55 no. Please!
Q.55. The inequality $\left(x-1\right)\ln \left(2-x\right)<0$ holds if x satisfies:
(A) 1 < x < 2
(B) x > 0
(C) x < 2
(D) x > 2

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \left(\mathrm{x}-1\right)\ln \left(2-\mathrm{x}\right)<0 \\ \mathrm{Now} \mathrm{first} \mathrm{of} \mathrm{all} \mathrm{quantity} \mathrm{inside} \log \mathrm{must} \mathrm{be} \mathrm{positive}, \mathrm{hence} \\ 2-\mathrm{x}>0 \\ \Rightarrow \mathrm{x}<2 \\ \mathrm{which} \mathrm{eliminates} \mathrm{option} \mathrm{B} \mathrm{and} \mathrm{D} \\ \mathrm{Now} \mathrm{if} \mathrm{x}=0 , \mathrm{then} \mathrm{inequality} \mathrm{becomes} \\ \left(0-1\right)\ln \left(2-0\right)<0 \\ \Rightarrow -\ln 2<0 \mathrm{which} \mathrm{is} \mathrm{true}, \mathrm{hence} \mathrm{x}=0 \mathrm{must} \mathrm{be} \mathrm{in} \mathrm{our} \mathrm{answer} \mathrm{that} \mathrm{eliminates} \\ \mathrm{A} \\ \mathrm{Hence} \mathrm{option}\left(\mathrm{C}\right) \mathrm{is} \mathrm{correct}. \end{gathered}$$

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