KCET · Maths · Functions
If \([x]^2-5[x]+6=0\), where \([x]\) denotes the greatest integer function, then
- A \(x \in[3,4]\)
- B \(x \in[2,4)\)
- C \(x \in[2,3]\)
- D \(x \in(2,3]\)
Answer & Solution
Correct Answer
(B) \(x \in[2,4)\)
Step-by-step Solution
Detailed explanation
\([x]^2-5[x]+6=0\)
\(\Rightarrow \quad[x]^2-3[x]-2[x]+6=0\)
\(\Rightarrow \quad[x]([x]-3)-2([x]-3)=0\)
\(\Rightarrow \quad([x]-3)([x]-2)=0\)
\(\Rightarrow \quad[x]=2\) or 3
\([x]=2\) or \([x]=3\)
\(x=[2,3)\) or \(x=[3,4)\)
Hence, \(x \in[2,4)\).
\(\Rightarrow \quad[x]^2-3[x]-2[x]+6=0\)
\(\Rightarrow \quad[x]([x]-3)-2([x]-3)=0\)
\(\Rightarrow \quad([x]-3)([x]-2)=0\)
\(\Rightarrow \quad[x]=2\) or 3
\([x]=2\) or \([x]=3\)
\(x=[2,3)\) or \(x=[3,4)\)
Hence, \(x \in[2,4)\).
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