A curve has polar equation \(r = a \tan \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\), where \(a\) is a positive constant.
Sketch the curve.
Find the area of the region between the curve and the line \(\theta = \frac { 1 } { 4 } \pi\). Indicate this region on your sketch.
Find the eigenvalues and corresponding eigenvectors for the matrix \(\mathbf { M }\) where
$$\mathbf { M } = \left( \begin{array} { l l }
0.2 & 0.8
0.3 & 0.7
\end{array} \right)$$
Give a matrix \(\mathbf { Q }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } = \mathbf { Q D } \mathbf { Q } ^ { - 1 }\).
Section B (18 marks)