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) .
The given matrix is in standard Echelon form,
rank of matrix if or
rank of matrix = 4 if or and
Thus, (i), (iii), (iv) satisfy the echelon form requirement
Consider the following series:
(i)
(ii)
(iii)
Choose the correct option.
(i) (where )
By -test,
series is divergent.
(ii)
series is convergent.
(iii)
By ratio test,
series is convergent.
Hence, option (b) is correct.
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?
Red balls , Blue balls
Total balls .
Total No. of ways of selecting 2 balls
No. of ways of withdrawn 2 balls of different colours
Hence, option (a) is correct.
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 __________.
By comparison with standard angle modulated signal expression:
Clearly, change in frequency is given by derivative of argument of cosine with respect to time.
Here,
change in angular frequency (in rad/s)
Note in question that unit asked is Hz, thus we have to divide by to get
Max value of summation of sinusoids of different frequency is summation of amplitudes, thus peak frequency deviation
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)?
By Shanon's Channel Capacity Theorem,
Noise Power (double side band PSD) .
And given No are constant,
constant
Putting (ii) & (iii) in (i)
Taking
limit (note the limit)
By L-Hopital's rule constant
Since only (A) matches the limits, (A) is the correct graph
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?
is the gain crossover frequency and is the phase crossover frequency
is the gain crossover frequency and is the phase crossover frequency
is the gain crossover frequency and is the phase crossover frequency
is the gain crossover frequency and is the phase crossover frequency
Gain cross over frequency is one at which or
This is obtained by intersection of Polar or Nyquist Plot with the unit circle in GH Plane.
Clearly in Nyquist plot, at
Phase crossover frequency is one at which ,
i.e., Nyquist plot crosses (-) ve real axis
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 _________.
We see that:
o/p has a constant term in addition, system fails Superposition (additivity), & hence is non linear
{Reason: is linear and is non-linear}
- Delay by
Thus, system is time variant
Consider a continuous-time, real-valued signal whose Fourier transform exists.
Which one of the following statements is always TRUE?
Hence, option (a) is correct.
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?
Admittances:
Hence, option (a) is correct.
A simplified small-signal equivalent circuit of a BJT-based amplifier is given below.
The small-signal voltage gain in is _________.
and
Small-signal voltage gain,
Hence, option (A) is correct.
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 _________.
Hence, option (c) is correct.
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 ?
F=f(X,Y, Z)
F=1 for Two or More inputs are 1.
F=0 for Two or More inputs are 0.
Truth Table:
Hence, option (a) is correct.
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.
When one input of XOR Gate is 1 then output is complement of other input.
→ 2’s Complement
Consider the function , defined as
Which of the following statements is/are correct?
(Here, is the set of real numbers.)
has no global maximizer
has no global minimizer
is a local minimizer of
is a local maximizer of
Stationary points:
Local Maxima
So, function has neither global maxima nor global minima.
Hence, option (a) and (b) are correct.
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 ?
For no positive value of K
For all positive of K
From root locus, it is observed that pole is not lies on the root locus.
Thus, no positive value of exist for system pole at , as no such pole exist for system at .
Hence, option (c) is only correct answer.
Let be a discrete-time signal whose -transform is .
Which of the following statements is/are TRUE?
The discrete-time Fourier transform (DTFT) of always exists
The region of convergence (RoC) of contains neither poles nor zeros
The discrete-time Fourier transform (DTFT) exists if the region of convergence (RoC) contains the unit circle
If , where is the unit impulse and is a scalar, then the region of convergence is the entire -plane
Option (A): The DTFT of exists if is absolutely Summable, ie, if .
This is not true for all signals. For example, False statement.
Option (B): The ROC of a z-transform is defined as the set of points in the z-plane where the z-transform converges. It can contain zero but not poles, as poles would causes divergence.
False statement.
Option (C): The DTFT of exists if the unit circle lies within the of the -transform.
True statement.
Option (D): For , which is a constant.
This converges everywhere in the -plane, making the ROC the entire complex plane.
True Statement.
Hence, options (c) and (d) are correct.
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?
