The masses and radii of the earth and moon are \(\left({M}_{1}, {R}_{1}\right)\) and \(\left({M}_{2}, {R}_{2}\right)\) respectively. Their centres are at a distance ' \({r}\) ' apart. Find the minimum escape velocity for a particle of mass ' \({m}\) ' to be projected from the middle of these two masses:

List-\(I\)	List-\(II\)
\((a)\) \(MI\) of the rod (length \({L}\), Mass \({M}\), about an axis \(\perp\) to the rod passing through the midpoint)	\((i)\;\frac {8 {ML}^{2}}{3}\)
\((b)\) MI of the rod (length \(L\), Mass \(2 M\), about an axis \(\perp\) to the rod Passing through one its end)	\((ii)\;\frac {{ML}^{2}}{3}\)
\((c)\) MI of the rod (length \(2 {L}\), Mass \({M}\), about an axis \(\perp\) to the rod  Passing through its midpoint)	\((iii)\;\frac {{ML}^{2}}{12}\)
\((d)\) MI of the rod (length \(2 {L}\), Mass \(2 {M}\), about an axis \(\perp\) to the rod  passing through one of its end)	\((iv)\;\frac {2 {ML}^{2}}{3}\)

 Choose the correct answer from the options given below

Statement\(-I :\) To get a steady de output from the pulsating voltage received from a full wave rectifier we can connect a capacitor across the output parallel to the load \({R}_{{L}}\) Statement\(-II :\) To get a steady de output from the pulsating voltage received from a full wave rectifier we can connect an inductor in series with \({R}_{{L}}\). In the light of the above statements, choose the most appropriate answer from the options given below:

There are two identical chambers, completely thermally insulated from surroundings. Both chambers have a partition wall dividing the chambers in two compartments. Compartment \(1\) is filled with an ideal gas and Compartment \(3\) is filled with a real gas. Compartments \(2\) and \(4\) are vacuum . A small hole (orifice) is made in the partition walls and the gases are allowed to expand in vacuum Statement \(-1\) : No change in the temperature of the gas takes place when ideal gas expands in vacuum. However, the temperature of real gas goes down (cooling) when it expands in vacuum Statement \(-2\) : The internal energy of an ideal gas is only kinetic. The internal energy of a real gas is kinetic as well as potential

Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures, \(\mathrm{T}_{1}\) and \(\mathrm{T}_{2} .\) The temperature of the hot reservoir of the first engine is \(\mathrm{T}_{1}\) and the temperature of the cold reservoir of the second engine is \(\mathrm{T}_{2} . T\) is temperature of the sink of first engine which is also the source for the second engine. How is \(T\) related to \(\mathrm{T}_{1}\) and \(\mathrm{T}_{2}\), if both the engines perform equal amount of work?

Consider a spherical shell of radius \(R\) at temperature \(T\). The black body radiation inside it can be considered as an ideal gas of photons with internal energy per unit volume\(E=\) \(\frac{U}{V} \propto {T^4}\) and pressure \(P = \frac{1}{3}\left( {\frac{U}{V}} \right)\) If the shell now undergoes an adiabatic expansion the relation between \(T\) and \(R\) is

A particle of mass \(20\,g\) is released with an initial velocity \(5\,m/s\) along the curve from the point \(A,\) as shown in the figure. The point \(A\) is at height \(h\) from point \(B.\) The particle slides along the frictionless surface. When the particle reaches point \(B,\) its angular momentum about \(O\) will be ......... \(kg - m^2/s\). [Take \(g = 10\,m/s^2\) ]

The sum of an infinite geometric series with positive terms is \(3\) and the sum of the cubes of its terms is \(\frac {27}{19}\). Then the common ratio of this series is

The mean of the data set comprising of \(16\) observations is \(16.\) If one of the observation valued \(16\) is deleted and three new observations valued \(3, 4\) and \(5\) are added to the data, then the mean of the resultant data, is:

If the coefficients of \(x^{-2}\) and \(x^{-4}\) in the expansion of \({\left( {{x^{\frac{1}{3}}} + \frac{1}{{2{x^{\frac{1}{3}}}}}} \right)^{18}}\,,\,\left( {x > 0} \right),\) are \(m\) and \(n\)  respectively, then \(\frac{m}{n}\) is equal to 

Let \(d\) be the distance between the foot of perpendiculars of the points \(P (1,2-1)\) and \(Q (2,-\) 1,3 ) on the plane \(- x + y + z =1\). Then \(d ^{2}\) is equal to

A square loop of side \(10 \mathrm{~cm}\) and resistance \(0.7\  \Omega\) is placed vertically in east-west plane. A uniform magnetic field of \(0.20 \mathrm{~T}\) is set up across the plane in north east direction. The magnetic field is decreased to zero in \(1 \mathrm{~s}\) at a steady rate. Then, magnitude of induced emf is \(\sqrt{\mathrm{x}} \times 10^{-3} \mathrm{~V}\). The value of \(x\) is _______.

The percentage decrease in the weight of a rocket, when taken to a height of \(32 km\) above the surface of earth will, be\(.....\%\) (Radius of earth \(=6400\,km\) )