GATE 2024 Electronics & Communication

The general form of the complementary function of a differential equation is given by $y(t) = (At + B)e^{-2t}$, where $A$ and $B$ are real constants determined by the initial condi...

In the context of Bode magnitude plots, 40 dB/decade is the same as ______.

For the Boolean function $F(A, B, C, D) = \sum m(0,2,5,7,8,10,12,13,14,15)$, the essential prime implicants are _________.

Let $\rho(x, y, z, t)$ and $u(x, y, z, t)$ represent density and velocity, respectively, at a point $(x, y, z)$ and time $t$. Assume $\frac{\partial \rho }{\partial t}$ is continuo...

A machine has a 32-bit architecture with 1-word long instructions. It has 24 registers and supports an instruction set of size 40. Each instruction has five distinct fields, namely...

In a number system of base $r$, the equation $x^2 - 12x + 37 = 0$ has $x = 8$ as one of its solutions. The value of $r$ is _______.

Let $\mathbb{R}$ and $\mathbb{R}^3$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$ for which the set of vectors...

Suppose $X$ and $Y$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0,1]$. The probability that $X \geq Y$ is _______...

Consider the Earth to be a perfect sphere of radius $R$. Then the surface area of the region, enclosed by the 60°N latitude circle, that contains the north pole in its interior is...

A 4-bit priority encoder has inputs $D_3, D_2, D_1,$ and $D_0$ in descending order of priority. The two-bit output $AB$ is generated as 00, 01, 10, and 11 corresponding to inputs $...

Let $z$ be a complex variable. If $f(z)=\frac{\sin(\pi z)}{z^{2}(z-2)}$ and $C$ is the circle in the complex plane with $|z|=3$ then $\oint\limits_{C} f(z)dz$ is _______.

A full scale sinusoidal signal is applied to a 10-bit ADC. The fundamental signal component in the ADC output has a normalized power of 1 W, and the total noise and distortion norm...

Which of the following statements is/are true for a BJT with respect to its DC current gain $\beta$?

Consider a system $S$ represented in state space as $$\frac{dx}{dt} = \begin{bmatrix} 0 & -2 \\ 1 & -3 \end{bmatrix}x + \begin{bmatrix} 1 \\ 0 \end{bmatrix}r , \quad y = \begin{bma...

Let $F_1$, $F_2$, and $F_3$ be functions of $(x, y, z)$. Suppose that for every given pair of points A and B in space, the line integral $\int\limits_C (F_1 dx + F_2 dy + F_3 dz)$...

Consider the matrix $\begin{bmatrix}1 & k \\ 2 & 1\end{bmatrix}$, where $k$ is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?

An NMOS transistor operating in the linear region has $I_{D}$ of 5 $\mu$A at $V_{DS}$ of 0.1 V. Keeping $V_{GS}$ constant, the $V_{DS}$ is increased to 1.5 V. Given that $\mu_{n}C_...

A digital communication system transmits through a noiseless bandlimited channel $[-W, W]$. The received signal $z(t)$ at the output of the receiving filter is given by $z(t) = \su...

Let $\hat{i}$ and $\hat{j}$ be the unit vectors along $x$ and $y$ axes, respectively and let $A$ be a positive constant. Which one of the following statements is true for the vecto...

A white Gaussian noise $w(t)$ with zero mean and power spectral density $\frac{N_0}{2}$, when applied to a first-order RC low pass filter produces an output $n(t)$. At a particular...

A uniform plane wave with electric field $\vec{E}(x)=A_y \hat{a}_y e^{-j \frac{2 \pi x}{3}} \mathrm{~V} / \mathrm{m}$ is travelling in the air (relative permittivity, $\dot{o}_r=1$...

An amplitude modulator has output (in Volts) $$s(t) = A \cos(400 \pi t) + B \cos(360 \pi t) + B \cos(440 \pi t)$$. The carrier power normalized to $1\Omega$ resistance is 50 Watts....

A source transmits symbols from an alphabet of size 16. The value of maximum achievable entropy (in bits) is _______ .

A source transmits a symbol $s$, taken from $\\{-4, 0, 4\\}$ with equal probability, over an additive white Gaussian noise channel. The received noisy symbol $r$ is given by $r = s...

Let $X(t) = A\cos(2\pi f_0 t+\theta)$ be a random process, where amplitude $A$ and phase $\theta$ are independent of each other, and are uniformly distributed in the intervals $[-2...

A causal and stable LTI system with impulse response h(t) produces an output y(t) for an input signal x(t) . A signal x(0.5t) is applied to another causal and stable LTI system wit...

