The denominator of a fraction is $16$ more than the square of the numerator. Find the least value of the fraction.

$A$ cone of maximum volume is inscribed in a sphere of radius $R$. The ratio of the height of the cone to the diameter of the sphere is:

Let $f(x) = 2 \cos^{-1} x + 4 \cot^{-1} x - 3x^2 - 2x + 10$,where $x \in [-1, 1]$. If $[a, b]$ is the range of the function,then $4a - b$ is equal to:

Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t) = \frac{|t+1|}{t^2}, t < 0$,is strictly decreasing. Then the local maximum value of the function $g(x) = 2\log_e(x-2) + \alpha x^2 + 4x - \alpha, x > 2$,is

If the function $f(x)=2x^3-9ax^2+12a^2x+1, a>0$ has a local maximum at $x=\alpha$ and a local minimum at $x=\alpha^2$,then $\alpha$ and $\alpha^2$ are the roots of the equation:

$A$ rectangular sheet of tin $45 \, cm$ by $24 \, cm$ is to be made into a box without a top by cutting off a square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum (in $, cm$)?