Question

Find the term independent of x in the expansion of `(sqrt(x)/sqrt(3) + sqrt(3)/(2x^2))^10`.

Answer 1

Let (r + 1)^th term be independent of x which is given by

T_r+1 = `""^10"C"_r  sqrt(x/3)^(10 - r)  sqrt(3)^r/(2x^2)`

= `""^10"C"_r  x^((10 - r)/2)/3  3^(r/2)  1/(2^r  x^(2r))`

= `""^10"C"_r  3^(r/2 - (10 - r)/2)  2^(-r)  x^((10 - r)/2 - 2r)` 

Since the term is independent of x, we have

`(10 - r)/2 - 2r` = 0

⇒ r = 2

Hence 3^rd term is independent of x and its value is given by

T₃ = `""^10"C"_2  (3^(-3))/4`

= `(10 xx 9)/(2 xx 1) xx 1/(9 xx 12)`

= `5/12`