If $f: R \rightarrow R$ is defined by $f(x)=[2x]-2[x]$ for $x \in R$,where $[x]$ is the greatest integer function,then the range of $f$ is:

The ratio of the lengths of two wires $A$ and $B$ of the same material is $1 : 2$ and the ratio of their diameters is $2 : 1$. If they are stretched by the same force,then the ratio of the increase in their lengths will be:

Two soap bubbles combine to form a single bubble. In this process, the change in volume and surface area are respectively $V$ and $A$. If $P$ is the atmospheric pressure, and $T$ is the surface tension of the soap solution, the following relation is true :

Which one of the following is the ratio of the lowering of vapour pressure of $0.1 \ M$ aqueous solutions of $BaCl_2, NaCl$ and $Al_2(SO_4)_3$ respectively?

The point $P$ is equidistant from $A(1, 3)$,$B(-3, 5)$,and $C(5, -1)$. Then $PA =$