The interval of $k$ for which the smaller root of the equation $x^2+kx-4=0$ lies in the interval $(-1, 2)$ is:

Solve the given two equations and select the correct answer from the given options.
$I.$ $\frac{9}{\sqrt{x}} + \frac{19}{\sqrt{x}} = \sqrt{x}$
$II.$ $y^{5} - \frac{(2 \times 14)^{11/2}}{\sqrt{y}} = 0$

If $\cos^4 \theta + \alpha$ and $\sin^4 \theta + \alpha$ are the roots of the equation $x^2 + 2bx + b = 0$,and $\cos^2 \theta + \beta$ and $\sin^2 \theta + \beta$ are the roots of the equation $x^2 + 4x + 2 = 0$,then find the value of $b$.

If $\alpha$ and $\beta$ are two roots of the equation $x^{2}-64x+256=0$,then the value of $\left(\frac{\alpha^{3}}{\beta^{5}}\right)^{\frac{1}{8}}+\left(\frac{\beta^{3}}{\alpha^{5}}\right)^{\frac{1}{8}}$ is

Solve the given two equations and select the correct answer from the given options.
$I. \quad x - 7\sqrt{x} + 12 = 0$
$II. \quad y - 5\sqrt{y} + 6 = 0$

Let $S$ be the set of all $\alpha \in R$ such that the equation $\cos 2x + \alpha \sin x = 2\alpha - 7$ has a solution. Then $S$ is equal to