C4 June 2011 Q2
2.\[\mathrm{f}(x) = \frac{1}{\sqrt{(9+4x^2)}}, \qquad |x| \lt \frac{3}{2}\]
Find the first three non-zero terms of the binomial expansion of \(\mathrm{f}(x)\) in ascending powers of \(x\). Give each coefficient as a simplified fraction. (6)
| Scheme | Marks |
|---|---|
| \(\mathrm{f}(x) = (\ \ldots + \ldots\ )^{-\frac{1}{2}}\) | M1 |
| \(= 9^{-\frac{1}{2}}(\ \ldots + \ldots\ )^{\cdots}\) \(3^{-1},\ \dfrac{1}{3}\) or \(\dfrac{1}{9^{\frac{1}{2}}}\) | B1 |
| \((1 + kx^2)^n = 1 + nkx^2 + \ldots\) \(n\) not a natural number, \(k \neq 1\) | M1 |
| \((1 + kx^2)^{-\frac{1}{2}} = \ldots + \dfrac{\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{2}(kx^2)^2\) ft their \(k \neq 1\) | A1 ft |
| \(\left(1 + \dfrac{4}{9}x^2\right)^{-\frac{1}{2}} = 1 - \dfrac{2}{9}x^2 + \dfrac{2}{27}x^4\) | A1 |
| \(\mathrm{f}(x) = \dfrac{1}{3} - \dfrac{2}{27}x^2 + \dfrac{2}{81}x^4\) | A1 |
| (6) | |
| (6 marks) |