CAT Preparation - DILR Solution for IIMCAT 2017 - Slot 1 ( Set 7 and Set 8)

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[iimcat]

Set 2 - Car Routes :

Four cars need to travel from Akala (A) to Bakala (B). Two routes are available, one via Mamur (M) and the other via Nanur (N). The roads from A to M, and from N to B, are both short and narrow. In each case, one car takes 6 minutes to cover the distance, and each additional car increases the travel time per car by 3 minutes because of congestion. (For example, if only two cars drive from A to M, each car takes 9 minutes.) On the road from A to N, one car takes 20 minutes, and each additional car increases the travel time per car by 1 minute. On the road from M to B, one car takes 20 minutes, and each additional car increases the travel time per car by 0.9 minute.

The police department orders each car to take a particular route in such a manner that it is not possible for any car to reduce its travel time by not following the order, while the other cars are following the order.

Questions

1. How many cars would be asked to take the route A-N-B, that is Akala-Nanur-Bakala route, by the police department? (TITA)

2. If all the cars follow the police order, what is the difference in travel time (in minutes) between a car which takes the route A-N-B and a car that takes the route A-M-B?
1 B) 0.1 C) 0.2 D) 0.9

3. A new one-way road is built from M to N. Each car now has three possible routes to travel from A to B: A-M-B, A-N-B and A-M-N-B. On the road from M to N, one car takes 7 minutes and each additional car increases the travel time per car by 1 minute. Assume that any car taking the A-M-N-B route travels the A-M portion at the same time as other cars taking the A-M-B route, and the N-B portion at the same time as other cars taking the A-N-B route.
How many cars would the police department order to take the A-M-N-B route so that it is not possible for any car to reduce its travel time by not following the order while the other cars follow the order? (Assume that the police department would never order all the cars to take the same route.)
(TITA)

4. A new one-way road is built from M to N. Each car now has three possible routes to travel from A to B: A-M-B, A-N-B and A-M-N-B. On the road from M to N, one car takes 7 minutes and each additional car increases the travel time per car by 1 minute. Assume that any car taking the A-M-N-B route travels the A-M portion at the same time as other cars taking the A-M-B route, and the N-B portion at the same time as other cars taking the A-N-B route.
If all the cars follow the police order, what is the minimum travel time (in minutes) from A to B? (Assume that the police department would never order all the cars to take the same route.)
A) 26 B) 32 C) 29.9 D) 30

[iimcat]

Set 1 - Airline Company :

A new airlines company is planning to start operations in a country. The company has identified ten different cities which they plan to connect through their network to start with. The flight duration between any pair of cities will be less than one hour. To start operations, the company has to decide on a daily schedule.

The underlying principle that they are working on is the following:

Any person staying in any of these 10 cities should be able to make a trip to any other city in the morning and should be able to return by the evening of the same day.

Questions

1. If the underlying principle is to be satisfied in such a way that the journey between any two cities can be performed using only direct (non-stop) flights, then the minimum number of direct flights to be scheduled is:
45 B) 90 C) 180 D) 135

2. Suppose three of the ten cities are to be developed as hubs. A hub is a city which is connected with every other city by direct flights each way, both in the morning as well as in the evening. The only direct flights which will be scheduled are originating and/or terminating in one of the hubs. Then the minimum number of direct flights that need to be scheduled so that the underlying principle of the airline to serve all the ten cities is met without visiting more than one hub during one trip is:
A) 54 B) 120 C) 96 D) 60

3. Suppose the 10 cities are divided into 4 distinct groups G1, G2, G3, G4 having 3, 3, 2 and 2 cities respectively and that G1 consists of cities named A, B and C. Further, suppose that direct flights are allowed only between two cities satisfying one of the following:
1. Both cities are in G1
2. Between A and any city in G2
3. Between B and any city in G3
4. Between C and any city in G4
Then the minimum number of direct flights that satisfies the underlying principle of the airline is: (TITA)

4. Suppose the 10 cities are divided into 4 distinct groups G1, G2, G3, G4 having 3, 3, 2 and 2 cities respectively and that G1 consists of cities named A, B and C. Further, suppose that direct flights are allowed only between two cities satisfying one of the following:
1. Both cities are in G1
2. Between A and any city in G2
3. Between B and any city in G3
4. Between C and any city in G4
However, due to operational difficulties at A, it was later decided that the only flights that would operate at A would be those to and from B. Cities in G2 would have to be assigned to G3 or to G4.
What would be the maximum reduction in the number of direct flights as compared to the situation before the operational difficulties arose? (TITA)

[aktake]

In the airlines company question, you could have made a figure showing connections via each flight path (road, jetstream, whatever you call it). The number of paths is then multiplied times 4 to get all flights (2 in the morning and 2 in the evening). Using this, we could have made connection diagrams for each sub-question and multiplied it times 4 to get the result.

I feel it makes intuitive sense, but I'm worried it may be too elegant and perhaps therefore wrong. Have you tried it, sir?

[figuring_out]

Why we have Taken no. Of flights as 4 instead of 2 A to B 1 flight in morning and 1 in evening.. irrespective of person is living in A or B can take the flight and come back

[anabilkanungoe]

Cars travel Route Q4: 51:22
2(A-M-B)+0(A-M-N-B)+2(A-N-B)
its taking 29.9 mins and 30 mins respectively
please clarify

[pranuthigodse]

this is helpful thanks a lot! i think the car travel 4 answer is 26?

[chhavidua-il]

If It's a one-way road from M to N then why are we considering cases in which cars travelling on A-N-B route might switch to M-N road to save time?

[shruthinarayanan]

Wonderful sets coupled with wonderful techniques of solving

[komaltuli]

Sir, in the second set, u didn't consider the possibilities of zero cars in more than two cases, as in only 0-3-1 and 0-2-2 were considered, though they aren't the final answers, but why haven't all possibilities been taken into account?

[nafimohammad]

In 2IIM DILR course, How many sets were there? I am talking about paid course?

[AbhishekMishra-xxsq]

Yes, the solution was helpful but why is the solution of 2 sets over 50 min long? arent we supposed to attempt the whole section in under 40 mins now? The time limit is absurd on cat papers!

[sahilahmed]

Why don't they implement this algorithm to Bangalore tanked up traffic????!!! *Google maps do something!!*

[invest_with_kartik]

Thank you so much sir.. your Di-Lr cat 2017 solutions are very helpful.. nicely explained