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- 2 \end{array} \right) .$$ 6 It is given that \(y = \mathrm { e } ^ { x } u\), where \(u\) is a function of \(x\). The \(r\) th derivatives \(\frac { \mathrm { d } ^ { r } y } { \mathrm {~d} x ^ { r } }\) and \(\frac { \mathrm { d } ^ { r } u } { \mathrm {~d} x ^ { r } }\) are denoted by \(y ^ { ( r ) }\) and \(u ^ { ( r ) }\) respectively. Prove by mathematical induction that, for all positive integers \(n\), $$y ^ { ( n ) } = \mathrm { e } ^ { x } \left( \binom { n } { 0 } u + \binom { n } { 1 } u ^ { ( 1 ) } + \binom { n } { 2 } u ^ { ( 2 ) } + \ldots + \binom { n } { r } u ^ { ( r ) } + \ldots + \binom { n } { n } u ^ { ( n ) } \right)$$ [You may use without proof the result \(\binom { k } { r } + \binom { k } { r - 1 } = \binom { k + 1 } { r }\).]
7 Let $$S _ { N } = \sum _ { r = 1 } ^ { N } ( 3 r + 1 ) ( 3 r + 4 ) \quad \text { and } \quad T _ { N } = \sum _ { r = 1 } ^ { N } \frac { 1 } { ( 3 r + 1 ) ( 3 r + 4 ) } .$$

1. Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = N \left( 3 N ^ { 2 } + 12 N + 13 \right)$$
2. Use the method of differences to show that $$T _ { N } = \frac { 1 } { 12 } - \frac { 1 } { 3 ( 3 N + 4 ) } .$$
3. Deduce that \(\frac { S _ { N } } { T _ { N } }\) is an integer.
4. Find \(\lim _ { N \rightarrow \infty } \frac { S _ { N } } { N ^ { 3 } T _ { N } }\).
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5. By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 6 }\), where \(z = \cos \theta + i \sin \theta\), express \(\cos ^ { 6 } \theta\) in the form $$\frac { 1 } { 32 } ( p + q \cos 2 \theta + r \cos 4 \theta + s \cos 6 \theta ) ,$$ where \(p , q , r\) and \(s\) are integers to be determined.
6. Hence find the exact value of $$\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 6 } \left( \frac { 1 } { 2 } x \right) \mathrm { d } x$$