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)

$10$

Given binomial is $[x+x^{{\log }_{10}x}{]}^5$

Also given, $T_3={10}^6$

Put ${\log }_{10}x=y$, the given expression becomes $(x+x^y{)}^5$.
$T_3{=}^5{C}_2.x^3(x^y{)}^2=10x^{3+2y}={10}^6$ ....(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 $x=(10{)}^{-5/2}$