Consider the matrix  below:

For which of the following combinations of , and , is the rank of  at least three?

(i)  and .

(ii) .

(iii)  and .

(iv) .

Consider the following series:

(i)

(ii)

(iii)

Choose the correct option.

A pot contains two red balls and two blue balls. Two balls are drawn from this pot randomly without replacement.

What is the probability that the two balls drawn have different colours?

Consider a frequency-modulated (FM) signal

where  and  are, respectively, the amplitude and frequency (in Hz) of the carrier waveform. The frequency  is in Hz, and assume that .

The peak frequency deviation of the FM signal in Hz is __________.

Consider an additive white Gaussian noise (AWGN) channel with bandwidth  and noise power spectral density . Let  denote the average transmit power constraint.

Which one of the following plots illustrates the dependence of the channel capacity  on the bandwidth  (keeping  and  fixed)?

The Nyquist plot of a system is given in the figure below. Let , and  be the positive frequencies at the points , and , respectively.

Which one of the following statements is TRUE?

Consider the discrete-time system below with input  and output . In the figure,  and  denote the impulse responses of LTI Subsystems 1 and 2, respectively. Also,  is the unit impulse, and .

Assuming , the overall system (denoted by the dashed box) is _________.

Consider a continuous-time, real-valued signal  whose Fourier transform  exists.

Which one of the following statements is always TRUE?

Consider a part of an electrical network as shown below. Some node voltages, and the current flowing through the  resistor are as indicated.

The voltage (in Volts) at node  is ________.

Let , and  be the currents flowing through the capacitor, inductor, and resistor, respectively, in the circuit given below. The AC admittances are given in Siemens (S).

Which one of the following is TRUE?

A simplified small-signal equivalent circuit of a BJT-based amplifier is given below.

The small-signal voltage gain  in  is _________.

The ideal BJT in the circuit given below is biased in the active region with a  of 100.

If  is , then  (in Volts, rounded off to two decimal places) is _________.

A 3-input majority logic gate has inputs , and . The output F of the gate is logic '' if two or more of the inputs are logic ''. The output  is logic '  ' if two or more of the inputs are logic '0'.

Which one of the following options is a Boolean expression of the output F ?

A full adder and an XOR gate are used to design a digital circuit with inputs , and , and output , as shown below. The input  is connected to the carry-in input of the full adder.

If the input  is set to logic '  ', then the circuit functions as _________ with  and  as inputs.

Consider the function , defined as

Which of the following statements is/are correct?

(Here,  is the set of real numbers.)

Consider the unity-negative-feedback system shown in Figure (i) below, where gain . The root locus of this system is shown in Figure (ii) below.

For what value(s) of  will the system in Figure (i) have a pole at  ?

Let  be a discrete-time signal whose -transform is .

Which of the following statements is/are TRUE?

Consider a message signal  which is bandlimited to , where  is in Hz. Consider the following two modulation schemes for the message signal:

• Double sideband-suppressed carrier (DSB-SC):

• Amplitude modulation (AM):

Here,  and  are the amplitude and frequency (in Hz) of the carrier, respectively. In the case of AM,  denotes the modulation index.

Consider the following statements:

(i) An envelope detector can be used for demodulation in the DSB-SC scheme if  for all .

(ii) An envelope detector can be used for demodulation in the AM scheme only if  for all .

Which of the following options is/are correct?

Which of the following statements is/are TRUE with respect to an ideal opamp?

Which of the following statements is/are TRUE with respect to ideal MOSFET based DC-coupled single-stage amplifiers having finite load resistors?

Which of the following can be used as an n-type dopant for silicon?

Select the correct option(s).

The function  satisfies

where  and  denote the first and second derivatives of  respectively.

Given  and , the maximum value of  over  is _________ (rounded off to two decimal places).

The generator matrix of a  binary linear block code is given by

The minimum Hamming distance  between codewords equals _________ (answer in integer).

All the components in the bandpass filter given below are ideal. The lower -3 dB frequency of the filter is 1 MHz.

The upper -3 dB frequency (in MHz, rounded off to the nearest integer) is _________.

A 4-bit weighted-resistor DAC with inputs , and  (MSB to LSB) is designed using an ideal opamp, as shown below. The switches are closed when the corresponding input bits are logic ' 1 ' and open otherwise.

When the input  changes from 1110 to 1101, the magnitude of the change in the output voltage , rounded off to the nearest integer) is ___________ .

Let  be the transfer function of a second-order system. A controller  is connected to the system  in the configuration shown below.

Consider the following statements.

(i) There exists no controller of the form , where  is a positive real number, such that the closed loop system is stable.

(ii) There exists at least one controller of the form , where  and  are positive real numbers, such that the closed loop system is stable.

Which one of the following options is correct?

Consider the polynomial

. Let  be defined as follows.

 is the number of roots of  with negative real parts.

 is the number of roots of  that are purely imaginary.

 is the number of roots of  with positive real parts.

Which one of the following options is correct?

Consider a continuous-time finite-energy signal  whose Fourier transform vanishes outside the frequency interval , where  is in .

The signal  is uniformly sampled to obtain . Here,

with  being the Dirac impulse, , and . The sampled signal  is passed through an ideal lowpass filter  with cutoff frequency  and passband gain .

The output of the filter is given by ___________.

In the circuit below,  is an ideal AC voltmeter and  is an ideal AC ammeter. The source voltage (in Volts) is .

What should be the value of the variable capacitor  such that the RMS readings on  and  are 25 V and 5 A, respectively?

