Q65 / 150 · MHT CET 2024 (11 May Shift 1) · Mathematics

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

  • A
    $x \in[2,3)$
  • B
    $x \in[2,3]$
  • C
    $\quad x \in[2,4]$
  • Correct answer
    D
    $\quad x \in[2,4)$

Solution

Given: $\begin{aligned} & {[x]^2-5[x]+6=0} \\ & \text { Let }[x]=\mathrm{a} \\ & \Rightarrow \mathrm{a}^2-5 \mathrm{a}+6=0 \\ & \Rightarrow(\mathrm{a}-2)(\mathrm{a}-3)=0 \\ & \Rightarrow \mathrm{a}=2 \text { or } \mathrm{a}=3 \\ & \Rightarrow[x]=2 \text { or }[x]=3 \\ & \Rightarrow x \in[2,3) \text { or } x \in[3,4) \\ & \Rightarrow x \in[2,4) \end{aligned}$

Hence, the correct answer is option: 'D'.

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