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`

Answer 1

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 = 6^th 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`