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Yes, your mother would pay income tax on the gain from the sale of the farm land so the general concept does make sense.
The gain for your mother is computed based on the sales price less the fair market value of the land on the date of death when it was inherited (presuming it was all inherited) and less any costs of selling the land. Since she has owned the land for more than one year she will qualify for the long term capital gain rate on the sale of 15%.
The $1750 per acre on the contract may or may not be the value at the time of inheritance as it is not clear from the information given if that value was at or near the date of death.
You may want to consider an installment sale over several years so the gain is not all reported in one year and the tax is not all due in one year. Any interest on the installment sale would be additional income reported each year; but your cousin might as well pay the interest to your mother rather than to the bank.
I hope this helps in regard to the sale of inherited land.