Question

If the $4^{th}$ term of the expansion ${[x^{\frac{1}{(1+\log x)}}+x^{\frac{1}{12}}]}^6$ is $200$, then the values of x are:

• A. $0.001,0.1$
• B. $0.0001,10$
• C. $0.01,100$
• D. $0.1,10000$

Answer 1

The correct option is B $0.0001,10$

$200=t_4={{}^{6}C}_3{\left[\sqrt{x^{\frac{1}{(1+\log x)}}}\right]}^{6-3}{{(x}^{\frac{1}{12}})}^3$
$\Rightarrow 10=x^{(\frac{3}{1+\log x}+\frac{1}{4})}\Rightarrow 1=(\frac{3}{1+t}+\frac{1}{4})t$
where $t=\log x$
$\Rightarrow t^2+3t-4=0$
$\Rightarrow t=-4,1\Rightarrow x={10}^{-4},10$