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 $\left( x + x ^{y}\right)^{5}$.
$T _{3}={ }^{5} C _{2} \cdot x ^{3}\left( x ^{y}\right)^{2}=10 x ^{3+2 y }=10^{6} \ldots .($ given $)$
$\Rightarrow (3+2 y) \log _{10} x=5 \log _{10} 10=5$
$\Rightarrow (3+2 y ) y =5$
$\Rightarrow y =1,-\frac{5}{2}$
$\Rightarrow \log _{10} x =1$ or $\log _{10} x =-\frac{5}{2}$
Therefore, $x =10$ or $x =(10)^{-5 / 2}$