If $1, \alpha_1, \alpha_2, \ldots, \alpha_{n-1}$ are the $n^{\text{th}}$ roots of unity,then $\sum_{1 \leq i < j \leq n-1} \alpha_i \alpha_j =$

If $\omega_0, \omega_1, \ldots, \omega_{n-1}$ are the $n$-th roots of unity,then $(1+2 \omega_0)(1+2 \omega_1)(1+2 \omega_2) \ldots (1+2 \omega_{n-1})=$

The number of complex roots of the equation $x^{11}-x^7+x^4-1=0$ whose arguments lie in the first quadrant is

If $1, \omega, \omega^2$ are the cube roots of unity,then the roots of the equation $8z^3 - 12z^2 + 6z - 28 = 0$ are

If $x$ is a cube root of unity other than $1$,then $\left(x+\frac{1}{x}\right)^2+\left(x^2+\frac{1}{x^2}\right)^2+\ldots+\left(x^{12}+\frac{1}{x^{12}}\right)^2=$

If $\omega$ is an imaginary cube root of unity,then the value of $(1-\omega+\omega^{2}) \cdot(1-\omega^{2}+\omega^{4}) \cdot(1-\omega^{4}+\omega^{8}) \cdot \ldots$ ($2n$ factors) is