All 65 questions of GATE 2022 EC

Mr. X speaks ________ Japanese ________ Chinese.

A sum of money is to be distributed among P, Q, R, and S in the proportion 5 : 2 : 4 : 3, respectively.

If R gets ₹ 1000 more than S, what is the share of Q (in ₹)?

A trapezium has vertices marked as P, Q, R and S (in that order anticlockwise). The side PQ is parallel to side SR.

Further, it is given that, PQ = 11 cm, QR = 4 cm, RS = 6 cm and SP = 3 cm.

What is the shortest distance between PQ and SR (in cm)?

The figure shows a grid formed by a collection of unit squares. The unshaded unit square in the grid represents a hole.

What is the maximum number of squares without a “hole in the interior” that can be formed within the 4 × 4 grid using the unit squares as building blocks?

An art gallery engages a security guard to ensure that the items displayed are protected. The diagram below represents the plan of the gallery where the boundary walls are opaque. The location the security guard posted is identified such that all the inner space (shaded region in the plan) of the gallery is within the line of sight of the security guard.

If the security guard does not move around the posted location and has a 360° view, which one of the following correctly represents the set of ALL possible locations among the locations P, Q, R and S, where the security guard can be posted to watch over the entire inner space of the gallery.

Mosquitoes pose a threat to human health. Controlling mosquitoes using chemicals may have undesired consequences. In Florida, authorities have used genetically modified mosquitoes to control the overall mosquito population. It remains to be seen if this novel approach has unforeseen consequences.

Which one of the following is the correct logical inference based on the information in the above passage?

Consider the following inequalities.

(i) $2x - 1 > 7$ (ii) $2x - 9 < 1$

Which one of the following expressions below satisfies the above two inequalities?

Four points P(0, 1), Q(0, −3), R(−2, −1), and S(2, −1) represent the vertices of a quadrilateral.

What is the area enclosed by the quadrilateral?

In a class of five students P, Q, R, S and T, only one student is known to have copied in the exam. The disciplinary committee has investigated the situation and recorded the statements from the students as given below.

Statement of P: R has copied in the exam. Statement of Q: S has copied in the exam. Statement of R: P did not copy in the exam. Statement of S: Only one of us is telling the truth. Statement of T: R is telling the truth.

The investigating team had authentic information that S never lies.

Based on the information given above, the person who has copied in the exam is

Consider the following square with the four corners and the center marked as P, Q, R, S and T respectively.

Let X, Y and Z represent the following operations:

X: rotation of the square by 180 degree with respect to the S-Q axis. Y: rotation of the square by 180 degree with respect to the P-R axis. Z: rotation of the square by 90 degree clockwise with respect to the axis perpendicular, going into the screen and passing through the point T.

Consider the following three distinct sequences of operation (which are applied in the left to right order).

(1) XYZZ (2) XY (3) ZZZZ

Which one of the following statements is correct as per the information provided above?

Consider the two-dimensional vector field $\vec{F}(x,y) = x\,\vec{i} + y\,\vec{j}$, where $\vec{i}$ and $\vec{j}$ denote the unit vectors along the $x$-axis and the $y$-axis, respectively. A contour $C$ in the $x$-$y$ 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

$\oint_C \vec{F}(x,y)\cdot(dx\,\vec{i} + dy\,\vec{j})$

is ________.

Consider a system of linear equations $Ax = b$, where

$A = \begin{bmatrix} 1 & -\sqrt{2} & 3 \\ -1 & \sqrt{2} & -3 \end{bmatrix}, \quad b = \begin{bmatrix} 1 \\ 3 \end{bmatrix}.$

This system of equations admits ________.

The current $I$ in the circuit shown is ________.

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

The Fourier transform $X(j\omega)$ of the signal

$x(t) = \frac{t}{(1 + t^2)^2}$

is ________.

Consider a long rectangular bar of direct bandgap $p$-type semiconductor. The equilibrium hole density is $10^{17}$ cm⁻³ and the intrinsic carrier concentration is $10^{10}$ cm⁻³. Electron and hole diffusion lengths are 2 µm and 1 µm, respectively.

The left side of the bar ($x = 0$) is uniformly illuminated with a laser having photon energy greater than the bandgap of the semiconductor. Excess electron-hole pairs are generated ONLY at $x = 0$ because of the laser. The steady state electron density at $x = 0$ is $10^{14}$ cm⁻³ 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 $x = 2$ µm, is ________.

