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JEE Main202224 Jun 2022Morning ShiftMathematicsInverse Trigonometric FunctionsActual

The set of all values of k for which tan - 1 x 3 + cot - 1 x 3 = k π 3 , x ∈ R , is the interval

Options

  1. A1 32 , 7 8
  2. B1 24 , 13 16
  3. C1 48 , 13 16
  4. D1 32 , 9 8

Correct answer

A. 1 32 , 7 8

Step-by-step solution

Let T = tan - 1 x 3 + cot - 1 x 3 = tan - 1 x + cot - 1 x 3 - 3 tan - 1 x · cot - 1 x tan - 1 x + cot - 1 x = π 3 8 - 3 π 2 tan - 1 x π 2 - tan - 1 x ∵ tan - 1 x + cot - 1 x = π 2 = 3 π 2 tan - 1 x 2 - 3 π 2 4 tan - 1 x + π 3 8 T = 3 π 2 tan - 1 x - π 4 2 + π 3 32 We know than tan - 1 x ∈ - π 2 , π 2 Minimum value of T is at tan - 1 x = π 4 and maximum value at tan - 1 x = - π 2 ⇒ π 3 32 ≤ T < 7 8 π 3 =

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