If $z=3+5i$,then $z^3+\bar{z}+198$ is equal to

If ${z_r} = \cos \frac{{r\alpha }}{{{n^2}}} + i\sin \frac{{r\alpha }}{{{n^2}}}$,where $r = 1, 2, 3, \dots, n$,then $\mathop {\lim }\limits_{n \to \infty } {z_1}{z_2}{z_3} \dots {z_n}$ is equal to

Let $A = \left\{ \frac{1967 + 1686 i \sin \theta}{7 - 3 i \cos \theta} : \theta \in R \right\}$. If $A$ contains exactly one positive integer $n$,then the value of $n$ is

If $a, b, c$ and $d \in \mathbb{R}$ such that $a^2+b^2=4$ and $c^2+d^2=2$ and if $(a+ib)^2=(c+id)^2(x+iy)$,then $x^2+y^2$ is equal to

The values of $x$ for which $\sin x + i \cos 2x$ and $\cos x - i \sin 2x$ are conjugate to each other are

If $z=\cos 6^{\circ}+i \sin 6^{\circ}$,then $\sum_{n=1}^{20} \operatorname{Im}\left(z^{2 n-1}\right)=$