Complex Number — NDA Mathematics PYQs

Consider the following inequalities :
I. $1+4i>3+2i$
II. $2+3i<3+4i$
III. $4+3i>3+4i$
where $i=\sqrt{-1}$
How many of the above are valid ?

Let $Z_1$ and $Z_2$ be complex numbers such that $\frac{3Z_1}{4Z_2}$ is purely imaginary. What is $\left|\frac{Z_1+Z_2}{Z_1-Z_2}\right|$ equal to ?

If $\alpha$, $\beta$, $\gamma$ are cube roots of $-8$, then what is $\frac{{\alpha }^2{p}^2+{\beta }^2{q}^2+{\gamma }^2{r}^2}{{\beta }^2{p}^2+{\gamma }^2{q}^2+{\alpha }^2{r}^2}$ equal to ?

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?

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