Mathematics - Binomial Theorem Question with Solution | TestHub

General term for the given binomial is
$T_{r+1}={}^6{C}_r{\left(\sqrt{x^{\frac{1}{1+{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}x}}}\right)}^{6-r}{\left(x^{\frac{1}{12}}\right)}^r$
Given$T_4=200,$putting$r=3$
$\Rightarrow T_4={}^6{C}_3{\left(\sqrt{x^{\frac{1}{1+{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}x}}}\right)}^3{\left(x^{\frac{1}{12}}\right)}^3=200$
$\Rightarrow {\left(x^{\frac{1}{1+{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}x}}\right)}^{\frac{3}{2}}\cdot \left(x^{\frac{1}{4}}\right)=10$
$\Rightarrow x^{\frac{3}{2\left(1+{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}x\right)}+\frac{1}{4}}=10$
$\Rightarrow \left(\frac{3}{2\left(1+{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}x\right)}+\frac{1}{4}\right){\mathrm{l}\mathrm{o}\mathrm{g}}_{10}x=1$
$\Rightarrow x=10,{10}^{-4}$
$\Rightarrow x=10$$($as$x>1)$