Given that z is the complex number \(x + {\text{i}}y\) and that \(\left| {\,z\,} \right| + z = 6 - 2{\text{i}}\) , find the value of x

and the value of y .

\(\sqrt {{x^2} + {y^2}} + x + y{\text{i}} = 6 - 2{\text{i}}\)     (A1)

equating real and imaginary parts     M1

\(y = - 2\)     A1

\(\sqrt {{x^2} + 4} + x = 6\)     A1

\({x^2} + 4 = {(6 - x)^2}\)     M1

\( - 32 = - 12x \Rightarrow x = \frac{8}{3}\)     A1

[6 marks]

There were some good solutions to this question, but those who failed to complete the question failed at a variety of different points. Many did not know the definition of the modulus of a complex number and so could not get started at all. Many then did not think to equate real and imaginary parts, and then many failed to solve the resulting irrational equation to be able to find x.