4 \end{array} \right) \quad \overrightarrow { B C } = \left( \begin{array} { r } q
- 3 p
2 \end{array} \right)$$ where \(p\) and \(q\) are constants.
Given that \(\overrightarrow { A C }\) is parallel to \(\left( \begin{array} { r } 3
- 4
3 \end{array} \right)\), find the value of \(p\) and the value of \(q\).
2. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by $$\begin{aligned} a _ { 1 } & = 3
a _ { n + 1 } & = \frac { a _ { n } - 3 } { a _ { n } - 2 } , \quad n \in \mathbb { N } \end{aligned}$$

1. Find \(\sum _ { r = 1 } ^ { 100 } a _ { r }\)
2. Hence find \(\sum _ { r = 1 } ^ { 100 } a _ { r } + \sum _ { r = 1 } ^ { 99 } a _ { r }\)
   3. Using algebraic integration and making your method clear, find the exact value of $$\int _ { 1 } ^ { 5 } \frac { 4 x + 9 } { x + 3 } d x = a + \ln b$$ where \(a\) and \(b\) are constants to be found
   (4)
   4.
3. Given that \(\theta\) is small, use the small angle approximation for \(\cos \theta\) to show that $$1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }$$ Adele uses \(\theta = 5 ^ { \circ }\) to test the approximation in part (a). Adele's working is shown below. \begin{displayquote} Using my calculator, \(1 + 4 \cos \left( 5 ^ { \circ } \right) + 3 \cos ^ { 2 } \left( 5 ^ { \circ } \right) = 7.962\), to 3 decimal places.
   Using the approximation \(8 - 5 \theta ^ { 2 }\) gives \(8 - 5 ( 5 ) ^ { 2 } = - 117\) \end{displayquote} \begin{displayquote} Therefore, \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta \approx 8 - 5 \theta ^ { 2 }\) is not true for \(\theta = 5 ^ { \circ }\)
4. 1. Identify the mistake made by Adele in her working.
   2. Show that \(8 - 5 \theta ^ { 2 }\) can be used to give a good approximation to \(1 + 4 \cos \theta + 3 \cos ^ { 2 } \theta\) for an angle of size \(5 ^ { \circ }\) \end{displayquote}