The root-locus plot of a closed-loop system with unity negative feedback and transfer function  in the forward path is shown in the figure. Note that  is varied from 0 to .                                        

Select the transfer function  that results in the root-locus plot of the closed-loop system as shown in the figure.

Consider an even polynomial  given by

where  is an unknown real parameter. The complete range of  for which  has all its roots on the imaginary axis is

The transition diagram of a discrete memoryless channel with three input symbols and three output symbols is shown in the figure. The transition probabilities are as marked.

The parameter  lies in the interval . The value of  for which the capacity of this channel is maximized, is ________ (rounded off to two decimal places).        

Consider the circuit shown with an ideal OPAMP. The output voltage  is _______ V (rounded off to two decimal places).

Consider communication over a memoryless binary symmetric channel using a  Hamming code. Each transmitted bit is received correctly with probability (  ), and flipped with probability . For each codeword transmission, the receiver performs minimum Hamming distance decoding, and correctly decodes the message bits if and only if the channel introduces at most one bit error.

For , the probability that a transmitted codeword is decoded correctly is ________ (rounded off to two decimal places).

A simple closed path  in the complex plane is shown in the figure. If

  where , then the value of  is ________ (rounded off to two decimal places).

 Let  and , where  denotes the unit step function.

If  denotes the convolution of  and , then  ________(rounded off to one decimal place).

The outputs of four systems , and  corresponding to the input signal , for all time , are shown in the figure. Based on the given information, which of the four systems is/are definitely NOT LTI (linear and time. invariant)?

Consider the following partial differential equation (PDE)

where  and  are distinct positive real numbers. Select the combination(s) of values of the real parameters  and  such that  is a solution of the given PDE.

Consider the following wave equation,

Which of the given options is/are solution(s) to the given wave equation?

A waveguide consists of two infinite parallel plates (perfect conductors) at a separation of , with air as the dielectric. Assume the speed of light in air to be . The frequency/ frequencies of  waves which can propagate in this waveguide is/are _______.

The bar graph shows the frequency of the number of wickets taken in a match by a bowler in her career. For example, in 17 of her matches, the bowler has taken 5 wickets each. The median number of wickets taken by the bowler in a match is __________ (rounded off to one decimal place).

linear 2-port network is shown in Fig. (a). An ideal DC voltage source of  is connected across Port 1. A variable resistance  is connected across Port 2. As  is varied, the measured voltage and current at Port 2 is shown in Fig. (b) as a  versus  plot. Note that for , , and for .

When the variable resistance  at  ort 2 is replaced by the load shown in Fig. (c), the current  is _________  (rounded off to one decimal place).         

A state transition diagram with states , and  and transition probabilities  is shown in the figure (e.g.,  denotes the probability of transition from state A to B ). For this state diagram, select the statement(s) which is/are universally true.

Select the correct statement(s) regarding CMOS implementation of NOT gates.

Consider the 2-bit multiplexer (MUX) shown in the figure. For OUTPUT to be the XOR of  and , the values for  and  are ________.        

Two linear time-invariant systems with transfer functions

have unit step responses  and , respectively. Which of the following statements is/are true?

The block diagram of a closed-loop control system is shown in the figure. , and  are the Laplace transforms of the time-domain signals , and , respectively. Let the error signal be defined as . Assuming the reference input  for all , the steady state error , due to a unit step disturbance , is __________(rounded off to two decimal places).

Consider a system of linear equations , where

This system of equations admits

Let  be two non-zero real numbers and  be two non-zero real vectors of size . Suppose that  and  satisfy , and . Let  be the  matrix given by: 

The eigen values of  are

Consider the CMOS circuit shown in the figure (substrates are connected to their respective sources). The gate width (  ) to gate length  ratios  of the transistors are as shown. Both the transistors have the same gate oxide capacitance per unit area. For the pMOSFET, the threshold voltage is  and the mobility of holes is  and the mobility of electrons is . The steady state output voltage  is ________.

The ideal long channel nMOSFET and pMOSFET devices shown in the circuits have threshold voltages of  and , respectively. The MOSFET substrates are connected to their respective sources. Ignore leakage currents and assume that the capacitors are initially discharged. For the applied voltages as shown, the steady state voltages are ________.

An ideal MOS capacitor ( -type semiconductor) is shown in the figure. The MOS capacitor is under strong inversion with . The corresponding inversion charge density  is . Assume oxide capacitance per unit area as . For , the value of  is __________  (rounded off to one decimal place).

Consider an ideal long channel nMOSFET (enhancement-mode) with gate length  and width . The product of electron mobility  and oxide capacitance per unit area  is . The threshold voltage of the transistor is 1V. For a gate-to-source voltage  and drain-to source voltage  (substrate connected to the source), the maximum value of the drain-to-source current is__________.

The Fourier transform  of the signal        

 is ________.

In a circuit, there is a series connection of an ideal resistor and an ideal capacitor. The conduction current (in Amperes) through the resistor is  ). The displacement current (in Amperes) through the capacitor is ________.

Consider the circuit shown with an ideal long channel nMOSFET (enhancement mode, substrate is connected to the source). The transistor is appropriately biased in the saturation region with  and  such that it acts as a linear amplifier.  is the small-signal ac input voltage.  and  represent the small-signal voltages at the nodes A and B, respectively. The value of   is ________ (rounded off to one decimal place).

