JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

If the domain of the function $f(x) = \frac{[x]}{1 + x^2}$ , where $[x]$ is the greatest integer $\leq x$ , is $[2, 6)$ , then its range is:

Choose the correct answer:

Explanation

Solution

Since the domain is $[2, 6)$ , we evaluate the function in integer intervals:

For $x \in [2, 3)$ , $[x] = 2 \implies f(x) = \frac{2}{1+x^2}$ . Range: $(\frac{2}{10}, \frac{2}{5}] = (\frac{1}{5}, \frac{2}{5}]$ .
For $x \in [3, 4)$ , $[x] = 3 \implies f(x) = \frac{3}{1+x^2}$ . Range: $(\frac{3}{17}, \frac{3}{10}]$ .
For $x \in [4, 5)$ , $[x] = 4 \implies f(x) = \frac{4}{1+x^2}$ . Range: $(\frac{4}{26}, \frac{4}{17}] = (\frac{2}{13}, \frac{4}{17}]$ .
For $x \in [5, 6)$ , $[x] = 5 \implies f(x) = \frac{5}{1+x^2}$ . Range: $(\frac{5}{37}, \frac{5}{26}]$ .

Combining these, the overall range is the union of these intervals.

Final Range: $(\frac{5}{37}, \frac{2}{5}]$ .
Correct Answer: Option (3).