Black feathers are dominant over white feathers. Black hen mated with a black rooster...

It's (D) has to be

The question states that black feathers are dominant over white feathers.  This questions deals with simple Mendelian inheritance, in particular a monohybrid cross.  

A black hen is mated with a white rooster and the result was 31 black and 11 white.  

So, let's define our alleles:

B = black (dominant)
b = white (recessive)

So, the genotypes in the cross will be as follows:

Bb x Bb (both parents must be heterozygous since the recessive trait appears in the F1 generation).  If one of the parents were to be homozygous dominant, then the offspring will all be black: 1/2 Bb and 1/2 BB.  The question says there were 11 WHITE in the F1 generation.  If both of the parents were homozygous dominant, the F1 generation would still be all black.  So, the only combination of the two parents that fits with your results are: Bb x Bb.

So, the answer is D

Hope that helps!