A capacitor of capacitance \(100\  \mu \mathrm{F}\) is charged to a potential of \(12  \mathrm{~V}\) and connected to a \(6.4 \ \mathrm{mH}\) inductor to produce oscillations. The maximum current in the circuit would be :

A sphere of radius \('a'\) and mass \('m'\) rolls along a horizontal plane with constant speed \(v_{0}\). It encounters an inclined plane at angle \(\theta\) and climbs upward. Assuming that it rolls without slipping, how far up the sphere will travel?

Two blocks of masses \(3 \,{kg}\) and \(5\, {kg}\) are connected by a metal wire going over a smooth pulley. The breaking stress of the metal is \(\frac{24}{\pi} \times 10^{2}\, {Nm}^{-2}\). What is the minimum radius of the wire? (Take \(\left.g=10\, {ms}^{-2}\right)\) (in \(cm\))

For a transverse wave travelling along a straight line, the distance between two peaks (crests) is \(5 \,m ,\) while the distance between one crest and one trough is \(1.5 \,m\) The possible wavelengths (in \(m\) ) of the waves are

Nucleus a having \(Z=17\) and equal number of protons and neutrons has \(1.2\,MeV\) binding energy per nucleon.Another nucleus \(B\) of \(Z=12\) has total \(26\) nucleons and \(1.8\,MeV\) binding energy per nucleons.The difference of binding energy of \(B\) and \(A\) will be \(...........MeV\).

If the maximum value of accelerating potential provided by a ratio frequency oscillator is \(12\, {kV}\). The number of revolution made by a proton in a cyclotron to achieve one sixth of the speed of light is ....... . \(\left[{m}_{{p}}=1.67 \times 10^{-27} {kg}, {e}=1.6 \times 10^{-19} {C},\right.\) Speed of light \(\left.=3 \times 10^{8} {m} / {s}\right]\)

Given below are two statements :
Statement (I) : The dimensions of Planck's constant and angular momentum are same.
Statement (II) : In Bohr's model electron revolve around the nucleus only in those orbits for which angular momentum is integral multiple of Planck's constant.
In the light of the above statements, choose the most appropriate answer from the options given below :

An \(\alpha\)-particle, a proton and an electron have the same kinetic energy. Which one of the following is correct in case of their De-Broglie wavelength:

The explosive in a Hydrogen bomb is a mixture of \({ }_1 \mathrm{H}^2,{ }_1 \mathrm{H}^3\) and \({ }_3 \mathrm{Li}^6\) in some condensed form. The chain reaction is given by \( { }_3 \mathrm{Li}^6+{ }_0 \mathrm{n}^1 \rightarrow{ }_2 \mathrm{He}^4+{ }_1 \mathrm{H}^3 \) \( { }_1 \mathrm{H}^2+{ }_1 \mathrm{H}^3 \rightarrow{ }_2 \mathrm{He}^4+{ }_0 \mathrm{n}^1\) During the explosion the energy released is approximately [Given : \(\mathrm{M}(\mathrm{Li})=6.01690\  \mathrm{amu} . \mathrm{M}\left({ }_1 \mathrm{H}^2\right)=2.01471 \  amu.\) \(\mathrm{M}\left({ }_2 \mathrm{He}^4\right)=4.00388\  \mathrm{amu}\),\( and\ \)\(1 \ \mathrm{amu}=931.5\) \(\mathrm{MeV}]\)

Two conductors have the same resistances at \(0^{\circ} \mathrm{C}\) but their temperature coefficients of resistance are \(\alpha_1\) and \(\alpha_2\). The respective temperature coefficients for their series and parallel combinations are _______.

A series combination of resistor of resistance \(100\,\Omega\), inductor of inductance \(1\,H\) and capacitor of capacitance \(6.25\,\mu F\) is connected to an ac source. The quality factor of the circuit will be \(.............\).