The work done by a force $\vec F = (-6x^3 \hat i) \, N$,in displacing a particle from $x = 4 \, m$ to $x = -2 \, m$ is ................ $J$.

The ratio of rates of diffusion of gases $X$ and $Y$ is $1:5$ and that of $Y$ and $Z$ is $1:6$. The ratio of rates of diffusion of $Z$ and $X$ is

It is now possible to breed plants and animals with desired characters through

$A$ thin hollow sphere of mass $m$ is completely filled with a liquid of mass $m$. When the sphere rolls with a velocity $v$,the kinetic energy of the system is (neglect friction).

If $(-2, 6)$ is the image of the point $(4, 2)$ with respect to line $L = 0$,then $L =$

$A$ bimetallic strip is formed out of two identical strips,one of copper and the other of brass. The coefficients of linear expansion of the two metals are $\alpha_C$ and $\alpha_B$. On heating,the temperature of the strip increases by $\Delta T$ and the strip bends to form an arc of radius $R$. Then $R$ is proportional to