If $z = \sqrt{2} - i\sqrt{2}$ is rotated through an angle $45^{\circ}$ in the anti-clockwise direction about the origin,then the coordinates of its new position are

If complex numbers $z_1$ and $z_2$ both satisfy $z + \overline{z} = 2 |z - 1|$ and $\arg(z_1 - z_2) = \frac{\pi}{3},$ then the value of $\text{Im}(z_1 + z_2)$ is,where $\text{Im}(z)$ denotes the imaginary part of $z$.

For $a \in \mathbb{C}$, let $A = \{z \in \mathbb{C} : \operatorname{Re}(a + \bar{z}) > \operatorname{Im}(\bar{a} + z)\}$ and $B = \{z \in \mathbb{C} : \operatorname{Re}(a + \bar{z}) < \operatorname{Im}(\bar{a} + z)\}$. Then among the two statements:
$(S1) : \text{If } \operatorname{Re}(a), \operatorname{Im}(a) > 0, \text{ then the set } A \text{ contains all the real numbers.}$
$(S2) : \text{If } \operatorname{Re}(a), \operatorname{Im}(a) < 0, \text{ then the set } B \text{ contains all the real numbers.}$

Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + iy$ $(x, y \in \mathbb{R}, i = \sqrt{-1})$ of the equation $sz + t\bar{z} + r = 0$,where $\bar{z} = x - iy$. Then,which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ If $L$ has exactly one element,then $|s| \neq |t|$
$(B)$ If $|s| = |t|$,then $L$ has infinitely many elements
$(C)$ The number of elements in $L \cap \{z : |z - 1 + i| = 5\}$ is at most $2$
$(D)$ If $L$ has more than one element,then $L$ has infinitely many elements

If $P$ is a complex number whose modulus is $1$,then the equation $\left(\frac{1+iz}{1-iz}\right)^4=P$ has

If $|z_1|=1, |z_2|=2, |z_3|=3$ and $|9z_1z_2 + 4z_1z_3 + z_2z_3| = 12$,then the value of $|z_1 + z_2 + z_3|$ is equal to :-