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The arithmetic mean of two positive numbers a and b exceeds their geometric mean by 2 and the harmonic mean is one-fifth of the greater of a and b , such that α = a + b and β = a - b , then the value of α + β 2 is equal to

Options

  1. A96
  2. B234
  3. C74
  4. D84

Correct answer

C. 74

Step-by-step solution

A = G + 2 H = b 5 (let b > a ) G 2 = A H ⇒ a b = a + b 2 b 5 ⇒ 10 a = a + b ⇒ b = 9 a A = a + b 2 = 5 a , G = 9 a . a = 3 a ⇒ 5 a = 3 a + 2 ⇒ a = 1 ⇒ b = 9 ⇒ α = 10 and β = 8 ⇒ α + β 2 = 7 4

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