3 A capacitor is an electrical component which stores charge. The value of the charge stored by the capacitor, in suitable units, is denoted by \(Q\). The capacitor is placed in an electrical circuit. At any time \(t\) seconds, where \(t \geqslant 0 , Q\) can be modelled by the differential equation \(\frac { \mathrm { d } ^ { 2 } Q } { \mathrm {~d} t ^ { 2 } } - 2 \frac { \mathrm {~d} Q } { \mathrm {~d} t } - 15 Q = 0\). Initially the charge is 100 units and it is given that \(Q\) tends to a finite limit as \(t\) tends to infinity.

1. Determine the charge on the capacitor when \(t = 0.5\).
2. Determine the finite limit of \(Q\) as \(t\) tends to infinity. The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 0.6 & 2.4
   - 0.8 & 1.8 \end{array} \right)\).
3. Find \(\operatorname { det } \mathbf { A }\). The matrix \(\mathbf { A }\) represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.
4. By considering the determinants of these transformations, determine the scale factor of the stretch.
5. Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer.
6. Find the angle of rotation.