For a train engine moving with speed of \(20 \;ms ^{-1}\). the driver must apply brakes at a distance of \(500 \;m\) before the station for the train to come to rest at the station. If the brakes were applied at half of this distance, the train engine would cross the station with speed \(\sqrt{ x }\; ms ^{-1}\). The value of \(x\) is \(..............\) (Assuming same retardation is produced by brakes)

Consider following statements:
A. Surface tension arises due to extra energy of the molecules at the interior as compared to the molecules at the surface, of a liquid.
B. As the temperature of liquid rises, the coefficient of viscosity increases.
C. As the temperature of gas increases, the coefficient of viscosity increases
D. The onset of turbulence is determined by Reynold's number.
E. In a steady flow two stream lines never intersect.
Choose the correct answer from the options given below:

Every planet revolves around the sun in an elliptical orbit : \(A.\) The force acting on a planet is inversely proportional to square of distance from sun. \(B.\) Force acting on planet is inversely proportional to product of the masses of the planet and the sun \(C.\) The centripetal force acting on the planet is directed away from the sun. \(D.\) The square of time period of revolution of planet around sun is directly proportional to cube of semi-major axis of elliptical orbit. Choose the correct answer from the options given below :

From a solid sphere of mass \(M\) and radius \(R\) a cube of maximum possible volume is cut. Moment of inertia of cube about an axis passing through its center and perpendicular to one of its faces is

Match List-\(I\) with List-\(II\) Choose the correct answer from the options given below :

Two identical tennis balls each having mass \(m\) and charge \(q\) are suspended from a fixed point by threads of length \(l\). What is the equilibrium separation when each thread makes a small angle \(\theta\) with the vertical?

Two concentric circular loops of radii \(r _{1}=30\,cm\) and \(r_{2}=50\,cm\) are placed in \(X - Y\) plane as shown in the figure. \(A\) current \(I=7\,A\) is flowing through them in the direction as shown in figure. The net magnetic moment of this system of two circular loops is approximately.

In a reactor, \(2\, kg\) of \({ }_{92} U ^{235}\) fuel is fully used up in \(30\) days. The energy released per fission is \(200\, MeV.\) Given that the Avogadro number, \(N =6.023 \times 10^{26}\) per kilo mole and \(1\, eV =1.6 \times 10^{-19}\, J .\) The power output of the eactor is close to\(.....MW\)

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 

The magnitude of torque on a particle of mass \(1\,kg\) is \(2.5\, Nm\) about the origin. If the force acting on it is \(1\,N\), and the distance of the particle from the origin is \(5\,m\), the angle between the force and the position vector is (in radians)

A galvanometer of resistance \(100 \Omega\) when connected in series with \(400 \Omega\) measures a voltage of upto \(10 \mathrm{~V}\). The value of resistance required to convert the galvanometer into ammeter to read upto \(10 \mathrm{~A}\) is \(\mathrm{x} \times 10^{-2} \Omega\). The value of \(\mathrm{x}\) is _______.

A magnetic field vector in an electromagnetic wave is represented by \(\vec{B}=B_0\sin\left(2\pi vt-\dfrac{2\pi x}{\lambda}\right)\hat{j}\). Its associated electric field vector is ______.

From a sphere of mass \(M\) and radius \(R\), a smaller sphere of radius \(\frac{R}{2}\) is carved out such that the cavity made in the original sphere is between its centre and the periphery (See figure) . For the configuration in the figure where the distance between the centre of the original sphere and the removed sphere is \(3R\), the gravitational force between the two sphere is