a) Divide the board into 2 by 2 squares and color them alternately white and black in a checkerboard pattern. Each type 1 piece covers 2 white and 2 black squares while each type 2 piece covers three of one and one of another. There are an odd number of type 2 pieces; contradiction.
Oddly enough, I can't find a coloring that works for part (b). I'll think about this later.