Your classmate says that if all the molecules in a particular liquid had the same speed, and some were able to evaporate, the remaining liquid would not undergo cooling. do you agree or disagree and what is your explanation to support your answer.

in an ideal system, your classmate is right

Explanation:

Here I suppose that we are in a closed environment.

If all the particles had the same speed, and some of them are able to evaporate, then you have coexistence between liquid state and gas state, In this coexistence state, there is a line of pressure as a function of temperature where the coexistence is possible.

As some of the liquid evaporates, the volume of the gas is bigger than the one of the liquid, so the pressure in the system will increment a little bit, and the temperature will answer to it until the equilibrium is reached.

In equilibrium, the temperature will not change if we don't do anything to the system (assuming adiabatic walls).

here your classmate may be right

A thing you may know is that the temperature is an intensive quantity, which means that it does not depend on the number of particles in the mixture, so the fact that some of the particles "leave" the liquid does not imply that the liquid changes their temperature. And there is also useful to notice that the change of phase is at a constant temperature.