In a dilute gas at pressure P and temperature T, the mean time between successive collisions of a molecule varies with T as :

A fluid is flowing through a horizontal pipe of varying cross-section, with
speed v ms^–1 at a point where the pressure is P pascal. At another point where pressure is $${P \over 2}$$ Pascal its speed is V ms^–1. If the density of the fluid is $$\rho $$ kg m^–3 and the flow is streamline, then V is equal to :

Assuming the nitrogen molecule is moving with r.m.s. velocity at 400 K, the de-Broglie wavelength of nitrogen molecule is close to :
(Given : nitrogen molecule weight : 4.64 $$ \times $$ 10^–26 kg,
Boltzman constant: 1.38 $$ \times $$ 10^–23 J/K,
Planck constant : 6.63 $$ \times $$ 10^–34 J.s)

The linear mass density of a thin rod AB of length L varies from A to B as
$$\lambda \left( x \right) = {\lambda _0}\left( {1 + {x \over L}} \right)$$, where x is the distance from A. If M is the mass of the rod then its moment of inertia about an axis passing through A and perpendicular to the rod is :