81. If φ(x, y) and ψ (x, y) are functions with continuous second derivatives, then φ(x, y) + iψ(x, y)

1. 1. δφ	= -	δψ	,	δφ	=	δψ
      δx		δx		δy		δy

   
   
   

   2. δφ	= -	δψ	,	δφ	=	δψ
      δy		δx		δx		δy

   
   
   

   3. δ²φ	= -	δ²ψ	,	δ²φ	=	δ²ψ	= 1
      δx²		δy²		δx²		δy²

   
   
   

   4. δφ	= -	δφ	,	δ²φ	=	δ²ψ	= 0
      δx²		δy²		δx²		δy²

      
      

   
   
   

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1. View Hint View Answer Discuss in Forum

   Can be expressed as an analytic function of
   φ + iψ(i = √-1) , when The necessary condition for a function
   f (z) = φ(x,y) + i ψ (x, y) to be analytic
   

   (i)	δφ	=	δψ
   	δx		δy

   
   

   (ii)	δφ	=	-δψ
   	δy		δx

   
   are known as Cauchy Rieman equations, provided
   

   δφ	,	δφ	,	δψ	,	δψ	exists.
   δx		δy		δx		δy

   
   

   Correct Option: B

   Can be expressed as an analytic function of
   φ + iψ(i = √-1) , when The necessary condition for a function
   f (z) = φ(x,y) + i ψ (x, y) to be analytic
   

   (i)	δφ	=	δψ
   	δx		δy

   
   

   (ii)	δφ	=	-δψ
   	δy		δx

   
   are known as Cauchy Rieman equations, provided
   

   δφ	,	δφ	,	δψ	,	δψ	exists.
   δx		δy		δx		δy

   
   

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82. The function f(t) satisfies the differential equation (d²f / dt²) + f = 0 and the auxilliary conditions, f(0) = 0, (df / dt) (o) = 4. The Laplace transform of f(t) is given by

1. 1. 	2
      	s + 1

   
   
   

   2. 	4
      	s + 1

   
   
   

   3. 	4
      	s² + 1

   
   
   

   4. 	2
      	s² + 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   Check by options
   

   L-1		4		= 4 sint satisfies	d²f	+ f = 0
   		s² + 1			dt²

   
   

   Correct Option: C

   Check by options
   

   L-1		4		= 4 sint satisfies	d²f	+ f = 0
   		s² + 1			dt²

   
   

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83. If F(s) is the Laplace transform of function f(t), then Laplace transform of
    

1. 1. 	1	F(s)
      	s

   
   
   

   2. 	1	F(s) - f(o)
      	s

   
   
   

   3. sF(s) - f(o)

   
   
   

   4. ∫ F(s) ds

   
   
   

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1. View Hint View Answer Discuss in Forum

   
   

   Correct Option: B

   
   

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84. 

1. 1. a.e.^-as

   
   
   

   2. 	e-as
      	s

   
   
   

   3. 	eas
      	s

   
   
   

   4. 	eas
      	a

   
   
   

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1. View Hint View Answer Discuss in Forum

   
   

   Correct Option: B

   
   

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85. Laplace transform of the function sin ωt is

1. 1. 	s
      	s² + ω²

   
   
   

   2. 	ω
      	s² + ω²

   
   
   

   3. 	s
      	s² - ω²

   
   
   

   4. 	ω
      	s² - ω²

   
   
   

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1. View Hint View Answer Discuss in Forum

   Laplace transforms,
   

   L[sin(ωt)]	ω
   	S² + ω²

   
   

   L[cos(ωt)]	ω
   	S² + ω²

   
   

   Correct Option: B

   Laplace transforms,
   

   L[sin(ωt)]	ω
   	S² + ω²

   
   

   L[cos(ωt)]	ω
   	S² + ω²

   
   

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