The ratio of rates of diffusion of gases $X$ and $Y$ is $1:5$ and that of $Y$ and $Z$ is $1:6$. The ratio of rates of diffusion of $Z$ and $X$ is

At a certain temperature and at infinite dilution,the equivalent conductances of sodium benzoate,hydrochloric acid,and sodium chloride are $240$,$349$,and $229 \ \Omega^{-1} \ cm^2 \ \text{equiv}^{-1}$ respectively. The equivalent conductance of benzoic acid in $\Omega^{-1} \ cm^2 \ \text{equiv}^{-1}$ at the same conditions is:

$A$ vertical bar magnet is dropped from a position on the axis of a fixed metallic coil as shown in fig-$I$. In fig-$II$,the magnet is fixed and a horizontal coil is dropped. If the acceleration of the magnet and the coil are $a_1$ and $a_2$ respectively,then:

$\sqrt[3]{61 - 46\sqrt{5}} = $

The solution of $\frac{dy}{dx} = \frac{x \log x^2 + x}{\sin y + y \cos y}$ is

The negation of the conditional statement: "If it rains,$I$ shall go to school" is