# If the fourth term of the binomial expansion $\left ( \sqrt{\frac{1}{x^{1+\log_{10}x}}}+x^{\frac{1}{12}} \right )^{6}$ is equal to $200,$ and $x>1$, then the value of x is :  Option 1) $100$ Option 2) $10$ Option 3) $10^{3}$ Option 4) $10^{4}$

$\left ( \sqrt{\frac{1}{x^{1+\log_{10}x}}}+x^{\frac{1}{12}} \right )^{6}$

Fourth term is given

So, $n=3$

$^6C_3\left ( \sqrt{\frac{1}{x^{1+\log_{10}x}}} \right )^{3}\left ( \frac{1}{x^{12}} \right )^{6-3}=200$

$\Rightarrow x^{^{\frac{3}{2\left(1+\log \:_{10}\left(x\right)\right)}+\frac{1}{4}}}=10$

take log both side.

$\left ( \frac{1}{4}+\frac{3}{2\left ( 1+\log_{10}x \right )} \right )\log_{10}x=\log_{10}10$

put $\log_{10}x=t$

$\left ( \frac{1}{4}++\frac{3}{2\left ( 1 +t\right )} \right )t=1$

$t^{2}+3t-4=0$

$\Rightarrow t=1,-4$

$\log_{10}x=1\Rightarrow x=10$

$\log_{10}x=-4\Rightarrow x=10^{-4}$

$x=10\: \: \: as\: \: x>1$

Option 1)

$100$

Option 2)

$10$

Option 3)

$10^{3}$

Option 4)

$10^{4}$

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