An air bubble of volume \(1\,cm ^3\) rises from the bottom of a lake \(40\,m\) deep to the surface at a temperature of \(12^{\circ}\,C\). The atmospheric pressure is \(1 \times 10^5 Pa\), the density of water is \(1000\,kg / m ^3\) and \(g =10\,m / s ^2\). There is no difference of the temperature of water at the depth of \(40\,m\) and on the surface. The volume of air bubble when it reaches the surface will be \(..........\,cm^{3}\)

A gun mounted on the ground fires bullets in all directions with same speed. The farthest distance the bullets could reach is \(6.4\) m. The speed of the bullets from the gun is ______ m/s.
(take \(g=10\) m/s\(^2\))

Vectors \(a \hat{i}+b \hat{j}+\hat{k}\) and \(2 \hat{i}-3 \hat{j}+4 \hat{k}\) are perpendicular to each other when \(3 a+2 b=7\), the ratio of a to \(b\) is \(\frac{x}{2}\). The value of \(x\) is \(..............\)

A wooden block floating in a bucket of water has \(\frac{4}{5}\) of its volume submerged. When certain amount of an oil is poured into the bucket, it is found that the block is just under the oil surface with half of its volume under water and half in oil. The density of oil relative to that of water is

A particle executes simple harmonic motion. Its amplitude is \(8 \,cm\) and time period is \(6 \,s\). The time it will take to travel from its position of maximum displacement to the point corresponding to half of its amplitude, is ............. \(s\)

When photon of energy \(4.0\; eV\) strikes the surface of a metal \(\mathrm{A}\), the ejected photoelectrons have maximum kinetic energy \(\mathrm{T}_{\mathrm{A}}\; eV\) end de-Broglie wavelength \(\lambda_{\mathrm{A}} .\) The maximum kinetic energy of photoelectrons liberated from another metal \(\mathrm{B}\) by photon of energy \(4.50 \;eV\) is \(\mathrm{T}_{\mathrm{B}}=\left(\mathrm{T}_{\mathrm{A}}-1.5\right)\;eV\). If the de-Broglie wavelength of these photoelectrons \(\lambda_{B}=2 \lambda_{A}\) then the work function of metal \(\mathrm{B}\) is ............. \(eV\)

A travelling microscope is used to determine the refractive index of a glass slab. If \(40\) divisions are there in \(1 \; cm\) on main scale and \(50\) Vernier scale divisions are equal to \(49\) main scale divisions, then least count of the travelling microscope is \(\dots \; \times 10^{-6} \; m\)

A roller is mad e by joining  together two cones at their vertices \(O\). It is kept on two rails \(AB\) and \(CD\), which are placed asymmetrically (see figure) , with its axis perpendicular to \(CD  \) and its centre \(O\) at the centre ofline joining \(AB\) and \(CD\) (see figure) . It is given a light push so that it \(A\) starts rolling with its centre \(O\) moving parallel to \(CD\) in the direction shown. As it moves, the roller will tend to

The mean free path and the average speed of oxygen molecules at 300 K and 1 atm are \(3 \times 10^{-7} \mathrm{~m}\) and \(600 \mathrm{~m} / \mathrm{s}\) respectively. Find the frequency of its collisions.

Electric field in a region is given by \( \vec{E}=Ax\hat{i}+By\hat{j} \), where \( A=10~V/m^{2} \) and \( B=5~V/m^{2} \). If the electric potential at a point (10, 20) is 500 V, then the electric potential at origin is __________ V.

For the given cyclic process \(CAB\) as shown for a gas, the work done is ..... \(J\)

Time period of a simple pendulum is \(T\) inside a lift when the lift is stationary. If the lift moves upwards with an acceleration \(g / 2,\) the time period of pendulum will be

Given below are two statements: Statement \(I:\) Rotation of the earth shows effect on the value of acceleration due to gravity \((g)\). Statement \(II:\) The effect of rotation of the earth on the value of \(g\) at the equator is minimum and that at the pole is maximum. In the light of the above statements, choose the correct answer from the options given below.