A) (4, 5)
B) (\[-5,\,\,-4\])
C) (3, 4)
D) (5, 6)
Correct Answer: A
Solution :
\[{{x}^{2}}-mx+4=0~\] Case-I: \[D>0\] \[{{m}^{2}}-16>0\] \[\Rightarrow \,\,\,\,m\,\,\in \text{ }\left( -\infty ,\,-4 \right),\text{ }\left( 4,\text{ }\infty \right)\] Case-II: \[\Rightarrow \,\,\,1<\frac{-b}{2a}\,\,<\,\,5\] \[\Rightarrow \,\,\,1<\frac{m}{2}\,\,<\,\,5\,\,\,\,\Rightarrow \,\,m\in \,(2,\,\,10)\] Case-III: \[f\left( 1 \right)\,\,>\,\,0~~~~~~and\text{ }f\left( 5 \right)>0\] \[1-m+4>0\,\,\,\,\text{ }and\text{ }25-5m+4>0\] \[m<5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,and\,\,m<\frac{29}{5}\] Case-IV: Let one root is \[x=1\] \[1-m+4=0\] \[m=5\] Now equation \[{{x}^{2}}-5x+4=0\] \[\left( x-1 \right)\left( x-4 \right)=0\] \[x=1\text{ }i.e.\text{ }m=5\] is also included hence \[m\,\,\in \,\,\left( 4,\text{ }5 \right]\] So given option is \[\left( 4,\text{ }5 \right)\]You need to login to perform this action.
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