The equation , where denotes the greatest integer function, has:
- A.a unique solution in
- B.exactly two solutions in
- C.a unique solution in
- D.no solution
Show answer and solution
Correct option: A
Solution
Rearrange: x² - 4x + [x] + 3 = x[x], so x² - 4x + 3 = x[x] - [x] = [x](x-1), giving (x-1)(x-3) = [x](x-1).
Either x = 1 (which satisfies the equation directly), or x ≠ 1 and we can divide: x - 3 = [x].
For x - 3 = [x]: since [x] is always an integer, x - 3 must be an integer, so x is an integer. If x = n (integer), then n - 3 = n gives -3 = 0, a contradiction.
So the only solution is x = 1, which is unique in (-∞, ∞).
Either x = 1 (which satisfies the equation directly), or x ≠ 1 and we can divide: x - 3 = [x].
For x - 3 = [x]: since [x] is always an integer, x - 3 must be an integer, so x is an integer. If x = n (integer), then n - 3 = n gives -3 = 0, a contradiction.
So the only solution is x = 1, which is unique in (-∞, ∞).
[JEE Main 2023 · 24 Jan · Shift 1 · Paper-1(B.E./B. Tech.)]