The modulus of the square root of the conjugate of $-7+24 \sqrt{-1}$ is ....

The value of $\left| \frac{1+i \sqrt{3}}{\left(1+\frac{1}{i+1}\right)^{2}} \right|$ is

If $a > 0$ and $z = \frac{(1 + i)^2}{a - i}$,where $i = \sqrt{-1}$,has magnitude $\sqrt{\frac{2}{5}}$,then $\overline{z}$ is equal to

Let $z_{1} = 2 - i$ and $z_{2} = -2 + i$. Find $\operatorname{Re}\left(\frac{z_{1} z_{2}}{\bar{z}_{1}}\right)$.

If $(\sqrt{8} + i)^{50} = 3^{49}(a + ib)$,then $a^2 + b^2$ is

If $z = \frac{1}{2} - 2i$ is such that $|z+1| = \alpha z + \beta(1+i)$,where $i = \sqrt{-1}$ and $\alpha, \beta \in \mathbb{R}$,then $\alpha + \beta$ is equal to