For non-degenerately doped n-type silicon, which one of the following plots represents the temperature ($T$) dependence of free electron concentration ($n$)?

For a causal discrete-time LTI system with transfer function $H(z) = \frac{2z^2 + 3}{\left(z + \frac{1}{3}\right)\left(z - \frac{1}{3}\right)}$ which of the following statements is...

A continuous time signal $x(t) = 2 \cos(8 \pi t + \frac{\pi}{3})$ is sampled at a rate of 15 Hz. The sampled signal $x_s(t)$ when passed through an LTI system with impulse response...

The radian frequency value(s) for which the discrete time sinusoidal signal $x[n] = A \cos(\Omega n + \pi/3)$ has a period of 40 is/are __.

The relationship between any N-length sequence $x[n]$ and its corresponding N-point discrete Fourier transform $X[k]$ is defined as $X[k] = \mathcal{F}\{x[n]\}$. Another sequence $...

Consider a MOS capacitor made with p-type silicon. It has an oxide thickness of 100 nm, a fixed positive oxide charge of $10^{-8}$ C/cm 2 at the oxide-silicon interface, and a meta...

A non-degenerate n-type semiconductor has 5 % neutral dopant atoms. Its Fermi level is located at 0.25 eV below the conduction band ($E_C$) and the donor energy level ($E_D$) has a...

The photocurrent of a PN junction diode solar cell is 1 mA. The voltage corresponding to its maximum power point is 0.3 V. If the thermal voltage is 30 mV, the reverse saturation c...

Two identical sheets A and B, of dimensions 24 cm × 16 cm, can be folded into half using two distinct operations, FO1 or FO2. In FO1, the axis of folding remains parallel to the in...

Sequence the following sentences (P, Q, R, S) in a coherent passage: P: Shifu’s student exclaimed, “Why do you run since the bull is an illusion?” Q: Shifu said, “Surely my running...

The greatest prime factor of $(3^{199} - 3^{196})$ is

For a real number $x > 1$ , $$ \frac{1}{\log_{2}x} + \frac{1}{\log_{3}x} + \frac{1}{\log_{4}x} = 1$$ The value of $x$ is

Five years ago, the ratio of Aman’s age to his father’s age was 1:4, and five years from now, the ratio will be 2:5. What was his father’s age when Aman was born?

P, Q, R, S, and T have launched a new startup. Two of them are siblings. The office of the startup has just three rooms. All of them agree that the siblings should not share the sa...

If ‘→’ denotes increasing order of intensity, then the meaning of the words [charm → enamor → bewitch] is analogous to [bored → _______ → weary]. Which one of the given options is...

A lossless transmission line with characteristic impedance $Z_0 = 50 \Omega$ is terminated with an unknown load. The magnitude of the reflection coefficient is $|\Gamma| = 0.6$. As...

The information bit sequence {1 1 1 0 1 0 1 0 1} is to be transmitted by encoding with Cyclic Redundancy Check 4 (CRC-4) code, for which the generator polynomial is $C(x) = x^4 + x...

If '→' denotes increasing order of intensity, then the meaning of the words [charm → enamor → bewitch] is analogous to [bored → _______ → weary]. Which one of the given options is...

P, Q, R, S, and T have launched a new startup. Two of them are siblings. The office of the startup has just three rooms. All of them agree that the siblings should not share the sa...

Five years ago, the ratio of Aman's age to his father's age was 1:4, and five years from now, the ratio will be 2:5. What was his father's age when Aman was born?

For a real number x > 1, $\frac{1}{\log_2 x} + \frac{1}{\log_3 x} + \frac{1}{\log_4 x} = 1$. The value of x is

The greatest prime factor of (3^199 – 3^196) is

Sequence the following sentences (P, Q, R, S) in a coherent passage: P: Shifu's student exclaimed, "Why do you run since the bull is an illusion?" Q: Shifu said, "Surely my running...

Four identical cylindrical chalk-sticks, each of radius r = 0.5 cm and length l = 10 cm, are bound tightly together using a duct tape as shown in the following figure. The width of...

The bar chart shows the data for the percentage of population falling into different categories based on Body Mass Index (BMI) in 2003 and 2023. Based on the data provided, which o...

Examples of mirror and water reflections are shown in the figures below: An object appears as the following image after first reflecting in a mirror and then reflecting on water. T...

Two identical sheets A and B, of dimensions 24 cm x 16 cm, can be folded into half using two distinct operations, FO1 or FO2. In FO1, the axis of folding remains parallel to the in...

