In a vernier callipers, 50 vernier scale divisions are equal to 48 main scale divisions. If one main scale division \(=0.05 mm\), then the least count of the vernier callipers is _________ mm.

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).

Assertion (A) : With the increase in the pressure of an ideal gas, the volume falls off more rapidly in an isothermal process in comparison to the adiabatic process.

Reason (R) : In isothermal process, \(\mathrm{PV}=\) constant, while in adiabatic process \(\mathrm{PV}^\gamma=\) constant. Here \(\gamma\) is the ratio of specific heats, P is the pressure and V is the volume of the ideal gas.

In the light of the above statements, choose the correct answer from the options given below :

A point source of light is placed at the centre of curvature of a hemispherical surface. The source emits a power of \(24\,W\) The radius of curvature of hemisphere is \(10\,cm\) and the inner surface is completely reflecting. The force on the hemisphere due to the light falling on it is \(..........\times 10^{-8}\,N\).

A \(600\,pF\) capacitor is charged by \(200\,V\) supply. It is then disconnected from the supply and is connected to another uncharged \(600\,pF\) capacitor. Electrostatic energy lost in the process is \(.........\,\mu J\).

Two capacitors of capacities \(2 {C}\) and \({C}\) are joined in parallel and charged up to potential \({V}\). The battery is removed and the capacitor of capacity \(C\) is filled completely with a medium of dielectric constant \({K}\). The potential difference across the capacitors will now be

An electron, a doubly ionized helium ion \(\left( He ^{++}\right)\) and a proton are having the same kinetic energy. The relation between their respective de\(-\)Broglie wavelengths \(\lambda_{ e }, \lambda_{ He ^{++}}\) and \(\lambda_{ P }\) is

An ac current is represented as
\(\mathrm{i}=5 \sqrt{2}+10 \cos \left(650 \pi \mathrm{t}+\frac{\pi}{6}\right) \mathrm{Amp}\)
The r.m.s value of the current is

An object is taken to a height above the surface of earth at a distance \(\frac{5}{4} R\) from the centre of the earth. Where radius of earth, \(R=6400\,km\). The percentage decrease in the weight of the object will be \(....\%\)

A hoop of radius \(r\) and mass \(m\) rotating with an angular velocity \(\omega_0\) is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

A body of mass \(0.5\, kg\) travels on straight line path with velocity \(v =\left(3 x ^{2}+4\right) m / s\). The net workdone by the force during its displacement from \(x =0\) to \(x =2\, m\) is\(.......J\)