The domain of the function $\mathrm{f}(\mathrm{x})=\frac{1}{\sqrt{[\mathrm{x}]^{2}-3[\mathrm{x}]-10}}$ is (where $[\mathrm{x}]$ denotes the greatest integer less than or equal to x)

(C) $(-\infty,-2) \cup[6, \infty)$

(A) $(-\infty,-2) \cup(5, \infty)$

(B) $(-\infty,-3] \cup[6, \infty)$

(D) $(-\infty,-3] \cup(5, \infty)$

JEE Main · 2023 · 11 Apr 2023 · Shift 2 · Sets Relations and Functions

The domain of the function

f (x) = \frac{1}{[ x ] ^{2} - 3 [ x ] - 10}

is (where

[x]

denotes the greatest integer less than or equal to x)

Answer: (C)

The expression under the square root sits in a denominator, so it must be strictly positive:

[x]^{2} - 3 [x] - 10 > 0

. Factoring gives

([x] - 5) ([x] + 2) > 0

, which holds when

[x] < - 2

[x] > 5

, that is

[x] \leq - 3

[x] \geq 6

. Now

[x] \leq - 3

is equivalent to

x < - 2

, giving the interval

(- \infty, - 2)

, while

[x] \geq 6

is equivalent to

x \geq 6

, giving

[6, \infty)

. Hence the domain of

f

(- \infty, - 2) \cup [6, \infty)

JEE Main · 2023 · 11 Apr 2023 · Shift 2 · Single correct option