JEE Main 2026 Jan 22 Shift 2, Mathematics Q18: Some Important Properties Of Functions

JEE Main2026Jan 22, Shift 2Mathematics

Let $f (x) = [x]^{2} - [x + 3] - 3$ , $x \in R$ where $[∙]$ is the greatest integer function. Then

Solution

Since $[x + 3] = [x] + 3$ , we get $f (x) = [x]^{2} - [x] - 6 = ([x] + 2) ([x] - 3)$ .

This is positive when $[x] \leq - 3$ or $[x] \geq 4$ , i.e., $x \in (- \infty, - 2) \cup [4, \infty)$ , so (1) is false.

It is negative when $[x] \in {- 1, 0, 1, 2}$ , i.e., $x \in [- 1, 3)$ , so (2) is correct.

$\int_{0}^{2} f (x) d x = \int_{0}^{1} (- 6) d x + \int_{1}^{2} (- 6) d x = - 6 - 6 = - 12$ , so (3) is false.

$f (x) = 0$ when $[x] = 3$ or $[x] = - 2$ , each giving an interval — infinitely many solutions, so (4) is false.

Concept behind this question

Concept-wise practice with instant solutions on Fundamenthol.