The general form of the complementary function of a differential equation is given by y(t) = (At+B)e^(-2t), where A and B are real constants determined by the initial condition. Th...

In the context of Bode magnitude plots, 40 dB/decade is the same as ________.

In the feedback control system shown in the figure below $G(s) = \frac{6}{s(s+1)(s+2)}$. $R(s), Y(s)$, and $E(s)$ are the Laplace transforms of $r(t), y(t)$, and $e(t)$, respective...

A digital communication system transmits through a noiseless bandlimited channel [-W W]. The received signal z(t) at the output of the receiving filter is given by $z(t) = \sum_n b...

Consider a lossless transmission line terminated with a short circuit as shown in the figure below. As one moves towards the generator from the load, the normalized impedances $z_{...

Let $\hat{i}$ and $\hat{j}$ be the unit vectors along x and y axes, respectively and let A be a positive constant. Which one of the following statements is true for the vector fiel...

In the circuit below, assume that the long channel NMOS transistor is biased in saturation. The small signal trans-conductance of the transistor is $g_m$. Neglect body effect, chan...

For the closed loop amplifier circuit shown below, the magnitude of open loop low frequency small signal voltage gain is 40. All the transistors are biased in saturation. The curre...

For the Boolean function F(A, B, C, D) = ∑m(0,2,5,7,8,10,12,13,14,15), the essential prime implicants are ____________.

A white Gaussian noise w(t) with zero mean and power spectral density $\frac{N_0}{2}$, when applied to a first-order RC low pass filter produces an output n(t). At a particular tim...

A causal and stable LTI system with impulse response h(t) produces an output y(t) for an input signal x(t). A signal x(0.5t) is applied to another causal and stable LTI system with...

For non-degenerately doped n-type silicon, which one of the following plots represents the temperature (T) dependence of free electron concentration (n)?

In the circuit shown, the n: 1 step-down transformer and the diodes are ideal. The diodes have no voltage drop in forward biased condition. If the input voltage (in Volts) is $V_s(...

For a causal discrete-time LTI system with transfer function $H(z) = \frac{2z^2+3}{(z+\frac{1}{3})(z-\frac{1}{3})}$ which of the following statements is/are true?

Let $\rho(x, y, z, t)$ and $u(x, y, z, t)$ represent density and velocity, respectively, at a point $(x, y, z)$ and time $t$. Assume $\frac{\partial \rho}{\partial t}$ is continuou...

The free electron concentration profile n(x) in a doped semiconductor at equilibrium is shown in the figure, where the points A, B, and C mark three different positions. Which of t...

A machine has a 32-bit architecture with 1-word long instructions. It has 24 registers and supports an instruction set of size 40. Each instruction has five distinct fields, namely...

An amplitude modulator has output (in Volts) s(t) = A cos(400πt) + B cos(360πt) + B cos(440πt). The carrier power normalized to 1Ω resistance is 50 Watts. The ratio of the total si...

In a number system of base r, the equation x² - 12x + 37 = 0 has x = 8 as one of its solutions. The value of r is _________.

Let ℝ and ℝ³ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of α for which the set of vectors {[2 -3 α], [3 -1 3], [1 -5 7]}...

In the given circuit, the current Ix (in mA) is _________.

In the circuit given below, the switch S was kept open for a sufficiently long time and is closed at time t = 0. The time constant (in seconds) of the circuit for t > 0 is ________...

Suppose X and Y are independent and identically distributed random variables that are distributed uniformly in the interval [0,1]. The probability that X≥Y is _________.

A source transmits symbols from an alphabet of size 16. The value of maximum achievable entropy (in bits) is _________.

As shown in the circuit, the initial voltage across the capacitor is 10 V, with the switch being open. The switch is then closed at $t = 0$. The total energy dissipated in the idea...

Consider the Earth to be a perfect sphere of radius R. Then the surface area of the region, enclosed by the 60°N latitude circle, that contains the north pole in its interior is __...

Consider a unity negative feedback control system with forward path gain G(s) = K / ((s+1)(s+2)(s+3)) as shown. The impulse response of the closed-loop system decays faster than e⁻...

A satellite attitude control system, as shown below, has a plant with transfer function $G(s) = \frac{1}{s^2}$ cascaded with a compensator $C(s) = \frac{K(s+\alpha)}{s+4}$, where K...

A uniform plane wave with electric field $\vec{E}(x) = A_y\hat{a}_y e^{-j\frac{2\pi x}{3}}$ V/m is travelling in the air (relative permittivity, $\epsilon_r = 1$ and relative perme...

