2020 ACJC P1 Q3

It is given that \(y=\mathrm{f}\left( x \right)\), where \(\mathrm{f}\left( x \right)={{\mathrm{e}}^{3x}}\sec 2x\).

1. Show that \(\mathrm{{f}'}\left( x \right)=y\left( 3+2\tan 2x \right)\).[1]
2. Find the Maclaurin series for \(\mathrm{f}\left( x \right)\), up to and including the term in \({{x}^{2}}\).[3]
3. Hence find the first two non-zero terms in the Maclaurin series for \(\left( {{\mathrm{e}}^{3x}}\sec 2x \right)\left( 3+2\tan 2x \right)\).[1]
4. Using the standard series from the List of Formulae (MF26), verify the result found in part (ii).[2]