At cross purposes with means having different aims or misunderstanding each other. The correct answer is this meaning because the expression is used figuratively, not only by its literal words.

Examples:

• The two departments were working at cross purposes with each other.
• In an HSC essay, At cross purposes with can express this meaning: having different aims or misunderstanding each other.
• When this expression appears in a passage, connect At cross purposes with with the idea of having different aims or misunderstanding each other.