8. A linear transformation of the plane is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 1 & - 2
\lambda & 3 \end{array} \right)\), where \(\lambda\) is a
constant.
- Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin.
- Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines.
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\section*{9. In this question you must show detailed reasoning.}
The complex number \(- 4 + \mathrm { i } \sqrt { 48 }\) is denoted by \(z\). - Determine the cube roots of \(z\), giving the roots in exponential form.
The points which represent the cube roots of \(z\) are denoted by \(A , B\) and \(C\) and these form a triangle in an Argand diagram.
- Write down the angles that any lines of symmetry of triangle \(A B C\) make with the positive real axis, justifying your answer.
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