Question

Determine the value of $x$ in the expression ${(x+x^t)}^5$, if the third term in the expression is 10,00,000 where $t={\log }_{10}x$.

• A. $10$
• B. ${10}^{-5/2}$
• C. Both (A)and (B)
• D. None of these

Answer 1

The correct option is C Both (A)and (B)

$T_3$ $={}^5{C}_2{x}^{3+2t}$
$={10}^6$
$x^{3+2t}={10}^5$
$\ln \left(x\right)(3+2t)=5\ln \left(10\right)$
${\log }_{10}\left(x\right)(3+2t)=5$
$t(3+2t)=5$
$2t^2+3t-5=0$
$t=1$ and $t=\frac{-5}{2}$
Hence, $x=10$ and $x={10}^{\frac{-5}{2}}$.