The correct answer is option B) "To play an instrument requires a lot of determination and practice."
Infinitive phrases have different functions within a sentence. In this case, the infinitive "to play" is acting as the noun ans subject of the sentence.
In option A) the infinitive "to relax" functions as an adjective since it is modifying the noun "vacation".
In option C) the infinitive "to study" functions as an adjective as it is modifying the noun "time".
And finally, in option D) the infinitive "to avoid" functions as an adverb because it explains the "why".
So, the only case in which the infinitive acts as a noun is option B).