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JEE Main202117 Mar 2021Morning ShiftMathematicsInverse Trigonometric FunctionsActual

The sum of possible values of x for tan - 1 x + 1 + cot - 1 1 x - 1 = tan - 1 8 31 is:

Options

  1. A- 32 4
  2. B- 31 4
  3. C- 30 4
  4. D- 33 4

Correct answer

A. - 32 4

Step-by-step solution

Given tan - 1 x + 1 + cot - 1 1 x - 1 = tan - 1 8 31 ⇒ tan - 1 x + 1 + tan - 1 x - 1 = tan - 1 8 31 ,       ∴ tan - 1 x = cot - 1 1 x ,   x > 0 ⇒ tan - 1 x + 1 + x - 1 1 - x + 1 x - 1 = tan - 1 8 31 ⇒ tan - 1 2 x 1 - x + 1 x - 1 = tan - 1 8 31 Taking tangent both the sides we get, ( x + 1 ) + ( x - 1 ) 1 - x 2 - 1 = 8 31 ⇒ 2 x 2 - x 2 = 8 31 ⇒ 4 x 2 + 31 x - 8 = 0 ⇒ x = - 8 , 1 4 But, if x = 1 4 tan - 1 x + 1 ∈ 0 , π 2 and cot - 1 1 x - 1

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