The conjugate of the complex number $\frac{2 - 3i}{4 - i}$ is

If $Z = \alpha + i \beta$ satisfies the equation $|Z| - Z = 1 + 2i$ and $|Z| = \sqrt{\alpha^2 + \beta^2}$,then $Z \bar{Z} = $

If $z$ is a complex number such that $|z| \geqslant 2$,then the minimum value of $|z + \frac{1}{2}|$ is:

If $x$ and $y$ are real numbers,which of the following is always true?

Find the conjugate of $\frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)}$.

If $(1 + i) \cdot (1 + 2i) \dots \dots \dots (1 + ni) = x + iy$ (where $i = \sqrt{-1}$), then find the value of $(2) \cdot (5) \cdot (10) \dots \dots \dots (1 + n^2)$.