The work function of a certain metal is $3.31 \times 10^{-19} \, J$. The maximum kinetic energy of photoelectrons emitted by incident radiation of wavelength $5000 \, \text{Å}$ is: (Given: $h = 6.62 \times 10^{-34} \, J \cdot s$, $c = 3 \times 10^8 \, m/s$, $e = 1.6 \times 10^{-19} \, C$) (in $\text{eV}$)

$A$ non-conducting body floats in a liquid at $20^{\circ} C$ with $\frac{2}{3}$ of its volume immersed in the liquid. When the liquid temperature is increased to $100^{\circ} C$, $\frac{3}{4}$ of the body's volume is immersed in the liquid. Then the coefficient of real expansion of the liquid is (neglecting the expansion of the container of the liquid):

The sum of the infinite series ${\tan ^{ - 1}}\left( {\frac{2}{{1 - {1^2} + {1^4}}}} \right) + {\tan ^{ - 1}}\left( {\frac{4}{{1 - {2^2} + {2^4}}}} \right) + {\tan ^{ - 1}}\left( {\frac{6}{{1 - {3^2} + {3^4}}}} \right) + \dots$ is:

If the two circles $2x^2 + 2y^2 - 3x + 6y + k = 0$ and $x^2 + y^2 - 4x + 10y + 16 = 0$ cut orthogonally,then the value of $k$ is

Two soap bubbles of radii $r_1 = 4 \, cm$ and $r_2 = 5 \, cm$ are touching each other over a common surface $S_1S_2$ (as shown in the figure). The radius of curvature of this common surface will be ....... $cm$.

There is a small hole in a hollow sphere. Water enters it when it is taken to a depth of $40 \,cm$ under water. The surface tension of water is $0.07 \,N/m$. The diameter of the hole is .......... $mm$.