Do predators really kill only the old and sick prey? what evidence is there for this statement

No, but it is known that old/sick prey is much easier to catch. The evidence is that, say, there's no old and sick prey. The predator has to eat regardless, and whatever is there for it to take, it will. Think of a lion and a zebra, and the zebra is eating grass in the middle of a field. The lion is going to take that chance.

Nope predator catches the preys they find not only the old and sick one but the old and sick prey would be catch more easily as compared to the healthy and strong ones

Active site is where they touch.