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# A linear, time - invarient, casual continuous time system has a rational transfer function with simple poles at s=−2 and s=−4, and one simple zero at s=−1.A unit step u(t) is applied at the input of the system. At steady state, the output has constant value of 1. The impulse response of this system is

A
[e2t+e4t]u(t)
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B
[4e2t+12e4tet]u(t)
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C
[4e2t+12e4t]u(t)
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D
[0.5e2t+1.5e4t]u(t)
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Solution

## The correct option is C [−4e−2t+12e−4t]u(t)Transfer function can be written as, T(s)=K(s+1)(s+2)(s+4) Input,R(s)=1s Output,C(s)=R(s)H(s) Given : lims→0sC(s)=1 or lims→0s.K(s+1)s(s+2)(s+4)=1 or k8=1 ⇒K=8 H(s)=8(s+1)(s+2)(s+4) =−4s+2+12s+4 so, h(t)=(−4e−2t+12e−4t)u(t) Which is also required impulse response of the system

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