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 

For the given circuit, in the steady state, \(\left| V _{ B }- V _{ D }\right|=.......V.\)

A cylinder of mass \(M_c\) and sphere of mass \(M_s\) are placed at points \(A\) and \(B\) of two inclines, respectively (See Figure). If they roll on the incline without sipping such that their accelerations are the same, then the ratio \(\frac{{\sin \,{\theta _c}}}{{\sin \,{\theta _s}}}\) is

One of the two small circular coils, (none of them having any self- inductance) is suspended with a \(V-\) shaped copper wire, with plane horizontal. The other coil is placed just below the first one with plane horizontal. Both the coils are connected in series with a \(dc\) supply. The coils are found to attract each other with a force. Which one of the following statements is incorrect ?

The equivalent resistance between \(A\) and \(B\) as shown in figure is \(........\,k\Omega\)

A particle starts from origin at \(t=0\) with a velocity \(5 \hat{i} \mathrm{~m} / \mathrm{s}\) and moves in \(x-y\) plane under action of a force which produces a constant acceleration of \((3 \hat{i}+2 \hat{j}) \mathrm{m} / \mathrm{s}^2\). If the \(x\)-coordinate of the particle at that instant is \(84 \mathrm{~m}\), then the speed of the particle at this time is \(\sqrt{\alpha} \mathrm{m} / \mathrm{s}\). The value of \(\alpha\) is___________.

The distance between charges \(+\mathrm{q}\) and \(-\mathrm{q}\) is \(2 l\) and between \(+2 \mathrm{q}\) and \(-2 \mathrm{q}\) is \(4 l\). The electrostatic potential at point \(P\) at a distance \(r\) from centre \(O\) is \(-\alpha\left[\frac{q l}{r^2}\right] \times 10^9 \mathrm{~V}\), where the value of \(\alpha\) is _______. (Use \(\left.\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-2}\right)\)

The height of victoria falls is \(63\, {m}\). What is the difference in temperature of water at the top and at the bottom of fall ? (In \(^{\circ} {C}\)) [Given \(1\, {cal}=4.2\, {J}\) and specific heat of water \(\left.=1 \,{cal} \,{g}^{-1}\;{ }^{\circ}\, {C}^{-1}\right]\)

Consider two ideal diatomic gases \(\mathrm{A}\) and \(\mathrm{B}\) at some temperature \(T\). Molecules of the gas \(A\) are rigid, and have a mass \(m\). Molecules of the gas \(\mathrm{B}\) have an additional vibrational mode, and have a mass \(\frac{\mathrm{m}}{4} .\) The ratio of the specific heats \((\mathrm{C}_{\mathrm{v}}^{\mathrm{A}}\) and \(\mathrm{C}_{\mathrm{v}}^{\mathrm{B}})\) of gas \(\mathrm{A}\) and \(\mathrm{B}\), respectively is