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Question
The position of the term independent of x in the expansion of `(sqrt(x/3) + 3/(2x^2))^10` is ______.
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Solution
The position of the term independent of x in the expansion of `(sqrt(x/3) + 3/(2x^2))^10` is 3rd term.
Explanation:
The given expansion is `(sqrt(x/3) + 3/(2x^2))^10`
Tr+1 = `""^10"C"_r (sqrt(x/3))^(10 - r) (3/(2x^2))^r`
= `""^10"C"_r (x/3)^((10 - r)/2) (3/2)^r * 1/x^(2r)`
= `""^10"C"_r (1/3)^((10 - r)/2) * x^((10 - r)/2) (3/2)^r * 1/x^(2r)`
= `""^10"C"_r (1/3)^((10 - r)/2) * x^((10 - r)/2 - 2r) * (3/2)^r`
= `""^10"C"_r (1/3)^((10 - r)/2) * x^((10 - r - 4r)/2) (3/2)^r`
For independent of x, we get
`(10 - r - 4r)/2` = 0
10 – 5r = 0
r = 2
So, the position of the term independent of x is 3rd term.
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