A telegraph line of length \(100 \,km\) has a capacity of \(0.01\,\mu F / km\) and it carries an alternating current at \(0.5 \,kilo \,cycle \,per \,second\). If minimum impedance is required, then the value of the inductance that needs to be introduced in series is__________ \(m H\). \(\text { (if } \pi=\sqrt{10})\)

A block of mass \(2\,kg\) is attached with two identical springs of spring constant \(20\,N / m\) each. The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see figure). When the mass is displaced from its equilibrium position, it executes a simple harmonic motion. The time period of oscillation is \(\frac{\pi}{\sqrt{x}}\) in SI unit. The value of \(x\) is \(..........\)

The intensity of the light from a bulb incident on a surface is \(0.22 \,W / m ^{2}\). The amplitude of the magnetic field in this light-wave is_______ \(\times 10^{-9} \,T\). (Given : Permittivity of vacuum \(\epsilon_{0}=8.85 \times 10^{-12} \,C ^{2} N ^{-1} m ^{-2}\), speed of light in vacuum \(c =3 \times 10^{8} \,ms ^{-1}\) )

The engine of a train moving with speed \(10\,ms ^{-1}\) towards a platform sounds a whistle at frequency \(400\,Hz\). The frequency heard by a passenger inside the train is \(........\,Hz\) (neglect air speed. Speed of sound in air \(330\,ms ^{-1}\) )

The focal length of the objective and the eyepiece of a telescope are \(50\,cm\) and \(5\,cm\) respectively. If the telescope is focussed for distinct vision on a scale distant \(2\,m\) from its objective, then its magnifying power will be

The dimensions of angular momentum , latent heat and capacitance are, respectively.

In the given circuit diagram when the current reaches steady state in the circuit, the charge on the capacitor of capacitance \(C\) will be

A \(d.c.\) main supply of \(e.m.f.\, 220\, V\) is connected across a storage battery of \(e.m.f.\, 200\, V\) through a resistance of \(1\,\Omega \). The battery terminals are connected to an external resistance \('R'\) . The minimum value of \('R'\), so that a current passes through the battery to charge it is ............... \(\Omega\)

In the given electromagnetic wave \(E_y=600 \sin (\omega t-k x) \mathrm{Vm}^{-1}\), intensity of the associated light beam is what? (in \(\mathrm{W} / \mathrm{m}^2\) ); (Given \(\epsilon_0=\) \(\left.9 \times 10^{-12} \mathrm{C}^{-2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}\right)\)

A particle is moving in a circular path of radius \(a,\) with a constant velocity \(v\) as shown in the figure.The centre of circle is marked by \('C'\). The angular momentum from the origin \(O\) can be written as

To find the spring constant \((k)\) of a spring experimentally, a student commits \(2 \%\) positive error in the measurement of time and \(1 \%\) negative error in measurement of mass. The percentage error in determining value of \(\mathrm{k}\) is _______.

In \(LC\) circuit the inductance \(\mathrm{L}=40\; \mathrm{mH}\) and capacitance \(\mathrm{C}=100\; \mu \mathrm{F} .\) If a voltage \(\mathrm{V}(\mathrm{t})=10 \sin (314 \mathrm{t})\) is applied to the circuit, the current in the circuit is given as 

A balloon was moving upwards with a uniform velocity of \(10\, {m} / {s}\). An object of finite mass is dropped from the balloon when it was at a height of \(75\, {m}\) from the ground level. The height of the balloon from the ground when object strikes the ground was around.(In \({m}\)) (Takes the value of \(g\) as \(10 \,{m} / {s}^{2}\) )