$A$ calcite crystal is placed over a dot on a piece of paper and rotated. On seeing through the calcite,one will see:

$\int_0^1 \sin \left(2 \tan ^{-1} \sqrt{\frac{1+x}{1-x}}\right) d x$ is equal to :

$A$ Zener diode,having a breakdown voltage equal to $15\, V$,is used in a voltage regulator circuit as shown in the figure. The current through the Zener diode is.....$mA$.

$A$ mixture of $2$ moles of helium gas (atomic mass $= 4$ $amu$) and $1$ mole of argon gas (atomic mass $= 40$ $amu$) is kept at $300$ $K$ in a container. The ratio of the rms speeds $(\frac{V_{rms}(helium)}{V_{rms}(argon)})$ is

........ of a reaction cannot be determined experimentally.

The inverse point of $(1,2)$ with respect to the circle $x^2+y^2-4x-6y+9=0$ is