JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank Self Evaluation Test - Binomial Theorem

If the fourth term in the expansion of \[{{\left( \sqrt{{{x}^{\left( \frac{1}{\log \,x+1} \right)}}}+{{x}^{1/12}} \right)}^{6}}\] is equal to 200 and \[\operatorname{x} > 1\], then x is equal to \[(log=lo{{g}_{10}})\]

A) \[{{10}^{\sqrt{2}}}\]

B) 10

C) \[{{10}^{4}}\]

D) None of these

Correct Answer: B

Solution :

[b] Given, \[{{T}_{4}}=200\]

\[\Rightarrow {{\,}^{6}}{{C}_{3}}{{\left( \sqrt{{{x}^{\left( \frac{1}{\log \,\,x+1} \right)}}} \right)}^{3}}{{({{x}^{1/12}})}^{3}}=200\]

\[\Rightarrow 20.{{x}^{\frac{3}{2(log\,x+1)}+\frac{1}{4}=}}200\]

\[\Rightarrow {{x}^{\left\{ \frac{3}{2(log\,x+1)}+\frac{1}{4} \right\}}}=10\]

\[\Rightarrow \,\,\,\frac{3}{2(\log \,x+1)}+\frac{1}{4}\,\,=\,{{\log }_{x}}10=\frac{1}{{{\log }_{10}}x}\]

\[\Rightarrow \frac{3}{2(y+1)}+\frac{1}{4}=\frac{1}{y}\] where \[y={{\log }_{10}}x\]

\[\Rightarrow y=-4\] or \[y=1\]

\[\Rightarrow {{\log }_{10}}x=-4\] or \[\Rightarrow {{\log }_{10}}x=1\]

\[\Rightarrow x={{10}^{-4}}\] or \[10\Rightarrow x=10(\because \,x>1)\]