Let $$S = \{z = x + iy: \frac{2z - 3i}{4z + 2i} \text{ is a real number}\}$$. Then which of the following is NOT correct?

$$\frac{2z-3i}{4z+2i}$$ is real. $$z = x+iy$$:

$$\frac{2x+i(2y-3)}{4x+i(4y+2)}$$. Rationalize: imaginary part of numerator × real of denom - real of num × imaginary of denom = 0.

$$(2y-3)(4x) - (2x)(4y+2) = 0$$

$$8xy - 12x - 8xy - 4x = 0$$

$$-16x = 0$$, so $$x = 0$$.

Also need $$4z + 2i \neq 0$$, i.e., $$4iy + 2i \neq 0$$, so $$y \neq -1/2$$.

S = {iy: y ∈ R, y ≠ -1/2}. Option 2 says (0, -1/2) ∈ S, but y = -1/2 is excluded.

The correct answer (NOT correct statement) is Option 2.