N2024 P1 Q8

It is given that \(y = \cos(1 - \mathrm{e}^{2x})\).

1. Show that \(\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = k\left(\frac{\mathrm{d}y}{\mathrm{d}x} - 2y\mathrm{e}^{4x}\right)\), where \(k\) is a constant to be found.[3]

2. By differentiation of the result in part (a), find the first three non-zero terms of the Maclaurin expansion of \(\cos(1 - \mathrm{e}^{2x})\).

   [4]

3. The first two non-zero terms of the Maclaurin expansion of \(\cos(1 - \mathrm{e}^{2x})\) are equal to the first two non-zero terms of the series expansion of \(\frac{1}{\sqrt{a + bx^2}}\), where \(a\) and \(b\) are constants.

   Using standard series from the List of Formulae (MF26), find the values of \(a\) and \(b\).

   [2]