The correct answer is C) maintain a vacuum in the shell of the feed water heater. The connections labeled "A" in the illustration are used to maintain a vacuum in the shell of the feed water heater. This is necessary to improve the efficiency of the heat transfer process and prevent condensation issues. Maintaining a vacuum helps draw the steam through the feed water heater, allowing it to effectively heat the feed water before it enters the boiler. The other answer choices are incorrect because they do not accurately describe the purpose of the "A" connections in the context of a feed water heater system. Option A relates to the LP (low pressure) bleed steam, option B describes draining condensate, and option D is about admitting steam air heater drains, none of which are the primary function of the "A" connections as described in the question.