91. I started climbing up the hill at 6 a.m. and reached the temple at the top at 6 p.m. Next day I started coming down at 6 a.m. and reached the foothill at 6 p.m. I walked on the same road. The road is so short that only one person can walk on it. Although I varied my pace on my way, I never stopped on my way. Then which of the following must be true.

(a) My average speed downhill was greater than that uphill.

(b) At noon, I was at the same spot on both the days.

(c) There must be a point where I reached at the same time on both the days.

(d) There cannot be a spot where I reached at the same time on both the days.

92. What is the digit in the unit place of 2^{51}?

(a) 2

(b) 8

(c) 1

(d) 4

93. There are two containers: the first contains 500 ml. of alcohol, while the second contains 500ml. of water. Three cups of alcohol from the first container is removed and is mixed well in the second container. Then three cups of this mixture is removed and is mixed in the first container. Let A denote the proportion of water in the first container and B denote the proportion of alcohol in the second container. Then,

(a) A > B

(b) A < B

(c) A = B

(d) cannot be determined

94. A hundred digit number is formed by writing first 54 natural numbers in front of each other as 12345678910111213â€¦… Find the remainder when this number is divided by 8.

(a) 1

(b) 7

(c) 2

(d) 0

95. A, B, C, D, â€¦..X, Y, Z are the players who participated in a tournament. Everyone played with every other player exactly once. A win scores 2 points, a draw scores 1 point and a lose scores 0 points. None of the matches ended in a draw. No two players scored the same score. At the end of the tournament, by ranking list is published which is in accordance with the alphabetical order. Then

(a) M wins over N.

(b) N wins over M.

(c) M does not play with N.

(d) None of these.

**Answers:**

91 | (c) |

92 | (b) |

93 | (c) |

94 | (a) |

95 | (a) |

91. Since the distance travelled was the same both ways and also it was covered in the same time, the average speed will be the same uphill and downhill. Hence statement (a) is false. Statement (b) need not be true. It would be true had the speeds (and not average speed) been the same both ways. But it is clearly indicated that he varied his pace throughout. Now, it has to be noted that the journey uphill and the journey downhill started at the same time i.e. 6.00 am and also ended at the same time 6.00 pm (though on different days). So if we were to assume a hypothetical case in which one person starts down hill at 6.00 am and other one starts uphill at 6.00 am, along the same path, then there would be a point on the path where they would meet (i.e. they would reach at the same time), irrespective of their speeds. Our case is similar to that, except for the fact that here we have only one person moving both ways. So there has to a point on the path, where he reached at the same time on both days. Hence the answer is (c).

92. Let us find a pattern. 2^{1} = 2, 2^{2} = 4, 2^{3} = 8, 2^{4} = 16, 2^{5} = 32, 2^{6} = 64 â€¦ Thus we see that depending on to which power 2 is raised to, the digit in the units place varies as 2, 4, 8, 6, 2, 4, 8, 6 â€¦.. So this cycle of last digits goes on repeating after every 4^{th} power of 2. In other words, 24, 28, 28, 216â€¦will all have the same last digit i.e. 6. So if 2 is raised to any power that is a multiple of 4, we will have a last digit of 6. So 2^{48} should also have the same last digit i.e. 6. So 2^{49} will have last digit 2, 2^{50} will have last digit 4 and 2^{51} will have last digit 8.

93. Let the capacity of each cup be 100 ml. So 300 ml. of alcohol is removed from the first container and poured into the second one. So the first vessel will have 200 ml. of alcohol and the second one will have 500 ml of water and 300 ml. of alcohol. So the ratio of Water : Alcohol in the second vessel is 5 : 3. Hence, proportion of alcohol = B = 3 : 8. Now if 300 ml of mixture is removed from the second container, it will have (300 x 5/8) = 187.5 ml of water and (300 x 3/8) = 112.5 ml of alcohol. Now if this mixture is poured in the second vessel, that vessel would have (200 + 112.5) = 312.5 ml of alcohol and 187.5 ml of water. Hence ratio of Alcohol : Water in this container = 312.5 : 187.5 = 5 : 3. Hence proportion of water = A = 3 : 8. Hence we find that A = B. Note : This result will be independent of the capacity of the cup.

94. To find out whether a number is divisible by 8, we simply have to find whether the number formed by the last three digits of the main number is divisible by 8. Hence the same rule holds good for finding the remainder when divided by 8. So we will have to find out which are the last 3 digits of our number. If we write first 9 natural numbers viz. 123456789, we would get a 9-digit number. Since there are 100 digits in all, we will have to write 91 more digits ahead of this. But after 9, all the natural number are 2 digit numbers. Hence to write 91 more digits, we will have to write 45 more numbers and only the first digit of the 46^{th} number. 45 more number after 9 would mean the number is 54. In addition we will have to write the first digit of 55 i.e. 5. Hence the number formed by the last three digits would be 545. This number when divided by 8, gives a remainder of 1. Hence the answer is 1.

95. It can be seen that each of the 26 players played 25 matches. Since none of the matches ended in a draw, the scores for each of the players has to be even (since a win gives 2 points). So the highest score posible for a player would be 50 and the lowest would be 0. Since all 26 of them had different scores varying between 0 and 50, the scores should indeed by all the even number between 0 and 50. And since the ranks obtained by players are in alphabetical order, it can be concluded that A scored 50, B scored 48, C scored 46 and so on â€¦.and Z scored 0. Now the only way A can score 50, is if he wins all his matches i.e. he defeats all other players. Now B has scored 48. So he has lost only 1 of his matches, which incidentally is against A. He must have defeated all other players. Similarly C has scored 46 matches. So he must have lost 2 matches, (i.e. to A & B) and defeated all other players. So we conclude that a player whose name appears alphabetically higher up in the order has defeated all the players whose name appear alphabetically lower down. Hence M should win over N.

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