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

If $(\sqrt{3}+i)^{100}=2^{99}(p+iq)$,then $p$ and $q$ are roots of the equation :

If $\alpha$ satisfies the equation $x^2+x+1=0$ and $(1+\alpha)^7=A+B\alpha+C\alpha^2$,where $A, B, C \geq 0$,then $5(3A-2B-C)$ is equal to:

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 $1, \omega, \omega^2, \omega^3, \dots, \omega^{n-1}$ are the $n^{th}$ roots of unity,then $(1 - \omega)(1 - \omega^2) \dots (1 - \omega^{n-1})$ equals

Let $A_r = \left(x+\frac{1}{x}\right)^3 \cdot \left(x^2+\frac{1}{x^2}\right)^3 \cdot \left(x^3+\frac{1}{x^3}\right)^3 \cdots \left(x^r+\frac{1}{x^r}\right)^3$. If $x^2+x+1=0$,then $\frac{1}{A_3}+\frac{1}{A_6}+\frac{1}{A_9}+\frac{1}{A_{12}}+\cdots \infty =$