In a non-degenerate bulk semiconductor with electron density $n = 10^{16}$ cm⁻³, the value of $E_C - E_{Fn} = 200$ meV, where $E_C$ and $E_{Fn}$ denote the bottom of the conduction band energy and electron Fermi level energy, respectively. Assume thermal voltage as 26 meV and the intrinsic carrier concentration is $10^{10}$ cm⁻³. For $n = 0.5 \times 10^{16}$ cm⁻³, the closest approximation of the value of $(E_C - E_{Fn})$, among the given options, is ________.

Consider the CMOS circuit shown in the figure (substrates are connected to their respective sources). The gate width ($W$) to gate length ($L$) ratios $\left(\frac{W}{L}\right)$ of the transistors are as shown. Both the transistors have the same gate oxide capacitance per unit area. For the $p$MOSFET, the threshold voltage is −1 V and the mobility of holes is 40 $\frac{\text{cm}^2}{\text{V.s}}$. For the $n$MOSFET, the threshold voltage is 1 V and the mobility of electrons is 300 $\frac{\text{cm}^2}{\text{V.s}}$. The steady state output voltage $V_O$ is ________.

Consider the 2-bit multiplexer (MUX) shown in the figure. For OUTPUT to be the XOR of C and D, the values for $A_0$, $A_1$, $A_2$, and $A_3$ are ________.

The ideal long channel $n$MOSFET and $p$MOSFET devices shown in the circuits have threshold voltages of 1 V and −1 V, 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 ________.

Consider a closed-loop control system with unity negative feedback and $KG(s)$ in the forward path, where the gain $K = 2$. The complete Nyquist plot of the transfer function $G(s)$ is shown in the figure. Note that the Nyquist contour has been chosen to have the clockwise sense. Assume $G(s)$ 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 ________.

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

Select the transfer function $G(s)$ that results in the root-locus plot of the closed-loop system as shown in the figure.

The frequency response $H(f)$ 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 $-\alpha \le f \le \alpha$.

Statement II: For any wide-sense stationary input process with power spectral density $S_X(f)$, the output power spectral density $S_Y(f)$ obeys $S_Y(f) = S_X(f)$ for $-\alpha \le f \le \alpha$.

Which one of the following combinations is true?

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 $2\sin(t + \pi/2)$. The displacement current (in Amperes) through the capacitor is ________.

Consider the following partial differential equation (PDE)

$a\frac{\partial^2 f(x,y)}{\partial x^2} + b\frac{\partial^2 f(x,y)}{\partial y^2} = f(x,y),$

where $a$ and $b$ are distinct positive real numbers. Select the combination(s) of values of the real parameters $\xi$ and $\eta$ such that $f(x,y) = e^{(\xi x + \eta y)}$ is a solution of the given PDE.

An ideal OPAMP circuit with a sinusoidal input is shown in the figure. The 3 dB frequency is the frequency at which the magnitude of the voltage gain decreases by 3 dB from the maximum value. Which of the options is/are correct?

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

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

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 the following wave equation,

$\frac{\partial^2 f(x,t)}{\partial t^2} = 10000\,\frac{\partial^2 f(x,t)}{\partial x^2}$

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

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).

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

$\oint_C \frac{2^z}{z^2 - 1}dz = -i\pi A,$

where $i = \sqrt{-1}$, then the value of $A$ is ______ (rounded off to two decimal places).

Let $x_1(t) = e^{-t}u(t)$ and $x_2(t) = u(t) - u(t-2)$, where $u(\cdot)$ denotes the unit step function.

If $y(t)$ denotes the convolution of $x_1(t)$ and $x_2(t)$, then $\lim_{t\to\infty} y(t) =$ ________ (rounded off to one decimal place).

An ideal MOS capacitor ($p$-type semiconductor) is shown in the figure. The MOS capacitor is under strong inversion with $V_G = 2$ V. The corresponding inversion charge density ($Q_{IN}$) is 2.2 µC/cm². Assume oxide capacitance per unit area as $C_{OX} = 1.7$ µF/cm². For $V_G = 4$ V, the value of $Q_{IN}$ is ______ µC/cm² (rounded off to one decimal place).

