Complex Numbers · Mathematics · NDA

If $x, y$ and $z$ are the cube roots of unity, then what is the value of $xy + yz + zx$?

Let $z_1$ and $z_2$ be two complex numbers such that $\left|\frac{z_1 + z_2}{z_1 - z_2}\right| = 1$, then what is $\operatorname{Re} \left(\frac{z_1}{z_2}\right) + 1$ equal to?

If $\omega \neq 1$ 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 $i=\sqrt{-1}$?

What is the modulus of z?

What is angle θ such that z is purely real ?

where n is an integer

What is angle θ such that z is purely imaginary ?

where n is an integer

Consider the following in respect of a complex number z:

1. $\rm {\overline{\left(z^{-1}\right)}}=(\bar{z})^{-1}$

2. zz^-1 = |z|²

Which of the above is/are correct?

1. The difference of Z and its conjugate is an imaginary number.

2. The sum of Z and its conjugate is a real number.

The smallest positive integer n for which

$\rm \left(\frac{1-i}{1+i}\right)^{n^2}=1$

where i = √-1, is