Complex Numbers - Algebra - Mathematics Previous Year Questions

If $\omega$ is the complex cube root of unity and

$\left(\frac{a+b \omega+c \omega^{2}}{c+a \omega+b \omega^{2}}\right)^{k}+\left(\frac{a+b \omega+c \omega^{2}}{b+a \omega^{2}+c \omega}\right)^{l}=2$, then $2 k+l$ is always

If $n, K \in N$ such that $n \neq 3 K$, then $(\sqrt{3}+i)^{2 n}+(\sqrt{3}-i)^{2 n}=$

$\omega$ is a complex cube root of unity and $Z$ is a complex number satisfying $|Z-1| \leq 2$. The possible values of $r$ such that $|Z-1| \leq 2$ and $\left|\omega Z-1-\omega^2\right|=r$ have no common solution are

If $|Z|=2, Z_1=\frac{Z}{2} e^{i \alpha}$ and $\theta$ is the $\operatorname{amp}(Z)$, then $\frac{Z_1^n-Z_1^{-n}}{Z_1^n+Z_1^{-n}}=$

In argand plane, no value of $\sqrt[3]{1-i \sqrt{3}}$ lie in

If $\omega$ is a complex cube root of unity and $x=\omega^2-\omega+2$, then

Let $z=x+i y$ and $P(x, y)$ be a point on the argand plane. If $z$ satisfies the condition $\arg \left(\frac{z-3 i}{z+2 i}\right)=\frac{\pi}{4}$, then the locus of $P$ is