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# Quantitative Reasoning || Logical Reasoning || CAT 2021 || 9 October

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Question 1

Direction: Read the following information carefully and answer the questions that follow.

Mathematicians are assigned a number called Erdos number (named after the famous mathematician, paul Erdos). Only Paul Erdos himself has an Erdos number of zero. Any mathematician who has written a research paper with Erdos has an Erdos number of 1. For other mathematicians, the calculation of his/her Erdos number is illustrated below:
Suppose that a mathematician X has co-authored papers with several other mathematicians From among them, mathematician Y has the smallest Erdos number. Let the Erdos number of Y be y. Then X has an Erdos number of y+1. Hence any mathematician with no co-authorship chain connected to Erdos has an Erdos number of infinity.
In a seven day long mini-conference organized in memory of Paul Erdos, a close group of eight mathematicians, call them A, B, C, D, E, F, G and H, discussed some research problems, At the beginning of the conference. A was the only participant who had an infinite Erdos number. Nobody had an Erdos number less that that of F.
• On the third day of the conference F co-authored a paper jointly with A and C. this reduced the average Erdos number of the group of eight mathematicians to 3. The Erdos numbers of B, D, E, G and H remained unchanged with the writing of this paper. Further no other co-authorship among any three members would have reduced the average Erdos number of the group of eight to as low as 3.
• At the end of the third day, five members of this group had identical Erdos numbers while the other three had Erdos numbers distinct from each other.
• On the fifth day, E co-authored a paper with F which reduced the group’s average Erdos number by 0.5. The Erdos numbers of the remaining six were unchanged with the writing of this paper
• No other paper was written during the conference.
The person having the largest Erdos number at the end of the conference must have had Erdos number (at that time):

Question 2

Direction: Read the following information carefully and answer the questions that follow.

Mathematicians are assigned a number called Erdos number (named after the famous mathematician, paul Erdos). Only Paul Erdos himself has an Erdos number of zero. Any mathematician who has written a research paper with Erdos has an Erdos number of 1. For other mathematicians, the calculation of his/her Erdos number is illustrated below:
Suppose that a mathematician X has co-authored papers with several other mathematicians From among them, mathematician Y has the smallest Erdos number. Let the Erdos number of Y be y. Then X has an Erdos number of y+1. Hence any mathematician with no co-authorship chain connected to Erdos has an Erdos number of infinity.
In a seven day long mini-conference organized in memory of Paul Erdos, a close group of eight mathematicians, call them A, B, C, D, E, F, G and H, discussed some research problems, At the beginning of the conference. A was the only participant who had an infinite Erdos number. Nobody had an Erdos number less that that of F.
• On the third day of the conference F co-authored a paper jointly with A and C. this reduced the average Erdos number of the group of eight mathematicians to 3. The Erdos numbers of B, D, E, G and H remained unchanged with the writing of this paper. Further no other co-authorship among any three members would have reduced the average Erdos number of the group of eight to as low as 3.
• At the end of the third day, five members of this group had identical Erdos numbers while the other three had Erdos numbers distinct from each other.
• On the fifth day, E co-authored a paper with F which reduced the group’s average Erdos number by 0.5. The Erdos numbers of the remaining six were unchanged with the writing of this paper
• No other paper was written during the conference.
How many participants in the conference did not change their Erdos number during the conference?

Question 3

Direction: Read the following information carefully and answer the questions that follow.

Mathematicians are assigned a number called Erdos number (named after the famous mathematician, paul Erdos). Only Paul Erdos himself has an Erdos number of zero. Any mathematician who has written a research paper with Erdos has an Erdos number of 1. For other mathematicians, the calculation of his/her Erdos number is illustrated below:
Suppose that a mathematician X has co-authored papers with several other mathematicians From among them, mathematician Y has the smallest Erdos number. Let the Erdos number of Y be y. Then X has an Erdos number of y+1. Hence any mathematician with no co-authorship chain connected to Erdos has an Erdos number of infinity.
In a seven day long mini-conference organized in memory of Paul Erdos, a close group of eight mathematicians, call them A, B, C, D, E, F, G and H, discussed some research problems, At the beginning of the conference. A was the only participant who had an infinite Erdos number. Nobody had an Erdos number less that that of F.
• On the third day of the conference F co-authored a paper jointly with A and C. this reduced the average Erdos number of the group of eight mathematicians to 3. The Erdos numbers of B, D, E, G and H remained unchanged with the writing of this paper. Further no other co-authorship among any three members would have reduced the average Erdos number of the group of eight to as low as 3.
• At the end of the third day, five members of this group had identical Erdos numbers while the other three had Erdos numbers distinct from each other.
• On the fifth day, E co-authored a paper with F which reduced the group’s average Erdos number by 0.5. The Erdos numbers of the remaining six were unchanged with the writing of this paper
• No other paper was written during the conference.
The Erdos number of C at the end of the conference was:

Question 4

Direction: Read the following information carefully and answer the questions that follow.

Mathematicians are assigned a number called Erdos number (named after the famous mathematician, paul Erdos). Only Paul Erdos himself has an Erdos number of zero. Any mathematician who has written a research paper with Erdos has an Erdos number of 1. For other mathematicians, the calculation of his/her Erdos number is illustrated below:
Suppose that a mathematician X has co-authored papers with several other mathematicians From among them, mathematician Y has the smallest Erdos number. Let the Erdos number of Y be y. Then X has an Erdos number of y+1. Hence any mathematician with no co-authorship chain connected to Erdos has an Erdos number of infinity.
In a seven day long mini-conference organized in memory of Paul Erdos, a close group of eight mathematicians, call them A, B, C, D, E, F, G and H, discussed some research problems, At the beginning of the conference. A was the only participant who had an infinite Erdos number. Nobody had an Erdos number less that that of F.
• On the third day of the conference F co-authored a paper jointly with A and C. this reduced the average Erdos number of the group of eight mathematicians to 3. The Erdos numbers of B, D, E, G and H remained unchanged with the writing of this paper. Further no other co-authorship among any three members would have reduced the average Erdos number of the group of eight to as low as 3.
• At the end of the third day, five members of this group had identical Erdos numbers while the other three had Erdos numbers distinct from each other.
• On the fifth day, E co-authored a paper with F which reduced the group’s average Erdos number by 0.5. The Erdos numbers of the remaining six were unchanged with the writing of this paper
• No other paper was written during the conference.
How many participants had the same Erdos number at the beginning of the conference?

Question 5

There are 10 people in a queue waiting to enter a hall. The hall has 10 seats numbered from 1 to 10. The first person in the queue enters the hall, chooses any seat and sits there. The (p)th person in the queue, where p can be 2, 3, 4, …, 10, enters the hall after the (p-1)th person is seated. He sits in the seat number p if he finds it vacant; otherwise he takes any unoccupied seat.

What is the total number of ways in which 10 seats can be filled up, provided the 10th person occupies seat number 10?

Question 6

There are 10 people in a queue waiting to enter a hall. The hall has 10 seats numbered from 1 to 10. The first person in the queue enters the hall, chooses any seat and sits there. The (p)th person in the queue, where p can be 2, 3, 4, …, 10, enters the hall after the (p-1)th person is seated. He sits in the seat number p if he finds it vacant; otherwise he takes any unoccupied seat.

What is the total number of ways in which 10 seats can be filled up?

Question 7

There are 10 people in a queue waiting to enter a hall. The hall has 10 seats numbered from 1 to 10. The first person in the queue enters the hall, chooses any seat and sits there. The (p)th person in the queue, where p can be 2, 3, 4, …, 10, enters the hall after the (p-1)th person is seated. He sits in the seat number p if he finds it vacant; otherwise he takes any unoccupied seat.

What is the total number of ways in which 10 seats can be filled up, provided the 1st person occupies seat number 1?

Question 8

In a box there are some candies of three colours: pink, green and red. There are candies of 3 sizes in each colour: large, medium and small. There is at least one candy of each size in each colour. Also, it is known that:

1. The total number of large candies is smallest and that of small candies is largest.

2. The number of candies of three sizes is in an A.P. in all three colours, in any order. The common difference of these A.P.'s is 2 in one of the colours, 3 in another colour and 4 in the third colour.

3. Medium red candies are 12 and are the largest in number whereas medium pink candies are 2 and the smallest in number.

4. Among green candies, small candies and large candies differ by 8.

5. Large red candies are equal in number to small pink candies.

Answer the following questions on the basis of information given above.

What is the total number of candies in the box?

Question 9

In a box there are some candies of three colours: pink, green and red. There are candies of 3 sizes in each colour: large, medium and small. There is at least one candy of each size in each colour. Also, it is known that:

1. The total number of large candies is smallest and that of small candies is largest.

2. The number of candies of three sizes is in an A.P. in all three colours, in any order. The common difference of these A.P.'s is 2 in one of the colours, 3 in another colour and 4 in the third colour.

3. Medium red candies are 12 and are the largest in number whereas medium pink candies are 2 and the smallest in number.

4. Among green candies, small candies and large candies differ by 8.

5. Large red candies are equal in number to small pink candies.

Answer the following questions on the basis of information given above.

Which of the following can uniquely be determined?

Question 10

In a box there are some candies of three colours: pink, green and red. There are candies of 3 sizes in each colour: large, medium and small. There is at least one candy of each size in each colour. Also, it is known that:

1. The total number of large candies is smallest and that of small candies is largest.

2. The number of candies of three sizes is in an A.P. in all three colours, in any order. The common difference of these A.P.'s is 2 in one of the colours, 3 in another colour and 4 in the third colour.

3. Medium red candies are 12 and are the largest in number whereas medium pink candies are 2 and the smallest in number.

4. Among green candies, small candies and large candies differ by 8.

5. Large red candies are equal in number to small pink candies.

Answer the following questions on the basis of information given above.

If there are 9 small red candies then how many large pink candies are there?
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