A2 June 2019 Paper 1 Q5
5 Using the Maclaurin series for \(\cos 2x\), show that, for small values of \(x\),
\[\sin^2 x \approx ax^2 + bx^4 + cx^6,\]where the values of \(a\), \(b\) and \(c\) are to be given in exact form. [5]
| Scheme | Marks | AO |
|---|---|---|
| \(\cos 2x = 1 - \dfrac{(2x)^2}{2!} + \dfrac{(2x)^4}{4!} - \dfrac{(2x)^6}{6!} + \ldots\) | M1 | 1.1a |
| \(= 1 - 2x^2 + \dfrac{2}{3}x^4 - \dfrac{4}{45}x^6 + \ldots\) | A1 | 1.1b |
| \(\sin^2 x = \frac{1}{2}(1 - \cos 2x)\) \(= \dfrac{1}{2}\left(1 - 1 + 2x^2 - \dfrac{2}{3}x^4 + \dfrac{4}{45}x^6 + \ldots\right)\) | M1 | 3.1a |
| \(= x^2 - \dfrac{1}{3}x^4 + \dfrac{2}{45}x^6 + \ldots\) so \(a = 1,\ b = -\dfrac{1}{3},\ c = \dfrac{2}{45}\) | A2,1,0 | 1.1b |
| [5] |
Notes
M1: at least 3 terms correct, or good attempt from 1st principles
A1: Allow unsimplified fractions (without factorials)