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      Question

      Average of five two-digit consecutive even natural

      numbers is (x + 10), and the average of five two-digit consecutive odd natural numbers is (y - 5). When each set is arranged in increasing order, the sum of the second even number and the fourth odd number is 71. The fifth odd number is nine less than the second even number. Which of the following statement(s) is/are true? I. The difference between the second even number and the fourth odd number is 1. II. The fourth even number is not a perfect square. III. The first odd number is 31.
      A Both I and III Correct Answer Incorrect Answer
      B Both I and II Correct Answer Incorrect Answer
      C All I, II and III Correct Answer Incorrect Answer
      D Only II Correct Answer Incorrect Answer
      E Both II and III Correct Answer Incorrect Answer

      Solution

      Five even numbers: (x + 6), (x + 8), (x + 10), (x + 12), (x + 14) Five odd numbers: (y - 9), (y - 7), (y - 5), (y - 3), (y - 1) Given, (x + 8) + (y - 3) = 71 Or, x + y = 66 ------ (I) (y - 1) = (x + 8) - 9 Or, x - y = -2 --------- (II) On adding equation I and II, We get, x + y + x - y = 66 - 2 Or, 2x = 64 Or, x = 32 On putting value of x in equation I, We get, y = 66 - 32 = 34 So, the five even numbers: 38, 40, 42, 44, 46 And the five odd numbers: 25, 27, 29, 31, 33 Statement I: Required difference = 40 - 31 = 9 (Statement is False) Statement II: 44 is not a perfect square. (Statement is True) Statement III: First odd number is 25. (Statement is False) Hence, option D.

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