If the third term in expansion of `(1+x^(log_2x))^5` is `2560` then `x` is equal to

Options

A. `4 sqrt(2)`
B. `(1)/(4)`
C. `(1)/(8)`
D. `2 sqrt(2)`

Answer

B

Explanation

The `(r + 1)` th term in the expansion of `(a + x)^(n)` is given by `T_(r + 1) = .^(n)C_(r ) a^(n - r) x^(r )`
`:. 3^(rd)` term in the expansion of `(1 + x^(log_(2)x))^(5)` is
`.^(5)C_(2) (1)^(5 - 2) (x^(log_(2)x))^(2)`
`implies .^(5)C_(2) (1)^(5 - 2) (x^(log_(2)x))^(2) = 2560` given
`implies 10 (x^(log_(2)x))^(2) = 2560`
`implies x^((2log_(2)x)) = 256`
`implies log_(2) x^(2log_(2)x) = log_(2) = 256`
(taking `log_(2)` on both sides )
`implies 2(log_(2)x)(log_(2)x) = 8`
`(log_(2)x) = 4`
`implies log_(2) = +- 2`
`implies log_(2) x = 2` or `log_(2) x = - 2`
`implies x = 4` or `x = 2^(-2) = (1)/(4)`