If $z_n = (1 + i \sqrt{2})^n$, $n \in Z$, then $\frac{1}{9} \operatorname{Re}(z_4 \bar{z}_5) = $

Consider the following two statements:
Statement $I$: For any two non-zero complex numbers $z_1, z_2$,
$(\left|z_1\right|+\left|z_2\right|)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2(\left|z_1\right|+\left|z_2\right|)$
Statement $II$: If $x, y, z$ are three distinct complex numbers and $a, b, c$ are three positive real numbers such that $\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}$,then
$\frac{a^2}{y-z}+\frac{b^2}{z-x}+\frac{c^2}{x-y}=1$
Between the above two statements,

Which of the following are correct for any two complex numbers $z_1$ and $z_2$?

The product of the real roots of the equation $4x^4 - 24x^3 + 57x^2 + 18x - 45 = 0$,given that one of the roots is $3 + i\sqrt{6}$,is:

If $e^{it} = \cos t + i \sin t$ and $e^{-it} = \cos t - i \sin t$,then $\cosh(x + iy) - \cosh(x - iy) =$

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} = $