"The chairman of the board, sensing that any effort was futile, became $\qquad$ the company's plight, and only shrugged his shoulders when the shareholders notified him that the corporation was insolvent."
- The chairman feels that efforts are futile.
- His reaction is to "shrug his shoulders," which typically implies a lack of concern or care.

Now, let's consider the options:
A. wary of - means cautious or suspicious of; does not fit well since "wary" suggests concern or vigilance.
B. apathetic to - means showing no interest or concern; fits the idea of shrugging off the problem.
C. irascible towards - means easily angered; doesn't fit the idea of shrugging.
D. hopeless regarding - means feeling despair or despairing; this fits the sense of futility, but "shrugged his shoulders" implies more indifference than despair.
E. pugnacious towards - means eager to fight or argue; doesn't fit the context.
F. indifferent to - means having no particular interest or sympathy; fits well with shrugging shoulders in response.

Given this, the two best answers are:
B. apathetic to
F. indifferent to

Both suggest lack of concern or care, which matches the chairman's shrug.
Final answer: $B$ and $F$.