CAT 2021 Slot 3 — DILR Question 11
Answer the next 6 questions based on the information given below:
10 players – P1, P2, … , P10 - competed in an international javelin throw event. The number (after P) of a player reflects his rank at the beginning of the event, with rank 1 going to the topmost player. There were two phases in the event with the first phase consisting of rounds 1, 2, and 3, and the second phase consisting of rounds 4, 5, and 6. A throw is measured in terms of the distance it covers (in meters, up to one decimal point accuracy), only if the throw is a ‘valid’ one. For an invalid throw, the distance is taken as zero. A player’s score at the end of a round is the maximum distance of all his throws up to that round. Players are re-ranked after every round based on their current scores. In case of a tie in scores, the player with a prevailing higher rank retains the higher rank. This ranking determines the order in which the players go for their throws in the next round.
In each of the rounds in the first phase, the players throw in increasing order of their latest rank, i.e. the player ranked 1 at that point throws first, followed by the player ranked 2 at that point and so on. The top six players at the end of the first phase qualify for the second phase. In each of the rounds in the second phase, the players throw in decreasing order of their latest rank i.e. the player ranked 6 at that point throws first, followed by the player ranked 5 at that point and so on. The players ranked 1, 2, and 3 at the end of the sixth round receive gold, silver, and bronze medals respectively.
All the valid throws of the event were of distinct distances (as per stated measurement accuracy). The tables below show distances (in meters) covered by all valid throws in the first and the third round in the event.
Distances covered by all the valid throws in the first round
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Distances covered by all the valid throws in the third round
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The following facts are also known.
- Among the throws in the second round, only the last two were valid. Both the throws enabled these players to qualify for the second phase, with one of them qualifying with the least score. None of these players won any medal.
- If a player throws first in a round AND he was also the last (among the players in the current round) to throw in the previous round, then the player is said to get a double. Two players got a double.
- In each round of the second phase, exactly one player improved his score. Each of these improvements was by the same amount.
- The gold and bronze medalists improved their scores in the fifth and the sixth rounds respectively. One medal winner improved his score in the fourth round.
- The difference between the final scores of the gold medalist and the silver medalist, as well as the difference between the final scores of the silver medalist and the bronze medalist was 1.0 m.
Which two players got the double?
Answer & solution
- A
P1, P8
- B
P2, P4
- C
P1, P10
P8, P10
Let us arrange the players in the order in which they throw in each round.
Round 1: Here the players throw in order of their initial seeds so the order is as follows:
P1 → P2 → P3 → P4 → P5 → P6 → P7 → P8 → P9 → P10
In the first round only 6 players had a valid throw: P1, P3, P5, P6, P7 and P9
∴ Ranking at the end of 1st round.
1 P7 87.2
2 P5 86.4
3 P9 84.1
4 P1 82.9
5 P6 82.5
6 P3 81.5
7 P2 -
8 P4 -
9 P8 -
10 P10 -
P7, P5, P9, P1, P6, P3 P2, P4, P8, P10
Now, in round 2, only last two throws i.e., of P8 and P10 were valid throws. Hence, their order will change at the start of Round 3, however, the remaining order stays the same. That is, P8 and P10 will move up in the table and occupy some higher places, whereas some of the others may move down consequently.
Round 3: In round 3, we can see that P1 improved his score from 82.9 to 88.6. The other 2 participants did not improve their scores. Also, after round 3, P8 and P10 qualify, where one of P8 or P10 is at the sixth position. So, at the end of round 3, we can say that P6, P3, P2 and P4 are at the bottom 4 positions. One of P8 or P10 is at the sixth positions. P1 > P7 > P5 > P9.
∴ Their ranks at the end of round 3.
1/2 P1 88.6
2/3 P7 87.2
3/4 P5 86.4
4/5 P9 84.1
6 P8/P10
The other person between P8 / P10 can go anywhere between rank 1 and 5.
Now let us consider the two players who got a double. Doubles happen in the transition between rounds.
Round 1 → Round 2: Double is not possible as P10 (last to throw in Round 1) did not have a valid throw.
Round 2 → Round 1: Possible if P10 (last to throw in Round 1) reaches Rank 1 in round 2.
Round 3 → Round 4: P8/P10 who is the last among qualifying will be the first to throw. So, here it definitely happens.
Round 4 → Round 5: Not possible since only one player improves his/her rank in round 4.
Round 5 → Round 6: Not possible since only one player improves his/her rank in round 5.
∴ After Round 2 P10 definitely reaches top of the ranking i.e., P10 throws more than 87.2.
∴ Ranks at the end of round 3.
Case 1
1 P1 88.6
2 P10 ?
3 P7 87.2
4 P5 86.4
5 P9 84.1
6 P8
Case 2
1 P10 ?
2 P1 88.6
3 P7 87.2
4 P5 86.4
5 P9 84.1
6 P8
P10 and P8 do not win any medals.
Case 2: 3 of P1, P7, P5 and P9 need to score better than P10. Each of these 3 will improve their score by same amount say x.
Possible scores after improvement:
P1: 88.6 + x
P7: 87.2 + x
P5: 86.4 + x
P9: 84.1 + x
No value of x will satisfy the 5th condition given in the question. Hence, this case is rejected.
Case 1: 2 or 3 of P1, P7, P5 and P9 need to score better than P10. Each of these 3 will improve their score by same amount say x.
Possible scores after improvement:
P1: 88.6 + x
P7: 87.2 + x
P5: 86.4 + x
P9: 84.1 + x
If P1 does not win any medal, P7, P5 and P9 will have to improve their scores. But this will not satisfy conditions 3 and 5 simultaneously. Hence P1 has to be one of the medalists.
P10 has scored more than 87.2, hence the three medalists need to score more than 87.2.
The only way this is possible is when P7 improves his score twice by 1.2 while P5 improves his score once by 1.2.
Scores at the end of round 5.
P7: 89.6
P1: 88.6
P5: 87.6
∴ P8 and P10 get the doubles.
Hence, option (d).