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 \; (As \; x > 1)$