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).

The function $f(x) = 8\log_e x - x^2 + 3$ attains its minimum over the interval $[1, e]$ at $x =$ ________.

(Here $\log_e x$ is the natural logarithm of $x$.)

Let $\alpha, \beta$ be two non-zero real numbers and $v_1, v_2$ be two non-zero real vectors of size 3 × 1. Suppose that $v_1$ and $v_2$ satisfy $v_1^Tv_2 = 0$, $v_1^Tv_1 = 1$, and $v_2^Tv_2 = 1$. Let $A$ be the 3 × 3 matrix given by:

$A = \alpha v_1v_1^T + \beta v_2v_2^T$

The eigenvalues of $A$ are ________.

For the circuit shown, the locus of the impedance $Z(j\omega)$ is plotted as $\omega$ increases from zero to infinity. The values of $R_1$ and $R_2$ are:

Consider the circuit shown in the figure with input $V(t)$ in volts. The sinusoidal steady state current $I(t)$ flowing through the circuit is shown graphically (where $t$ is in seconds). The circuit element $Z$ can be ________.

Consider an ideal long channel $n$MOSFET (enhancement-mode) with gate length 10 µm and width 100 µm. The product of electron mobility ($\mu_n$) and oxide capacitance per unit area ($C_{OX}$) is $\mu_nC_{OX} = 1$ mA/V². The threshold voltage of the transistor is 1 V. For a gate-to-source voltage $V_{GS} = [2 - \sin(2t)]$ V and drain-to-source voltage $V_{DS} = 1$ V (substrate connected to the source), the maximum value of the drain-to-source current is ________.

For the following circuit with an ideal OPAMP, the difference between the maximum and the minimum values of the capacitor voltage ($V_c$) is ________.

A circuit with an ideal OPAMP is shown. The Bode plot for the magnitude (in dB) of the gain transfer function $(A_V(j\omega) = V_{out}(j\omega)/V_{in}(j\omega))$ of the circuit is also provided (here, $\omega$ is the angular frequency in rad/s). The values of $R$ and $C$ are ________.

For the circuit shown, the clock frequency is $f_0$ and the duty cycle is 25%. For the signal at the Q output of the Flip-Flop, ________.

Consider an even polynomial $p(s)$ given by

$p(s) = s^4 + 5s^2 + 4 + K,$

where $K$ is an unknown real parameter. The complete range of $K$ for which $p(s)$ has all its roots on the imaginary axis is ________.

Consider the following series:

$\sum_{n=1}^{\infty}\frac{n^d}{c^n}$

For which of the following combinations of $c, d$ values does this series converge?

The outputs of four systems ($S_1$, $S_2$, $S_3$, and $S_4$) corresponding to the input signal $\sin(t)$, for all time $t$, are shown in the figure.

Based on the given information, which of the four systems is/are definitely NOT LTI (linear and time-invariant)?

Select the CORRECT statement(s) regarding semiconductor devices.

A state transition diagram with states $A, B$, and $C$, and transition probabilities $p_1, p_2, \dots, p_7$ is shown in the figure (e.g., $p_1$ denotes the probability of transition from state $A$ to $B$). For this state diagram, select the statement(s) which is/are universally true.

Consider a Boolean gate (D) where the output $Y$ is related to the inputs $A$ and $B$ as, $Y = A + \overline{B}$, where $+$ denotes logical OR operation. The Boolean inputs ‘0’ and ‘1’ are also available separately. Using instances of only D gates and inputs ‘0’ and ‘1’, ________ (select the correct option(s)).

Two linear time-invariant systems with transfer functions

$G_1(s) = \frac{10}{s^2 + s + 1} \quad \text{and} \quad G_2(s) = \frac{10}{s^2 + s\sqrt{10} + 10}$

have unit step responses $y_1(t)$ and $y_2(t)$, respectively. Which of the following statements is/are true?

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

$H_{pe}(\omega) = 1 + \frac{j\omega}{\omega_0},$

where $\omega_0 = 10^4$ rad/sec.

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

A waveguide consists of two infinite parallel plates (perfect conductors) at a separation of $10^{-4}$ cm, with air as the dielectric. Assume the speed of light in air to be $3 \times 10^8$ m/s. The frequency/frequencies of TM waves which can propagate in this waveguide is/are ________.

