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}

 

Answers (1)

\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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