The correct option is
A perfectly
In order to fill in the blanks with the most suitable adjective and adverb, let's understand what adjectives and adverbs are.Adjectives are the words that define/explain nouns or pronouns. For example: Roshni is pretty. Here, 'Roshni' is the noun and 'pretty' explains how she is, hence 'pretty' is the adjective.
Adverbs are the words that describe action words (verbs). For example: Ram fought bravely. Here, 'Ram' is the noun and 'fought' is the verb. 'Bravely' describes how he fought, hence it is the adverb.
NOTE: We need a word to describe the way the person knows French. Since, 'know' is a verb, the word that describes it is called an adverb. Hence, we need an adverb to fill in the blank.
Option B - 'Perfect' is an adjective and not an adverb. Thus option B is incorrect.
Option C - 'Perfectness' is a noun and not an adverb. Hence, option C is incorrect.
Option D - 'Perfectfully' is a wrong word. Thus option D is incorrect.
Option A - 'Perfectly' is an adverb which can be used to describe the way the person in the sentence knows French. Hence, option A is the correct answer.
'My mother speaks French. Hence, I know French perfectly.'