Question

If the third term in the expansion of $[x+x^{{\log }_{10}x}{]}^5$ is $10^6$, then x may be

• A. 1
• B. $\sqrt{10}$
• C. 10
• D. $10^{-2/5}$

Answer 1

Answer: (C)

Given binomial is ${\left[x+x^{{\log }_{10}x}\right]}^5$
Also given, $T_3=10^6$
Put ${\log }_{10}x=y$, the given expression becomes ${\left(x+x^y\right)}^5$.
$T_3={}^5{C}_2\cdot x^3{\left(x^y\right)}^2=10x^{3+2y}=10^6\dots .($ given $)$
$\Rightarrow (3+2y){\log }_{10}x=5{\log }_{10}10=5$
$\Rightarrow (3+2y)y=5$
$\Rightarrow y=1,-\frac{5}{2}$
$\Rightarrow {\log }_{10}x=1$ or ${\log }_{10}x=-\frac{5}{2}$
Therefore, $x=10$ or $x=(10{)}^{-5/2}$