When average , worst and best case time complexity happens in quick sort?
Remember, for big-O we care about the asymptotic limit. If you choose the $k$th largest (or smallest) element, for constant $k$, then when you make $n$ big enough it's as though you chose the single largest or smallest. The $k$ elements on the other side just aren't enough to matter any more. So statement 2 is true.

Statement 1 is also true; you say you understand the proof of it, so I'm not sure how much detail to go into here. But intuitively, the amount on each side of the pivot grows proportionally to $n$. That's what makes it different from the constant $k$.