Correct Option: D

Here, we will calculate the value of ξ for each case. For the system to be underdamped ξ should be less than one, i.e.,
ξ < 1

(A) Given H(s) =	1
	s2 + 4s + 4

C.E. = s² + 4s + 4
On comparing this C.E. with standard equation
s² + 2ξω_n + ω²_n = 0
we get, 2ξ ω_n = 4
and ω²_n = 4
or ω_n = ± 2
or 2ξ2=4
or ξ = 1

(B) H(s) =	1
	s2 + 5s + 4

Here, 2ξ2=5
and ω²_n = 4
or ω_n = ± 2
or 2ξ2=5

or ξ =	5	= 1·25
	4

(C) H(s) =	1
	s2 + 4·5s + 4

Here, 2ξωn = 4·5
and ω²_n = 4
or ω_n = ± 2
or 2ξω_n = 4·5

or ξ =	4·5	= 1·125
	4

(D) H(s) =	1
	s2 + 3s + 4

Here, 2ξω_n = 3,
ω²_n = 4
or ω_n = ± 2
or 2ξ2=3

ξ =	3	= ·75 < 1
	4

Hence, alternative (D) is the correct choice.