Question

The value of $x$ in the expression ${(x+x^{{\log }_{10}x})}^5$, if the third term in the expansion is $1,000,000$, is

• A. $10,{10}^{-3/2}$
• B. $100$ or ${10}^{-3/2}$
• C. $10$ or ${10}^{-5/2}$
• D. None of these

Answer 1

The correct option is C $10$ or ${10}^{-5/2}$

$\log x$ is defined only when $x>0$
Now, the $3^{rd}$ term in the expansion
$T_{2+1}{=}^5{C}_2\cdot x^{5-2}\cdot {\left(x^{{\log }_{10}x}\right)}^2=1,000,000$ ....(Given)
$\Rightarrow x^{3+2{\log }_{10}x}={10}^5$
Taking logarithm of both sides, we get
$(3+2{\log }_{10}x)\cdot {\log }_{10}x=5$
$\Rightarrow 2y^2+3y-5=0$
where ${\log }_{10}x=y$
$\Rightarrow (y-1)(2y+5)=0$
$\Rightarrow y=1$ or $-5/2$
$\Rightarrow {\log }_{10}x=1$ or $-5/2$
$\Rightarrow x={10}^1=10$ or ${10}^{-5/2}$