$\frac{\sqrt{5 + 12i} + \sqrt{5 - 12i}}{\sqrt{5 + 12i} - \sqrt{5 - 12i}} = $

The value of $4x^3 - 4x^2 - 7x + 127$ when $x = \frac{4 + 5i}{2}$ is

If $\cos (u + iv) = \alpha + i\beta ,$ then ${\alpha ^2} + {\beta ^2} + 1$ equals

If $z_1=1-2 i$,$z_2=1+i$,and $z_3=3+4 i$,then $\left(\frac{1}{z_1}+\frac{3}{z_2}\right) \frac{z_3}{z_2}=$

Let $Z$ and $W$ be complex numbers such that $|Z| = |W|$,and $\text{arg } Z$ denotes the principal argument of $Z$.
Statement $1$: If $\text{arg } Z + \text{arg } W = \pi$,then $Z = -\overline{W}$.
Statement $2$: $|Z| = |W|$ implies $\text{arg } Z - \text{arg } \overline{W} = \pi$.

Convert the complex number $z = \frac{i-1}{\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}}$ into polar form.