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QUESTION: 1

**Directions for Questions:** 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.

(2017)

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

Solution:

As there are four cars and as the time through each route is nearly the same, two cars should go through A–M–B and the other two through A–N–B. In case three cars are directed to go through any of the routes, one of the three cars can break the police order and reduce its travel time.

QUESTION: 2

**Directions for Questions:** 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.

(2017)

**Q. If all the car s follow the police order, what is the differen ce 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?**

Solution:

According to the police order 2 cars each would pass through A–M–B and A–N–B.

Then time taken through A–M–B = 29.9 and time taken through A–N–8 = 30.0

∴ Difference = 0.1

QUESTION: 3

**Directions for Questions:** 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.

(2017)

**Q. 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 AM-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.)**

Solution:

Original time (A–N–B) = 21 + 12 = 33 (three cars) New time = 12 + 20.9 = 32.9

If one car is directed through M–N, one of the car directed through M–B can break the Police order and go through M–N and save time as shown– original time (A–M–B) = 12 (3 cars) + 20.9 = 32.9

New time (A–M–N–B) = 12 + 8 + 12 = 32 Minutes

∴ Only two cars must be directed through M–N Such that any car breaking the Police order can not reduce the travel time.

QUESTION: 4

**Directions for Questions:** 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.

(2017)

**Q. 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 AM-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.)**

Solution:

When all cars follow the Police order the time taken would be A–M–B (1 car)

= 12 + 20 = 32 minutes.

A–M–N–B (2 cars) = 12 + 8 + 12

= 32 minutes.

A–N–B (1 car) = 20 + 12

= 32 minutes.

QUESTION: 5

**Directions for Questions:** 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.

(2017)

**Q. If the underlying principle is to 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.**

Solution:

Number of ways of selecting two cities from ten cities -^{10}C_{2}

= 10 x 9 / 2

Hence the minimum no. of flights that must be scheduled = 45 × 4 = 180

QUESTION: 6

**Directions for Questions:** 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.

(2017)

**Q. 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.**

Solution:

Let the ten cities be represented by A through J.

Among these ten cities, consider A, B and C to be hubs and the other seven cities to be non-hub cities.

It is given that any direct flight should originate and terminate at a hub.

Total flight from D = 4 × 3 = 12

[∵ Only for A, B, C]

Non hub city flight = 12 × 7 = 84

[∵ for other then A, B, C]

Total flight = 12 + 84 = 96

QUESTION: 7

**Directions for Questions:** 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.

(2017)

**Q. 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.**

(I) Both cities are in G1

(II) Between A and any city in G2

(III) Between B and any city in G3

(IV) Between C and any city in G4

Then the minimum number of direct flights that satisfies the underlying principle of the airline is.

Solution:

Given that G1 has the cities A, B and C. G2, G3 and G4 have 3, 2 and 2 cities respectively. From the given conditions, we can see that a city in G2 cannot be connected by a direct flight to a city in G3 or G4. Hence, for a person to travel from a city in G2 to a city in G3 or G4, all the citres in G2 must be connected to A and from A, he can travel to B or C to travel to a city G3 or G4 respectively.

Hence, the 3 cities in G2 must be connected to A between each pair of cities there must be four flights.

Hence, there must be 4 × 3 = 12 flights between cities in G2 and A.

Since there are 2 cities in G3, there must be 2 × 4 = 8 flights between cities in G3 and 8.

Since there are 2 cities in G4, there must be 2× 4 = 8 flights between cities in G4 and C.

Also, the cities in G1, i.e., A, B and C must be connected to each other Hence, there must be an additional 4 × 3 = 12 flights between these three cities.

Therefore, the total minimum number of direct flights that must be scheduled =12 + 8 + 8 + 12 = 40

QUESTION: 8

**Directions for Questions:** 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.

(2017)

**Q.** 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.

(A) Both cities are in G1

(B) Between a and any city in G2

(C) Between b and any city in G3

(D) 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 G4.

**What would be the maximum reduction in the number of direct flights as compared to the situation before the operational difficulties arose?**

Solution:

It is given that the cities in G2 will be assigned to G3 or G4. However, this, by itself, will not result in any reduction in the number of flights because the cities in G2 will still have to be connected to either 8 or C.

However, it is also given that there are now flights between A and C. Hence, the 4 flights that would have been scheduled in the previous case, will now not be scheduled. Hence, the reduction in the number of flights can be a maximum of 4.

QUESTION: 9

**Directions for Questions:** answer the questions on the basis of the information given below.

A team must be selected from ten probable – A, B, C, D, E, F, G, H, I and J. Of these, A, C, E and are forwards, B, G and H are point guards and D, F and I are defenders.

Further, the following conditions need to be observed:

(i) The team must have at least one forward, one point guard and one defender.

(ii) If the team includes J, it must also include F.

(iii) The team must include E or B, but not both.

(iv) If the team includes G, it must also include F.

(v) The team must include exactly one among C, G and I.

(vi) C and F cannot be members of the same team

(vii) D and H cannot be members of the same team

(viii) The team must include both A and D or neither of them.

There is no restriction on the number of members in the team.

(2016)

**Q. What could be the size of the team that includes G?**

Solution:

The conditions related to G are, if G is included, F also must be included and C and I cannot be included. There are no other conditions related to G. Hence, a team that includes G can be

(a) AEJGDF

(b) ADGFE

(c) EGFH

More arrangement are possible. Hence, the number of members in the team is 4 or 5.

QUESTION: 10

**Directions for Questions:** answer the questions on the basis of the information given below.

A team must be selected from ten probable – A, B, C, D, E, F, G, H, I and J. Of these, A, C, E and are forwards, B, G and H are point guards and D, F and I are defenders.

Further, the following conditions need to be observed:

(i) The team must have at least one forward, one point guard and one defender.

(ii) If the team includes J, it must also include F.

(iii) The team must include E or B, but not both.

(iv) If the team includes G, it must also include F.

(v) The team must include exactly one among C, G and I.

(vi) C and F cannot be members of the same team

(vii) D and H cannot be members of the same team

(viii) The team must include both A and D or neither of them

There is no restriction on the number of members in the team.

(2016)

**Q. What would be the size of the largest possible team?**

Solution:

The largest possible team can have six members. As one of, (E, B) and (D, H) and two of (C, G, I) must definitely be excluded. Hence, a miniumum of four members must be excluded. So size of the largest possible team is 6.

QUESTION: 11

**Directions for Questions:** answer the questions on the basis of the information given below.

A team must be selected from ten probable – A, B, C, D, E, F, G, H, I and J. Of these, A, C, E and are forwards, B, G and H are point guards and D, F and I are defenders.

Further, the following conditions need to be observed:

(i) The team must have at least one forward, one point guard and one defender.

(ii) If the team includes J, it must also include F.

(iii) The team must include E or B, but not both.

(iv) If the team includes G, it must also include F.

(v) The team must include exactly one among C, G and I.

(vi) C and F cannot be members of the same team

(vii) D and H cannot be members of the same team

(viii) The team must include both A and D or neither of them

There is no restriction on the number of members in the team.

(2016)

**Q. Who cannot be included in a team of size 6?**

Solution:

If a team includes H, neither A nor D can be included in that team and further one of (E, B) and two of (C, G, I) must be excluded making the size of the team as 5.

QUESTION: 12

**Directions for Questions:** answer the questions on the basis of the information given below.

A team must be selected from ten probable – A, B, C, D, E, F, G, H, I and J. Of these, A, C, E and are forwards, B, G and H are point guards and D, F and I are defenders.

Further, the following conditions need to be observed:

(i) The team must have at least one forward, one point guard and one defender.

(ii) If the team includes J, it must also include F.

(iii) The team must include E or B, but not both.

(iv) If the team includes G, it must also include F.

(v) The team must include exactly one among C, G and I.

(vi) C and F cannot be members of the same team

(vii) D and H cannot be members of the same team

(viii) The team must include both A and D or neither of them

There is no restriction on the number of members in the team.

(2016)

**Q. What can be the size of the team that includes C?**

Solution:

If C is included, G, I and F must be excluded.

As F is excluded, J also must be excluded.

As F and I are excluded, the only remaining defender D must be included.

As D is included, A must be included and thus H must be excluded.

As G and H are excluded, the only remaining point guard B must be selected.

As B is included, E must be excluded.

