If $(1 + x)^n = C_0 + C_1x + C_2x^2 + ..... + C_nx^n$,then the value of $C_0 - C_2 + C_4 - C_6 + .....$ is

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

If $z = \frac{1 + i\sqrt{3}}{\sqrt{3} + i}$,then $(\bar{z})^{100}$ lies in

Let $\omega=\operatorname{cis}\left(\frac{2 \pi}{3}\right)=\cos \left(\frac{2 \pi}{3}\right)+i \sin \left(\frac{2 \pi}{3}\right)$ and $f(x)=x^7-2 x^4-4 x^3+8$. Which of the following options is correct?

If $z = \sin \theta - i \cos \theta,$ then for any integer $n$

If $f(x)$ and $g(x)$ are two polynomials such that the polynomial $P(x) = f(x^3) + xg(x^3)$ is divisible by $x^2 + x + 1$,then $P(1)$ is equal to ....... .