Which of the following is related to Cauchy-Riemann equations?

If z is non zero complex number such that $\mathrm{Re}\left(\mathrm{z}\right)=0$ , then _____ .

If z is non-zero complex number such that $\mathrm{Arg}\left(\mathrm{z}\right)=\frac{\mathrm{\pi }}{4}$ , then _____ .

Let $\mathrm{z}=\mathrm{x}+ \mathrm{i} \mathrm{y}$ be a complex number and $\overline{\mathrm{z}}=\mathrm{z}$ , then _____ .

Let $\mathrm{z}=\left|\begin{array}{cc}1& 1-\mathrm{i}\\ 1+\mathrm{i}& -2\end{array}\right|$ , then ______ .

Let $\mathrm{z}=\mathrm{x}+ \mathrm{i} \mathrm{y}$ be a complex number. If $\left|z+4\right|\le 3$ , then the maximum value of $\left|z+1\right|$ is _____ .

If $\left|w\right|=2$ , then the set of points $z=w-\frac{1}{w}$ lie on _____ .

The value of $\sum _{\mathrm{n}=1}^3\left[ {\mathrm{i}}^{\mathrm{n}} + {\mathrm{i}}^{\mathrm{n}+1} \right]$ , where $\mathrm{i}=\sqrt{-1}$ , equals to _____ .

The smallest positive integer n for which ${\left(\frac{1+\mathrm{i}}{1-\mathrm{i}}\right)}^{\mathrm{n}}=1$ is _____ .

If $\left|z_1\right|=1, \left|z_2\right|=1, z_1+\overline{z_2}$ is non-zero and $z=\frac{z_1 \overline{z_2} + 1}{z_1 + \overline{z_2}}$ , then conjugate of z equals to _____ .