In Young's double slit experiment, two slits \(S_1\) and \(S_2\) are ' \(d\) ' distance apart and the separation from slits to screen is \(D\) (as shown in figure). Now if two transparent slabs of equal thickness \(0.1\,mm\) but refractive index \(1.51\) and \(1.55\) are introduced in the path of beam \((\lambda=4000 \mathring A)\) from \(S_1\) and \(S_2\) respectively. The central bright fringe spot will shift by  \(..........\) number of fringes.

A particle of mass \(m\) is moving along the \(x\) -axis with initial velocity \(u \hat i\). It collides elastically with a particle of mass \(10\, m\) at rest and then moves with half its initial kinetic energy (see figure). If \(\sin \theta_{1}=\sqrt{n} \sin \theta_{2}\) then value of \(n\) is.....

Given below are two statements: Statement \(I\) : In hydrogen atom, the frequency of radiation emitted when an electron jumps from lower energy orbit \(\left( E _{1}\right)\) to higher energy orbit \(\left( E _{2}\right)\), is given as \(hf = E _{1}- E _{2}\). Statement \(II\) : The jumping of electron from higher energy orbit \(\left(E_{2}\right)\) to lower energy orbit \(\left(E_{1}\right)\) is associated with frequency of radiation given as \(f\) \(=\left( E _{2}- E _{1}\right) / h\) This condition is Bohr's frequency condition. In the light of the above statements, choose the correct answer from the options given below

Shown in the figure are two point charges \(+Q\) and \(-Q\) inside the cavity of a spherical shell. The charges are kept near the surface of the cavity on opposite sides of the centre of the shell. If \(\sigma _1\) is the surface charge on the inner surface and \(Q_1\) net charge on it and \(\sigma _2\) the surface charge on the outer surface and \(Q_2\) net charge on it then

Three particles \({P}, {Q}\) and \({R}\) are moving along the vectors \({A}=\hat{{i}}+\hat{{j}}, {B}=\hat{{j}}+\hat{{k}}\) and \({C}=-\hat{{i}}+\hat{{j}}\) respectively. They strike on a point and start to move in different directions. Now particle \(P\) is moving normal to the plane which contains vector \(\vec{A}\) and \(\vec{B} .\) Similarly particle \(Q\) is moving normal to the plane which contains vector \(\vec{A}\) and \(\vec{C} .\) The angle between the direction of motion of \(P\) and \(Q\) is \(\cos ^{-1}\left(\frac{1}{\sqrt{x}}\right)\). Then the value of \(x\) is ...... .

A solid sphere of radius \(R\) gravitationally attracts a particle placed at \(3 R\) form its centre with a force \(F _{1}\). Now a spherical cavity of radius \(\left(\frac{ R }{2}\right)\) is made in the sphere (as shown in figure) and the force becomes \(F _{2}\). The value of \(F _{1}: F _{2}\) is

A \(25\times10^{-3}\, m^3\) volume cylinder is filled with \(1\, mol\) of \(O_2\) gas at room temperature \((300\, K)\). The molecular diameter of \(O_2\), and its root mean square speed, are found to be \(0.3\, nm\) and \(200\, m/s\), respectively. What is the average collision rate (per second) for an \(O_2\) molecule?

Given below are two statements: \(Statement\) \(I\) : If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in hot water. \(Statement\) \(II\) : If a capillary tube is immersed first in cold water and then in hot water, the height of capillary rise will be smaller in cold water. In the light of the above statements, choose the most appropriate from the options given below

There is a small source of light at some depth below the surface of water (refractive index \(=\frac{4}{3}\) ) in a tank of large cross sectional surface area. Neglecting any reflection from the bottom and absorption by water, percentage of light that emerges out of surface is (nearly)...........\(\%\) [Use the fact that surface area of a spherical cap of height \(\mathrm{h}\) and radius of curvature \(\mathrm{r} \text { is } 2 \pi \mathrm{rh}]\)

A transverse wave on a string is described by \(y = 3\sin(36t + 0.018x + \pi/4)\), where \(x, y\) are in cm and \(t\) in seconds. The least distance between the two successive crests in the wave is _____ cm. (Nearest integer)
(\(\pi = 3.14\))