JEE Main 2023 · 24 January · morning · Single correct · +4 / −1
The equation x² - 4x + [x] + 3 = x[x], where [x] denotes the greatest integer function, has :
- (a) exactly two solutions in (-∞,∞)
- (b) no solution
- (c) a unique solution in (-∞,∞)
- (d) a unique solution in (-∞,1)
Answer and solution
Answer (c) a unique solution in (-∞,∞)
Solution
Maths sir · 4 steps
Move everything around and you get (x-1) on both sides. Factor it out, do not cancel it.
- Regroup the terms. Keep the terms without [x] on the left and move the [x] terms to the right: x² - 4x + 3 = x[x] - [x].
- Factor both sides. The left side is (x-1)(x-3) and the right side is [x](x-1). Bring everything to one side and take out the common factor.
(x-1)(x - 3 - [x]) = 0
- Check each factor. x - 1 = 0 gives x = 1. Check it: the left side is 1 - 4 + 1 + 3 = 1 and the right side is 1 × 1 = 1. The other factor needs x - [x] = 3, that is {x} = 3. This is impossible because 0 ≤ {x} < 1.
- So. x = 1 is the only solution, so the equation has a unique solution in (-∞, ∞). That is option (c).
The idea to keep. When a common factor like (x-1) appears on both sides, factor it out instead of dividing by it. Then use 0 ≤ x - [x] < 1 to kill any impossible branch.
Answer from the NTA final answer key. Solution by PrepWiser.