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JEE MainMathematicsInverse Trigonometric Functions

Let S be the set of all real solutions of the equation ⁻¹ ( x-1 x+1 ) + ⁻¹ ( 2x-1 2x+1 ) = ⁻¹ ( 7 6 ) . The value of _ x S 12x is equal to

Options

  1. A21
  2. B3
  3. C0
  4. D24

Correct answer

D. 24

Step-by-step solution

Let A = x-1 x+1 and B = 2x-1 2x+1 . Applying the inverse tangent addition formula, the argument on the left-hand side becomes A+B 1-AB . A+B = (x-1)(2x+1) + (2x-1)(x+1) (x+1)(2x+1) = 2x^2 - x - 1 + 2x^2 + x - 1 (x+1)(2x+1) = 4x^2 - 2 (x+1)(2x+1) 1-AB = 1 - (x-1)(2x-1) (x+1)(2x+1) = 2x^2 + 3x + 1 - (2x^2 - 3x + 1) (x+1)(2x+1) = 6x (x+1)(2x+1) Thus, A+B 1-AB = 4x^2 - 2 6x = 2x^2 - 1 3x . Equating this to the argument on the right-hand side: 2x^2 - 1 3x = 7 6 12x^2 - 6 = 21x 12x^2 - 21x - 6 = 0 4x^2 - 7x - 2 = 0 (4x+1

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