CAT 2025 Slot 1 QA Question 7: Let $3 \leq x \leq 6$ and $[x^{2}] = {[x]}^{2}$, where $[x]$ is the greatest integer not exceeding $x$. If set $S$ represents all feasible values of $x$, then a possible subset of $S$ is

Question

Let $3 \leq x \leq 6$ and $[x^{2}] = {[x]}^{2}$, where $[x]$ is the greatest integer not exceeding $x$. If set $S$ represents all feasible values of $x$, then a possible subset of $S$ is

Accepted Answer

Let $[x] = n$. Since $3 \leq x \leq 6$, the possible integer values of $n$ are 3, 4, 5 and 6.

For a fixed $n$, $[x] = n$ means $n \leq x 
$n^{2} \leq x^{2} 
$n \leq x

For $n = 3$: $x \in [3, \sqrt{10})$
For $n = 4$: $x \in [4, \sqrt{17})$
For $n = 5$: $x \in [5, \sqrt{26})$
For $n = 6$: $x = 6$

Let 3≤x≤63 \leq x \leq 63≤x≤6 and [x2]=[x]2[x^{2}] = {[x]}^{2}[x2]=[x]2, where [x][x][x] is the greatest integer not exceeding xxx. If set SSS represents all feasible values of xxx, then a possible subset of SSS is

Let $3 \leq x \leq 6$ and $[x^{2}] = {[x]}^{2}$ , where $[x]$ is the greatest integer not exceeding $x$ . If set $S$ represents all feasible values of $x$ , then a possible subset of $S$ is