$A$ wire of resistance $4\,\Omega$ is stretched to twice its original length. The resistance of the stretched wire would be ................ $\Omega$.

$A$ frictionless track $ABCDE$ ends in a circular loop of radius $R$. $A$ body slides down the track from point $A$ which is at a height $h = 5 \, cm$. The maximum value of $R$ for the body to successfully complete the loop is .................. $cm$.

If ${a^x} = {b^y} = {(ab)^{xy}},$ then $x + y = $

If $y = \sin(m \sin^{-1} x)$,then $(1 - x^2) y_2 - x y_1$ is equal to (Here,$y_n$ denotes $\frac{d^n y}{dx^n}$)

Focal length of a convex lens will be maximum for

Two monoatomic ideal gases $1$ and $2$ of molecular masses $M_1$ and $M_2$ respectively are enclosed in separate containers kept at the same temperature. The ratio of the speed of sound in gas $1$ to that in gas $2$ is