1 The complex number \(z _ { 1 }\) is \(1 + \mathrm { i }\) and the complex number \(z _ { 2 }\) has modulus 4 and argument \(\frac { \pi } { 3 }\).
- Express \(z _ { 2 }\) in the form \(a + b \mathrm { i }\), giving \(a\) and \(b\) in exact form.
- Express \(\frac { z _ { 2 } } { z _ { 1 } }\) in the form \(c + d i\), giving \(c\) and \(d\) in exact form.
- Describe fully the transformation represented by the matrix \(\left( \begin{array} { l l } 1 & 2
0 & 1 \end{array} \right)\). - A triangle of area 5 square units undergoes the transformation represented by the matrix \(\left( \begin{array} { l l } 1 & 2
0 & 1 \end{array} \right)\).
Explaining your reasoning, find the area of the image of the triangle following this transformation.