66. A linear time invariatiant system has an impulse response e^2t, t > 0. If the initial conditions are zero and the input is e^3t, the output for t > 0 is

1. 1. e^3t – e^2t

   
   
   

   2. e^5t

   
   
   

   3. e^3t + e^2t

   
   
   

   4. None of these

   
   
   

---

1. View Hint View Answer Discuss in Forum

   Output C(s) = H (s). R (s) =	1	.	1	=	(s - 2) - (s - 3)	then C (t) = L – 1 [C(s)]
   	s - 2		s - 3		(s - 2)(s - 3)

   
   

   = L-1		1		- L-1		1		= e3t - e2t
   		(s - 3)				(s - 2)

   Correct Option: A

   Output C(s) = H (s). R (s) =	1	.	1	=	(s - 2) - (s - 3)	then C (t) = L – 1 [C(s)]
   	s - 2		s - 3		(s - 2)(s - 3)

   
   

   = L-1		1		- L-1		1		= e3t - e2t
   		(s - 3)				(s - 2)

---

67. The impulse response of an R– L circuit is a

1. 1. rising exponential function

   
   
   

   2. decaying exponential function

   
   
   

   3. step function

   
   
   

   4. parabolic function

   
   
   

---

1. View Hint View Answer Discuss in Forum

   I(s) =	V(s)	, and Z(s) = R + Ls
   	Z(s)

   
   

   I(s) =	V(s)
   	R + Ls

   
   Also, V(s) = 1
   

   I(s) =	1
   	R + Ls

   
   

   i(t) =		V(s)		eR/Lt
   		R + Ls

   
   a decaying exponential function

   Correct Option: B

   I(s) =	V(s)	, and Z(s) = R + Ls
   	Z(s)

   
   

   I(s) =	V(s)
   	R + Ls

   
   Also, V(s) = 1
   

   I(s) =	1
   	R + Ls

   
   

   i(t) =		V(s)		eR/Lt
   		R + Ls

   
   a decaying exponential function

---

---

68. A system is represented ^dy/_dt + 2y = 4tu(t). The ramp constant in the forced response will be

1. 1. t.u(t)

   
   
   

   2. 2t u (t)

   
   
   

   3. 3t u(t)

   
   
   

   4. 4t u(t)

   
   
   

---

1. View Hint View Answer Discuss in Forum

   dy/dt + 2y = 4 t u (t)
   Laplace transform is, 5y(s) + 2y(s) = 4/s²
   or y(s) [s + 2] = 4/s²
   or y (s)= 4/s² [s + 2]
   

   ∴ y(s) = -	1	+	2	+	1
   	s		s²		s + 2

   
   = – u (t) + 2 tu (t) + e^{– 2t}
   forced response terms.

   Correct Option: B

   dy/dt + 2y = 4 t u (t)
   Laplace transform is, 5y(s) + 2y(s) = 4/s²
   or y(s) [s + 2] = 4/s²
   or y (s)= 4/s² [s + 2]
   

   ∴ y(s) = -	1	+	2	+	1
   	s		s²		s + 2

   
   = – u (t) + 2 tu (t) + e^{– 2t}
   forced response terms.

---

69. Unit impulse response of a system is given as c(t) = – 4e^–t + 6 e^–2t The step response of the same system for t ≥ 0 is equal to

1. 1. – 3e^{– 2t} + 4 e^{– t} – 2

   
   
   

   2. – 3e^{– 2t} + 4 e^{– t} + 1

   
   
   

   3. – 3e^{– 2t} + 4e^{– t} – 1

   
   
   

   4. 3e^{– 2t} + 4e^{– t} – 1

   
   
   

---

1. View Hint View Answer Discuss in Forum

   Step response is integral of unit impulse response
   
   = 4e^{– t} – 3e^{– 2t} – 1

   Correct Option: C

   Step response is integral of unit impulse response
   
   = 4e^{– t} – 3e^{– 2t} – 1

---

---

70. The laplace transform of (t² – 2t) u (t – 1) is

1. 1. 2	e-s -	2	e-s
      s³		s²

   
   
   

   2. 2	e-2s -	2	e-s
      s³		s²

      
      

   
   
   

   3. 2	e-s -	1	e-s
      s³		s

   
   
   

   4. none of these

   
   
   

---

1. View Hint View Answer Discuss in Forum

   [(t² - 2t)u(t -1)] = (t - 1)² u (t - 1) - u(t - 1)]
   

   =	2	e-s -	e-s
   	s³		s

   Correct Option: C

   [(t² - 2t)u(t -1)] = (t - 1)² u (t - 1) - u(t - 1)]
   

   =	2	e-s -	e-s
   	s³		s

---