If the fourth term in the binomial expansion of

Question:

If the fourth term in the binomial expansion of

$\left(\sqrt{\frac{1}{x^{1+\log _{10} x}}}+x^{\frac{1}{12}}\right)^{6}$ is equal to 200 , and $x>1$,

then the value of x is :

1. $10^{3}$

2. 100

3. $10^{4}$

4. 10

Correct Option: , 4

Solution:

$200={ }^{6} \mathrm{C}_{3}\left(\mathrm{x}^{\frac{1}{x+\log _{10} x}}\right)^{\frac{3}{2}} \times \mathrm{x}^{\frac{1}{4}}$

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

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

where $t=\log _{10} x$

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

$\Rightarrow t=1,-4$

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

$\Rightarrow x=10($ As $x>1)$