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JEE Main202324 Jan 2023Morning ShiftMathematicsFunctionsActual

The equation x 2 – 4 x + [ x ] + 3 = x [ x ] , where [ x ] denotes the greatest integer function, has:

Options

  1. Aexactly two solutions in ( - ∞ ,   ∞ )
  2. Bno solution
  3. Ca unique solution in ( - ∞ ,   1 )
  4. Da unique solution in ( - ∞ ,   ∞ )

Correct answer

D. a unique solution in ( - ∞ ,   ∞ )

Step-by-step solution

Given: x 2 - 4 x + x + 3 = x x ⇒ x 2 - 4 x + 3 = x x - x ⇒ x - 1 x - 3 = x x - 1 ⇒ x = 1 or x - 3 = x If x - 3 = x ⇒ x – x = 3 ⇒ x = 3 which is not Possible, since x ∈ [ 0 , 1 ) . Hence, x = 1 is the only one solution in - ∞ , ∞ .

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