Complex Number

Area of the triangle in the argand diagram formed by the complex numbers $z, i z, z+i z$, where $z=x+i y$ is

The complex number $z=x+i y$ which satisfy the equation $\left|\frac{z-5 i}{z+5 i}\right|=1$, lie on

If $\omega$ is a cube root of unity, then $\left(1+\omega-\omega^2\right)\left(1-\omega+\omega^2\right)$ is

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

Let $z_1, z_2, z_3$ be three vertices of an equilateral triangle circumscribing the circle $|z|=\frac{1}{2}$. If $z_1=\frac{1}{2}+\frac{i \sqrt{3}}{2}$ and $z_1, z_2$, $z_3$ were in anticlockwise sense, then $z_2$ is

If $\mathbf{i} z^4+1=0$, then $z$ can take the value

If $z_1$ and $z_2$ are two complex numbers such that $\left|z_1\right|=\left|z_2\right|+\left|z_1-z_2\right|$, then

If $Z_1=1+i, Z_2=-2+3 i$ and $Z_3=\frac{a l}{3}$ are collinear, where $i^2=-1$, then the value of $a$ will be

If $\frac{(3-i)^2}{2+i}=A+i B$, where $A$ and $B$ are real numbers, then $A$ and $B$ are equal to

If $z=x+i y, z^{1 / 3}=a-i b$, then $\frac{x}{a}-\frac{y}{b}=k\left(a^2-b^2\right)$, where $k$ is equal to:

If $\cos \alpha+\cos \beta+\cos \gamma=\sin \alpha+\sin \beta+\sin \gamma=0$, then the value of $\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma$ is

If $(\sqrt{3}-i)^{50}=2^{48}(x-i y)$, then $x^2+y^2$ is equal to