Question

If $[x{]}^2–5[x]+6=0,$where $[.]$ denote the greatest integer function, then

• A. $xϵ[2,3]$
• B. $xϵ[2,4]$
• C. $xϵ[3,4]$
• D. $xϵ[2,3]$

Answer 1

The correct option is B $xϵ[2,4]$

Given:$[x{]}^2–5[x]+6=0$ … (i)
Let $y=\left[x\right]$
Then equation (i) becomes
$y2–5y+6=0$
$\Rightarrow y2–3y–2y+3\times 2=0$
$\Rightarrow y(y–3)-2(y–3)=0$
$\Rightarrow (y–3)(y-2)=0$
$\Rightarrow y=3ory=2$
Since,$y=\left[x\right]$
Therefore,$\left[x\right]=3or\left[x\right]=2\Rightarrow xϵ[3,4)orxϵ[2,3)\Rightarrow xϵ[2,4)$

Hence, the correct option is (D)