Find a truth assignment of 2SAT that has the most number of true variables? Given a 2SAT instance in CNF where each clause has at most two literals. Let $m$ be the number of clauses and $n$ be the number of variables et let $k$ be a positive number.

Question: Is there a truth assignment such that the number of variables that are true is $k$ or more?

I cannot find the name of this problem, if it was already studied? If so, is it NP-hard?
Your problem is known as Weighted-2-satisfiability, and is known to be NP-complete. The easiest way to see that is by reduction from Vertex Cover (exercise).

Note that the link above is about whether there is a satisfying assignment having at most $k$ true variables. This is equivalent to your problem (exercise).