111. Nyquist plot of the functions G₁ (s) and G₂ (s) are shown in figure.
     
     Nyquist plot of the product of G₁ (s) and G₂ (s) is
     

1. 1. 
      

   
   
   

   2. 
      

   
   
   

   3. 
      

   
   
   

   4. 

   
   
   

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1. View Hint View Answer Discuss in Forum

   
   

   G1(s) =	1	and G2(s) = s
   	s

   
   From the Nquist plot,
   

   ∴ G1 G2(s) =	1	× s = 1
   	s

   Correct Option: B

   
   

   G1(s) =	1	and G2(s) = s
   	s

   
   From the Nquist plot,
   

   ∴ G1 G2(s) =	1	× s = 1
   	s

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112. An open loop transfer function G(s) of a system is
     

     G(s) =	K
     	s (s + 1)(s + 2)

     
     For a unity feedback system, the breakaway point of the root loci on the real axis occurs at,
     

1. 1. – 0.42

   
   
   

   2. – 1.58

   
   
   

   3. – 0.42 and – 1.58

   
   
   

   4. None of the above

   
   
   

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1. View Hint View Answer Discuss in Forum

   The characterstic equation of given function is
   1 + G(s) H(s) = 0
   ∴ s(s² + 3s + 2) + K = 0 ( &becuse; H(s) = 1)
   ∴ – k = s³ + 3s² +2s
   In order to find break away point
   

   We have	dK	= 0
   	ds

   
   ∴ 3s² + 6s + 2 = 0
   ∴ S = – 0.42 is the solution that makes k > 0

   Correct Option: A

   The characterstic equation of given function is
   1 + G(s) H(s) = 0
   ∴ s(s² + 3s + 2) + K = 0 ( &becuse; H(s) = 1)
   ∴ – k = s³ + 3s² +2s
   In order to find break away point
   

   We have	dK	= 0
   	ds

   
   ∴ 3s² + 6s + 2 = 0
   ∴ S = – 0.42 is the solution that makes k > 0

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113. In the formation of Routh-Hurwitz array for a polynomial, all the elements of a row have zero values. This premature termination of the array indicates the presence of
     

1. 1. Only one root at the origin

   
   
   

   2. Imaginary roots

   
   
   

   3. Only positive real roots

   
   
   

   4. Only negative real roots

   
   
   

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1. View Hint View Answer Discuss in Forum

   When all elements of row have zero values which leads to auxiliary equation. This premature termination of the array indicates the presence of imaginary roots.
   

   Correct Option: B

   When all elements of row have zero values which leads to auxiliary equation. This premature termination of the array indicates the presence of imaginary roots.
   

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114. The root locus of a unity feedback system is shown in the figure
     
     The closed loop transfer function of the system is

1. 1. 	C(s)	=	K
      	R(s)		(s + 1)(s + 2)

   
   
   

   2. 	C(s)	=	-K
      	R(s)		(s + 1)(s + 2) + K

   
   
   

   3. 	C(s)	=	K
      	R(s)		(s + 1)(s + 2) - K

   
   
   

   4. 	C(s)	=	K
      	R(s)		(s + 1)(s + 2) + K

      
      

   
   
   

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1. View Hint View Answer Discuss in Forum

   	C(s)	=	k	will give the root locus given the diagram.
   	R(s)		(s + 1)(s + 2) - k

   Correct Option: C

   	C(s)	=	k	will give the root locus given the diagram.
   	R(s)		(s + 1)(s + 2) - k

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115. The state transition matrix for the system
     

     	x1̇		=		1			0			x1		+		1		u is
     	x2̇				1			1			x2				1

1. 1. 		et			0
      		et			et

   
   
   

   2. 		et			0
      		t2 et			et

   
   
   

   3. 		et			0
      		t et			et

   
   
   

   4. 		et			t et
      		0			et

   
   
   

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1. View Hint View Answer Discuss in Forum

   State transition matrix,
   φ (t) = L^-1 [sI - A]^-1
   

   = L-1		s - 1			0		-1
   		-1			s - 1

   
   

   = L-1		1		s - 1			0
   		(s - 1)2		1			s - 1

   
   
   

   Correct Option: C

   State transition matrix,
   φ (t) = L^-1 [sI - A]^-1
   

   = L-1		s - 1			0		-1
   		-1			s - 1

   
   

   = L-1		1		s - 1			0
   		(s - 1)2		1			s - 1

   
   
   

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