CAT 2024 Slot 3 — DILR Question 8

Mixed PracticeEasy

Passage / Data

Answer the following questions based on the information given below.

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.

Line graph of inside temperature from 23:00 to 2:00

Time	23:00	23:30	0:00	0:30	1:00	1:30	2:00
Inside temperature (°C)	38	32	26	31	26	30	28

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.

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?

Answer & solution

A
Either 2 or 3
B
More than 3
Exactly 3
D
Exactly 2

Solution

Hard

Identify each mode from the slope, then use the rule “POWER used longer than REGULAR.” The straight 23:00→0:00 fall is ambiguous; the rule forces it to be a two-stage POWER run, revealing a hidden setting change at 23:30.

Cooling segments: 23:00→0:00 (drop $12$ in 1 hr), 0:30→1:00 (drop $5$ in 30 min), 1:30→2:00 (drop $2$ ). Rises (AC off): 0:00→0:30 and 1:00→1:30.

REGULAR reaches the set temperature in 1 hr; POWER in 30 min.

The 23:00→0:00 fall must be POWER to satisfy the duration rule. A single REGULAR run would make REGULAR last 60 min, but only $\le 60$ min of POWER is available elsewhere — POWER could never exceed REGULAR. So this hour is POWER set to $32$ (reached at 23:30), then the setting is altered at 23:30 to POWER set $26$ ; both stages have the same slope, so the line looks straight.

\begin{aligned} &23{:}00\ \text{turn on (POWER, set }32)\\ &\Rightarrow\ 23{:}30\ \text{setting altered (POWER, set }26)\ \text{— inside }11{:}01\text{ pm–}1{:}59\text{ am} \end{aligned}

List the remaining on/alter events inside the window. Turning off (at 0:00 and 1:00) is neither “turning on” nor “altering the set temperature,” so it is excluded.

\begin{aligned} &0{:}30\ \text{turn on (POWER cooling }31\to26)\\ &\Rightarrow\ 1{:}30\ \text{turn on (POWER cooling, starts }30)\\ &\Rightarrow\ \text{events}=\{23{:}30,\ 0{:}30,\ 1{:}30\} \end{aligned}

Count. Three turn-on/alter events lie strictly between 11:01 pm and 1:59 am.

\begin{aligned} &\text{turn-on / alter events}=3 \quad\text{[(1)+(2)]} \end{aligned}

\text{Exactly }3