If $x$ is numerically so small that $x^2$ and higher powers of $x$ can be neglected,then $\left(1+\frac{2 x}{3}\right)^{3 / 2} \cdot(32+5 x)^{-1 / 5}$ is approximately equal to

Arrange $Ce^{+3}, La^{+3}, Pm^{+3}$ and $Yb^{+3}$ in increasing order of their ionic radii.

$A$ point source emits sound equally in all directions in a non-absorbing medium. Two points $P$ and $Q$ are at a distance of $9 \ m$ and $25 \ m$ respectively from the source. The ratio of the amplitudes of the waves at $P$ and $Q$ is

$A$ body is projected vertically upwards from the surface of the earth with a velocity equal to half the escape velocity. If $R$ is the radius of the earth,the maximum height attained by the body from the surface of the earth is

Five capacitors,each of capacitance $C$,are connected as shown in the figure. The ratio of the equivalent capacitance between $P$ and $R$ to the equivalent capacitance between $P$ and $Q$ is:

The displacements of two particles of same mass executing $SHM$ are represented by the equations $x_1 = 4 \sin \left(10 t + \frac{\pi}{6}\right)$ and $x_2 = 5 \cos (\omega t)$. The value of $\omega$ for which the energies of both the particles remain same is (in $\text{ unit}$)