Mathematics - Binomial Theorem Question with Solution | TestHub

Put ${\log }_{10}x=y,$ the given expression becomes ${\left(x+x^y\right)}^5$ ${\mathrm{T}}_3={}^5{\mathrm{C}}_2\cdot {\mathrm{x}}^3{\left({\mathrm{x}}^{\mathrm{y}}\right)}^2=10{\mathrm{x}}^{3+2\mathrm{y}}=10^6$ (given) $\Rightarrow (3+2\mathrm{y}){\log }_{10}\mathrm{x}=5{\log }_{10}10=5$ $\Rightarrow (3+2\mathrm{y})\mathrm{y}=5$ $\Rightarrow \mathrm{y}=1,-\frac{5}{2}$ $\Rightarrow {\log }_{10}\mathrm{x}=1$ or ${\log }_{10}\mathrm{x}=-\frac{5}{2}$ $\therefore \mathrm{x}=10$ or $\mathrm{x}=(10{)}^{-5/2}$