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Practice Question - 23 (Routes and Network) | 100 DILR Questions for CAT Preparation PDF Download

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.

Q1: How many cars would be asked to take the route A-N-B, that is Akala-Nanur-Bakala route, by the police department?

Practice Question - 23 (Routes and Network) | 100 DILR Questions for CAT PreparationView Answer  Practice Question - 23 (Routes and Network) | 100 DILR Questions for CAT Preparation

Ans: 2
Since there are two routes i.e A-M-B and A-N-B and four cars, then 2 cars must be allowed to take each route. In case if one car tried to break rule, then its travel time will increase. Now assume that on route A-M-B three cars are allowed and on route A-N-B one car is allowed, then one car running on A-M-B can break the rule and reduce its travel time. Hence, two cars must be allowed on each route.

Q2: 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?
(a) 1
(b) 0.1
(c) 0.2
(d) 0.9

Practice Question - 23 (Routes and Network) | 100 DILR Questions for CAT PreparationView Answer  Practice Question - 23 (Routes and Network) | 100 DILR Questions for CAT Preparation

Ans: (b) 
Since, two cars are allowed on each route, The A-M part and N-B will take same time. The difference will be in travelling M-B part and A-N part, and that difference is 0.1 minute. In route M-B car will take 20+0.9 = 20.9 min and in route A-N car will take 20+1 = 21 min. So difference = 0.1 min

Q3: 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.)

Practice Question - 23 (Routes and Network) | 100 DILR Questions for CAT PreparationView Answer  Practice Question - 23 (Routes and Network) | 100 DILR Questions for CAT Preparation

Ans: 2
Case 1:  Let us assume 1 car takes AMB route, 3 cars take ANB route
Then travel time of AMB will be A-M + M-B = 6 + 20 = 26
Then travel time of ANB will be A-N + N-B = (20+2) + (6+3*2) = 34
Now, one car(A) travelling on ANB broke the rule and decided to move on AMB route. then

Case 2: Let us assume 2 cars take AMB route, 2 cars take ANB route
Then travel time of AMB will be A-M + M-B =(6+3) + (20+0.9) = 29.9
Then travel time of ANB will be A-N + N-B = (20+1) + (6+3) = 30
Since the car A reduced its time from 34 to 29.9, the case-1 route is not optimal, hence case 1 is invalid.
Now, one car(B) travelling on ANB broke the rule and decided to move on AMB route. then, case

Case 3:  Let us assume 3 cars take AMB route, 1 car take ANB route
Then travel time of AMB will be A-M + M-B = (6+3*2) + (20+0.9*2) = 33.8
Then travel time of ANB will be A-N + N-B = (20) + (6) = 29
Now, one car(C) travelling on AMB broke the rule and decided to move on AMNB route. then

Case 4:  Let us assume 2 cars take AMB route, 1 car take AMNB route and other take ANB route. 
Then travel time of AMB will be A-M + M-B = (6+3*2) + (20+0.9) = 32.9
Then travel time of AMNB will be A-M + M-N + N-B = (6+3*2) + (7) + (6+3) = 28
Then travel time of ANB will be A-N + N-B = (20) + (6+3) = 29
Since this car(C) reduced its time from 33.8 to 28, the route is not optimal, hence case 3 is invalid.
Since this car(B) reduced its time from 30 to 28, the route is not optimal, hence case 2 is invalid.
Now, one car(D) travelling on AMB broke the rule and decided to move on AMNB route. then

Case 5:   Let us assume 1 car takes AMB route, 2 cars take AMNB route and other takes ANB route.
Then, the portion A-M will be travelled by 3 cars, M-B by one car, M-N by 2 cars, A-N by 1 car and N-B by 3 cars. 
Then, travel time of AMB will be A-M + M-B = (6+3*2) + (20) = 32
Then, travel time of AMNB will be A-M + M-N + N-B = (6+3*2) + (7+1) + (6+3*2) = 32
Then, travel time of ANB will be A-N + N-B = (20) + (6+3*2) = 32
It is clear that the car D reduced its time from 32.9 min to 32 min if it broke the rule. Hence, case 4 is invalid.
In this arrangement of case 5, no car can improve their travel time by changing their path
Hence, the optimal allocation will be to order 2 cars on A-M-N-B route.

Q4: 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 j. 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

Practice Question - 23 (Routes and Network) | 100 DILR Questions for CAT PreparationView Answer  Practice Question - 23 (Routes and Network) | 100 DILR Questions for CAT Preparation

Ans: (b)
From the previous question we have found that
1 car take AMB route, 2 cars take AMNB route and other take ANB route. 
Then the portion A-M will be travelled by 3 cars, M-B by one car, M-N by 2 cars, A-N by 1 car and N-B by 3 cars. 
Then travel time of AMB will be A-M + M-B = (6+3*2) + (20) = 32
Then travel time of AMNB will be A-M + M-N + N-B = (6+3*2) + (7+1) + (6+3*2) = 32
Then travel time of ANB will be A-N + N-B = (20) + (6+3*2) = 32
The minimum travel time from A to B is 32 min.

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FAQs on Practice Question - 23 (Routes and Network) - 100 DILR Questions for CAT Preparation

1. What are the key concepts related to routes and networks that are important for competitive exams?
Ans. Key concepts include understanding different types of routes (direct, indirect, and network routes), network topology (star, bus, ring, and mesh), graph theory basics, optimization techniques for routing, and real-world applications of routing algorithms in transportation and telecommunications.
2. How can I improve my problem-solving skills for routes and network questions in competitive exams?
Ans. To improve problem-solving skills, practice a variety of route and network problems regularly, familiarize yourself with common algorithms (like Dijkstra's and A*), work on understanding graphs and their properties, and participate in mock tests that simulate exam conditions to build speed and accuracy.
3. What types of questions can I expect regarding routes and networks in competitive exams?
Ans. You can expect questions that involve calculating shortest paths, determining the most efficient routes, analyzing network connectivity, and applying graph theory concepts to real-life scenarios. Questions may also include theoretical aspects, like definitions and applications of key terms.
4. Are there any specific strategies to tackle difficult routes and network questions during the exam?
Ans. Yes, effective strategies include breaking down complex problems into smaller parts, sketching diagrams to visualize routes, identifying key information upfront, using elimination methods for multiple-choice questions, and managing your time wisely to ensure you can attempt all questions.
5. How important is understanding network topology for routes and network questions in competitive exams?
Ans. Understanding network topology is crucial as it provides insights into how different network structures affect routing efficiency and connectivity. Familiarity with various topologies allows you to analyze problems more effectively and apply the right strategies for optimization and problem-solving.
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