The real part of ${e^{e^{i\theta }}}$ is

In the complex plane $\mathbb{C}$,the set $\{z \in \mathbb{C} : \arg \left(\frac{z-1}{z+1}\right) = \frac{\pi}{4}\}$ represents

If the amplitude of $z-2-3i$ is $\pi/4$,then the locus of $z=x+iy$ is

Let $O$ be the origin and $A$ be the point $z_{1} = 1 + 2i$. If $B$ is the point $z_{2}$ with $\operatorname{Re}(z_{2}) < 0$, such that $\triangle OAB$ is a right-angled isosceles triangle with $OB$ as the hypotenuse, then which of the following is $NOT$ true?

$A$ function $f$ is defined on the complex numbers by $f(z) = (a + ib)z$,where $a, b \in \mathbb{R}^+$. This function has the property that the $f$-image of any point in the complex plane is equidistant from that point and the origin. If $|a + bi| = 10$ and $b^2 = \frac{p}{q}$,where $p, q \in \mathbb{Z}$ and $\text{gcd}(p, q) = 1$,then $p + q$ is:

$PQ$ and $PR$ are two infinite rays. $QAR$ is an arc. $A$ point lying in the shaded region,excluding the boundary,satisfies: