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Question
Find the term independent of x in the expansion of `(sqrt(x)/sqrt(3) + sqrt(3)/(2x^2))^10`.
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Solution
Let (r + 1)th term be independent of x which is given by
Tr+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 3rd 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`
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