Shown in the figure is rigid and uniform one meter long rod \(AB\) held in horizontal position by two strings tied to its ends and attached to the ceiling. The rod is of mass \('m'\) and has another weight of mass \(2m\) hung at a distance of \(75\, cm\) from \(A\). The tension in the string at \(A\) is\(....mg\)

Match list-\(I\) with list-\(II\):

List-\(I\)	List-\(II\)
\((A)\)Kinetic energy of plant	\((I)\) \(-\frac{\mathrm{GMm}}{\mathrm{a}}\)
\((B)\) Gravaitatioin potentiyal energy of sun -plant system	\((II)\) \(\frac{\mathrm{GMm}}{2 \mathrm{a}}\)
\((C)\)Total mecaniacal energy of palnt	\((III)\) \(\frac{\mathrm{Gm}}{\mathrm{r}}\)
\((D)\)Escap energyat the surface of plant for unit mass object	\((IV)\) \(-\frac{\mathrm{GMm}}{2 \mathrm{a}}\)

(Where \(\mathrm{a}=\) radius of planet orbit, \(\mathrm{r}=\) radius of planet, \(M=\) mass of Sun, \(m=\) mass of planet)
Choose the correct answer from the options given below:

A parallel plate capacitor consisting of two circular plates of radius 10 cm is being charged by a constant current of 0.15 A . If the rate of change of potential difference between the plates is \(7 \times 10^8 \mathrm{~V} / \mathrm{s}\) then the integer value of the distance between the parallel plates is \(\left(\right.\) Take, \(\left.\epsilon_0=9 \times 10^{-12} \frac{\mathrm{~F}}{\mathrm{~m}}, \pi=\frac{22}{7}\right)\) ________ \(\mu \mathrm{m}\).

In the figure given, the electric current flowing through the \(5\, k \Omega\) resistor is \('x' mA.\) The value of \(x\) to the nearest integer is .........

A butterfly is flying with a velocity \(4 \sqrt{2} \,{m} / {s}\) in North-East direction. Wind is slowly blowing at \(1\) \({m} / {s}\) from North to South. The resultant displacement of the butterfly in \(3\, seconds\) is \(....\,{m}.\)

Due to cold weather a \(1\, {m}\) water pipe of cross-sectional area \(1\, {cm}^{2}\) is filled with ice at \(-10^{\circ} {C}\). Resistive heating is used to melt the ice. Current of \(0.5\, {A}\) is passed through \(4\, {k} \Omega\) resistance. Assuming that all the heat produced is used for melting, what is the minimum time required ? (In \({s}\)) (Given latent heat of fusion for water/ice \(=3.33 \times 10^{5}\, {J} {kg}^{-1}\), specific heat of ice \(=2 \times 10^{3}\, {J}\) \({kg}^{-1}\) and density of ice \(=10^{3}\, {kg} / {m}^{3}\)

A closed vessel contains \(0.1\) mole of a monoatomic ideal gas at \(200\, K\). If \(0.05\) mole of the same gas at \(400\, K\) is added to it, the final  equilibrium temperature (in \(K\) ) of the gas in the vessel will be closed to

At a certain depth "\(d\)" below surface of earth. value of acceleration due to gravity becomes four times that of its value at a height \(3\,R\) above earth surface. Where \(R\) is Radius of earth (Take \(R =6400\,km\) ). The depth \(d\) is equal to \(............\,km\)

The volume \(V\) of a given mass of monoatomic gas changes with temperature \(T\) according to the relation \(V = KT ^{2 / 3}\). The workdone when temperature changes by \(90\, K\) will be \(x\,R\). The value of \(x\) is \([ R =\) universal gas constant \(]\)

In an experiment with a convex lens. The plot of the image distance (\(v'\)) against the object distance \(\left(\mu^{\prime}\right)\) measured from the focus gives a curve \(v^{\prime} \mu^{\prime}=225\). If all the distances are measured in \(cm\). The magnitude of the focal length of the lens is \(cm\).

A cylinder with fixed capacity of \(67.2\, lit\) contains helium gas at \(STP\). The amount of heat needed to raise the temperature of the gas by \(20\,^oC\) is ..... \(J\) [Given that \(R = 8.31\, J\, mol^{-1}\, K^{-1}\)]

A satellite of mass \(M\) is in a circular orbit of radius \(R\) about the centre of the earth. A meteorite of the same mass, falling towards the earth collides with the satellite completely inelastically. The speeds of the satellite and the meteorite are the same, just before the collision. The subsequent motion of the combined body will be