The frequency response  of a linear time invariant system has magnitude as shown in the figure.

Statement I: The system is necessarily a pure delay system for inputs which are bandlimited to .

Statement II: For any wide-sense stationary input process with power spectral density , the output power spectral density  obeys   for .        

Which one of the following combinations is true?

The current I in the circuit shown is ________.         

In an electrostatic field, the electric displacement density vector, , is given by

where  are the unit vectors along -axis, -axis, and -axis, respectively. Consider a cubical region  centered at the origin with each side of length , and vertices at  . The electric charge enclosed within  is ________ C (rounded off to two decimal places).

For a vector , the 8-point discrete Fourier transform (DFT) is denoted by , where

Here, . If  and , then the value of  is __________(rounded off to one decimal place).        

Consider the circuit shown in the figure. The current I flowing through the 10Ω resistor is ________.         

A circuit and the characteristics of the diode (D) in it are shown. The ratio of the minimum to the maximum small signal voltage gain  is __________ (rounded off to two decimal places).

Select the Boolean function(s) equivalent to  , where , and  are Boolean variables, and + denotes logical OR operation.

Consider a Boolean gate  where the output  is related to the inputs  and  as, , where + denotes logical OR operation. The Boolean inputs ' 0 ' and ' 1 ' are also available separately. Using instances of only  gates and inputs ' 0 ' and ' 1 ', (select the correct options)

Consider a long rectangular bar of direct bandgap p-type semiconductor. The equilibrium hole density is  and the intrinsic carrier concentration is . Electron and hole diffusion lengths are  and , respectively. The left side of the bar  is uniformly illuminated with a laser having photon energy greater than the bandgap of the semiconductor. Excess electron-hole pairs are generated ONLY at  because of the laser. The steady state electron density at  is  due to laser illumination. Under these conditions and ignoring electric field, the closest approximation (among the given options) of the steady state electron density at , is _______.

In a non-degenerate bulk semiconductor with electron density , the value of , where  and  denote the bottom of the conduction band energy and electron Fermi level energy, respectively. Assume thermal voltage as  and the intrinsic carrier concentration is . For  3 , the closest approximation of the value of , among the given options, is _________.

Select the CORRECT statement(s) regarding semiconductor devices.

A p-type semiconductor with zero electric field is under illumination (low level injection) in steady state condition. Excess minority carrier density is zero at , where  is the diffusion length of electrons. Assume electronic charge, . The profiles of photogeneration rate of carriers and the recombination rate of excess minority carriers  are shown. Under these conditions, the magnitude of the current density due to the photo-generated electrons at  is __________  (rounded off to two decimal places).

Consider a closed-loop control system with unity negative feedback and  in the forward path, where the gain . The complete Nyquist plot of the transfer function  is shown in the figure. Note that the Nyquist contour has been chosen to have the clockwise sense. Assume  has no poles on the closed right-half of the complex plane. The number of poles of the closed-loop transfer function in the closed right-half of the complex plane is ________.

For the circuit shown, the locus of the impedance  is plotted as  increases from zero to infinity. The values of  and  are:

For the circuit shown, the clock frequency is  and the duty cycle is . For the signal at the  output of the Flip-Flop, ________.

Consider the circuit shown in the figure with input  in volts. The sinusoidal steady state current  flowing through the circuit is shown graphically (where  is in seconds). The circuit element  can be _______.

Consider the two-dimensional vector field , where  and  denote the unit vectors along the -axis and the -axis, respectively. A contour  in the -y plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral   is __________.

The function  attains its minimum over the interval  at  __________.

(Here  is the natural logarithm of .)

Consider the following series:  

For which of the following combinations of  values does this series converge?

The value of the integral  where  is the shaded triangular region shown in the diagram, is ________ (rounded off to the nearest integer).

Let H(X) denote the entropy of a discrete random variable X taking K possible distinct real values. Which of the following statements is/are necessarily true?         

Consider a real valued source whose samples are independent and identically distributed random variables with the probability density function, , as shown in the figure.

Consider a 1 bit quantizer that maps positive samples to value  and others to value . If  and  are the respective choices for  and  that minimize the mean square quantization error, then  __________(rounded off to two decimal places).

An ideal OPAMP circuit with a sinusoidal input is shown in the figure. The  frequency is the frequency at which the magnitude of the voltage gain decreases by  from the maximum value. Which of the options is/are correct?

For the following circuit with an ideal OPAMP, the difference between the maximum and the minimum values of the capacitor voltage  is __________             

A circuit with an ideal OPAMP is shown. The Bode plot for the magnitude (in  ) of the gain transfer function  of the circuit is also provided (here,  is the angular frequency in . The values of R and  are ________.        

Consider an FM broadcast that employs the pre-emphasis filter with frequency response

Where .

For the network shown in the figure to act as a corresponding de-emphasis filter, the appropriate pair(s) of  values is/are ________.

A symbol stream contains alternate QPSK and 16-QAM symbols. If symbols from this stream are transmitted at the rate of 1 mega-symbols per second, the raw (uncoded) data rate is _________ mega-bits per second (rounded off to one decimal place).         

Consider a channel over which either symbol  or symbol  is transmitted. Let the output of the channel  be the input to a maximum likelihood (ML) detector at the receiver. The conditional probability density functions for  given  and  are:

where  is the standard unit step function. The probability of symbol error for this system is __________ (rounded off to two decimal places).