As long as , envelop detector can be used to detect and signals
mentions the word "can", this true as can at times ensure
(D) mention the word "only", it is not a must condition for , only the bound is on , thus: is wrong.
Which of the following statements is/are TRUE with respect to an ideal opamp?
Following are properties of ideal op-amp:
1. Infinite open-loop differential gain
2. Infinite input Impedance
3. Zero output Impedance
4. Zero open-loop common-mode gain
Thus, from given options (a) and (c) are correct.
Which of the following statements is/are TRUE with respect to ideal MOSFET based DC-coupled single-stage amplifiers having finite load resistors?
Concept of active Resistance in MOSFET:
Thus, input resistance will be infinite when terminal input. It is possible in Common-Source Amplifier.
Thus, option (A) is wrong, option (B) is correct.
Common-Source Amplifies produces a 180° phase shift
Common- drain Amplifier produces 0° phase shift.
Thus, input and output voltages of Common-drain Amplifies are in phase.
Option (C) is wrong, option (D) is correct.
Hence, option (B) ant (D) are correct.
Which of the following can be used as an n-type dopant for silicon?
Select the correct option(s).
The correct options are:
(A) Arsenic and (D) Phosphorous
These are both -type dopants for silicon because they have five valence electrons (group elements), which provide extra electrons to the silicon lattice, making it an n-type semiconductor.
while, Boron and Gallium are p-type dopants because they have only three valence electrons, creating holes in the silicon lattice.
Alternate Solution:
Arsenic (As) and Phosphorous (P) are Group 15 elements, they have an extra electrons (5 electrons in valence shell) to contribute to lattice when doped in silicon.
n-type means material has more electrons than holes,
n-type dopants are (a) and (d)
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).
Given:
Given differential equation is of the form of Cauchy-Euler equation.
The equation reduces to :
Its, auxiliary equation is
Put ,
stationary point:
point of Maxima
The generator matrix of a binary linear block code is given by
The minimum Hamming distance between codewords equals _________ (answer in integer).
Given code (n,k) = (6,3),
Where n total bits, k message bits
Corresponding codeword
d ranges from 000 to 111
C values by matrix multiplication
smallest homing weight (no. of non-zero bits) of non-zero codeword
=minimum Hamming Distance see column
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 ___________ .
Given op-amp circuit is inverting summing amplifier
for
For
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?
Given :
Characteristic equation:
Sign change (closed Loop system: Unstable)
(ii)
Characteristic equation:
Given that Real no's
All coefficient +ve in characteristic equation, hence closed loop system is stable.
Hence, correct answer is (D) Both (i) and (ii) are True.
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?
, and
, and
, and
, and
Routh Hurwitz criteria,
No. of sign change in column of RH table .
Auxiliary equation:
Imaginary Poles
and
Hence, option (a) 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 ___________.
if
Given: : Continuous-time finite-energy signal
exist in interval
FS-Coefficient calculation for p(t):
Let for one Time period
FS-Coefficient of
F.T. of periodic signal
Hence, option (A) is correct.
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?
Insufficient information to find C
Hence, option (a) is correct.
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
Hence, option (A) is correct.
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 ?
M.L. decoder is used, all symbols are equiprobable
Probability of error
where Probability of transmitting
Probability of receiving when was sent
is given by shaded region:
Thus
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.)
are independent
Given: constant
(i) is correct
Auto correlation
We can’t comment on ACF of however looking at (as no time data given)
we solve
dependence on is asked, lets fix for all z
as for to :
Clearly is a function
(ii) is false
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 __________.
Given: MOSFETS and : Ideal, Saturation region
DC Analysis:
AC Analysis-
DC Voltage Source = Short-Circuited
DC Current Source = Open-Circuited
In MOSFET based differential amplifier with two input and ,
If polarity of is consider such that , then is non-inverting terminal and is inverting terminal.
As the circuit is symmetrical,
And,
Thus,
Hence, option (d) is correct.
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.
Given: Resolution bits
sampling frequency
full scale voltage (FSV)
Input sinusoidal signal frequency
Maximum SNR (in dB) for -bit resolution,
Data Rate sampling frequency No. of bits per sample
Hence, option (A) is correct.
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 ________.
