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

For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^4 - |z|^4 = 4iz^2$, where $i = \sqrt{-1}$. Then the minimum possible value of $|z_1 - z_2|^2$, where $z_1, z_2 \in S$ with $\operatorname{Re}(z_1) > 0$ and $\operatorname{Re}(z_2) < 0$, is:

The number of solutions for $z^3+\bar{z}=0$ is

If $(3 + i)z = (3 - i)\bar{z}$,then the complex number $z$ is

In a cubic equation,the coefficient of $x^2$ is $0$ and the remaining coefficients are real. If one root is $\alpha = 3 + 4i$,and the remaining roots are $\beta$ and $\gamma$,then find the value of $\alpha \beta \gamma$.

If $a = \cos (2\pi /7) + i\sin (2\pi /7)$,then the quadratic equation whose roots are $\alpha = a + a^2 + a^4$ and $\beta = a^3 + a^5 + a^6$ is