Osmotic pressure is a colligative property of a solution, meaning it depends on the concentration of solute particles rather than on the specific identity of the solute. At constant temperature, osmotic pressure can be influenced by two factors: concentration and molecular weight of the solute.

1. Directly proportional to the concentration (Option A):
Osmotic pressure is directly proportional to the concentration of solute particles in a solution. This can be explained using the concept of osmosis. Osmosis is the movement of solvent molecules from an area of lower solute concentration to an area of higher solute concentration through a semipermeable membrane. The driving force for osmosis is the difference in solute concentration, which is reflected in the osmotic pressure. As the concentration of solute particles increases, the osmotic pressure also increases. Therefore, option A is correct.

2. Inversely proportional to the molecular weight of solute (Option B):
Osmotic pressure is inversely proportional to the molecular weight of the solute. This can be understood by considering the colligative nature of osmotic pressure. Colligative properties depend on the number of solute particles present in a solution, rather than on their individual identities. For example, if we compare two solutions with the same concentration but different solutes, the solution with a higher molecular weight solute will have a lower osmotic pressure. This is because a higher molecular weight solute will have fewer particles for the same concentration, resulting in a lower osmotic pressure. Therefore, option B is correct.

The other options, C and D, are incorrect because osmotic pressure is not directly proportional to the square of the concentration or the square root of the concentration. These relationships do not hold true for osmotic pressure and would not accurately describe its behavior.

In conclusion, at constant temperature, the osmotic pressure of a solution is directly proportional to the concentration of solute particles and inversely proportional to the molecular weight of the solute. These relationships can be explained by the principles of osmosis and the colligative nature of osmotic pressure.