11. If L[f(t)] =	2(s + 2)	, then ƒ(0+) and ƒ(∞) are given by
    	s² + 2s + 5

1. 1. 0, 2 respectively

   
   
   

   2. 2, 0 respectively

   
   
   

   3. 0, 1 respectively

   
   
   

   4. 2/5, 0 respectively

   
   
   

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1. View Hint View Answer Discuss in Forum

   2(s + 1)	=	2(s + 1)	= 2		s + 1
   (s² + 2s + 5)		(s² + 1)² + 5			(s + 1)² + 4

   
   This is standard Laplace Transform for the function
   

   L| exp (– at) cos ωt | =	(s + a)
   	(s + a)² + ω²

   
   where a = 1, ω = 2
   Hence ƒ(t) = 2e^{– t} cos 2t
   ƒ(0) = ƒ(0+) = 2e^{– 0} cos 2 × 0 = 2
   ƒ(∞) = 2e^{– ∞} t = 0

   Correct Option: B

   2(s + 1)	=	2(s + 1)	= 2		s + 1
   (s² + 2s + 5)		(s² + 1)² + 5			(s + 1)² + 4

   
   This is standard Laplace Transform for the function
   

   L| exp (– at) cos ωt | =	(s + a)
   	(s + a)² + ω²

   
   where a = 1, ω = 2
   Hence ƒ(t) = 2e^{– t} cos 2t
   ƒ(0) = ƒ(0+) = 2e^{– 0} cos 2 × 0 = 2
   ƒ(∞) = 2e^{– ∞} t = 0

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12. The unit impulse response of a linear time invariant system is unit step function u(t). For t > 0, the response of the system to an excitation e^{– at} u(t), a > 0 will be

1. 1. ae^{– at}
      

   
   
   

   2. 		1		(1 - ae-at)
      		a

   
   
   

   3. a(1 – ae^{– at})

   
   
   

   4. 1 – ae^{– at}

   
   
   

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1. View Hint View Answer Discuss in Forum

   Given: h(t) = u(t)
   x(t) = e^–at u(t)
   

   ∴ Y(s) = X(s) H(s) =	1	.	1	=	1		1	-	1
   	(s + a)		s		a		s		s + a

   
   y(t) = 1/a (1 – e^{– at}) u(t)

   Correct Option: B

   Given: h(t) = u(t)
   x(t) = e^–at u(t)
   

   ∴ Y(s) = X(s) H(s) =	1	.	1	=	1		1	-	1
   	(s + a)		s		a		s		s + a

   
   y(t) = 1/a (1 – e^{– at}) u(t)

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13. Given the transform pair
    
    Determine the laplace transform Y(s) of the given time signal in question and choose correct option. y(t) = x(t – 2)

1. 1. 2se-2e
      s² + 2

   
   
   

   2. 2se-2e
      s² + 2

   
   
   

   3. 2(s - 2)
      (s - 2)² + 1

   
   
   

   4. 2(s + 2)
      (s + 2)² + 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   

   Correct Option: A

   

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