\begin{enumerate}
  \item (i)
\end{enumerate}

$$\mathbf { P } = \left( \begin{array} { r r } 
0 & - 1 \\
- 1 & 0
\end{array} \right)$$

The matrix $\mathbf { P }$ represents a geometrical transformation $U$\\
(a) Describe $U$ fully as a single geometrical transformation.

The transformation $V$, represented by the $2 \times 2$ matrix $\mathbf { Q }$, is a rotation through $240 ^ { \circ }$ anticlockwise about the origin followed by an enlargement about ( 0,0 ) with scale factor 6\\
(b) Determine the matrix $\mathbf { Q }$, giving each entry in exact numerical form.

Given that $U$ followed by $V$ is the transformation $T$, which is represented by the matrix $\mathbf { R }$\\
(c) determine the matrix $\mathbf { R }$\\
(ii) The transformation $W$ is represented by the matrix

$$\left( \begin{array} { c c } 
- 2 & 2 \sqrt { 3 } \\
2 \sqrt { 3 } & 2
\end{array} \right)$$

Show that there is a real number $\lambda$ for which $W$ maps the point $( \lambda , 1 )$ onto the point ( $4 \lambda , 4$ ), giving the exact value of $\lambda$

$\_\_\_\_$ VIAV SIHI NI JIIHM ION OC\\
VILU SIHI NI JLIYM ION OC\\
VEYV SIHI NI ELIYM ION OC

\hfill \mbox{\textit{Edexcel F1 2023 Q7 [11]}}