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Another set of five questions from quant. See if you can crack this set in five minutes.

Q.1 Arun, Barun and Kiranmala start from the same place and travel in the same direction at speeds of 30, 40 and 60 km per hour respectively. Barun starts two hours after Arun. If Barun and Kiranmala overtake Arun at the same instant, how many hours after Arun did Kiranmala start? (CAT 2006)
(1) 3
(2) 3.5
(3) 4
(4) 4.5
(5) 5

Q.2 A shop stores x kg of rice. The first customer buys half this amount plus half a kg of rice. The second customer buys half the remaining amount plus half a kg of rice. Then the third customer also buys half the remaining amount plus half a kg of rice. Thereafter, no rice is left in the shop. Which of the following best describes the value of x? (CAT 2008)
(1) 2 ≤ x ≤ 6
(2) 5 ≤ x ≤ 8
(3) 9 ≤ x ≤ 12
(4) 11≤ x ≤ 14
(5) 13 ≤ x ≤ 18

Q.3 Four points A, B, C and D lie on a straight line in the X-Y plane, such that AB = BC = CD, and the length of AB is 1 m. An ant at A wants to reach a sugar particle at D. But there are insect repellents kept at points B and C. The ant would not go within one metre of any insect repellent. The minimum distance in metres the ant must traverse to reach the sugar particle is (CAT 2005)
(1) 3√2
(2) 1 + π
(3) 4 π/3
(4) 5

Q.4 Number S is equal to the square of the sum of the digits of a 2 digit number D. If the difference between S and D is 27 then D is (CAT 2002)
(1) 32
(2) 54
(3) 64
(4) 52

Q.5 The sum of four consecutive two digit odd numbers when divided by 10 becomes a perfect square. Which of the following can possibly be one of these four numbers? (CAT 2006)
(1) 21
(2) 25
(3) 41
(4) 67
(5) 73

Solution:

Q.1 Arun, Barun and Kiranmala start from the same place and travel in the same direction at speeds of 30, 40 and 60 km per hour respectively. Barun starts two hours after Arun. If Barun and Kiranmala overtake Arun at the same instant, how many hours after Arun did Kiranmala start? (CAT 2006)

Solution:

Barun starts two hours after Arun. In other words, when Barun starts, the distance between Arun and Barun is 60. To cover this 60 with a relative speed of 10, it will take Barun 6 hours. In these 6 hours, distance covered is 240 km. Kiranmala takes 4 hour to cover 240 km. Hence, Kiranmala left after 4 hours. Answer is 4 hours.

Q.2 A shop stores x kg of rice. The first customer buys half this amount plus half a kg of rice. The second customer buys half the remaining amount plus half a kg of rice. Then the third customer also buys half the remaining amount plus half a kg of rice. Thereafter, no rice is left in the shop. Which of the following best describes the value of x? (CAT 2008)

Solution:

Going in the reverse direction, as there was no rice left after the third sell, he should have received 0.5 + 0.5 = 1 kg rice. Which means in the earlier sell, it should have been 1 + 0.5 = 1.5 * 2 = 3 kg

While selling the first customer, it should have been 3 + 0.5 = 3.5 * 2 = 7 kg.

Hence, the answer is 5 ≤ x ≤ 8

Q.3 Four points A, B, C and D lie on a straight line in the X-Y plane, such that AB = BC = CD, and the length of AB is 1 m. An ant at A wants to reach a sugar particle at D. But there are insect repellents kept at points B and C. The ant would not go within one metre of any insect repellent. The minimum distance in metres the ant must traverse to reach the sugar particle is (CAT 2005)

Solution:

The diagram will look like this. To reach D from A, ant will cover half the circumference of the circle + the bold line (1m).

The minimum distance will be π/2 + 1 + π/2 = 1 + π

Q.4 Number S is equal to the square of the sum of the digits of a 2 digit number D. If the difference between S and D is 27 then D is (CAT 2002)

Solution:

Instead of solving the question by assuming D as xy and then using 10x + y and so on, the easier way is to use the answer options.

Option 1: If D is 32 then 3+2 =5. Square of 5 is 25. 32-25 is not 27. Hence this option is wrong
Option 2: If D is 54 then 5+4=9. Square of 9 is 81. 81-54 is 27. Hence this is the right option

This question can be solved in less than 10 seconds using the answer options.

Q.5 The sum of four consecutive two digit odd numbers when divided by 10 becomes a perfect square. Which of the following can possibly be one of these four numbers? (CAT 2006)

Solution:

Since the four numbers are odd numbers they would be ending in 1 3 5 7 or 9. Also since they are consecutive the ending digits would be 1 3 5 7 or 3 5 7 9 so on. And the sum of these numbers would end in a 0. Thus the only combination in which this is possible is 7 9 1 3. Use answer options:

Option 1: The numbers would be 17 19 21 and 23. The sum is 80. 80 divided by 10 is 8 which is not a perfect square
Option 2: Not possible as the ending digit is 5
Option 3: The numbers would be 37 39 41 and 43. The sum is 160. 160 divided by 10 is 16 which is a perfect square

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