The gas that cannot be collected over water is

$\int_0^{\pi / 2} \frac{d x}{1+\tan ^3 x}$ is equal to :

Two identical containers $A$ and $B$ with frictionless pistons contain an ideal gas at the same temperature and the same volume $V$. The mass of the gas in $A$ is $m_A$ and in $B$ is $m_B$. The gas in each cylinder is now allowed to expand isothermally to the same final volume $2V$. The changes in pressure in $A$ and $B$ are found to be $\Delta P$ and $1.5\Delta P$ respectively. Then:

If $\overrightarrow{a} \cdot \hat{i} = \overrightarrow{a} \cdot (2 \hat{i} + \hat{j}) = \overrightarrow{a} \cdot (\hat{i} + \hat{j} + 3 \hat{k}) = 1$,then $\overrightarrow{a}$ is equal to :

In Young's double slit experiment,one of the slits is wider than the other,so that the amplitude of the light from one slit is double that from the other slit. If $I_m$ be the maximum intensity,the resultant intensity when they interfere at phase difference $\phi$ is given by:

Among the following pairs of complexes,in which case is the $\Delta_0$ value higher for the first one than the second?