5 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & - \frac { 1 } { 2 } \sqrt { 2 } \\ \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right) \left( \begin{array} { c c } 1 & k \\ 0 & 1 \end{array} \right)$, where $k$ is a constant.\\
(a) The matrix $\mathbf { M }$ represents a sequence of two geometrical transformations.

State the type of each transformation, and make clear the order in which they are applied.\\

(b) The triangle $A B C$ in the $x - y$ plane is transformed by $\mathbf { M }$ onto triangle $D E F$.

Find, in terms of $k$, the single matrix which transforms triangle $D E F$ onto triangle $A B C$.\\

(c) Find the set of values of $k$ for which the transformation represented by $\mathbf { M }$ has no invariant lines through the origin.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2022 Q5}}