$A$ steel ring of radius $r$ and cross-section area $A$ is fitted onto a wooden disc of radius $R$ $(R > r)$. If Young's modulus is $E$,then the force with which the steel ring is expanded is:

$A$ 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.

$\sinh^{-1}(2^{3/2})$ is equal to :

Which of the following is an electrophile?

$A$ soap bubble,having a radius of $1 \ mm$,is blown from a detergent solution having a surface tension of $2.5 \times 10^{-2} \ N/m$. The pressure inside the bubble is equal to the pressure at a point $Z_0$ below the free surface of water in a container. Taking $g = 10 \ m/s^2$ and the density of water $\rho = 10^3 \ kg/m^3$,the value of $Z_0$ is: (in $cm$)

Ferrous ion changes to $X$ ion upon reacting with acidified hydrogen peroxide. The number of $d$-electrons present in $X$ and its magnetic moment (in $BM$) are,respectively: