Match series to function form

2 Given that the first three terms of the Maclaurin series for \(( 1 + \sin x ) \mathrm { e } ^ { 2 x }\) are identical to the first three terms of the binomial series for \(( 1 + a x ) ^ { n }\), find the values of the constants \(a\) and \(n\). (You may use appropriate results given in the List of Formulae (MF1).)

2

1. Write down the expansion of \(\sin 2 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 5 }\).
2. It is given that the first non-zero term in the expansion of $$\sin 2 x - 2 x \left( 1 - p x ^ { 2 } \right) \left( 1 - x ^ { 2 } \right) ^ { - 1 }$$ in ascending powers of \(x\) is \(q x ^ { 5 }\).
   Find the values of the rational numbers \(p\) and \(q\).

3 Which of the following functions has the fourth term \(- \frac { 1 } { 720 } x ^ { 6 }\) in its Maclaurin series expansion? Circle your answer.
[0pt] [1 mark]
\(\sin x\)
\(\cos x\)
\(\mathrm { e } ^ { x }\)
\(\ln ( 1 + x )\)

8 You are given that \(\mathrm { f } ( x ) = ( 1 - a \sin x ) \mathrm { e } ^ { b x }\) where \(a\) and \(b\) are positive constants. The first three terms in the Maclaurin expansion of \(\mathrm { f } ( x )\) are \(1 + 2 x + \frac { 3 } { 2 } x ^ { 2 }\).

1. Find the value of \(a\) and the value of \(b\).
2. Explain if there is any restriction on the value of \(x\) in order for the expansion to be valid.

2 The first two non-zero terms of the Maclaurin series expansion of \(\mathrm { f } ( x )\) are \(x\) and \(- \frac { 1 } { 2 } x ^ { 3 }\) Which one of the following could be \(\mathrm { f } ( x )\) ?
Circle your answer.
\(x \mathrm { e } ^ { \frac { 1 } { 2 } x ^ { 2 } }\)
\(\frac { 1 } { 2 } \sin 2 x\)
\(x \cos x\)
\(\left( 1 + x ^ { 3 } \right) ^ { - \frac { 1 } { 2 } }\)