When was the first bank of the united states formed by the federalists a. during the american revolution. b. in the late eighteenth century. c. in the first quarter of the nineteenth century. d. after the civil war.?

Correct  b. in the late 18th century

The specific year that the First Bank of the United States was formed was 1791.

Further details:

In his role as Secretary of the Treasury, federalist leader Alexander Hamilton worked to establish the First Bank of the United States.  The federalist argument for such a bank referenced the "necessary and proper" clause of Article I, Section 8 of the Constitution of the United States.   In the Constitution, after enumerating a number of the powers of Congress, including borrowing money, coining money, regulating commerce, etc, Section 8 of Article I closes by saying Congress shall have power "to make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof."  

A national bank was not strictly listed as something Congress could establish, but there was nothing in the Constitution to prohibit it.  And the "necessary and proper" clause gave leeway to create such an institution. The First Bank of the United States existed from 1791 till 1811, when its charter expired and was not renewed by Congress.

The Second Bank of the United States was formed in 1816, and also existed for a 20-year charter period, from 1816 to 1846.

using the scientific method.