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      Question

      Quantity-I: 'Arjun' and 'Bheem'

      are running on a circular track with a diameter of 'a' meters, at speeds of 25 m/s and 40 m/s, respectively. If they are moving in the same direction, determine the number of distinct points where they will meet. Quantity-II: Two different numbers have an LCM of 924 and an HCF of 11. Determine the total number of such possible pairs. In the question, two Quantities I and II are given. You have to solve both the Quantity to establish the correct relation between Quantity-I and Quantity-II and choose the correct option.
      A Quantity I < Quantity II Correct Answer Incorrect Answer
      B Quantity I ≤ Quantity II Correct Answer Incorrect Answer
      C Quantity I ≥ Quantity II Correct Answer Incorrect Answer
      D Quantity I > Quantity II Correct Answer Incorrect Answer
      E Quantity I = Quantity II or No relation can be established Correct Answer Incorrect Answer

      Solution

      ATQ, Quantity I: Ratio of speeds of 'Arjun' and 'Bheem' = 25:40 = 5:8 We know that when two people are racing on a circular track at 'x' m/s and 'y' m/s in same direction, the number of distinct meeting points will be |x - y|, where 'x' and 'y' are co-prime numbers. So, number of distinct meeting points = 8 - 5 = 3 So, Quantity I = 3 Quantity II: Let the two numbers be '11x' and '11y' where 'x' and 'y' are co-primes. So, 11x × 11y = 924 × 11 Or, xy = 84 So, xy = 84 = 22 × 3 × 7 If N = xp × yq X zr....., then, the number of ways of writing 'N' as a product of 2 co-primes is 2(n - 1), where 'n' is the number of distinct prime factors of the given number N. Number of ways in which 84 can be written as a product of two co-primes = 2 (3 - 1) = 4 So, Quantity II = 4 So, Quantity I < Quantity II

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