∴ The team is (A, B, C, D) So, size of the team that includes C is 4.

QUESTION: 13

**Directions for Questions: Answer the questions on the basis of the information given below.**

After facing yet another World Cup debacle, the Board of Cricket Control in India (BCCI) is in search of a new coach for the team. It shortlisted five persons – Anshuman, Buchanen, John, Whatmore and Chappel. Each of them is from a different country among Australia, India, Japan, Pakistan and Canada, not necessarily in that order. At present, each of them is coaching the team of a different country among Australia, Bangladesh, china, Wales and Bermuda, not necessarily in that order. The following details were also observed about their particulars:

(i) For any person, each of his three particulars – his name, the name of the country from which he is and the name of the country that he is coaching at present, starts with a different letter.

(ii) Whatmore is coaching Australia and John is from neither Australia nor Pakistan.

(iii) Buchanen is not coaching China and the person who is coaching Bermuda is from Canada.

(iv) Anshuman is neither from Canada nor from Pakistan and also the person from Pakistan is coaching Bangladesh.

(2016)

**Q. Whatmore is from which country?**

Solution:

Let the countries to which the persons belong i.e.

Australia, Canada, Pakistan, India and Japan be represented by A, C, P, I and J respectively. Let the countries that they are coaching i.e. Australia, Wales, Bangladesh, Bermuda and China be represented by Au, Wa, Ba, Be and Ch respectively.

What more can be from India on Japan, So, can be defermined.

QUESTION: 14

**Directions for Questions: Answer the questions on the basis of the information given below.**

After facing yet another World Cup debacle, the Board of Cricket Control in India (BCCI) is in search of a new coach for the team. It shortlisted five persons – Anshuman, Buchanen, John, Whatmore and Chappel. Each of them is from a different country among Australia, India, Japan, Pakistan and Canada, not necessarily in that order. At present, each of them is coaching the team of a different country among Australia, Bangladesh, china, Wales and Bermuda, not necessarily in that order. The following details were also observed about their particulars:

(i) For any person, each of his three particulars – his name, the name of the country from which he is and the name of the country that he is coaching at present, starts with a different letter.

(ii) Whatmore is coaching Australia and John is from neither Australia nor Pakistan.

(iii) Buchanen is not coaching China and the person who is coaching Bermuda is from Canada.

(iv) Anshuman is neither from Canada nor from Pakistan and also the person from Pakistan is coaching Bangladesh.

(2016)

**Q. Who is the person from Australia?**

Solution:

Let the countries to which the persons belong i.e.

Australia, Canada, Pakistan, India and Japan be represented by A, C, P, I and J respectively. Let the countries that they are coaching i.e. Australia, Wales, Bangladesh, Bermuda and China be represented by Au, Wa, Ba, Be and Ch respectively.

According to above data.

Buchanen is from Australia.

QUESTION: 15

**Directions for Questions: Answer the questions on the basis of the information given below.**

After facing yet another World Cup debacle, the Board of Cricket Control in India (BCCI) is in search of a new coach for the team. It shortlisted five persons – Anshuman, Buchanen, John, Whatmore and Chappel. Each of them is from a different country among Australia, India, Japan, Pakistan and Canada, not necessarily in that order. At present, each of them is coaching the team of a different country among Australia, Bangladesh, china, Wales and Bermuda, not necessarily in that order. The following details were also observed about their particulars:

(i) For any person, each of his three particulars – his name, the name of the country from which he is and the name of the country that he is coaching at present, starts with a different letter.

(ii) Whatmore is coaching Australia and John is from neither Australia nor Pakistan.

(iii) Buchanen is not coaching China and the person who is coaching Bermuda is from Canada.

(iv) Anshuman is neither from Canada nor from Pakistan and also the person from Pakistan is coaching Bangladesh.

(2016)

**Q. The person from Japan is definitely not coaching**

Solution:

Let the countries to which the persons belong i.e.

Australia, Canada, Pakistan, India and Japan be represented by A, C, P, I and J respectively. Let the countries that they are coaching i.e. Australia, Wales, Bangladesh, Bermuda and China be represented by Au, Wa, Ba, Be and Ch respectively.

Wales had Buchanen, who from Australia as their coach.

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