If $z_1 = 2 - 3i$ and the roots of the equation $z^3 + bz^2 + cz + d = 0$ are $i$,$z_1$,and $\bar{z}_1$,then $b + c + d =$

The least value of $|z|$ where $z$ is a complex number satisfying the inequality $\exp \left(\frac{(|z|+3)(|z|-1)}{|z|+1} \log _{ e } 2\right) \geq \log _{\sqrt{2}}|5 \sqrt{7}+9 i |$,where $i=\sqrt{-1}$,is equal to:

The expression $\frac{(1+i)^{n}}{(1-i)^{n-2}}$ equals

If $a, b \in \mathbb{R}$ and $i=\sqrt{-1}$,then the number of ordered pairs of real numbers $(a, b)$ satisfying the condition $(a+bi)^3 = a-bi$ is

If $\frac{2z_1}{3z_2}$ is a purely imaginary number,then $\left| \frac{z_1 - z_2}{z_1 + z_2} \right| =$

Among the statements:
$(S1) :$ The set $\{z \in \mathbb{C} - \{-i\} : |z|=1 \text{ and } \frac{z-i}{z+i} \text{ is purely real}\}$ contains exactly two elements,and
$(S2) :$ The set $\{z \in \mathbb{C} - \{-1\} : |z|=1 \text{ and } \frac{z-1}{z+1} \text{ is purely imaginary}\}$ contains infinitely many elements.