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
$$\begin{aligned}
a _ { 1 } & = 3
a _ { n + 1 } & = \frac { a _ { n } - 3 } { a _ { n } - 2 } , \quad n \in \mathbb { N }
\end{aligned}$$
- Find \(\sum _ { r = 1 } ^ { 100 } a _ { r }\)
- 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. - 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 \(\theta = 5 ^ { \circ }\) - Identify the mistake made by Adele in her working.
- 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}