Showing all necessary working, solve the equation \(\mathrm { i } z ^ { 2 } + 2 z - 3 \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact.
On a sketch of an Argand diagram, show the locus representing complex numbers satisfying the equation \(| z | = | z - 4 - 3 \mathrm { i } |\).
Find the complex number represented by the point on the locus where \(| z |\) is least. Find the modulus and argument of this complex number, giving the argument correct to 2 decimal places.