Capital gain on sale of inherited property

"If the son gift or relinquish money in favour of their mother now whether cap gain arises in case of son.?"

When mother pays total tax liability, no tax liability arises on sons.

"Another situation, in absence of power of attorney, if three separately issued cheques deposited in joint account with their mother, and mother issued cheques separately to them, whether and how much  capital gain arises in case of son."

Depends upon their share in the capital gains. In any option, total tax liability remains same, and department would be more interested in the tax amount, rather than who pays it.

"How FMV as on 1/4/81 be determined to find cost " GOVERNMENT REGISTERED VALUER'S VALUATION REPORT.

Thanks for your reply.

Who is actually liable to payment of capital gain tax in both the situations?

Mother has no taxable income after considering capital gain of other two sons , as such no tax is payable.

If one of the sons considers  share of capital in his account , he has to pay tax.

Will it be proper to shift mother with their share of capital gain to avoid tax in their hand in both the situations and if not, is there way out considering both situations separately?

Relinquishment deed is a legal document/instrument where a person legally or formally gives up or releases his legal rights of the said property being relinquished in the name of some other person. If both/any son relinqishes his/their rights in favour of mother by registered deed, mother will be owner of the share, and the corresponding responsibilty of (if any) tax liability will rest on herself.