4 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l } \mathrm { a } & \mathrm { b } ^ { 2 } \\ \mathrm { c } ^ { 2 } & \mathrm { a } \end{array} \right)$, where $a , b , c$ are real constants and $b \neq 0$.\\
(a) Show that $\mathbf { M }$ does not represent a rotation about the origin.\\

(b) Find the equations of the invariant lines, through the origin, of the transformation in the $x - y$ plane represented by $\mathbf { M }$.\\

It is given that $\mathbf { M }$ 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 $( 5,1 )$.\\
(c) Find $\mathbf { M }$.\\

(d) The triangle $D E F$ in the $x - y$ plane is transformed by $\mathbf { M }$ onto triangle $P Q R$.

Given that the area of triangle $D E F$ is $12 \mathrm {~cm} ^ { 2 }$, find the area of triangle $P Q R$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2023 Q4}}