If $\log _{27}(\log _3 x) = \frac{1}{3}$, then the value of $x$ is

The centroid of the triangle formed by the pair of straight lines $12x^2 - 20xy + 7y^2 = 0$ and the line $2x - 3y + 4 = 0$ is:

The figure shows a transparent sphere of radius $R$ and refractive index $\mu$. An object $O$ is placed at a distance $x$ from the pole of the first surface so that a real image is formed at the pole of the exactly opposite surface. If $x = \infty$,then the value of $\mu$ is

$\frac{1}{2 \cdot 3} + \frac{1}{4 \cdot 5} + \frac{1}{6 \cdot 7} + \frac{1}{8 \cdot 9} + \dots$ is equal to

During the depression of freezing point in a solution,which of the following are in equilibrium?

If $\int \frac{\sqrt{1-x^2}}{x^4} ~d x=A(x)\left(\sqrt{1-x^2}\right)^m+C$ for a suitable chosen integer $m$ and a function $A(x)$,where $C$ is a constant of integration,then $(A(x))^m$ equals