To cut the mustard is to meet a set of expectations either of a person or organization. The idiom is listed in most dictionaries under the verb cut. The verb can be conjugated through all its forms, and the article is always the.
Cut the mustard has been around for a long time in the Unites States. By the 1890s it was used with this definition in rural newspapers. While the origins of the idiom are unknown, there are plenty of guesses. It is highly unlikely it stems from literally attempting to cut mustard in some form. Mustard is sometimes mixed or cut with other ingredients to make it taste more appetizing. Though it is unlikely that this knowledge was something commonly known in the rural United States in the 1890s.
One thing is known, that it is not related to the phrase pass muster, since cutting muster would be a bad thing and not a good thing.
Strapping herself to a pick-up truck on a runway in Jordan and being chased by an aeroplane – all in a day’s work for British filmmaker Vicky Jewson, determined to cut the mustard as a member of the small but growing list of female action film directors. [Huffington Post UK]
With a chance of cutting the mustard at a world-class eatery, frying bin lid burgers in laybys may become a distant memory. [The Guardian]
Van Winkle does not get a pass either and must participate in semi-annual auditions to ensure he cuts the mustard as a second tenor. [North Jersey]