Question

If the third term in the expansion of $\left(\frac{1}{x}+x^{log_{10^x}}\right)^{5}$ is $1000$, then $x =$

• A. $10^2$
• B. $10^3$
• C. $10$
• D. $10^4$

Answer 1

Answer: (A)

Given, $T_3 = 1000$ in the expansion of
$\left(\frac{1}{x}+x^{log_{10^x}}\right)^{5}$
$\therefore \,{}^{5}C_{2}\left(\frac{1}{x}\right)^{3}\,\left(x^{log_{10^x}}\right)^{2} = 1000$
$\Rightarrow 10x^{-3} \times x^{2log_{10^x}} = 1000$
$\Rightarrow x^{2log_{10^{x-3}}} = 10^{2}$
$\Rightarrow \left(2\,log_{10}\,x-3\right) = log_{x}\,10^{2}$
(Taking logarithm on both sides to the base $x$)
$\left(2\,log_{10}\,x-3\right) = \frac{2}{log_{10}\,x}$
$\Rightarrow 2t - 3 = \frac{2}{t}$, where $t = log_{10}x$
$\Rightarrow \left(2t + 1\right) \left(t- 2\right) = 0$
$\Rightarrow t= 2$ ($t \ne-1/2$, neglected)
$\therefore log_{10}x = 2$
$\Rightarrow x = 10^{2} = 100$