Problem

6. (i) $A = \begin{pmatrix} 1 & 0 \\ 0 & 3 \end{pmatrix}$

(a) Describe fully the single transformation represented by the matrix $A$.

The matrix $B$ represents a rotation of $45°$ clockwise about the origin.

(b) Write down the matrix $B$, giving each element of the matrix in exact form.

The transformation represented by matrix $A$ followed by the transformation represented by matrix $B$ is represented by the matrix $C$.

(c) Determine $C$.

(ii) The trapezium $T$ has vertices at the points $(-2, 0)$, $(-2, k)$, $(5, 8)$ and $(5, 0)$, where $k$ is a positive constant. Trapezium $T$ is transformed onto the trapezium $T'$ by the matrix

$\begin{pmatrix} 5 & 1 \\ -2 & 3 \end{pmatrix}$

Given that the area of trapezium $T'$ is $510$ square units, calculate the exact value of $k$.