The -parameter matrix of a two port network relates the port voltages and port currents as follows:

The -parameter matrix (with each entry in Ohms) of the network shown below is

A source transmits symbol  that takes values uniformly at random from the set . The receiver obtains , where  is a zero-mean Gaussian random variable independent of . The receiver uses the maximum likelihood decoder to estimate the transmitted symbol .

Suppose the probability of symbol estimation error  is expressed as follows:

where  denotes the probability that  exceeds 1.

What is the value of  ?

Consider a real-valued random process

where  and  is a positive integer. Here,  for  and 0 otherwise. The coefficients  are pairwise independent, zero-mean unit variance random variables.

Read the following statements about the random process and choose the correct option.

(i) The mean of the process  is independent of time .

(ii) The autocorrelation function  is independent of time  for all .

(Here,  is the expectation operation.)

The identical MOSFETs  and  in the circuit given below are ideal and biased in the saturation region.  and  have a transconductance  of 5 mS .

The input signals (in Volts) are:

The output signal  (in Volts) is __________.

A 10-bit analog-to-digital converter (ADC) has a sampling frequency of 1 MHz and a full scale voltage of 3.3 V.

For an input sinusoidal signal with frequency 500 kHz, the maximum SNR (in dB, rounded off to two decimal places) and the data rate (in Mbps) at the output of the ADC are _________, respectively.

A positive-edge-triggered sequential circuit is shown below. There are no timing violations in the circuit. Input  is set to logic ' 0 ' and  is set to logic ' 1 ' at all times. The timing diagram of the inputs  and  are also shown below.

The sequence of output  from time  to  is ________.

The intrinsic carrier concentration of a semiconductor is  at 300 K.

If the electron and hole mobilities are  and , respectively, then the intrinsic resistivity of the semiconductor (in  ) at 300 K is _________.

(Charge of an electron .)

In the circuit shown, the identical transistors Q1 and Q2 are biased in the active region with . The Zener diode is in the breakdown region with  and .

If  and , then the values of  and  (in , rounded off to one decimal place) are __________, respectively.

The electron mobility  in a non-degenerate germanium semiconductor at 300 K is .

The electron diffusivity  at 300 K (in , rounded off to the nearest integer) is _________.

(Consider the Boltzmann constant  and the charge of an electron .)

A square metal sheet of  is placed on the  plane as shown in the figure below.

If the surface charge density (in  ) on the sheet is , then the total charge (in , rounded off to the nearest integer) on the sheet is _________.

An electric field of  is applied along the length of a copper wire of circular cross-section with diameter 1 mm. Copper has a conductivity of .

The current (in Amperes, rounded off to two decimal places) flowing through the wire is _________.

Consider a non-negative function  which is continuous and bounded over the interval . Let  and  denote, respectively, the maximum and the minimum values of  over the interval.

Among the combinations of  and  given below, choose the one(s) for which the inequality   is guaranteed to hold.

Which of the following statements involving contour integrals (evaluated counterclockwise) on the unit circle C in the complex plane is/are TRUE?

Consider a system where , and  are three internal state signals and  is the input signal. The differential equations governing the system are given by

Which of the following statements is/are TRUE?

The random variable  takes values in  with probabilities  and , where . Let  denote the entropy of  (in bits), parameterized by . Which of the following statements is/are TRUE?

Let  be a periodic signal with fundamental period . Consider the signal , where .

The Fourier series expansions of  and  are given by

Which of the following statements is/are TRUE?

Consider a system represented by the block diagram shown below. Which of the following signal flow graphs represent(s) this system? Choose the correct option(s).

All the diodes in the circuit given below are ideal.

Which of the following plots is/are correct when  (in Volts) is swept from - M to M?

Two fair dice (with faces labeled , and 6) are rolled. Let the random variable  denote the sum of the outcomes obtained.

The expectation of  is _________ (rounded off to two decimal places).

Consider the vectors

For real-valued scalar variable , the value of

  is _________ (rounded off to two decimal places).

 denotes the Euclidean norm, i.e., for .

 and  are Bernoulli random variables taking values in . The joint probability mass function of the random variables is given by:

The mutual information  is _________ (rounded off to two decimal places).

The diode in the circuit shown below is ideal. The input voltage (in Volts) is given by , where time t is in seconds.

The time duration (in ms, rounded off to two decimal places) for which the diode is forward biased during one period of the input is ________ .

In the circuit shown below, the AND gate has a propagation delay of 1 ns. The edge-triggered flip-flops have a set-up time of 2 ns, a hold-time of 0 ns, and a clock-to-Q delay of 2 ns.

The maximum clock frequency (in MHz, rounded off to the nearest integer) such that there are no setup violations is ________

An ideal p-n junction germanium diode has a reverse saturation current of  at 300 K.

The voltage (in Volts, rounded off to two decimal places) to be applied across the junction to get a forward bias current of 100 mA at 300 K is _______.

(Consider the Boltzmann constant  and the charge of an electron .)

A  lossless transmission line is terminated with a load  of .

If the average incident power on the line is 10 mW, then the average power delivered to the load (in mW, rounded off to one decimal place) is ________.

Two resistors are connected in a circuit loop of area , as shown in the figure below. The circuit loop is placed on the  plane.

When a time-varying magnetic flux, with flux-density  (in Tesla), is applied along the positive -axis, the magnitude of current  (in Amperes, rounded off to two decimal places) in the loop is _________.