Question

If  \[\left[ x \right]^2 - 5\left[ x \right] + 6 = 0\], where [.] denotes the greatest integer function, then 

 

पर्याय

• (a) x ∈ [3, 4]   

•    (b) x ∈ (2, 3]           

•   (c) x ∈ [2, 3]      

•   (d) x ∈ [2, 4)

Answer 1

The given equation is \[\left[ x \right]^2 - 5\left[ x \right] + 6 = 0\]

\[\left[ x \right]^2 - 5\left[ x \right] + 6 = 0\]
\[ \Rightarrow \left[ x \right]^2 - 3\left[ x \right] - 2\left[ x \right] + 6 = 0\]
\[ \Rightarrow \left[ x \right]\left( \left[ x \right] - 3 \right) - 2\left( \left[ x \right] - 3 \right) = 0\]
\[ \Rightarrow \left( \left[ x \right] - 2 \right)\left( \left[ x \right] - 3 \right) = 0\]

\[\Rightarrow \left[ x \right] - 2 = 0 \text{ or } \left[ x \right] - 3 = 0\]
\[ \Rightarrow \left[ x \right] = 2\text{  or }  \left[ x \right] = 3\]

⇒ x ∈ [2, 3) or x ∈ [3, 4)
⇒ x ∈ [2, 4)