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MHT CET202618 April 2026Morning ShiftMathematicsInverse Trigonometric FunctionsActual

If 0 x 1 and ( ⁻¹x)^3 + ( ⁻¹x)^3 = a ^3 then

Options

  1. Aa 1 32
  2. Ba 1 16
  3. Ca 1 32
  4. Da 1 16

Correct answer

A. a 1 32

Step-by-step solution

Let ⁻¹x = y . Since 0 x 1 , we have 0 y 2 . We know that ⁻¹x = 2 - ⁻¹x = 2 - y . The given equation is: y^3 + ( 2 - y )^3 = a ^3 Using the identity A^3 + B^3 = (A+B)(A^2 - AB + B^2) , we get: y^3 + ( 2 - y )^3 = (y + 2 - y ) ( y^2 - y ( 2 - y ) + ( 2 - y )^2 ) = 2 ( y^2 - 2 y + y^2 + ^2 4 - y + y^2 ) = 2 ( 3y^2 - 3 2 y + ^2 4 ) = 3 2 ( y^2 - 2 y + ^2 12 ) Completing the square for the quadratic in y : = 3 2 ( (y - 4 )^2 - ^2 16 + ^2 12 ) = 3 2 ( (y - 4 )^2 + ^2 48 ) = 3 2 (y - 4 )^2 + ^3 32 Since 0 y 2 , the minimu

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