96. The auto-correlation function R_x(τ) of a random process has the property that R_x(0) is equal to—

1. 1. The square of the mean value of the process

   
   
   

   2. The mean square value of the process

   
   
   

   3. The smallest value of the process

   
   
   

   4. 	1	[Rx(τ) + Rx(– τ)]
      	2

   
   
   

---

1. View Hint View Answer Discuss in Forum

   R_x(0) = is called the mean square value of random process. Furthermore, this is maximum value of the process.

   Correct Option: B

   R_x(0) = is called the mean square value of random process. Furthermore, this is maximum value of the process.

---

97. The graph shown below represents a waveform
    
    obtained by convolving two rectangular waveforms of duration—

1. 1. Four units each

   
   
   

   2. Four and two units respectively

   
   
   

   3. Six and three units respectively

   
   
   

   4. Six and two units respectively

   
   
   

---

1. View Hint View Answer Discuss in Forum

   NA

   Correct Option: D

   NA

---

---

98. The system represented by equations—
    

    (i)	d	y(t) + ty(t) = x(t)
    	dt

    
    

    (ii) x(z) =	d2	+ y(t)	d	y(t) + y(t) = x(t)
    	dt2		dt

1. 1. (i) is linear, (ii) is non-linear

   
   
   

   2. (i) is non-linear, (ii) is linear

   
   
   

   3. Both (i) and (ii) are linear

   
   
   

   4. Both (i) and (ii) are non-linear

   
   
   

---

1. View Hint View Answer Discuss in Forum

   (i)	dt(t)	+ ty(t) = x(t)
   	dt

   
   

   F[x1(t) + x2(t)] =	dy1(t)	+	dy2(t)	+ t[y1(t) + y2(t)]
   	dt		dt

   
   

   F[x1(t)] + F[x2(t)] =	dy1(t)	+	dy2(t)	+ t[y1(t) + y2(t)]
   	dt		dt

   
   Here, F[x₁(t) + x₂(t)] = F[x₁(t)] + F[x₂(t)]
   Hence, the system is linear.
   

   (ii)	d2y(t)	+ y(t)	dy(t)	+ y(t) = x(t)
   	dt2		dt

   
   

   F[x1(t)] =	d2	y1(t) + y1(t)	dy1(t)	+ y1(t)
   	dt2		dt

   
   

   F[x2(t)] =	d2	y2(t) + y2(t)	dy2(t)	+ y2(t)
   	dt2		dt

   
   Therefore,
   

   aF[x1(t)] + bF[x1(t)] =	d2y1(t)	+ b	d2y2(t)
   	dt2		dt2

   
   

   + ay1(t)	dy1(t)	+ by2(t)	dy2(t)	+ ay1(t) + by2(t)
   	dt		dt

   
   

   aF[ax1(t) + bx2(t) =	d2	[ay1(t) + by2(t)] + [ay1(t)
   	dt2

   
   

   + by2(t)]			d	y1(t) +	d	y2(t)		+ by2 [ay1(t) + by2(t) + 1]
   			dt		dt

   
   Since here,
   aF[x₁(t)] + bF[x₂(t)] ≠ F[ax₁(t) + bx₂(t)]
   Hence, the system is non-linear.

   Correct Option: A

   (i)	dt(t)	+ ty(t) = x(t)
   	dt

   
   

   F[x1(t) + x2(t)] =	dy1(t)	+	dy2(t)	+ t[y1(t) + y2(t)]
   	dt		dt

   
   

   F[x1(t)] + F[x2(t)] =	dy1(t)	+	dy2(t)	+ t[y1(t) + y2(t)]
   	dt		dt

   
   Here, F[x₁(t) + x₂(t)] = F[x₁(t)] + F[x₂(t)]
   Hence, the system is linear.
   

   (ii)	d2y(t)	+ y(t)	dy(t)	+ y(t) = x(t)
   	dt2		dt

   
   

   F[x1(t)] =	d2	y1(t) + y1(t)	dy1(t)	+ y1(t)
   	dt2		dt

   
   

   F[x2(t)] =	d2	y2(t) + y2(t)	dy2(t)	+ y2(t)
   	dt2		dt

   
   Therefore,
   

   aF[x1(t)] + bF[x1(t)] =	d2y1(t)	+ b	d2y2(t)
   	dt2		dt2

   
   

   + ay1(t)	dy1(t)	+ by2(t)	dy2(t)	+ ay1(t) + by2(t)
   	dt		dt

   
   

   aF[ax1(t) + bx2(t) =	d2	[ay1(t) + by2(t)] + [ay1(t)
   	dt2

   
   

   + by2(t)]			d	y1(t) +	d	y2(t)		+ by2 [ay1(t) + by2(t) + 1]
   			dt		dt

   
   Since here,
   aF[x₁(t)] + bF[x₂(t)] ≠ F[ax₁(t) + bx₂(t)]
   Hence, the system is non-linear.

---

99. What does the transfer function of a system describe for the system?

1. 1. Only zero-input response

   
   
   

   2. Only zero-state response

   
   
   

   3. Both zero-input and zero-state response

   
   
   

   4. Neither zero-input response nor zero-state response

   
   
   

---

1. View Hint View Answer Discuss in Forum

   The transfer function of a system describes for both zero input and zero state response.

   Correct Option: C

   The transfer function of a system describes for both zero input and zero state response.

---

---

100. Which one of the following represents the phase response of the function?
     

     H(s) =	s2 + ω02
     	s2 + (ω0/Q) s + ω2

1. 1. 

   
   
   

   2. 

   
   
   

   3. 

   
   
   

   4. 

   
   
   

---

1. View Hint View Answer Discuss in Forum

   The given transfer function
   

   H(s) =		s2 + ω20
   	s2 +	ω0	s + ω2
   		Q

   
   represents band reject or notch filter as given in alternative (A).

   Correct Option: A

   The given transfer function
   

   H(s) =		s2 + ω20
   	s2 +	ω0	s + ω2
   		Q

   
   represents band reject or notch filter as given in alternative (A).

---