The expression $\frac{(1+i)^{n}}{(1-i)^{n-2}}$ equals

If $z \in \mathbb{C}$,then the minimum value of $|z| + |2z - 3| + |z - 1|$ is

If $i z^3+z^2-z+i=0$ (where $z$ is a complex number),then the value of $|z|$ is

If $\omega$ is a complex cube root of unity,then $\left(\frac{1-\sqrt{3} i}{2}\right)^{2020}+\left(\frac{1+\sqrt{3} i}{2}\right)^{2026} +\sin \left(\sum_{j=1}^6(j+\omega)(j+\omega^2) \frac{3 \pi}{152}\right)=$

Let $f(x) = x^4 + ax^3 + bx^2 + c$ be a polynomial with real coefficients such that $f(1) = -9$. Suppose that $i\sqrt{3}$ is a root of the equation $4x^3 + 3ax^2 + 2bx = 0$,where $i = \sqrt{-1}$. If $\alpha_1, \alpha_2, \alpha_3$,and $\alpha_4$ are all the roots of the equation $f(x) = 0$,then $|\alpha_1|^2 + |\alpha_2|^2 + |\alpha_3|^2 + |\alpha_4|^2$ is equal to:

If ${x_n} = \cos \left( \frac{\pi }{3^n} \right) + i\sin \left( \frac{\pi }{3^n} \right)$,then the product ${x_1} \cdot {x_2} \cdot {x_3} \cdots {x_\infty }$ is equal to: