Complex Number — NDA Mathematics PYQs

What is ${\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)}^3$ equal to?

If $x^2-x+1=0$, then what is ${\left(x-\frac{1}{x}\right)}^2+{\left(x-\frac{1}{x}\right)}^4+{\left(x-\frac{1}{x}\right)}^8$ equal to?

If $z\ne 0$ is a complex number, then what is $\text{amp}(z)+\text{amp}(\bar{z})$ equal to?

If $\alpha =\frac{-1+\sqrt{-3}}{2}$, then what is the value of $(1+{\alpha }^{19}-{\alpha }^{35}{)}^{100}-(1-3{\alpha }^{25}+{\alpha }^{38}{)}^{50}$?

If ${\left(\frac{1-i}{1+i}\right)}^{2m}{\left(\frac{1+i}{1-i}\right)}^{2n}=1$ where $i=\sqrt{-1}$, then what is the smallest positive value of $(m-n)$?

Let $1,\omega ,{\omega }^2$ be three cube roots of unity. If $x=a+b$, $y=a\omega +b{\omega }^2$, $z=a{\omega }^2+b\omega$, then what is $x^2+y^2+z^2$ equal to?