If $x = \frac{1}{2} \left( \sqrt{7} + \frac{1}{\sqrt{7}} \right)$, then $\frac{\sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}}$ is equal to

$\alpha$ is the maximum value of $1-2x-5x^2$ and $\beta$ is the minimum value of $x^2-2x+r$. If $5\alpha x^2+\beta x+6>0$ for all real values of $x$,then the interval in which $r$ lies is

The number of ordered pairs $(x, y)$ of positive integers satisfying $2^x + 3^y = 5^{xy}$ is

Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2x^2 + (\cos \theta)x - 1 = 0$,where $\theta \in (0, 2\pi)$. If $m$ and $M$ are the minimum and the maximum values of $\alpha_\theta^4 + \beta_\theta^4$,then $16(M + m)$ equals:

If a polynomial $P(x)$ of degree $4$ is given by $P(x) = 2x^4 + ax^3 + bx^2 + cx + d$ such that $P(1) = 4, P(2) = 7, P(3) = 12$,and $P(4) = 19$,then find the value of $P(5)$.

The number of integral values of $a$ for which $x^2 - (a - 1)x + 3 = 0$ has both roots positive and $x^2 + 3x + 6 - a = 0$ has both roots negative is