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CAT-2006-Previous Years Paper

Question
13 out of 25
 

Mathematicians are assigned a number called Erdös number (named after the famous mathematician, Paul Erdös). Only Paul Erdös himself has an Erdös number of zero. Any mathematician who has written a research paper with Erdös has an Erdös number of 1. For other mathematicians, the calculation of his/her Erdös 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 Erdös number. Let the Erdös number of Y be y. Then X has an Erdös number of y + 1. Hence, any mathematician with no co-authorship chain connected to Erdös has an Erdös number of infinity.

In a seven day long mini-conference organized in memory of Paul Erdös, 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 Erdös number. Nobody had an Erdös number less than that of F.

·      On the third day of the conference F co-authored a paper jointly with A. and C. . This reduced the average Erdös number of the group of eight mathematicians to 3. The Erdös numbers of B. , D. Go through the options. and H remained unchanged with the writing of this paper. Further, no other co-authorship among any three members would have reduced the average Erdös 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 Erdös numbers while the other three had Erdös numbers distinct from each other.

·      On the fifth day, E co-authored a paper with F which reduced the group’s average Erdös number by 0.5. The Erdös 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 Erdös number at the beginning of the conference?



A 2

B 3

C 4

D 5

Following information is given:

i. There are eight participants—A, B, C, D, E, F, G and H.

ii. At the beginning A’s Erdös number is infinity.

iii. None of them is having an Erdös number less than that of F.

iv. On the 3rd day, (after F, A and C co-authored the paper) the average of their Erdös number is 3 (i.e., total Erdös number of the group is 4) that is the least possible.

v. No other combination of three participants was possible that reduces the average upto 3 at this time.

vi. At this point of time, 5 members have same Erdös numbers, and the other 3 have distinct Erdös number.

vii. On the 5th day, E and F co-authored the paper; this reduces the average Erdös number by 0.5 (i.e., the total is reduced by 4).

viii. No other paper was written during the conference.

ix. Each time the change in Erdös number occurs, it occurs only for those who co-authored a paper.

x. Each time a participant co-authors a paper, his Erdos number becomes exactly one more than that of the F.

Assume that the Erdos numbers of A, B, C, D, E, F, G and H be abcdefg and h.

Then from (iv) after the 3rd day:

= c = f + 1;

And, + b + c + d + e + f + g + h = 24

5 of them have same Erdös number, and other 3 have distinct numbers. So the other 5 members should have an Erdös number same as that of A and C, i.e., (F + 1).

Hence, 5(F + 1) + f + p + q = 24 (where p and q represents the Erdös numbers which are distinct)

= 6f + p + q = 19

Hence, the possible values of f are 1, 2 and 3 only. Now 3 can be discarded because it will give rise to p + q = 1.

Hence, one of the p and q will be 0 (which is not possible).

Let us check for f = 2

p + q = 7

Possible values of p and q are (1 + 6) (2 + 5) and (3 + 4). None of these is possible.

Hence, for f = 1 and a = = 2 and other 3 participants will also have their Erdös number as 2.

Now, if c = 2 at the beginning of the conference, then there are 3 other possibilities (F, A and any one of B, D, G and H; who does not have an Erdös number 7) that could have brought the total Erdös number equal to 24, which violates the condition (v) written above. So C must be greater than 2 at the starting.

So at the starting:

= infinity, c > 2, f = 1 and e = 6

And one of the bdg and h will be 7 and other three will be 2.

At the end:

a = c = 2, f = 1, e = 2

And one of the bdg and h will be 7 and other three will be 2.


Ans. B

CAT-2006-Previous Years Paper Flashcard List

25 flashcards
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In a Class X Board examination, ten papers are distributed over five groups—PCB, Mathematics, Social Science, Vernacular and English. Each of the ten papers is evaluated out of 100. The final score of a student is calculated in the following manner. First, the Group Scores are obtained by averaging marks in the papers within the Group. The final score is the simple average of the Group Scores. The data for the top ten students are presented below. (Dipan’s score in English Paper II has been intentionally removed in the table.) Name of the Student PCB Group Mathemetics Social Science Group Vernacular Group English Group Final Score Phy Chem Bio Hist Geo Paper I Paper II Paper I Paper II Ayesha (G) 98 96 97 98 95 93 94 96 96 98 96.2 Ram B. 97 99 95 97 95 95 96 94 96 98 96.1 Dipan B. 98 98 98 95 96 95 96 94 96 ?? 96.0 Sagnik B. 97 98 99 96 96 98 94 97 92 94 95.9 Sanjiv B. 95 96 97 98 97 96 92 93 95 96 95.7 Shreya (G) 96 89 85 100 97 98 94 95 96 95 95.5 Joseph B. 90 94 98 100 94 97 90 92 94 95 95 Agni B. 96 99 96 99 95 96 82 93 92 93 94.3 Pritam B. 98 98 95 98 83 95 90 93 94 94 93.9 Tima (G) 96 98 97 99 85 94 92 91 87 96 93.7 Note: B or G against the name of a student respectively indicates whether the student is a boy or a girl. Had Joseph, Agni, Pritam and Tirna each obtained Group Score of 100 in the Social Science Group, then their standing in decreasing order of final score would be: A Pritam, Joseph, Tirna, Agni B Joseph, Tirna, Agni, Pritam C Pritam, Agni, Tirna, Joseph D Joseph, Tirna, Pritam, Agni
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In a Class X Board examination, ten papers are distributed over five groups—PCB, Mathematics, Social Science, Vernacular and English. Each of the ten papers is evaluated out of 100. The final score of a student is calculated in the following manner. First, the Group Scores are obtained by averaging marks in the papers within the Group. The final score is the simple average of the Group Scores. The data for the top ten students are presented below. (Dipan’s score in English Paper II has been intentionally removed in the table.) Name of the Student PCB Group Mathemetics Social Science Group Vernacular Group English Group Final Score Phy Chem Bio Hist Geo Paper I Paper II Paper I Paper II Ayesha (G) 98 96 97 98 95 93 94 96 96 98 96.2 Ram B. 97 99 95 97 95 95 96 94 96 98 96.1 Dipan B. 98 98 98 95 96 95 96 94 96 ?? 96.0 Sagnik B. 97 98 99 96 96 98 94 97 92 94 95.9 Sanjiv B. 95 96 97 98 97 96 92 93 95 96 95.7 Shreya (G) 96 89 85 100 97 98 94 95 96 95 95.5 Joseph B. 90 94 98 100 94 97 90 92 94 95 95 Agni B. 96 99 96 99 95 96 82 93 92 93 94.3 Pritam B. 98 98 95 98 83 95 90 93 94 94 93.9 Tima (G) 96 98 97 99 85 94 92 91 87 96 93.7 Note: B or G against the name of a student respectively indicates whether the student is a boy or a girl. Students who obtained Group Scores of at least 95 in every group are eligible to apply for a prize. Among those who are eligible, the student obtaining the highest Group Score in Social Science Group is awarded this prize. The prize was awarded to: A Shreya B Ram C Ayesha D Dipan
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In a Class X Board examination, ten papers are distributed over five groups—PCB, Mathematics, Social Science, Vernacular and English. Each of the ten papers is evaluated out of 100. The final score of a student is calculated in the following manner. First, the Group Scores are obtained by averaging marks in the papers within the Group. The final score is the simple average of the Group Scores. The data for the top ten students are presented below. (Dipan’s score in English Paper II has been intentionally removed in the table.) Name of the Student PCB Group Mathemetics Social Science Group Vernacular Group English Group Final Score Phy Chem Bio Hist Geo Paper I Paper II Paper I Paper II Ayesha (G) 98 96 97 98 95 93 94 96 96 98 96.2 Ram B. 97 99 95 97 95 95 96 94 96 98 96.1 Dipan B. 98 98 98 95 96 95 96 94 96 ?? 96.0 Sagnik B. 97 98 99 96 96 98 94 97 92 94 95.9 Sanjiv B. 95 96 97 98 97 96 92 93 95 96 95.7 Shreya (G) 96 89 85 100 97 98 94 95 96 95 95.5 Joseph B. 90 94 98 100 94 97 90 92 94 95 95 Agni B. 96 99 96 99 95 96 82 93 92 93 94.3 Pritam B. 98 98 95 98 83 95 90 93 94 94 93.9 Tima (G) 96 98 97 99 85 94 92 91 87 96 93.7 Note: B or G against the name of a student respectively indicates whether the student is a boy or a girl. Each of the ten students was allowed to improve his/her score in exactly one paper of choice with the objective of maximizing his/her final score. Everyone scored 100 in the paper in which he or she chose to improve. After that, the topper among the ten students was A RamB AgniC PritamD Dipan
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Mathematicians are assigned a number called Erdös number (named after the famous mathematician, Paul Erdös). Only Paul Erdös himself has an Erdös number of zero. Any mathematician who has written a research paper with Erdös has an Erdös number of 1. For other mathematicians, the calculation of his/her Erdös 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 Erdös number. Let the Erdös number of Y be y. Then X has an Erdös number of y + 1. Hence, any mathematician with no co-authorship chain connected to Erdös has an Erdös number of infinity. In a seven day long mini-conference organized in memory of Paul Erdös, 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 Erdös number. Nobody had an Erdös number less than that of F. ·      On the third day of the conference F co-authored a paper jointly with A. and C. . This reduced the average Erdös number of the group of eight mathematicians to 3. The Erdös numbers of B. , D. Go through the options. and H remained unchanged with the writing of this paper. Further, no other co-authorship among any three members would have reduced the average Erdös 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 Erdös numbers while the other three had Erdös numbers distinct from each other. ·      On the fifth day, E co-authored a paper with F which reduced the group’s average Erdös number by 0.5. The Erdös 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 Erdös number during the conference? A 2 B 3 C 4 D 5
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Mathematicians are assigned a number called Erdös number (named after the famous mathematician, Paul Erdös). Only Paul Erdös himself has an Erdös number of zero. Any mathematician who has written a research paper with Erdös has an Erdös number of 1. For other mathematicians, the calculation of his/her Erdös 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 Erdös number. Let the Erdös number of Y be y. Then X has an Erdös number of y + 1. Hence, any mathematician with no co-authorship chain connected to Erdös has an Erdös number of infinity. In a seven day long mini-conference organized in memory of Paul Erdös, 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 Erdös number. Nobody had an Erdös number less than that of F. ·      On the third day of the conference F co-authored a paper jointly with A. and C. . This reduced the average Erdös number of the group of eight mathematicians to 3. The Erdös numbers of B. , D. Go through the options. and H remained unchanged with the writing of this paper. Further, no other co-authorship among any three members would have reduced the average Erdös 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 Erdös numbers while the other three had Erdös numbers distinct from each other. ·      On the fifth day, E co-authored a paper with F which reduced the group’s average Erdös number by 0.5. The Erdös 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 Erdös number at the end of the conference must have had Erdös number (at that time) A 5 B 7 C 9 D 14
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Mathematicians are assigned a number called Erdös number (named after the famous mathematician, Paul Erdös). Only Paul Erdös himself has an Erdös number of zero. Any mathematician who has written a research paper with Erdös has an Erdös number of 1. For other mathematicians, the calculation of his/her Erdös 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 Erdös number. Let the Erdös number of Y be y. Then X has an Erdös number of y + 1. Hence, any mathematician with no co-authorship chain connected to Erdös has an Erdös number of infinity. In a seven day long mini-conference organized in memory of Paul Erdös, 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 Erdös number. Nobody had an Erdös number less than that of F. ·      On the third day of the conference F co-authored a paper jointly with A. and C. . This reduced the average Erdös number of the group of eight mathematicians to 3. The Erdös numbers of B. , D. Go through the options. and H remained unchanged with the writing of this paper. Further, no other co-authorship among any three members would have reduced the average Erdös 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 Erdös numbers while the other three had Erdös numbers distinct from each other. ·      On the fifth day, E co-authored a paper with F which reduced the group’s average Erdös number by 0.5. The Erdös 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 Erdös number at the beginning of the conference? A 2 B 3 C 4 D 5
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Mathematicians are assigned a number called Erdös number (named after the famous mathematician, Paul Erdös). Only Paul Erdös himself has an Erdös number of zero. Any mathematician who has written a research paper with Erdös has an Erdös number of 1. For other mathematicians, the calculation of his/her Erdös 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 Erdös number. Let the Erdös number of Y be y. Then X has an Erdös number of y + 1. Hence, any mathematician with no co-authorship chain connected to Erdös has an Erdös number of infinity. In a seven day long mini-conference organized in memory of Paul Erdös, 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 Erdös number. Nobody had an Erdös number less than that of F. ·      On the third day of the conference F co-authored a paper jointly with A. and C. . This reduced the average Erdös number of the group of eight mathematicians to 3. The Erdös numbers of B. , D. Go through the options. and H remained unchanged with the writing of this paper. Further, no other co-authorship among any three members would have reduced the average Erdös 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 Erdös numbers while the other three had Erdös numbers distinct from each other. ·      On the fifth day, E co-authored a paper with F which reduced the group’s average Erdös number by 0.5. The Erdös numbers of the remaining six were unchanged with the writing of this paper. ·      No other paper was written during the conference.   The Erdös number of C at the end of the conference was A 1 B 2 C 3 D 4
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Mathematicians are assigned a number called Erdös number (named after the famous mathematician, Paul Erdös). Only Paul Erdös himself has an Erdös number of zero. Any mathematician who has written a research paper with Erdös has an Erdös number of 1. For other mathematicians, the calculation of his/her Erdös 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 Erdös number. Let the Erdös number of Y be y. Then X has an Erdös number of y + 1. Hence, any mathematician with no co-authorship chain connected to Erdös has an Erdös number of infinity. In a seven day long mini-conference organized in memory of Paul Erdös, 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 Erdös number. Nobody had an Erdös number less than that of F. ·      On the third day of the conference F co-authored a paper jointly with A. and C. . This reduced the average Erdös number of the group of eight mathematicians to 3. The Erdös numbers of B. , D. Go through the options. and H remained unchanged with the writing of this paper. Further, no other co-authorship among any three members would have reduced the average Erdös 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 Erdös numbers while the other three had Erdös numbers distinct from each other. ·      On the fifth day, E co-authored a paper with F which reduced the group’s average Erdös number by 0.5. The Erdös numbers of the remaining six were unchanged with the writing of this paper. ·      No other paper was written during the conference.   The Erdös number of E at the beginning of the conference was A 2 B 5 C 6 D 7
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