If $x = 3 + i$,then $x^3 - 3x^2 - 8x + 15 = $

The number of complex numbers $z$ satisfying $\overline{z} = i z^2$ is

If the set $R = \{(a, b) : a + 5b = 42, a, b \in N\}$ has $m$ elements and $\sum_{n=1}^m (1 - i^{n!}) = x + iy$,where $i = \sqrt{-1}$,then the value of $m + x + y$ is:

The complex numbers $\sin x + i \cos 2x$ and $\cos x - i \sin 2x$ (where $i = \sqrt{-1}$) are conjugate to each other for:

For a complex number $Z = a + ib$,let $\hat{Z} = b + ia$. If $Z_1$ and $Z_2$ are such complex numbers,then $\widehat{Z_1 Z_2} = $

If $\alpha+i \beta$ and $\gamma+i \delta$ are the roots of $x^2-(3-2 i) x-(2 i-2)=0$,where $i=\sqrt{-1}$,then $\alpha \gamma+\beta \delta$ is equal to :