6 The curve \(C\) has polar equation \(r ^ { 2 } = \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right)\), where \(0 \leqslant \theta \leqslant 2\).
- Sketch \(C\) and state, in exact form, the greatest distance of a point on \(C\) from the pole.
- Find the exact value of the area of the region bounded by \(C\) and the half-line \(\theta = 2\).
Now consider the part of \(C\) where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). - Show that, at the point furthest from the half-line \(\theta = \frac { 1 } { 2 } \pi\),
$$\left( \theta ^ { 2 } + 4 \right) \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right) \sin \theta - \cos \theta = 0$$
and verify that this equation has a root between 0.6 and 0.7 .
\(7 \quad\) The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 3
4 & k & 6
7 & 8 & 9 \end{array} \right)\). - Find the set of values of \(k\) for which \(\mathbf { A }\) is non-singular.
- Given that \(\mathbf { A }\) is non-singular, find, in terms of \(k\), the entries in the top row of \(\mathbf { A } ^ { - 1 }\).
- Given that \(\mathbf { B } = \left( \begin{array} { l l l } 1 & 0 & 0
0 & 1 & 0 \end{array} \right)\), give an example of a matrix \(\mathbf { C }\) such that \(\mathbf { B A C } = \left( \begin{array} { l l } 2 & 1
k & 4 \end{array} \right)\). - Find the set of values of \(k\) for which the transformation in the \(x - y\) plane represented by \(\left( \begin{array} { l l } 2 & 1
k & 4 \end{array} \right)\) has two distinct invariant lines through the origin.
If you use the following page to complete the answer to any question, the question number must be clearly shown.