Let $z \neq -i$ be any complex number such that $\frac{z - i}{z + i}$ is a purely imaginary number. Then $z + \frac{1}{z}$ is

Let $S = \{ z \in \mathbb{C} : |z - 2| \leq 1, z(1 + i) + \overline{z}(1 - i) \leq 2 \}$. Let $|z - 4i|$ attain minimum and maximum values,respectively,at $z_1 \in S$ and $z_2 \in S$. If $5(|z_1|^2 + |z_2|^2) = \alpha + \beta \sqrt{5}$,where $\alpha$ and $\beta$ are integers,then the value of $\alpha + \beta$ is equal to

Let $z$ and $w$ be two non-zero complex numbers such that $|z| = |w|$ and $arg(z) + arg(w) = \pi$. Then $z$ is equal to:

The locus of the point $z$ satisfying $arg\left( \frac{z - 1}{z + 1} \right) = k$ (where $k$ is non-zero) is

Match the statements in column-$I$ with those in column-$II$.
[Note: Here $z$ takes the values in the complex plane and $\operatorname{Im} z$ and $\operatorname{Re} z$ denote,respectively,the imaginary part and the real part of $z$]

column-$I$	column-$II$
$(A)$ The set of points $z$ satisfying $|z-i|z||=|z+i|z||$ is contained in or equal to	$(p)$ an ellipse with eccentricity $\frac{4}{5}$
$(B)$ The set of points $z$ satisfying $|z+4|+|z-4|=10$ is contained in or equal to	$(q)$ the set of points $z$ satisfying $\operatorname{Im} z=0$
$(C)$ If $|\omega|=2$,then the set of points $z=\omega-1/\omega$ is contained in or equal to	$(r)$ the set of points $z$ satisfying $|\operatorname{Im} z| \leq 1$
$(D)$ If $|\omega|=1$,then the set of points $z=\omega+1/\omega$ is contained in or equal to	$(s)$ the set of points $z$ satisfying $|\operatorname{Re} z| \leq 1$
	$(t)$ the set of points $z$ satisfying $|z| \leq 3$

Let $S = \{z \in \mathbb{C} : z^{2} + \bar{z} = 0\}$. Then $\sum_{z \in S} (\operatorname{Re}(z) + \operatorname{Im}(z))$ is equal to $......$