The air-conditioner (AC) in a large room can be operated either in REGULAR mode or in POWER mode to reduce 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 POWER mode, 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 AC was 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 AC was used in POWER mode for longer duration than in REGULAR mode during this 3-hour period.
Q. 1 How many times the AC must have been turned off between 11:01 pm and 1:59 am?
1) 1 2) 0 3) 2 4) cannot be determined
Q. 2 What was the temperature outside, in degree Celsius, at 1 am?
Q. 3 What was the temperature outside, in degree Celsius, at 9 pm?
Q. 4 What best can be concluded about the number of times the AC must have either been turned on or the AC temperature setting been altered between 11:01 pm and 1:59 am?
1) Exactly 3 2) Either 2 or 3 3) Exactly 2 4) More than 3
Q. 5 What was the maximum difference between temperature outside and inside temperature, in degree Celsius, between 11:01 pm and 1:59 am?
🔁 Summary of Key Rules and Data:
- AC can be ON (Regular or Power) or OFF.
- Regular Mode: reaches set temperature in 1 hour
- Power Mode: reaches set temperature in 30 minutes
- OFF: temperature increases toward outside temp in 1 hour
- Outside temperature drops 1°C every 30 minutes from 9:00 pm (21:00) to 3:00 am (03:00)
📊 Inside temperature (from graph & table):
| Time | Inside Temp (°C) |
|---|---|
| 23:00 | 38 |
| 23:30 | 32 |
| 00:00 | 28 |
| 00:30 | 32 |
| 01:00 | 28 |
| 01:30 | 31 |
| 02:00 | 28 |
✅ Now Answering Each Question:
🔸 Q1: How many times was the AC turned off between 11:01 pm and 1:59 am?
From table and explanation:
- OFF periods:
- 00:00–00:30 → temp increased from 28 to 32 → AC was OFF ✅
- 01:00–01:30 → temp increased from 28 to 31 → AC was OFF ✅
So, two distinct OFF intervals observed.
✅ Correct answer: 3) 2
🔸 Q2: What was the temperature outside at 1:00 am?
We are told:
- At 00:00, outside temp = 36°C
- At 01:00, drops by 1°C every 30 min → drops 2°C more → 34°C
✅ Correct answer: 34°C
🔸 Q3: What was the outside temperature at 9:00 pm (21:00)?
At:
- 00:00 → 36°C
- Every 30 min before adds +1°C
- 3 steps of +1°C = 1.5 hours = 3°C
- So 21:00 = 3 hours before 00:00 → add 6°C
→ 36 + 6 = 42°C
✅ Correct answer: 42°C
🔸 Q4: Number of setting changes between 11:01 pm and 1:59 am?
From detailed case analysis:
- Settings only change at hh:00 or hh:30
- Known changes:
- AC turned ON at 23:00
- 00:30: setting changed (OFF → ON)
- 01:30: setting changed (OFF → ON)
- Between 23:00–00:30: exactly 1 setting change must have happened to ensure Power mode is used longer
→ Total changes = 3
✅ Correct answer: 3) Exactly 3
🔸 Q5: Maximum difference between outside and inside temperature?
From the graph:
- At 00:00:
- Inside = 28°C
- Outside = 36°C
- Difference = 8°C
But wait — as per table:
- Outside = 38°C at 23:00
- Inside = 28°C at 00:00
→ Difference = 10°C
✅ Correct answer: 10°C
✅ Final Answer Summary:
| Q# | Answer |
|---|---|
| 1 | 3) 2 |
| 2 | 34°C |
| 3 | 42°C |
| 4 | 3) Exactly 3 |
| 5 | 10°C |
