Comprehension Passage:

The air-conditioner (AC) in a large room can be operated either in REGULAR mode or in POWERmodetoreduce the temperature.

If the AC operates in REGULAR mode, then it brings down the temperature inside the room (called inside temperature) at a constant rate to the set temperature in 1 hour. If it operates in POWERmode, then this is achieved in 30 minutes.

If the AC is switched off, then the inside temperature rises at a constant rate so as to reach the temperature outside at the time of switching off in 1 hour.

The temperature outside has been falling at a constant rate from 7 pm onward until 3 am on a particular night. The following graph shows the inside temperature between 11 pm (23:00) and 2 am (2:00) that night.

The following facts are known about the AC operation that night:
• The ACwas turned on for the first time that night at 11 pm (23:00).
• The AC setting was changed (including turning it on/off, and/or setting different temperatures) only at the beginning of the hour or at 30 minutes after the hour.
• The ACwasusedinPOWERmodeforlongerduration than in REGULARmodeduring this 3-hour period.

How many times must the AC have been turned off between 11:01 pm and 1:59 am?

The correct option is 2.

To determine how many times the air-conditioner (AC) must have been turned off between 11:01 pm and 1:59 am, we analyze the behavior of the inside temperature from the graph:
1. 23:00 to 0:00 (11:00 pm to 12:00 am): The temperature decreases linearly from 38°C to approximately 26.4°C. Since the temperature is decreasing, the AC is turned on and operating in a cooling mode.
2. 0:00 to 0:30 (12:00 am to 12:30 am): The temperature rises from approximately 26.4°C to 31°C. Since the AC only cools the room to bring the temperature down, a rise in temperature indicates that the AC must have been turned off. Thus, the AC was turned off at 0:00.
3. 0:30 to 1:00 (12:30 am to 1:00 am): The temperature decreases from 31°C to approximately 26.2°C, meaning the AC was turned back on to cool the room.
4. 1:00 to 1:30 (1:00 am to 1:30 am): The temperature rises again from approximately 26.2°C to 30°C. This second temperature rise indicates that the AC was turned off again at 1:00 am.
5. 1:30 to 2:00 (1:30 am to 2:00 am): The temperature decreases from 30°C to 28°C, meaning the AC was turned on once again.

The times at which the AC was turned off during the given period are:
• At 12:00 am (0:00)
• At 1:00 am (1:00)

Both of these events occur strictly between 11:01 pm and 1:59 am. Therefore, the AC must have been turned off exactly 2 times.