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 a non-real root of the equation $x^3 - 1 = 0$,then the value of $\sum_{r=1}^{5} (1 + \omega^r + \omega^{2r})$ is

If $z^2 + z + 1 = 0$,where $z$ is a complex number,then the value of $\left( z + \frac{1}{z} \right)^2 + \left( z^2 + \frac{1}{z^2} \right)^2 + \left( z^3 + \frac{1}{z^3} \right)^2 + \dots + \left( z^6 + \frac{1}{z^6} \right)^2$ is

If ${x_r} = \cos \left( \frac{\pi }{2^r} \right) + i\sin \left( \frac{\pi }{2^r} \right)$,then the product ${x_1} \cdot {x_2} \cdot {x_3} \cdots \infty$ is:

Let $a = \cos 1^{\circ}$ and $b = \sin 1^{\circ}$. We say that a real number is algebraic if it is a root of a polynomial with integer coefficients. Then,

Let $\alpha, \beta$ be the roots of the equation $x^2-\sqrt{2} x+2=0$. Then $\alpha^{14}+\beta^{14}$ is equal to