Find the coefficient of the term independent of x in the expansion of [ √(x/3) + √3/x]^10 .

Question

Find the coefficient of the term independent of x in the expansion of \(\big[\sqrt{\frac{x}{3}}+\frac{\sqrt{3}}{x}\big]^{10}\)

Answer 1

Given :

\(\big[\sqrt{\frac{x}{3}}+\frac{\sqrt{3}}{x}\big]^{10}\) ,

General term :

For the term independent of x,

\(\frac{10-r}{2}\) - 2r = 0 

⇒ \(\frac{10-r}{2}\) = 2r  ⇒ 10 - r = 4r ⇒ 5r = 10 ⇒ r = 2

∴ T_{2 + 1}, i.e., T₃ is the term independent of x and 

= \(\frac{10 \times 9}{2} \times (\frac{1}{3})^{4} \times 3\) 

= \(45 \times \frac{1}{27}\) 

= \(\frac{5}{3}\).