The matrix M is given by \(\mathbf{M} = \begin{pmatrix} a & b^2 \\ c^2 & a \end{pmatrix}\), where \(a, b, c\) are real constants and \(b \neq 0\).

1. Show that M does not represent a rotation about the origin.
2. Find the equations of the invariant lines, through the origin, of the transformation in the x–y plane represented by M.

It is given that M represents the sequence of two transformations in the x–y plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, x-axis fixed, with (0, 1) mapped to (5, 1).

3. Find M.
4. The triangle DEF in the x–y plane is transformed by M onto triangle PQR. Given that the area of triangle DEF is 12 cm², find the area of triangle PQR.