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

Suppose $z$ is any root of $11 z^8 + 21 i z^7 + 10 i z - 22 = 0$ where $i = \sqrt{-1}$. Then,$S = |z|^2 + |z| + 1$ satisfies

If $(3 + i)z = (3 - i)\bar{z}$,then the complex number $z$ is

Assertion $(A)$: If $z$ is a complex number such that $|z| \geq 3$,then the least value of $|z + \frac{3}{z}|$ is $1$.
Reason $(R)$: $|z_1 - z_2| \leq |z_1| + |z_2|$,for any two complex numbers $z_1, z_2$.
The correct option among the following is:

The product of the real roots of the equation $4x^4 - 24x^3 + 57x^2 + 18x - 45 = 0$,given that one of the roots is $3 + i\sqrt{6}$,is:

The value of the infinite product $(\cos \theta + i\sin \theta )(\cos \frac{\theta }{2} + i\sin \frac{\theta }{2})(\cos \frac{\theta }{2^2} + i\sin \frac{\theta }{2^2}) \dots$ is