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

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+3x^2-x-3=0$,then $(1+\alpha^2)(1+\beta^2)(1+\gamma^2) = $

For which value of $k$,are the roots of the equation $2x^2 + 5x + k = 0$ rational?

If both roots of the equation $x^2 - (p - 4)x + 2e^{2 \ln p} - 4 = 0$ are negative,then in which interval does $p$ lie?

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 $...........$

The number of real roots of the equation $5 + |2^x - 1| = 2^x(2^x - 2)$ is