The correct option is
A Rather
In order to fill in the blank, let's understand what degree of adverbs and quantity are. The degree of adverbs explains 'to what extent' and the quantity explains us 'how much.'Option B - 'A lot' is an adverb which means 'very often' or 'very much'. 'A lot' is never used before the positive degree of an adjective and a noun, as 'strict' is the positive degree of an adjective, we can't use 'a lo' here. Hence, option B is incorrect.
Option C - 'Most' is an adverb, which is used to indicate the highest degree (superlative) of an adjective or an adverb. 'Strict' is the positive degree of an adjective, and as there is a superlative degree: 'strict- strictest', we can't use 'most' here. Hence, option C is incorrect.
Option D - 'Much' is an adverb, which is used with singular uncountable nouns only. 'Much' means 'greatly'. As the sentence doesn't convey that the Principal is greatly strict, we can't use 'much' with 'strict'. Hence, option D is incorrect.
Option A - 'Rather' is an adverb, which is used to indicate negativity about the noun/subject, adjective or the adverb in a sentence. For example, 'She is behaving rather strangely.' It means the same as 'quite', and 'to a certain extent.' As the sentence indicates negativity (strict), we can use 'rather' here. Hence, option A is the correct answer.
'Raj: What's your new Principal like?
Maya: Well, he's rather strict.'