4
\end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r }
0
- 8
3
\end{array} \right) .$$
- Show that \(\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}\) is a basis for \(\mathbb { R } ^ { 3 }\).
- Express \(\mathbf { d }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).
2 The roots of the equation
$$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$
are \(\alpha , 2 \alpha , 4 \alpha\), where \(p , q , r\) and \(\alpha\) are non-zero real constants. - Show that
$$2 p \alpha + q = 0$$
- Show that
$$p ^ { 3 } r - q ^ { 3 } = 0$$
3 The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } < 3\) and, for \(n \geqslant 1\),
$$u _ { n + 1 } = \frac { 4 u _ { n } + 9 } { u _ { n } + 4 }$$
- By considering \(3 - u _ { n + 1 }\), or otherwise, prove by mathematical induction that \(u _ { n } < 3\) for all positive integers \(n\).
- Show that \(u _ { n + 1 } > u _ { n }\) for \(n \geqslant 1\).
4 A curve is defined parametrically by
$$x = t - \frac { 1 } { 2 } \sin 2 t \quad \text { and } \quad y = \sin ^ { 2 } t$$
The arc of the curve joining the point where \(t = 0\) to the point where \(t = \pi\) is rotated through one complete revolution about the \(x\)-axis. The area of the surface generated is denoted by \(S\). - Show that
$$S = a \pi \int _ { 0 } ^ { \pi } \sin ^ { 3 } t \mathrm {~d} t$$
where the constant \(a\) is to be found.
- Using the result \(\sin 3 t = 3 \sin t - 4 \sin ^ { 3 } t\), find the exact value of \(S\).