If the `4^(th)` term of `{sqrt(x^((1)/(1+log_(10)x)))+root(12)(x)}^(6)` is equal to `200`, `x gt 1`and the logarithm is common logarithm, then`x` is not divisible by
(a)`2` (b)`5` (c)`10` (d)`4`

Options

A. `2`
B. `5`
C. `10`
D. `4`

Answer

D

Explanation

`(d)` Given expression is
`{sqrt((1)/(x^(1+log_(10)x)))+root12x}^(6)={x^((1)/(2))((1)/(1+log_(10x)))+x^((1)/(12))}^(6)`
`T_(4)=200`
`implies^(6)C_(3)*x^((1)/(2)((1)/(1+log_(10)x))^(6-3))xx(x^((1)/(12)))^(3)=200`
`impliesx^((3)/(2)(1)/(1+log_(10)x)+(1)/(4))=10`
`=(3)/(2(1+log_(10)x))+(1)/(4)=log_(x)10`
`implies(3)/(2(1+log_(10)x))+(1)/(4)=(1)/(log_(10)x)`
Put `log_(10)x=y`
`implies(3)/(2(1+y))+(1)/(4)=(1)/(y)`
`impliesy^(2)+3y-4=0`
`impliesy=-4`, `y=1`
`implieslog_(10)x=-4`, `log_(10)x=1`
`impliesx=10` (as `x gt 1`)