Complex Number — Class 12 Mathematics PYQs

If $\omega \ne 1$ is a cube root of unity, then what is $(1+\omega -{\omega }^2{)}^{100}+(1-\omega +{\omega }^2{)}^{100}$ equal to?

If
$$Z=\frac{1}{3}\left|\begin{array}{ccc}i& 2i& 1\\ 2i& 3i& 2\\ 3& 1& 3\end{array}\right|=x+iy;i=\sqrt{-1}$$
then what is modulus of $Z$ equal to?

If $\omega \ne 1$ is a cube root of unity, then what are the solutions of $(z-100{)}^3+1000=0$?

What is $(1+i{)}^4+(1-i{)}^4$ equal to, where $i=\sqrt{-1}$?

If $z\bar{z}=|z+\bar{z}|$, where $z=x+iy$, $i=\sqrt{-1}$, then the locus of $z$ is a pair of :

If $\omega$ is a non-real cube root of $1$, then what is the value of $\left|\frac{1-\omega }{\omega +{\omega }^2}\right|$?

What is the value of $\sqrt{12+5i}+\sqrt{12-5i}$, where $i=\sqrt{-1}$ ?

If $z=\frac{1+i\sqrt{3}}{1-i\sqrt{3}}$ where $i=\sqrt{-1}$, then what is the argument of $z$?

Which one of the following is a square root of $-\sqrt{-1}$ ?

If $z$ is a complex number such that $\frac{z-1}{z+1}$ is purely imaginary, then what is $|z|$ equal to?