The frequencies at which the current amplitude in an \(LCR\) series circuit becomes \(\frac{1}{\sqrt{2}}\) times its maximum value, are \(212\,rad\,s ^{-1}\) and \(232\,rad\,s ^{-1}\). The value of resistance in the circuit is \(R =5 \Omega\). The self inductance in the circuit is \(.........mH\)

Three blocks \(\mathrm{M}_1, \mathrm{M}_2, \mathrm{M}_3\) having masses \(4 \mathrm{~kg}, 6 \mathrm{~kg}\) and \(10 \mathrm{~kg}\) respectively are hanging from a smooth pully using rope 1,2 and 3 as shown in figure. The tension in the rope \(1, \mathrm{~T}_1\) when they are moving upward with acceleration of \(2 \mathrm{~ms}^{-2}\) is _______ \(\mathrm{N}\) (if \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\) )

The velocity (\(v\)) versus time (\(t\)) plot of a particle is shown in the figure, for a time interval of \(40\text{ s}\). The total distance travelled by the particle and the average velocity during this period are, respectively _______.

A small toy starts moving from the position of rest under a constant acceleration. If it travels a distance of \(10 \,m\) in \(t \;s ,\) the distance travelled by the toy in the next \(t s\) will be .........\(m\)

The threshold frequency of a metal with work function \(6.63\  \mathrm{eV}\) is:

A person observes two moving trains, '\(A\)' reaching the station and '\(B\)' leaving the station with equal speed of \(30\,m / s\). If both trains emit sounds with frequency \(300\,Hz\), (Speed of sound : \(330\,m / s\) ) approximate difference of frequencies heard by the person will be \(..........Hz\)

If \(\varepsilon_0\) is the permittivity of free space and \(E\) is the electric field, then \(\varepsilon_0 E^2\) has the dimensions is _______.

A sportsman runs around a circular track of radius \(r\) such that he traverses the path \(A B A B\). The distance travelled and displacement, respectively, are

Pressure inside a soap bubble is greater than the pressure outside by an amount _______. (given : \(\mathrm{R}=\) Radius of bubble, \(\mathrm{S}=\) Surface tension of bubble)

On heating water, bubbles being formed at the bottom of the vessel detach and rise. Take the bubbles to be spheres of radius \(R\) and making a circular contact of radius \(r\) with the bottom of the vessel. If \(r < < R\), and the surface tension of water is \(T\), value of \(r\) just before bubbles detach is (density of water is \(\rho_{w}\))

In the Bohr 's model of hydrogen-like atom the force between the nucleus and the electron is modified as \(F = \frac{{{e^2}}}{{4\pi {\varepsilon _0}}}\left( {\frac{1}{{{r^2}}} + \frac{\beta }{{{r^3}}}} \right)\), where \(\beta \) is a constant. For this atom, the radius of the \(n^{th}\) orbit in terms of the Bohr radius \(\left( {{a_0} = \,\frac{{{\varepsilon _0}{h^2}}}{{m\pi {e^2}}}} \right)\) is

Two wires \(A\) and \(B\) are carrying currents \(I_1\) and \(I_2\) as shown in the figure. The separation between them is \(d\). A third wire \(C\) carrying a current \(I\) is to be kept parallel to them at a distance \(x\) from \(A\) such that the net force acting on it is zero. The possible values of \(x\) are