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 and $(1+\omega)^7=A+B \omega$,then the values of $A$ and $B$ are,respectively.

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 $1, \omega, \omega^2$ are the cube roots of unity,then their product is

If $n$ is an integer which leaves a remainder of $1$ when divided by $3$,then $(1+\sqrt{3}i)^n + (1-\sqrt{3}i)^n$ equals

If $1, \alpha_1, \alpha_2, \alpha_3, \alpha_4$ are the roots of $z^5-1=0$ and $\omega$ is a cube root of unity,then $(\omega-1)(\omega-\alpha_1)(\omega-\alpha_2)(\omega-\alpha_3)(\omega-\alpha_4)+\omega$ is equal to