The value of the integral

$\iint_D 3(x^2 + y^2)\,dx\,dy,$

where $D$ is the shaded triangular region shown in the diagram, is ______ (rounded off to the nearest integer).

A linear 2-port network is shown in Fig. (a). An ideal DC voltage source of 10 V is connected across Port 1. A variable resistance $R$ is connected across Port 2. As $R$ is varied, the measured voltage and current at Port 2 is shown in Fig. (b) as a $V_2$ versus $-I_2$ plot. Note that for $V_2 = 5$ V, $I_2 = 0$ mA, and for $V_2 = 4$ V, $I_2 = -4$ mA.

When the variable resistance $R$ at Port 2 is replaced by the load shown in Fig. (c), the current $I_2$ is ________ mA (rounded off to one decimal place).

For a vector $\bar{x} = [x[0], x[1], \dots, x[7]]$, the 8-point discrete Fourier transform (DFT) is denoted by $\bar{X} = DFT(\bar{x}) = [X[0], X[1], \dots, X[7]]$, where

$X[k] = \sum_{n=0}^{7} x[n]\exp\left(-j\frac{2\pi}{8}nk\right).$

Here, $j = \sqrt{-1}$. If $\bar{x} = [1, 0, 0, 0, 2, 0, 0, 0]$ and $\bar{y} = DFT(DFT(\bar{x}))$, then the value of $y[0]$ is ________ (rounded off to one decimal place).

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 $x = \pm 2l_n$, where $l_n = 10^{-4}$ cm is the diffusion length of electrons. Assume electronic charge, $q = -1.6 \times 10^{-19}$ C. The profiles of photo-generation rate of carriers and the recombination rate of excess minority carriers ($R$) are shown. Under these conditions, the magnitude of the current density due to the photo-generated electrons at $x = +2l_n$ is ________ mA/cm² (rounded off to two decimal places).

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 $\frac{\partial V_{out}}{\partial V_{in}}$ is ________ (rounded off to two decimal places).

Consider the circuit shown with an ideal OPAMP. The output voltage $V_0$ is ________ V (rounded off to two decimal places).

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

The block diagram of a closed-loop control system is shown in the figure. $R(s)$, $Y(s)$, and $D(s)$ are the Laplace transforms of the time-domain signals $r(t)$, $y(t)$, and $d(t)$, respectively. Let the error signal be defined as $e(t) = r(t) - y(t)$. Assuming the reference input $r(t) = 0$ for all $t$, the steady-state error $e(\infty)$, due to a unit step disturbance $d(t)$, is ________ (rounded off to two decimal places).

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 $\alpha$ lies in the interval [0.25, 1]. The value of $\alpha$ for which the capacity of this channel is maximized, is ________ (rounded off to two decimal places).

Consider communication over a memoryless binary symmetric channel using a (7, 4) Hamming code. Each transmitted bit is received correctly with probability $(1 - \epsilon)$, and flipped with probability $\epsilon$. 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 $\epsilon = 0.1$, the probability that a transmitted codeword is decoded correctly is ________ (rounded off to two decimal places).

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

$f_{Y|x_A}(y) = e^{-(y+1)}u(y+1),$ $f_{Y|x_B}(y) = e^{(y-1)}(1 - u(y-1)),$

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

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

Consider a 1 bit quantizer that maps positive samples to value $\alpha$ and others to value $\beta$. If $\alpha^*$ and $\beta^*$ are the respective choices for $\alpha$ and $\beta$ that minimize the mean square quantization error, then $(\alpha^* - \beta^*) =$ ________ (rounded off to two decimal places).

In an electrostatic field, the electric displacement density vector, $\vec{D}$, is given by

$\vec{D}(x,y,z) = (x^3\,\vec{i} + y^3\,\vec{j} + xy^2\,\vec{k})\ \text{C/m}^2,$

where $\vec{i}, \vec{j}, \vec{k}$ are the unit vectors along $x$-axis, $y$-axis, and $z$-axis, respectively. Consider a cubical region $R$ centered at the origin with each side of length 1 m, and vertices at (±0.5 m, ±0.5 m, ±0.5 m). The electric charge enclosed within $R$ is ________ C (rounded off to two decimal places).