If $(a + \sqrt{2} b \cos x)(a - \sqrt{2} b \cos y) = a^2 - b^2$,where $a > b > 0$,then $\frac{dx}{dy}$ at $(\frac{\pi}{4}, \frac{\pi}{4})$ is

If the angle $2 \theta$ is acute,then the acute angle between the pair of straight lines $x^2(\cos \theta - \sin \theta) + 2xy \cos \theta + y^2(\cos \theta + \sin \theta) = 0$ is:

If $\sin A + \sin B = \sqrt{3}(\cos B - \cos A)$,then $\sin 3A + \sin 3B$ is equal to:

$A$ transverse wave is described by the equation $y = y_0 \sin 2\pi \left( ft - \frac{x}{\lambda} \right)$. The maximum particle velocity is equal to four times the wave velocity if:

The refractive index of a material of a plano-concave lens is $5/3$,and the radius of curvature is $0.3 ~m$. The focal length of the lens in air is (in $~m$)

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: