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 mathematicians 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 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 organised 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 has an infinite Erdos number. Nobody had an Erdos 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 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, Eco-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.

1.How many participants in the conference did not change their Erdos number during the conference?

(1) 2

(2) 3

(3) 4

(4) 5

(5) 2

2. The person having the largest Erdos number at the end of the conference must have had Erdos number (at that time) :

(1) 5

(2) 7

(3) 9

(4) 14

(5) 15

3. How many participants had the same Erdos number at the beginning of the conference?

(1) 2

(2) 3

(3) 4

(4) 5

(5) cannot be determined

4. The Erdos number of C at the end of the conference was:

(1) 1

(2) 2

(3) 3

(4) 4

(5) 5

5. The Erdos number of E at the beginning of the conference was:

(1) 2

(2) 5

(3) 6

(4) 7

(5) 8

Answer the following questions on the basis of the information given below:

Two traders, Chetan and Michael, were involved in the buying and selling of MCS shares over five trading days. At the beginning of the first day, the MCS share was priced at Rs.100, while at the end of the fifth day it was priced at Rs.110. At the end of each day, the MCS share price either went up by Rs.10, or else, it came down by Rs.10. Both Chetan and Michael took buying and selling decisions at the end of each trading day. The beginning price of MCS share on a given day was the same as the ending price of the previous day. Chetan and Michael started with the same number of shares and amount of cash, and had enough of both below are some additional facts about how Chetan and Michael traded over the five trading days.

.Each day if the price went up, Chetan sold 10 shares of MCS at the closing price. On the other hand, each day if the price went down, he bought 10 shares at the closing price.

If on any day, the closing price was above Rs.110, then Michael sold 10 shares of MCS, while if it was below Rs.90, he bought 10 shares, all at the closing price.

6. If Chetan sold 10 shares of MCS on three consecutive days, while Michael sold 10 shares only once during the five days, what was the price of MCS at the end of day 3?

(1) Rs.90

(2) Rs.100

(3) Rs.110

(4) Rs.120

(5) Rs.130

7 If Chetan ended up with Rs.1300 more cash than Michael at the end of day 5, what was the price of the MCS share at the end of day 4?

(1) Rs.90

(2) Rs.100

(3) Rs.110

(4) Rs.120

(5) Not uniquely determinable

8. If Michael ended up with 20 more shares than Chetan at the end of days 5, what was the price of the shares at the and of day 3?

(1) Rs.90

(2) Rs.100

(3) Rs.110

(4) Rs.120

(5) Rs.130

9. If Michael ended up with Rs.100 less cash than Chetan at the end of day 5, what was the difference in the number of shares possessed by Michael and Chetan (at the end of day 5)?

(1) Michael had 10 less shares than Chetan.

(2) Michael had 10 more shares than Chetan.

(3) Chetan had 10 more shares than Michael.

(4) Chetan had 20 more shares than Michael.

(5) Both had the same number of shares.

10. What could have been the maximum possible increase in combined cash balance of Chetan and Michael at the end of the fifth day?

(1) Rs.3700

(2) Rs.4000

(3) Rs.4700

(4) Rs.5000

(5) Rs.6000

Directions (Q. 11-15): In the following questions, the symbol @, $, @, ? and ? are used with the following meanings: P @ Q means P is greater than Q. P $ Q means P is equal to Q. P @ Q means P is neither greater than nor equal to Q. P ? Q means P is either smaller than or equal to Q. P ? Q means P is either greater than or equal to Q. Now in each of the following questions assuming the given statements to be true, find which of the two conclusions I and II given below them is/are definitely true? Give answer 1) if only conclusion I is true; 2) if only conclusion II is true; 3) if either I or II is true; 4) if neither I nor II is true; and 5) if both I and II are true;

11.

Statement: M@ K, P ? K, K $ T

Conclusions: I. P $ TII. M @ T

12.

Statements: R @ M, M $ P, P ? T

Conclusions: I. R @ TII. R $ T

13.

Statements: T ? N, R @ N, R ? K

Conclusions: I. N @ K II. T ? K

14.

Statements: M @ R, R $ A, A ? K

Conclusions: I. K @ M II. K ? R

15.

Statements: P ? M, M @ K, K $ T

Conclusions: I. P $ M II. T @ P

Directions (Q. 16-20):In the following questions, four alternatives are given for the idiom/phrase printed in italics. Choose the alternative which best expresses the meaning of the idiom/phrase and mark it as your answer. If none of the alternatives are correct, mark 5),i.e. None of these, as your answer.

16. He knows all the ins and outs of the political system.

1) loopholes

2) details

3) problems

4) pressures

5) None of these

17. Before liberation they hardly earned enough to keep body and soul together.

1) maintain themselves comfortably

2) satisfy themselves

3) free themselves

4) remain alive

5) None of these

18.The economic problem was on the carpet most of the time today at the conference.

1) under discussion

2) dealt sternly

3) taken lightly

4) highly controversial

5) None of these

19. If he can keep from smoking his health will improve.

1) disagree to

2) minimise

3) restrain from

4) discontinue

5) None of these

20. Had he studied properly he would not have to eat the humble pie in the examination.

1) be disqualified

2) stand on trial

3) haunted by problems

4) suffer humiliation

5) None of these

Directions (Q. 21-25):Three of the following four words are almost SIMILAR in meaning and so form a group. Select the one that does not belong to that group. If all the four words have the same or similar meaning, mark 5), ie None of these, as your answer.

21. 1) Palpable 2) Obvious 3) Timid 4) Certain 5) None of these

22. 1) Remarkable 2) Sociable 3) Friendly 4) Convivial 5) None of these

23. 1) Speculate 2) Conjecture 3) Guess 4) Unfold 5) None of these

24. 1) Ungainly 2) Resonant 3) Awkward 4) Clumsy 5) None of these

25. 1) Intransigent 2) Uncompromising 3) Fatuous 4) Obstinate 5) None of these

ANSWERS : 1.(4) 2.(2) 3.(2) 4.(2) 5.(3) 6.(3) 7.(2) 8.(1) 9.(5) 10.(4) 11.(2) 12.(4) 13.(4) 14.(2) 15.(4) 16. (2) 17.(4)18.(1) 19.(3) 20.(4) 21.(3) 22.(1) 23.(4) 24. (2) 25.(3)