If y is directly proportional to √x ,will on increasing y by 200%, x will become 4 times

Answer 1

Assuming you mean that y changes from y to 2y (an increase of 100%), then since

y = k√x
2y = 2k√x = k√(4x)

so, yes x will be 4 times as big.

But doubling y means increasing by 100%, not 200%. "increasing by" means "adding to".

Think about it. Increasing 10 by 10% means 10(1+.1) = 11

Increasing 10 by 100% means 10(1+1.00) = 10(2)