Question

The value of $x$ in the expression ${(x+x^{{\log }_{10}x})}^5$, if the third term in the expansion is $10,00,000,$ is

• A. ${10}^{-1}$
• B. ${10}^1$
• C. ${10}^{-5/2}$
• D. ${10}^{5/2}$

Answer 1

Answer: (B), (C)

${10}^1$
${10}^{-5/2}$

Substitute ${\log }_{10}x=z$

Then, given expression $={(x+x^z)}^5$

Now, $T_3{=}^5{C}_2{x}^3{\left(x^z\right)}^2=10x^{3+2z}={10}^6$

$\therefore x^{3+2z}={10}^5$

Taking log. we get

$(3+2z){\log }_{10}x=5{\log }_{10}10$

$\Rightarrow (3+2z)z=5\Rightarrow 2z^2+3z-5=0$

$\Rightarrow (z-1)(2z+5)=0\Rightarrow z=1,-\frac{5}{2}$

$\therefore {\log }_{10}x=1$ or $-\frac{5}{2}$

$\therefore x={10}^1$ or ${10}^{-\frac{5}{2}}$