Use the correct form of the word "RESOLVE" to fill in the blank:-

Did you make any New Year's _______________ this year?

The correct option is resolution.

Step-by-step Explanation:
1. Identify the required part of speech: In the sentence, "Did you make any New Year's _______________ this year?", the blank space follows the modifier "New Year's" and the determiner "any". This position requires a noun to act as the object of the verb "make".
2. Analyze the word "RESOLVE": The base word "resolve" is a verb meaning to decide firmly on a course of action. We need to find the noun form that represents a firm decision or promise made for the new year.
3. Evaluate the given options:
• resolution: This is the noun form of "resolve" that means a firm decision to do or not to do something. The standard idiom for a promise made to oneself at the start of the year is a "New Year's resolution". Therefore, this is the correct word.
• resolve: Although "resolve" can be used as a noun, it refers to a state of mind, specifically determination or willpower (e.g., "to strengthen one's resolve"). It is an uncountable noun and does not refer to individual decisions.
• resolving: This is the present participle/gerund form of the verb, which is grammatically incorrect in this position.
• resolvation: This is not a standard English word.

Hence, resolution is the correct form to fill in the blank.