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Question
If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.
Options
`2npi + pi/6`
`npi + pi/6`
`npi + (-1)^n pi/6`
`npi + (-1)^n pi/3`
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Solution
If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is `npi + (-1)^n pi/6`.
Explanation:
Given expression is `(1/x + x sin x)^10`
Number of terms = 10 + 1 = 11 odd
∴ Middle term = `(11 + 1)/2` th term = 6th term
T6 = T5+1
= `""^10"C"_5 (1/x)^(10 - 5) (x sin x)^5`
∴ `""^10"C"_5 (1/x)^5 * x^5 * sin^5x = 7 7/8`
⇒ `""^10"C"_5 * sin^5x = 63/8`
⇒ `(10*9*8*7*6)/(5*4*3*2*1) * sin^5x = 63/8`
⇒ `252 * sin^5x = 63/8`
⇒ `sin^5x = 63/(8 xx 252)`
⇒ `sin^5x = 1/32`
⇒ `sin^5x = (1/2)^5`
⇒ sin x = `1/2`
⇒ sin x = `sin pi/6`
∴ x = `"n"pi + (-1)^"n" * pi/6`
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