A 4-bit priority encoder has inputs $D_3, D_2, D_1$, and $D_0$ in descending order of priority. The two-bit output AB is generated as 00, 01, 10, and 11 corresponding to inputs $D_...

The propagation delay of the 2×1 MUX shown in the circuit is 10 ns. Consider the propagation delay of the inverter as 0 ns. If S is set to 1 then the output Y is _________.

The sequence of states (Q1Q0) of the given synchronous sequential circuit is

Let z be a complex variable. If $f(z) = \frac{\sin(\pi z)}{z^2(z-2)}$ and C is the circle in the complex plane with $|z|=3$ then $\oint_C f(z)dz$ is _________.

Consider two continuous time signals $x(t)$ and $y(t)$ as shown below If $X(f)$ denotes the Fourier transform of $x(t)$, then the Fourier transform of $y(t)$ is ________.

A source transmits a symbol s, taken from {-4,0,4} with equal probability, over an additive white Gaussian noise channel. The received noisy symbol r is given by r = s + w, where t...

A full scale sinusoidal signal is applied to a 10-bit ADC. The fundamental signal component in the ADC output has a normalized power of 1 W, and the total noise and distortion norm...

The information bit sequence {111010101} is to be transmitted by encoding with Cyclic Redundancy Check 4 (CRC-4) code, for which the generator polynomial is C(x) = x⁴ + x + 1. The...

A continuous time signal x(t) = 2 cos(8πt + π/3) is sampled at a rate of 15 Hz. The sampled signal x_s(t) when passed through an LTI system with impulse response h(t) = (sin 2πt /...

The opamps in the circuit shown are ideal, but have saturation voltages of ±10 V. Assume that the initial inductor current is 0 A. The input voltage (Vi) is a triangular signal wit...

In the circuit below, the opamp is ideal. If the circuit is to show sustained oscillations, the respective values of R₁ and the corresponding frequency of oscillation are _________...

In the circuit shown below, the transistors M₁ and M₂ are biased in saturation. Their small signal transconductances are g_m1 and g_m2 respectively. Neglect body effect, channel le...

Which of the following statements is/are true for a BJT with respect to its DC current gain β?

Consider a system $S$ represented in state space as $\frac{dx}{dt} = \begin{bmatrix} 0 & -2 \\ 1 & -3 \end{bmatrix}x + \begin{bmatrix} 1 \\ 0 \end{bmatrix}r, y=[2 \ -5]x$. Which of...

Let F1, F2, and F3 be functions of (x, y, z). Suppose that for every given pair of points A and B in space, the line integral ∫(F₁dx+ F₂dy + F₃dz) evaluates to the same value along...

Consider the matrix $\begin{bmatrix} 1 & k \\ 2 & 1 \end{bmatrix}$, where k is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?

The radian frequency value(s) for which the discrete time sinusoidal signal $x[n] = A \cos(\Omega n + \pi/3)$ has a period of 40 is/are ________.

Let X (t) = A cos(2π f₀t+θ) be a random process, where amplitude A and phase θ are independent of each other, and are uniformly distributed in the intervals [-2,2] and [0,2π], resp...

A lossless transmission line with characteristic impedance Z₀ = 50 Ω is terminated with an unknown load. The magnitude of the reflection co-efficient is |Γ| = 0.6. As one moves tow...

The relationship between any N-length sequence x[n] and its corresponding N-point discrete Fourier transform X[k] is defined as X[k] = F{x[n]}. Another sequence y[n] is formed as b...

For the two port network shown below, the value of the Y21 parameter (in Siemens) is _________.

Consider a MOS capacitor made with p-type silicon. It has an oxide thickness of 100 nm, a fixed positive oxide charge of 10⁻⁸ C/cm² at the oxide-silicon interface, and a metal work...

In the network shown below, maximum power is to be transferred to the load RL. The value of RL (in Ω) is _________.

A non-degenerate n-type semiconductor has 5% neutral dopant atoms. Its Fermi level is located at 0.25 eV below the conduction band (Ec) and the donor energy level (ED) has a degene...

An NMOS transistor operating in the linear region has IDS of 5 µA at VDS of 0.1 V. Keeping VGS constant, the VDS is increased to 1.5 V. Given that \mu_n C_{ox} \frac{W}{L} = 50 \mu...

The photocurrent of a PN junction diode solar cell is 1 mA. The voltage corresponding to its maximum power point is 0.3 V. If the thermal voltage is 30 mV, the reverse saturation c...