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 the point $(x, y)$ satisfies the equation $\frac{x+i(x-2)}{3+i}-i=\frac{2y+i(1-3y)}{i-3}$,then $x+y=$

If $\cos \alpha+3 \cos 3 \beta+5 \cos 5 \gamma=0$,$\sin \alpha+3 \sin 3 \beta+5 \sin 5 \gamma=0$ and $\cos 3 \alpha+27 \cos 9 \beta+125 \cos 15 \gamma=\left(\lambda^2-4\right) \cos (\alpha+3 \beta+5 \gamma)$,then $\lambda$ is equal to

If $x=p+q$,$y=p \omega+q \omega^2$ and $z=p \omega^2+q \omega$,where $\omega$ is a complex cube root of unity,then $xyz$ is equal to

Let $\omega = e^{i \pi / 3}$,and $a, b, c, x, y, z$ be non-zero complex numbers such that $a+b+c = x$,$a+b \omega + c \omega^2 = y$,and $a+b \omega^2 + c \omega = z$. Then the value of $\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}$ is

If $z_{1}, z_{2}, z_{3}$ are complex numbers such that $|z_{1}|=|z_{2}|=|z_{3}|=|\frac{1}{z_{1}}+\frac{1}{z_{2}}+\frac{1}{z_{3}}|=1$, then $|z_{1}+z_{2}+z_{3}|$ is