\begin{enumerate}
  \item (a) Prove by induction that for $n \in \mathbb { Z } ^ { + }$
\end{enumerate}

$$\left( \begin{array} { l l } 
1 & r \\
0 & 2
\end{array} \right) ^ { n } = \left( \begin{array} { c c } 
1 & \left( 2 ^ { n } - 1 \right) r \\
0 & 2 ^ { n }
\end{array} \right)$$

where $r$ is a constant.

$$\mathbf { M } = \left( \begin{array} { l l } 
4 & 0 \\
0 & 5
\end{array} \right) \quad \mathbf { N } = \left( \begin{array} { r r } 
1 & - 2 \\
0 & 2
\end{array} \right) ^ { 4 }$$

The transformation represented by matrix $\mathbf { M }$ followed by the transformation represented by matrix $\mathbf { N }$ is represented by the matrix $\mathbf { B }$\\
(b) (i) Determine $\mathbf { N }$ in the form $\left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$ where $a , b , c$ and $d$ are integers.\\
(ii) Determine B

Hexagon $S$ is transformed onto hexagon $S ^ { \prime }$ by matrix $\mathbf { B }$\\
(c) Given that the area of $S ^ { \prime }$ is 720 square units, determine the area of $S$

\hfill \mbox{\textit{Edexcel F1 2024 Q6 [9]}}