If \(z\) is any complex number and \(iz^3 + z^2 - z + i = 0\), where \(i = \sqrt{-1}\), then what is the value of \((|z|+1)^2\)?

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 \(\text{Re}\left(\frac{z_1}{z_2}\right)+1\) 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 \(\omega \neq 1\) is a cube root of unity, then what is \((1 + \omega - \omega^2)^{100} + (1 - \omega + \omega^2)^{100}\) equal to?

What is the value of the sum
\(\sum_{n=1}^{20}(i^{n-1} + i^n + i^{n+1})\)
where \(i = \sqrt{-1}\)?

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

What is the value of \(\left|\frac{Z_1}{Z_2}\right|\)?

What is the value of \(\frac{1}{2} + \text{Re}\left(\frac{Z_1}{Z_2}\right)\)?

Consider the following complex numbers
$$ z_1 = r_1(\cos \theta_1 + i \, \sin \theta_1) $$
$$ z_2 = r_2(\cos \theta_2 + i \, \sin \theta_2) $$
where $r_1, r_2$ are real numbers, $0 \leq \theta_1 \leq \frac{\pi}{2}$, $0 \leq \theta_2 \leq \frac{\pi}{2}$, and $i = \sqrt{-1}$
If $|z_1 + z_2| = |z_1| + |z_2|$, which one of the following conditions is necessarily CORRECT?

What is the value of \((2 - 3i) + (3 + 5i)\)

What is the real part of the number \(15 + 2i\)

What is the real part of the number $5i$?