Hm, this doesn't look at all like a real GMAT question. Just a friendly reminder: please please please always include the source when you're posting a new question!!
A quick google search reveals that this is a bizarro ripoff of a 2007
Paul Krugman article in the NY Times Here's the original sentence: "...under the guise of promoting a conservative agenda, the Bush administration has created a supersized version of the 19th-century spoils system."
So the ripoff artist responsible for this question just swapped out "the Bush administration" with "many a modern day government." And I've never seen that particular colloquialism on the GMAT, and I really don't think you should waste any time thinking about it. I've never heard anybody under the age of 90 say anything like that, and I'd be shocked if you ever see it on the GMAT.
But for whatever it's worth: "many a _____" basically means
"a large number of _____" So "many a modern day government"
would be singular, in theory.
But again, if you ever use that expression in normal conversation with an American, you'll probably earn a confused look.
I hope this helps!
.
Example: A large number of people ARE sitting in the park.
Example: A large number of ducks ARE floating on the water.
Basically "a number" is plural while "the number" is singular. I am confused by your statement above sire. Please throw some light on this doubt.