question_answer

If \[f(x)=({{2}^{x}}-1)\,{{({{2}^{x}}-2)}^{2}},\] then \[f(x)\] decreases in the interval

A)  \[x\,\in \,\left( -\infty ,\,\,{{\log }_{2}}\,\frac{4}{3} \right)\,\cup \,(1,\,\,\infty )\]

B)  \[x\,\in \,\left( {{\log }_{2}}\,\left( \frac{4}{3} \right),\,\,0 \right)\,\]

C)  \[x\,\in \,\left( {{\log }_{2}}\,\left( \frac{4}{3} \right),\,\,1 \right)\,\]

D)  Cannot say

Correct Answer: C

Solution :

 Put      \[\alpha =\beta =\gamma \ne {{90}^{o}}\] \[{{p}_{1}}=\frac{nRT}{{{V}_{1}}}=\frac{2\times 0.0821\times 243.6}{20}=2\,\text{atm}\]              \[\Delta S=nR\,\,In\,\left( \frac{{{p}_{1}}}{{{p}_{2}}} \right)\] \[=2\times 2=\,In\,2\]   \[=2\times 2\times 0.693\]                 \[[\because \,I{{n}^{2}}\,or\,{{\log }_{e}}2=0.693]\]\[FeC{{l}_{3}}+3NaOH\xrightarrow{\,}\,Fe{{(OH)}_{3}}+3NaCl\] For f(t) to be decreasing, \[NaOH\xrightarrow{\,}N{{a}^{\oplus }}+O{{H}^{\odot -}}\]        \[O{{H}^{\odot -}}\] \[Fe{{(OH)}_{3}}\]           \[Fe{{(OH)}_{3}}\] \[P{{b}^{2+}}+2O{{H}^{\odot -}}\xrightarrow{\,}\,Pb{{(OH)}_{2}}\downarrow \,(\text{white}\,\text{ppt}\text{.})\]                \[P{{b}^{2+}}+CO_{3}^{2-}\xrightarrow{\,}\,PbC{{O}_{3}}\downarrow \,(\text{white}\,\text{ppt}\text{.})\] \[P{{b}^{2+}}+2C{{l}^{-}}\xrightarrow{\,}\,PbC{{l}_{2}}\downarrow \,(\text{white}\,\text{ppt}\text{.})\]                \[P{{b}^{2+}}+SO_{4}^{2-}\xrightarrow{\,}\,PbS{{O}_{4}}\downarrow \,(\text{white}\,\text{ppt}\text{.})\] \[\underset{0.15-x\sim 0.15}{\mathop{HCOOH}}\,\,\underset{x}{\mathop{HCO{{O}^{\odot -}}}}\,+\underset{0.1}{\mathop{{{H}^{\oplus }}}}\,\]        \[\therefore \]