The value of x, for which the ninth term in the
expansion of `{sqrt(10)/((sqrt(x))^(5log _(10)x ))+ x.x^(1/(2log_(10)x))}^(10)`
is 450 is equal to

Options

A. 10
B. `10^(2)`
C. `sqrt(10)`
D. `10^(-2//5)`

Answer

b, d

Explanation

Let `log_(10 )x =lambda rArr x = 10^(lambda) `
Given, `T_(9) =450`
`rArr ""^(10)C_(8) cdot (sqrt(10)/(10^(5(lambda^(2))/2)))^(2) cdot (10^(lambda) cdot10^(1//2))^(8) = 450`
`rArr ""^(10)C_(2)cdot 10/(10^(5lambda^(2))) cdot 10 ^(8^(lambda)) cdot 10^(4) = 450`
`rArr 18 ^(8lambda +4-5lambda^(2))=1=10^(0)`
`rArr 8 lambda + 4 - 5 lambda ^(2) = 0`
`rArr 5lambda^(2) - 8 lambda - 4 = 0 `
` rArr lambda = 2, -2 //5`
`rArr x= 10^(2), 10^(-2//5)` [ from Eq. (i)]