4 The matrix $\mathbf { M }$ represents the sequence of two transformations in the $x - y$ plane given by a rotation of $60 ^ { \circ }$ anticlockwise about the origin followed by a one-way stretch in the $x$-direction, scale factor $d ( d \neq 0 )$.\\
(a) Find $\mathbf { M }$ in terms of $d$.\\

(b) The unit square in the $x - y$ plane is transformed by $\mathbf { M }$ onto a parallelogram of area $\frac { 1 } { 2 } d ^ { 2 }$ units ${ } ^ { 2 }$. Show that $d = 2$.\\

The matrix $\mathbf { N }$ is such that $\mathbf { M N } = \left( \begin{array} { l l } 1 & 1 \\ \frac { 1 } { 2 } & \frac { 1 } { 2 } \end{array} \right)$.\\
(c) Find $\mathbf { N }$.\\

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

\hfill \mbox{\textit{CAIE Further Paper 1 2021 Q4}}