Question

If the fourth term in the binomial expansion of ${\left(\sqrt{\frac{1}{x^{1+{\log }_{10}x}}+x^{\frac{1}{12}}}\right)}^6$ is equal to $200$, and $x>1$, then the value of $x$ is :

• A. $10^3$
• B. 100
• C. $10^4$
• D. 10

Answer 1

Answer: (D)

$200{=}^6{C}_3{\left(x^{\frac{1}{x+{\log }_{10}x}}\right)}^{\frac{3}{2}}\times x^{\frac{1}{4}}$
$\Rightarrow 10=x^{\frac{3}{2\left(1+{\log }_{10}x\right)}+\frac{1}{4}}$
$\Rightarrow 1=\left(\frac{3}{2(1+t)}+\frac{1}{4}\right)t$
where $t={\log }_{10}x$
$\Rightarrow t^2+3t-4=0$
$\Rightarrow$ t = 1, - 4
$\Rightarrow x=10,10^{-4}$
$\Rightarrow x=10(Asx>1)$