8 A transformation T of the plane has matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { l l } \cos \theta & 2 \cos \theta - \sin \theta
\sin \theta & 2 \sin \theta + \cos \theta \end{array} \right)\).
- Show that T leaves areas unchanged for all values of \(\theta\).
- Find the value of \(\theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\), for which the \(y\)-axis is an invariant line of T .
The matrix \(\mathbf { N }\) is \(\left( \begin{array} { l l } 1 & 2
0 & 1 \end{array} \right)\). - Find \(\mathbf { M N } ^ { - 1 }\).
- Hence describe fully a sequence of two transformations of the plane that is equivalent to T .
\section*{END OF QUESTION PAPER}
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