C4 June 2010 Q5
5. \[\frac{2x^2 + 5x - 10}{(x - 1)(x + 2)} \equiv A + \frac{B}{x - 1} + \frac{C}{x + 2}\]
(a) Find the values of the constants \(A\), \(B\) and \(C\). (4)
(b) Hence, or otherwise, expand \(\dfrac{2x^2 + 5x - 10}{(x - 1)(x + 2)}\) in ascending powers of \(x\), as far as the term in \(x^2\). Give each coefficient as a simplified fraction. (7)
| Scheme | Marks |
|---|---|
| \(A = 2\) | B1 |
| \(2x^2 + 5x - 10 = A(x - 1)(x + 2) + B(x + 2) + C(x - 1)\) | |
| \(x \to 1\) \(-3 = 3B \Rightarrow B = -1\) | M1 A1 |
| \(x \to -2\) \(-12 = -3C \Rightarrow C = 4\) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\dfrac{2x^2 + 5x - 10}{(x - 1)(x + 2)} = 2 + (1 - x)^{-1} + 2\left(1 + \dfrac{x}{2}\right)^{-1}\) | M1 |
| \((1 - x)^{-1} = 1 + x + x^2 + \ldots\) | B1 |
| \(\left(1 + \dfrac{x}{2}\right)^{-1} = 1 - \dfrac{x}{2} + \dfrac{x^2}{4} + \ldots\) | B1 |
| \(\dfrac{2x^2 + 5x - 10}{(x - 1)(x + 2)} = (2 + 1 + 2) + (1 - 1)x + \left(1 + \dfrac{1}{2}\right)x^2 + \ldots\) | M1 |
| \(= 5 + \ldots\) ft their \(A - B + \tfrac{1}{2}C\) | A1 ft |
| \(= \ldots + \dfrac{3}{2}x^2 + \ldots\) \(0x\) stated or implied | A1 A1 |
| (7) | |
| (11 marks) |