Problem

7. In part (i), the elements of each matrix should be expressed in exact numerical form.

(i) (a) Write down the $2 \times 2$ matrix that represents a rotation of $210°$ anticlockwise about the origin.

(b) Write down the $2 \times 2$ matrix that represents a stretch parallel to the $y$-axis with scale factor $5$

The transformation $T$ is a rotation of $210°$ anticlockwise about the origin followed by a stretch parallel to the $y$-axis with scale factor $5$

(c) Determine the $2 \times 2$ matrix that represents $T$

(ii) $M = \begin{pmatrix} k & k+3 \\ -5 & 1-k \end{pmatrix}$ where $k$ is a constant

(a) Find $\det M$, giving your answer in simplest form in terms of $k$.

A closed shape $R$ is transformed to a closed shape $R'$ by the transformation represented by the matrix $M$.

Given that the area of $R$ is $2$ square units and that the area of $R'$ is $16k$ square units,

(b) determine the possible values of $k$.