Label left as and right
Y=1011
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 .)
Intrinsic Carrier Concentration,
Hence, option (B) is correct.
Alternate Solution:
charge of electron (magnitude)
intrinsic carrier concentration
mobility of holes
mobility of electrons
Upon substitution,
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.
Given: Transistors and : Active Region
Zener diode is in breakdown region: &
Hence, option (A) is Correct.
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 .)
Given:
Hence, option (B) is correct.
Alternate Solution:
By Einstein's relation, Thermal Voltage
Given and
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 _________.
Total charge,
Hence, option (c) is correct.
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 _________.
Given:
Area of conductor,
Hence, option (A) is correct.
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.
Given: for
Thus, m , also
And, , also
Given that
Hence, option (A) is correct.
Which of the following statements involving contour integrals (evaluated counterclockwise) on the unit circle C in the complex plane is/are TRUE?
, where is an even integer
Exponential, Sinusoidal and Polynomials in z are always analytic over entire z-plane
For analytic functions the integral over closed contour is always 0.
(i) is always analytic function
(i) is always analytic function
(ii) even integer is always analytic function
(iii) is also analytic function in contour .
(iv)
Poles:
Hence, no poles of lies in contour . Thus, is analytic in contour .
Hence, option (A) am (B) are correct.
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 signals , and are bounded for all bounded inputs
There exists a bounded input such that at least one of the signals , , and is unbounded
There exists a bounded input such that the signals , and are unbounded
The signals , and are unbounded for all bounded inputs
Option (A), If bounded input
Say unit step function,
False, statement.
Option (B), As check above for bounded i/p of unit step function, are unbounded.
Hence, True statement.
Option (C), if u(t) is bounded, then all poles of will lies in LHS of -plane.
Thus, all poles of will also lies in LHS of s-plane.
For bounded, always bounded.
False, statement.
option (D), As check above, if always bounded,
then also always bounded.
Hence, option (b) is correct.
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?
The entropy of a discrete random variable is given as,
The entropy function will thus be:
For maximum entropy,
Thus, at is near to Maximum.
Alternatively,
Hence, option (b) & (c) are correct.
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?
for all
is periodic with a fundamental period
for all
is periodic with a fundamental period
where, is fundamental Time period of
and is, fundamental Time period of .
Also,
Since, .
Thus, for all
& y(t) is periodic with fundamental period
Hence, option (a) & (d) are correct
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).
Option (A):
Option (B):
Option (C):
Option (D):
Hence, option (B) is correct.
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?
During +ve half cycle of ON & OFF
During -ve half cycle of ON & OFF
And as reverses, direction also reverses)
Hence, Option (A) and (D) is correct.
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).
Given that is a random variable, denote the sum of the outcomes when two fair dice is rolled.
Sum of two dice
Sum | Possible Pairs | No. of ways |
2 | 1 | |
3 | 2 | |
4 | 3 | |
5 | 4 | |
6 | 5 | |
7 | 6 | |
8 | 5 | |
9 | 4 | |
10 | 3 | |
11 | 2 | |
12 | 1 |
X | 2 | 3 | 4 | 5 | 6 |
X | 7 | 8 | 9 | 10 | 11 | 12 |
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 .
Given:
Let
For
should also minimum.
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 mutual information I between two random variables and is defined as:
where Joint Probability Mass function: of X and Y.
, Marginal Probability of
Marginal Probability of
Given:
Marginal Probabilities and :
The marginal probability of and :
The marginal probability of and :
= 0+0+0+0
= 0
Alternate Solution:
are Bernauli Random variables,
they are independent.
Thus,
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 ________ .
.
for ideal diode to be forward biased, volt.
for
Also,
Angle for which diode is FB doing one period of input
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 ________
Given:
For FF, set-up time
For FF-1,
For
Time required for clock,
Maximum clock frequency,
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 .)
Given: forward bias current,
Reverse saturation current,
Thermal Voltage,
for -diode, ideality factor
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 ________.
Given:
Reflection coefficient,
Average Power delivered to Load,
Alternate Solution:
Given:
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 _________.
Given: (Tesla), Area
EMF induced in the loop,
Magnitude of current
