I am not sure whether I understood your issue properly. It seems you sold the plot in the name of your mother through a GPA in Feb 2013 for Rs. 80 lakhs but the date of registration is not clear.
It also appears that you purchased another plot in March 2013 by making all the payments and even executing the sale deed in the joint name of yourself and your mother but this registration of this was done for Rs. 77 Lakhs in December 2013.
I hope my understanding is correct.
Now the following issues arise out of these transactions:
1 As the property was sold through a registered power of attoreny, the nature of power of attorney is important to determine whether there was any transfer of property on the execution of power of attorney. If the PA holder is only your agent, then the risks and rewards on the plot will be still with you till you actually transfer the property. If the PA holder acquired the property in his own right, then the property can be considered to be transferred to the PA holder in Feb 2013 itself. In such case, you will have to compute long term capital gains with the date of acquisition being in the year 1984 and the date of sale being in Feb 2013. In such case the start date will be from Feb 13 itself.
2 You have invested the sale proceeds in the new house in March 2013. Though you did make the investment, your construction should be completed within 3 years from Feb 2013. If the construction is not completed because it is not livable yet, you lose the exemption. Perhaps one solution could be to give it for construction to a co-operative society under self financing scheme and have a letter of allottment. Allotment of a flat under the self financing scheme of a co-operative society or other institution is treated as construction of the house as per Circular No. 471 dated 15th October 1986 and 672 dated 16th December 1993.
3 The new house shall be registered in your name and your mother can be a joint holder.