A wooden block of mass $m$ and density $\rho$ is tied to a string; the other end of the string is fixed to the bottom of a tank. The tank is filled with a liquid of density $\sigma$ with $\sigma >\rho$. What is the tension in the string.

Volume of the block $=\frac{m}{\rho }$. Now upthrust = weight of water displaced = weight of volume $m/\rho$ of liquid = mass of volume $m/\rho$ of liquid $\times g=\frac{m\sigma g}{\rho }$. This is the upward force on the block due to buoyancy. The downward force on the block = its weight =mg. The tension in the string is the net upward force on the block which is
$T=\frac{m\sigma g}{\rho }-mg=\left(\frac{\sigma -\rho }{\rho }\right)mg$
Hence the correct choice is (2).