Question

The term independent of x in the expansion of ${(\sqrt{\left(\frac{x}{3}\right)}+\sqrt{\left(\frac{3}{2x^2}\right)})}^{10}$ is

• A. $\frac{5}{12}$
• B. ${}^{10}{C}_1$
• C. $1$
• D. none of these

Answer 1

The correct option is D none of these

${(\sqrt{\frac{x}{3}}+\sqrt{\frac{3}{2x^2}})}^{10}$

$T_{r+1}={}^{10}{C}_r{\left(\sqrt{\frac{x}{3}}\right)}^{10-r}{\left(\sqrt{\frac{3}{2x^2}}\right)}^r$

For term independent of x,

The power of $x=0$

$x^{\frac{10-r}{2}}\cdot x^{-2r/2}$

$\frac{10-r-2r}{2}=0$

$3r=10$

$r=\frac{10}{3}\text{/}\in N$

Hence, there is no term independent of x.