Explanation

(b)
Let $z = x + iy$
Given, $|z| - z = 1 + 2i$
$\Rightarrow \ \ \ \ \ \sqrt{x^{2} + y^{2}} - (x + iy) = 1 + 2i$
$\Rightarrow \ \ \ \sqrt{x^{2} + y^{2}} - x = 1,\ \ \ y = - 2$
$\Rightarrow \ \ \ \sqrt{x^{2} + 4} - x = 1$
$\Rightarrow \ \ \ x^{2} + 4 = (1 + x)^{2}$
$\Rightarrow \ \ \ 2x = 3\ \ \ \Rightarrow \ \ x = \frac{3}{2}$
$\therefore\ \ \ \ \ z = \frac{3}{2} - 2i$