2023 MI PU3 P2 Q2

Given that \(\mathrm{f}\left( x \right)={{\sec }^{2}}x\), find \(\mathrm{f}'\mathrm{ }\left( x \right)\) and \(\mathrm{f}''\mathrm{ }\left( x \right)\). Hence, find the Maclaurin series for \(\mathrm{f}\left( x \right)\), up to and including the term in \({x}^{2}\).[5]

Using standard series from the List of Formulae (MF27), expand \({{\left( 1+ax \right)}^{-\,4}}\) as far as the term in \({{x}^{2}}\), where \(a\) is a non- zero constant. Hence find the value of \(a\) for which the coefficients of the \(x\) and \({{x}^{2}}\) terms in the expansion are equal.[3]