Final answer:
The correct adverb to complete the sentence is 'always' (Option D), indicating that the student habitually turns in homework late.
Step-by-step explanation:
The question asks to choose an adverb that best completes the sentence about the student's habit of turning in homework. The sentence 'She invariably leaves her homework and turns it in late.' suggests that the student always turns in their homework late. Hence, the correct word to describe how often this happens is 'always'. Option A ('purposefully') suggests intention, which is not implied in the sentence. Option B ('hated') is not an adverb and does not fit grammatically or contextually. Option C ('happily') implies a positive emotion which is contrary to the context of the sentence.
The best choice is option D 'always', which is an adverb of frequency, indicating that the action of turning in homework late happens every time. The additional context provided, where she skips houses and fills in forms by selecting random answers due to running late, supports the use of 'always' since it shows a pattern of habitual lateness.