$A(1,15), B(3,-12), C(6,12)$ are three consecutive turning points of a continuous curve $y=f(x)$. If $f(x)=0$ only for $x=\alpha$ and $x=\beta$,then $|\beta-\alpha| < $

$A$ cylindrical tank without a top lid is being manufactured to hold a fixed volume of $125\pi \text{ cm}^3$. The minimum surface area required to construct this tank is ............ $\text{cm}^2$. (in $\pi$)

The maximum value of $\frac{\log x}{x}, 0 < x < \infty$ is

The height of a right circular cone of maximum volume inscribed in a sphere of diameter $a$ is-

The minimum value of $e^{(2x^2 - 2x - 1)\sin^2 x}$ is .......

The radius of the cylinder of maximum volume,which can be inscribed in a sphere of radius $R$ is :-