Question

If the third term in the expansion of
${\left[\right(1/x)+x^{{\log }_{10}x}]}^5$ is $1000$, then $x$ is equal to

• A. $100$
• B. $10$
• C. $1$
• D. $\frac{1}{\sqrt{10}}$

Answer 1

Answer: (A), (D)

The correct options are
A $100$
D $\frac{1}{\sqrt{10}}$

$T_3={}^5{C}_2{x}^{-3+2lo{g}_{10}x}$

$=10x^{-3+2lo{g}_{10}x}$

$={10}^3$

Hence
$x^{-3+2{\log }_{10}x}={10}^2$

Taking $lo{g}_{10}$ on both sides we get

$\Rightarrow \log \left(x\right)\left(2\log \right(x)-3)=2$

$\Rightarrow 2\log (x{)}^2-3\log (x)-2=0$

$\Rightarrow 2\log (x{)}^2-4\log (x)+\log (x)-2=0$

$\Rightarrow \left(\log \right(x)-2)\left(2\log \right(x)+1)=0$

$\Rightarrow \log \left(x\right)=2$ and $\log \left(x\right)=\frac{-1}{2}$

$\therefore x=100$ and $x=\frac{1}{\sqrt{10}}$