Let `I_(1) : (log_(x)2) (log_(2x)2) (log_(2)4x)gt1`
`I_(2) : x^((log_(10)x)^(2)-3(log_(10)x)+1) gt 1000`
and solution of inequality `I_(1)` is `((1)/(a^(sqrt(a))),(1)/(b))cup(c, a^(sqrt(a)))`
and solution of inequality `I_(2)` is `(d, oo)` then answer the following
Sum of `'d'` term of a `GP` whose common ratio is `(1)/(a)` and first term is `c` is more than

Options

A. `1`
B. `2`
C. `(3)/(2)`
D. `0`

Answer

A, C, D

Explanation

`log_(2)x = t`
`(1)/(t).(1)/(1 + t).(2 + t) gt 1`
`(2 + t)/(t^(2) + t) gt 1`
`1 - (2 + t)/(t^(2) + t) lt 0`
`(t^(2) - 2 )/(t(t + 1)) lt 0`

`-sqrt(2) lt t lt -1` or `0 lt t lt sqrt(2)`
`2^(-sqrt(2)) lt x lt (1)/(2)` or `1 lt x lt 2^(sqrt(2))`
`x in((1)/(2^(sqrt(2))),(1)/(2))cup(1,2^(sqrt(2)))`
`I_(2) : log_(10)x = t`
`(t^(3) - 3t + 1)t gt 3`
`t^(3) - 3t^(2) + t - 3 gt 0`
`(t - 3) (t^(2) - 1) gt 0`
`t gt 3`
`x gt 1000`
`a = 2, b = 2, c = 1, d = 1000`
Sol.2. Sum of `GP = (1-(1/2)^(1000))/(1-1/2)`
`= 2 (1-1/2^(1000))`