If the fourth term of `(1/(x^(1+(log)_(10)x)+x 12))^6` is equal to 200 and `x >1,` then `x` is equal to

Options

A. 100
B. `10^(4)`
C. 10
D. `10^(3)`

Answer

C

Explanation

Given binominal is `(sqrt(x^((1)/(1 + log_(10) x))) + x^(1)/(12))^(6)`
Since, the fourth term in the given expansion is 200.
`:. .^(6)C_(3) (x^((1)/(1 + log_(10) x)))^((3)/(2)) (x^((1)/(12)))^(3) = 200`
`implies 20 xx x^([(3)/(2(1 + log_(10) x)) + (1)/(4)] = 200`
`implies x^((3)/(2(1 + log_(10) x)) + (1)/(4) = 10`
`implies [(3)/(2 (1 + log_(10) x)) + (1)/(4)] log_(10) x = 1`
[applying `log_(10)` both sides]
`implies [6 + (1 + log_(10 x)] log_(10) x = 4 (1 + log_(10) x)`
`implies (7 + log_(10) x) log_(10) x = 4 + 4 log_(10) x`
`impliess t^(2) + 7 t = 4 + 4t [ let `log_(1) x = t]`
`implies t^(2) + 3t - 4 = 0`
`implies t = 1, - 4 = log_(10) x`
`implies x = 10, 10^(-4)`
Since, `x gt 1 x = 10`