Two wires of the same material and length but diameters in the ratio $1:2$ are stretched by the same force. The elastic potential energy per unit volume for the wires, when stretched by the same force, will be in the ratio: (in $:1$)

$A$ block of $200 \text{ g}$ mass moves with a uniform speed in a horizontal circular groove with vertical side walls of radius $20 \text{ cm}$. If the block takes $40 \text{ s}$ to complete one round,the normal force exerted by the side walls of the groove is:

Radius of gyration of a body depends on

If $x_1, x_2, \ldots, x_n$ are $n$ observations such that $\sum_{i=1}^n x_i^2 = 400$ and $\sum_{i=1}^n x_i = 80$,then the least value of $n$ is

The density $($in $g cm ^{-3} )$ of the metal which forms a cubic close packed $(ccp)$ lattice with an axial distance $($edge length$)$ equal to $400 \ pm$ is $....$
Use: Atomic mass of metal $=105.6 \ amu$ and Avogadro's constant $=6 \times 10^{23} mol^{-1}$

The differential equation obtained by eliminating the arbitrary constants $a$ and $b$ from $xy = ae^x + be^{-x}$ is