A tennis ball (treated as hollow spherical shell) starting from \(O\) rolls down a hill. At point \(A\) the ball becomes air borne leaving at an angle of \(30^o\) with the horizontal. The ball strikes the ground at \(B\). What is the value of the distance \(AB\) ? (Moment of inertia of a spherical shell of mass \(m\) and radius \(R\) about its diameter .......... \(m\). \(= \frac {2}{3}\,mR^2\))

A quantity \(f\) is given by \(f=\sqrt{\frac{{hc}^{5}}{{G}}}\) where \(c\) is speed of light, \(G\) universal gravitational constant and \(h\) is the Planck's constant. Dimension of \(f\) is that of

Moment of inertia of a dise of mass \(M\) and radius ' \(R\) ' about any of its diameter is \(\frac{ MR ^2}{4}\). The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, \(\frac{ x }{2} MR ^2\). The value of \(x\) is \(..........\).

A big drop is formed by coalescing \(1000\) small droplets of water. The ratio of surface energy of \(1000\) droplets to that of energy of big drop is \(\frac{10}{x}\). The value of \(x\) is _______.

Seawater at a frequency \(f =9 \times 10^{2} \,Hz ,\) has permittivity \(\varepsilon=80\, \varepsilon_{0}\) and resistivity \(\rho=0.25\, \Omega m.\) Imagine a parallel plate capacitor is immersed in seawater and is driven by an alternating voltage source \(V ( t )= V _{0} \sin (2 \pi ft ) .\) Then the conduction current density becomes \(10^{x}\) times the displacement current density after time \(t =\frac{1}{800} \,s\). The value of \(x\) is ......... .
\(\left(\frac{1}{4 \pi \varepsilon_{0}}=9 \times 10^{9} Nm ^{2} C ^{-2}\right)\)

Given below are two statements: one is labelled as Assertion \((A)\) and the other is labelled as Reason \((R)\). \(Assertion\) \((A)\) : The angular speed of the moon in its orbit about the earth is more than the angular speed of the earth in its orbit about the sun. \(Reason\) \((R)\): The moon takes less time to move around the earth than the time taken by the earth to move around the sun. In the light of the above statements, choose the most appropriate answer from the options given below:

Starting from the origin at time \(t=0,\) with initial velocity \(5 \hat{ j }\, ms ^{-1},\) a particle moves in the \(x-y\) plane with a constant acceleration of \((10 \hat{ i }+4 \hat{ j })\, ms ^{-2}\). At time \(t\), its coordinates are \(\left(20\, m , y _{0}\, m \right) .\) The values of \(t\) and \(y _{0},\) are respectively

Two wires of same length and thickness having specific resistances \(6\, \Omega \,cm\) and \(3 \,\Omega\, cm\) respectively are connected in parallel. The effective resistivity is \(\rho\, \Omega \,cm\). The value of \(\rho\) to the nearest integer, is ..... .

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R

Assertion A: If the average kinetic energy of \(H_2\) and \(O_2\) molecules, kept in two different sized containers are same, then their temperatures will be same.

Reason R: The r.m.s. speed of \(H_2\) and \(O_2\) molecules are same at same temperature.

Choose the correct answer from the options given below

A ray of light is incident at an angle of \(60^o\) on one face of a prism of angle \(30^o\) . The emergent ray of light makes an angle of \(30^o\)  with incident ray. The angle made by the emergent ray with second face of prism will be....\(^o\)

The de Broglie wavelength of an oxygen molecule at \(27^{\circ} C\) is \(x \times 10^{-12} m\). The value of \(x\) is (take Planck's constant \(=6.63 \times 10^{-34} J . s\), Boltzmann constant \(=1.38 \times 10^{-23} J / K\), mass of oxygen. Molecule \(\left.=5.31 \times 10^{-26} kg\right)\).

The pitch of the screw gauge is \(1\, mm\) and there are \(100\) divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lies \(8\) divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is clearly visible while \(72^{\text {nd }}\) division on circular scale coincides with the reference line. The radius of the wire is.........\(mm\)