If \(\vec{a}\) and \(\vec{b}\) makes an angle \(\cos ^{-1}\left(\frac{5}{9}\right)\) with each other, then \(|\vec{a}+\vec{b}|=\sqrt{2}|\vec{a}-\vec{b}|\) for \(|\vec{a}|=n|\vec{b}|\) The integer value of \(n\) is _______.

A uniform wire (Young's modulus \(2 \times 10^{11}\, Nm^{-2}\) ) is subjected to longitudinal tensile stress of \(5 \times 10^7\,Nm^{-2}\) . If the over all volume change in the wire is \(0.02\%,\) the fractional decrease in the radius of the wire is close to

The magnetic susceptibility of a material of a rod is \(499 .\) Permeability in vaccum is \(4 \pi \times 10^{-7}\; {H} / {m}\). Absolute permeability of the material of the rod is \(....\,\pi \times 10^{-4} {H} / {m}\)

A large number \((n)\) of identical beads, each of mass \(m\) and radius \(r\) are strung on a thin smooth rigid horizontal rod of length \(L\, (L >> r)\) and are at rest at random positions. The rod is mounted between two rigid supports (see figure) . If one of the beads is now given a speed \(v\), the average force experienced by each support after a long time is (assume all collisions are elastic)

The gravitational potential at a point above the surface of earth is \(-5.12 \times 10^7 \mathrm{~J} / \mathrm{kg}\) and the acceleration due to gravity at that point is \(6.4 \mathrm{~m} / \mathrm{s}^2\). Assume that the mean radius of earth to be \(6400 \mathrm{~km}\). The height of this point above the earth's surface is _______.

A source supplies heat to a system at the rate of \(1000 \,W\). If the system performs work at a rate of \(200\,W\). The rate at which internal energy of the system increases \(.......\,W\)

A very long (length \(L\) ) cylindrical galaxy is made of uniformly distributed mass and has radius \(R(R < < L).\) A star outside the galaxy is orbiting the galaxy in a plane perpendicular to the galaxy and passing through its centre. If the time period of star is \(T\) and its distance from the galaxy's axis is \(r,\) then

In a Young's double slit experiment, the slits are separated by \(0.3\, {mm}\) and the screen is \(1.5\, {m}\) away from the plane of slits. Distance between fourth bright fringes on both sides of central bright is \(2.4\, {cm}\). The frequency of light used is \(..........\,\times 10^{14} {Hz}\)

Mass per unit area of a circular disc of radius \(\mathrm{a}\) depends on the distance \(\mathrm{r}\) from its centre as \(\sigma(\mathrm{r})=\mathrm{A}+\mathrm{Br} .\) The moment of inertia of the disc about the the axis, perpendicular to the plane and passing through its centre is

Two massless springs with spring constants \(2\,k\) and \(9\,k\), carry \(50\, g\) and \(100 \,g\) masses at their free ends. These two masses oscillate vertically such that their maximum velocities are equal. Then, the ratio of their respective amplitudes will be

A particle performs simple harmonic motion with a period of \(2\) second. The time taken by the particle to cover a displacement equal to half of its amplitude from the mean position is \(\frac{1}{ a } s .\) The value of \('a'\) to the nearest integer is.........

Given below are two statement  Statement \(-I\) : What \(\mu\) amount of an ideal gas undergoes adiabatic change from state \(\left( P _{1}, V _{1}, T _{1}\right)\) to state \(\left( P _{2}, V _{2}, T _{2}\right)\), the work done is \(W =\frac{1 R \left( T _{2}- T _{1}\right)}{1-\gamma}\), where \(\gamma=\frac{ C _{ P }}{ C _{ V }}\) and \(R =\) universal gas constant, Statement \(-II\) : In the above case. when work is done on the gas. the temperature of the gas would rise. Choose the correct answer from the options given below

The activity of a radioactive material is \(6.4 \times 10^{-4}\) curie. Its half life is \(5\; days\). The activity will become \(5 \times 10^{-6}\) curie after \(.......day\)