Showing all working and without using a calculator, solve the equation
$$( 1 + \mathrm { i } ) z ^ { 2 } - ( 4 + 3 \mathrm { i } ) z + 5 + \mathrm { i } = 0$$
Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
The complex number \(u\) is given by
$$u = - 1 - \mathrm { i }$$
On a sketch of an Argand diagram show the point representing \(u\). Shade the region whose points represent complex numbers satisfying the inequalities \(| z | < | z - 2 \mathrm { i } |\) and \(\frac { 1 } { 4 } \pi < \arg ( z - u ) < \frac { 1 } { 2 } \pi\).