Let $\omega$ be a cube root of unity not equal to $1$. Then,the maximum possible value of $|a + b\omega + c\omega^2|$,where $a, b, c \in \{+1, -1\}$ is

If $\omega$ is an imaginary cube root of unity,then the value of $\sin \left[ (\omega^{10} + \omega^{23})\pi - \frac{\pi}{4} \right]$ is

Let $z = \left( \frac{\sqrt{3}}{2} + \frac{i}{2} \right)^5 + \left( \frac{\sqrt{3}}{2} - \frac{i}{2} \right)^5$. If $R(z)$ and $I(z)$ respectively denote the real and imaginary parts of $z$,then:

If $n$ is a positive integer, then $(1 + i\sqrt{3})^{2n} + (1 - i\sqrt{3})^{2n}$ is equal to .........

If $\text{cis } \alpha$ is the common value of $(-1)^{1/4}$ and $(-i)^{1/2}$,then $\tan \alpha = $

If $\omega ( \neq 1)$ is a cube root of unity and $(1 + \omega )^7 = A + B\omega$,then $A$ and $B$ are respectively,the numbers