$A$ wire of length $L$ and radius $r$ is clamped rigidly at one end. When the other end of the wire is pulled by a force $f$,its length increases by $l$. Another wire of the same material of length $2L$ and radius $2r$ is pulled by a force $2f$. Find the increase in length of this wire.

The limiting points of the co-axial system containing the two circles $x^2+y^2+2x-2y+2=0$ and $25(x^2+y^2)-10x-80y+65=0$ are

If $R$ is a relation from a finite set $A$ having $m$ elements to a finite set $B$ having $n$ elements,then the number of relations from $A$ to $B$ is

In the reaction of $BaO_2$ with dilute $H_2SO_4$,what are the oxidation states of the most electronegative element in the products?

Which of the following reactions will yield $2-$propanol? Select the correct answer from $a, b, c$ and $d$.
$(I)$ $CH_2=CH-CH_3 + H_2O \xrightarrow{H^+} $
$(II)$ $CH_3CHO \xrightarrow[(ii) H_2O]{(i) CH_3MgI}$
$(III)$ $CH_2O \xrightarrow[(ii) H_2O]{(i) C_2H_5MgI}$
$(IV)$ $CH_2=CH-CH_3 \xrightarrow{\text{neutral } KMnO_4}$

Two photons of energy $2.5 \ eV$ and $3.5 \ eV$ fall on a metal surface of work function $1.5 \ eV$. The ratio of the maximum velocities of the photoelectrons emitted from the metal surface is