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If the middle term in the binomial expan...

If the middle term in the binomial expansion of `(1/x+xsinx^(10))` is equal to `(63)/8,` find the value of `xdot`

A

`2n pi + (pi)/(6)`

B

`n pi + (pi)/(6)`

C

`n pi + (-1)^(n) (pi)/(6)`

D

`n pi + (-1)^(n) (pi)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
C
Given expansion is `((1)/(x) + x sin x)^(10)`
Since, `n = 10` is even, so this expansion has only one middle term i.e., 6 th term.
`:. T_(6) = T_(5 + 1) = .^(10)C_(s) ((1)/(x))^(10 - 5) (x sin x)^(5)`
`rArr (63)/(8) = .^(10)C_(5) x^(-5) x^(5) sin^(5) x`
`rArr (63)/(8) = (10.9.8.7.6.5!)/(5.4.3.2.1.5!) sin^(5) x`
`rArr (63)/(8) = 2.9.2.7. sin^(5) x`
`rArr sin^(5) x = (1)/(32)`
`rArr sin^(5) x = ((1)/(2))^(5)`
`rArr sin x = (1)/(2)`
`:. x = n pi + (-1)^(n) pi//6`
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