A) \[m\in \left( \frac{1}{3},\infty \right)\]
B) \[m\in (1,\infty )\]
C) \[m\in \left( -\infty ,\frac{1}{3} \right)\cup (1,\infty )\]
D) \[m\in \left( \frac{1}{3},1 \right)\]
Correct Answer: D
Solution :
(d): inequality can also be written as \[\frac{1}{m-1}<\frac{-3}{1-3m}\Rightarrow \frac{1}{m-1}+\frac{-3}{1-3m}<0\] \[\Rightarrow \frac{1-3m+3m-3}{(m-1)(1-3m)}<0\Rightarrow \frac{-2}{(m-1)(1-3m)}<0\] In equation holds good if, \[(m-1)(1-3m)>0\Rightarrow (m-1)(3m-1)<0\]. \[\Rightarrow (m-1)\left( m-\frac{1}{3} \right)<0\]; \[\therefore \]The solution of the given in equation is \[\frac{1}{3}<m<1,\] i.e., \[m\in \left( \frac{1}{3},1 \right)\]
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