Let $\omega=\mathrm{z} \overline{\mathrm{z}}+\mathrm{k} 1 \mathrm{z}+\mathrm{k} 2 \mathrm{z}+\lambda(1+\mathrm{j}), \mathrm{k}_1, \mathrm{k} 2 \in \mathbb{R}$. Let $\operatorname{Re}(\omega)=0$ be the circle C of radius 1 in the first quadrant...

$$ \text { If the set }\left\{\operatorname{Re}\left(\frac{z-\bar{z}+z \bar{z}}{2-3 z+5 \bar{z}}\right): z \in \mathbb{C}, \operatorname{Re}(z)=3\right\} \text { is equal to the interval }(\alpha, \beta] \text {,...

Let C be the circle in the complex plane with centre $z_0=\frac{1}{2}(1+3 i)$ and radius $r=1$. Let $z_1=1+i$ and the complex number $z 2$ be...

If for $z=\alpha+i \beta=|z+2|=z+4(1+i)$, then $\alpha+\beta$ and $\alpha \beta$ are the roots of the equation

Let w1 be the point obtained by the rotation of z1 = 5 + 4i about the origin through a right angle in the anticlockwise...

Let the complex number $z=x+i y$ be such that $\frac{2 z-3 i}{2 z+i}$ is purely imaginary. If $x+y^2=0$, then $y^4+y^2-y$ is equal ton

The sum of the square of the modulus of the elements in the set {z=a+ib:a,b∈Z,z∈C,|z-1|≤1,|z-5|≤|z-5i|} is

If the set $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}) ; \mathrm{a}+5 \mathrm{~b}=42, \mathrm{a}, \mathrm{b} \in \mathbb{N}\}$ has $m$ elements and $\sum_{n=1}^m\left(1+i^{n!}\right)=x+i y$, where $\mathrm{I}=\sqrt{-1}$, then the value of $m+x+y$...

Let $z$ be a complex number such that $|z+2|=1$ and $\operatorname{lm}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$. Then the value of $|\operatorname{Re}(\overline{z+2})|$ is :

Consider the following two statements
Statement I: For any two non-zero complex numbers $z_1, z_2\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)$ and
Statement II: If $x, y, z$ are...