a\}\), Lowest \(< x <\) Highest \(0 < x < 1\) | M1 A1 |
| (4) | |
| (12 marks) |