If $x = 3 - 2\sqrt{3}i$,then $x^4 - 12x^3 + 54x^2 - 108x - 54 = $

Let $z = a + ib, b \neq 0$ be a complex number satisfying $z^{2} = \overline{z} \cdot 2^{1-|z|}$. Then the least value of $n \in N$ such that $z^{n} = (z + 1)^{n}$ is equal to:

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 the real part of the complex number $z = \frac{3 + 2i \cos \theta}{1 - 3i \cos \theta}$, where $\theta \in (0, \frac{\pi}{2})$, is zero, then the value of $\sin^2 3\theta + \cos^2 \theta$ is equal to:

If $i=\sqrt{-1}$,then $\sum_{n=0}^{\infty}\left(\frac{i}{3}\right)^n=$

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