CAT 2025 Question Paper | DILR Slot 2

CAT Questions

CAT 2025 DILR was similar to CAT 2022 in pattern. All the four sets had 5 questions this year too. There were 6 TITA Questions. Atleast two sets were doable in each slot and one of the four sets was extremely tough. Choice of Sets became a crucial factor. Overall this section was at a medium to high level of difficulty.

The two most populous cities and the non-urban region (NUR) of each of three states, Whimshire, Fogglia, and Humbleset, are assigned Pollution Measures (PMs). These nine PMs are all distinct multiples of 10, ranging from 10 to 90. The six cities in increasing order of their PMs are: Blusterburg, Noodleton, Splutterville, Quackford, Mumpypore, Zingaloo.

The Pollution Index (PI) of a state is a weighted average of the PMs of its NUR and cities, with a weight of 50% for the NUR, and 25% each for its two cities.

There is only one pair of an NUR and a city (considering all cities and all NURs) where the PM of the NUR is greater than that of the city. That NUR and the city both belong to Humbleset.

The PIs of all three states are distinct integers, with Humbleset and Fogglia having the highest and the lowest PI respectively.

Question 2 : What is the PI of Fogglia?

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Video Explanation

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Explanatory Answer

The statements The six cities in increasing order of their PMs are: Blusterburg, Noodleton, Splutterville, Quackford, Mumpypore, Zingaloo. and There is only one pair of an NUR and a city (considering all cities and all NURs) where the PM of the NUR is greater than that of the city. That NUR and the city both belong to Humbleset. are very powerful when combined.
It implies that two of the NURs have a PM that is less than the PM of Blusterburg and other NUR has a PM between that of Blusterburg and Noodleton.

So, now we know the PMs of NURs and the cities. Also, this implies that Blusterberg & NUR3 must belong to Humbleset.

Region/City	NUR-1	NUR-2	Blusterburg	NUR-3	Noodleton	Splutterville	Quackford	Mumpypore	Zingaloo
PM	10	20	30	40	50	60	70	80	90

The PI of a state is defined as follows:
\( \operatorname{PI}(\text { state })=\frac{\operatorname{PM}(N U R)}{2}+\frac{\operatorname{PM}(\text { City } 1)+\operatorname{PM}(\text { City } 2)}{4} \)
PMs of all the NURs are divisible by 2 and produce an integer. So, for the PI of a state to be an integer, the sum of the PMs of its cities must be divisible by 4.

Observe that any pair in (30, 50, 70, 90) add up to a multiple of 4.
(60, 80) pair up to a multiple of 4. No combination of numbers between these two sets pair up to be a multiple of 4.
This implies Splutterville (60) and Mumpypore (80) joined by NUR-1 (10) or NUR-2 (20) form a state. And that state will have a PI of 40 or 45.

\( \operatorname{PI}(\text { Humbleset })=\frac{\operatorname{PM}(NUR-3)}{2}+\frac{\operatorname{PM}(\text { Blusterburg })+\operatorname{PM}(\text { City } 2)}{4} \)
\( \operatorname{PI}(\text { Humbleset })=\frac{40}{2}+\frac{30+\operatorname{PM}(\text { City } 2)}{4} \)
For the \( \operatorname{PI}(\text { Humbleset }) \) to be an integer \(\operatorname{PM}(\text { City } 2) \) must belong to (50, 70, 90)
The PI of Humbleset cane be 40, 45 or 50 respectively.

It can't be 40, because the state containing Splutterville (60) and Mumpypore (80) has a minimum PI of 40. But we know that PI's should be distinct and Humbleset has the highest PI.
If the City 2 in Humbleset is Quackford (70) then its PI will be 45. In that case, Noodleton (50) and Zingaloo (90) will belong to the same state, joined by NUR-1 (10) or NUR-2 (20), the PI of this state will be 40 or 45.
Therefore, City 2 in Humbleset can't be Quackford (70), it has to be Zingaloo (90). And the PI of Humbleset is 50.

Splutterville (60) and Mumpypore (80) belong to the same state, while Noodleton (50) and Quackford (70) should belong to the other state.
In order to have distinct PIs, Splutterville (60), Mumpypore (80) and NUR-2 (20) with a PI of 45 must belong to a state while Noodleton (50), Quackford (70) and NUR-1 (10) with a PI of 35 must belong to the other state.
Having the least PI, Noodleton (50), Quackford (70) and NUR-1 (10) must belong to Fogglia, while Splutterville (60), Mumpypore (80) and NUR-2 (20) must belong to Whimshire.

State	Region/City	PI
Humbleset	Blusterburg (30), Zingaloo (90), NUR-3 (40)	50
Whimshire	Splutterville (60), Mumpypore (80) and NUR-2 (20)	45
Fogglia	Noodleton (50), Quackford (70) and NUR-1 (10)	35

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The question is " What is the PI of Fogglia? "

Hence, the answer is '35'

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