By what factor does the average velocity of a gaseous molecule increase when the temperature (in Kelvin) is doubled?

If the roots of the equation $x^2 - 8x + (a^2 - 6a) = 0$ are real,then:

If one of the lines in the pair of straight lines given by $4x^2+6xy+ky^2=0$ bisects the angle between the coordinate axes,then $k \in$

The average degree of freedom per molecule of a gas is $6$. The gas performs $25 \, J$ of work while expanding at constant pressure. The heat absorbed by the gas is ...... $J$.

If the foot of the perpendicular from $(0,0,0)$ to a plane is $(1,2,3)$,then the equation of the plane is

If $\bar{M}_w$ is the weight average molecular weight and $\bar{M}_n$ is the number average molecular weight of a polymer,the poly dispersity index $(PDI)$ of the polymer is given by