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

If $5 + ix^3y^2$ and $x^3 + y^2 + 6i$ are conjugate complex numbers and arg $(x + iy) = \theta $ , then ${\tan ^2}\,\theta $ is equal to

Let $\alpha=8-14 i , A=\left\{ z \in C : \frac{\alpha z -\bar{\alpha} \overline{ z }}{ z ^2-(\overline{ z })^2-112 i }=1\right\}$ and $B =\{ z \in C :| z +3 i |=4\}$ Then $\sum_{z \in A \cap B}(\operatorname{Re} z-\operatorname{Im} z)$ is equal to $...............$.

Find the modulus of $\frac{1+i}{1-i}-\frac{1-i}{1+i}$

For a non-zero complex number $z$, let $\arg ( z )$ denote the principal argument with $-\pi<\arg ( z ) \leq \pi$. Then, which of the following statement (s) is (are) $FALSE$ ?

$(A)$ $\arg (-1- i )=\frac{\pi}{4}$, where $i =\sqrt{-1}$

$(B)$ The function $f: R \rightarrow(-\pi, \pi]$, defined by $f(t)=\arg (-1+i t)$ for all $t \in R$, is continuous at all points of $R$, where $i=\sqrt{-1}$

$(C)$ For any two non-zero complex numbers $z_1$ and $z_2$, $\arg \left(\left(\frac{z_1}{z_2}\right)-\arg \left(z_1\right)+\arg \left(z_2\right)\right.$ is an integer multiple of $2 \pi$.

$(D)$ For any three given distinct complex numbers, $z_1, z_2$ and $z_3$, the locus of the point $z$ satisfying the condition $\arg \left(\frac{\left( z - z _1\right)\left( z _2- z _3\right)}{\left( z - z _3\right)\left( z _2- z _1\right)}\right)=\pi$, lies on a straight line

If a complex number $z$ statisfies the equation $x + \sqrt 2 \,\,\left| {z + 1} \right|\,+ \,i\, = \,0,$ then $\left| z \right|$ is equal to