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

$A$ body is projected vertically from the surface of the earth of radius $R$ with a velocity equal to half of the escape velocity. The maximum height reached by the body is

Assertion: One molal aqueous solution of glucose contains $180 \ g$ of glucose in $1 \ kg$ of water.
Reason: $A$ solution containing one mole of solute in $1000 \ g$ of solvent is called a one molal solution.

Which one of the following arrangements represents the correct order of electron gain enthalpy (with negative sign) of the given atomic species :-

The area (in sq units) of the region bounded by $x=-1, x=2, y=x^2+1$ and $y=2x-2$ is

A small block of mass $m$ slides down from the top of a frictionless inclined surface, while the inclined plane is moving towards left with constant acceleration $a_{0}$. The angle between the inclined plane and ground is $ heta$ and its base length is $L$. Assuming that initially the small block is at the top of the inclined plane, the time it takes to reach the lowest point of the inclined plane is . . . . . . . .