Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a = z^2 + z + 1$ is real. Then $a$ cannot take the value

The sum of all distinct roots of the equation $x^5-3x^4+5x^3-5x^2+3x-1=0$ is

Let $p, q \in \{1, 2, 3, 4\}$. The number of equations of the form $px^2 + qx + 1 = 0$ having real roots is

Let $S = \{ \alpha : \log_2(9^{2\alpha-4} + 13) - \log_2(\frac{5}{2} \cdot 3^{2\alpha-4} + 1) = 2 \}$. Then the maximum value of $\beta$ for which the equation $x^2 - 2(\sum_{\alpha \in S} \alpha)^2 x + \sum_{\alpha \in S} (\alpha+1)^2 \beta = 0$ has real roots,is $...........$

If $1$ is a multiple root of order $3$ for the equation $x^4-2x^3+2x-1=0$,then the other root is

If $x = \sqrt{\frac{2+\sqrt{3}}{2-\sqrt{3}}}$,then $x^2(x-4)^2$ is equal to :