`e^(|sin x|)+e^(-|sin x|)+4a=0` will have exactly four different solutions in `[0, 2pi]` if

Options

A. `a in [-(e)/(4),-(1)/(4)]`
B. `a in R`
C. `a in [-(-1-e^(2))/(4e),oo)`
D. none of these

Answer

D

Explanation

Let `t = e^(|sin x|) "Clearly", t in [1, e]`.
`therefore" "e^(|sin x|)+e^(-|sin x|) + 4a = 0`
`rArr" "t^(2) + 4at + 1 = 0 or f(t) = 0, "where" f(t) = t^(2) + 4at + 1`
This will have two distinct real roots in [1, e], if `f(1) gt 0, f(e) gt 0, 1 lt -2a lt e and 16 a^(2) - 4 gt 0`
`rArr" "4a+2 gt 0, e^(23)+4ae + 1 gt 0, -(e)/(2)lt a lt (-1)/(2) and a lt (-1)/(2) or a gt (1)/(2)`
`rArr" "a gt (-1)/(2), a gt -(1+e^(2))/(4e), -(e)/(2) lt a lt (-1)/(2) and a lt (-1)/(2) or a gt (1)/(2)`
Clearly, there is no value of a satisfying the above in equalities.