GATE Electrical Engineering

Consider a linear arrangement of seven bulbs, each of which can be in the ON or OFF states. The initial configuration of the bulbs is shown in the figure. In every Step, the states...

Consider the infinite-length, discrete-time sequence x[n] = 0.9|n|, where n is an integer. The region of convergence of its Z-transform X(Z) is given by: (Note: z is a complex vari...

Let $x_c(t)$ be any continuous-time periodic signal with period $T$. It is sampled uniformly with a sampling period $T_s$ where $T_s \neq T$, resulting in the discrete sequence $x[...

The Laplace transform of the step response of a system is given by Y(s) = 100 / (s(s + 100)) The rise time is defined as the time taken for the response to go from 0.1 to 0.9 of it...

Consider the following differential equation: $t^2 \frac{d^2y}{dt^2} + 7t \frac{dy}{dt} + 8ty = 10 \sin(t)$ Which one of the following options is correct?

A uniform ring charge of radius R carries a total charge Q. Which one of the following options correctly quantifies the magnitude of the force on a point charge of strength q kept...

A positive point charge with velocity $\vec{v} = 5 \hat{x}$ enters a region having electric field $\vec{E} = 4 \hat{y}$ and magnetic field $\vec{B} = -6 \hat{z}$. Which one of the...

A 15 kVA, 1100 V/220 V, single-phase two-winding transformer is configured as a 1.32 kV/1.1 kV autotransformer. What will be the rating of the autotransformer?

Consider a power system with N buses, of which P are generator buses and the remaining Q are load buses (where there is no generation). Assume that there are no reactive power-limi...

The initial three-phase voltage phasors ($\bar{V}_A$, $\bar{V}_B$, and $\bar{V}_C$) at a bus of a power network are as shown in Case-1. Due to a disturbance, the bus voltage phasor...

A certain application requires power at a frequency of 16.67 Hz, while the available grid frequency is 50 Hz. A 3-phase synchronous motor connected to the 50 Hz, 3-phase grid, driv...

Which one of the following options is correct regarding the typical double-squirrel-cage structure used in induction motors?

The figure shows the single-line diagram of a synchronous generator delivering P = 50 MW of power at unity power factor to an infinite bus. I_s denotes the stator current phasor. I...

A circuit with ideal elements is shown. Which one of the following options correctly identifies all the linear elements in the circuit?

For the circuit shown, which one of the following options correctly identifies the Thevenin's equivalent parameters between nodes Y and Z?

Refer to the four circuits shown. Which one of the following options for k1, k2, k3, and k4 makes all of them realizable?

A single-phase voltage source v_s = 325 sin(2π50t) V delivers a current, i = 12 sin(2π50t) + 9 sin(2π150t) A to a load. The load power factor, correct up to two decimal places, is:

The figure shows a straight-line approximation for the forward characteristics of a power diode. A continuous on-state current of 15 A is flowing through the diode. What is the pow...

Consider the circuit shown in Figure (a). A gate pulse Vg is applied between time instants t0 and t1. After t1, during the MOSFET turn OFF process, it experiences a voltage oversho...

The asymptotic Bode magnitude plot of a system is shown. Which one of the following options best represents the transfer function of the system?

Two $n \times n$ matrices $A$ and $B$ have a common eigenvalue 2, and the same corresponding nonzero eigenvector. Which of the following options is/are correct? (Note: $I$ is the $...

A 220V/12V single-phase transformer is designed for use in India and rated 100 VA at 50 Hz. Later, this unit is shipped to the USA where it is used as a 110V/6V transformer at 60 H...

Given that $\vec{F}(x, y, z) = \sin(y) \hat{x} + \cos(x) \hat{y} + 5 \hat{z}$, the integral $\oiint_S \vec{F}(x, y, z) \cdot d\vec{s}$ over the unit sphere S centered at the origin...

In the linear regulator circuit shown, the base to emitter voltage VBE of the BJT is 0.6 V. The Zener diode clamps the base voltage to 5.4 V. Ignore the biasing current of the Zene...

Consider the circuit shown. Assume that the diode D is ideal. The supply voltage $v_s = 325 \sin(2\pi50t)$ V, $L = 500 \mu H$, and $R = 10 \Omega$. The peak diode current (in amper...

A is an $m \times m$ skew-symmetric matrix with real-valued entries, and $x$ is an $m$-dimensional column vector with real-valued entries such that $x^T x = 1$. The quantity $x^T A...

A time-limited waveform g(x) is specified as follows: g(x) = { -k, -π < x ≤ 0 +k, 0 < x ≤ π 0, otherwise A new waveform f(x) is constructed from g(x) as follows: f(x) = Σ_{m=-∞}^{∞...

In the circuit shown, the open loop gain of the operational amplifier is A₀ = 105. What is the voltage gain of the circuit? (Round off to two decimal places)

Three single-phase 11kV/3.3kV transformers are connected to form a three-phase transformer bank with connections as shown. Considering ABC phase sequence, the vector group of the t...

In the circuit shown, the phase currents are $I_A = 572.812+ j50.115 A$ $I_B = -254.525- j459.175 A$ $I_C = -207.083+ j444.091 A$ Given that the CTs are ideal with no saturation, a...

The operating characteristic of a reactance relay is given by $X \le 1 \Omega$, where X is the reactance calculated by the relay. Its operating characteristic in the admittance pla...

An electrical component has voltage drop $v = V_m \sin (\omega t)$, when the current through it is $i = I_m \sin (\omega t-\theta)$. What is the average power dissipated over a hal...

The electrical network shown has an independent voltage source (10 V) and a current source (1 u(t) mA). The voltage across the capacitor at time instants (in seconds) t = 0+, t = 0...

The terminal voltage and current of a linear electrical network shown in Figure (a) are given in the table. The correct choice for the parameters (IN, RN) of the Norton equivalent...

The digital circuit shown has 3 inputs (x, y, and z). The simplified logical expression for the output (OUT) is:

The MOSFET switches shown in the circuit are ideal. Which of the following is the correct option for Boolean logical expression of the output (OUT), and the maximum possible power...

Consider the single-phase voltage source inverter circuit feeding an inductive load (L). Assume that the power MOSFET switches are ideal. S1 and S2 are switched on during the first...

Consider the boost converter circuit shown. Assume that the semiconductor devices are ideal. In steady state, the inductor current rises linearly from 0 A to 6 A in the first 10 µs...

Which one of the following statements is ALWAYS correct about a collection of $p$ column vectors, each having $n$ real-valued entries?

Consider the second-order differential equation $\frac{d^2y}{dx^2} + \frac{dy}{dx} + y = 0$ with initial conditions $y(0) = 1$, $\frac{dy}{dx}|_{x=0} = 1$ The solution is given by

The figure shows an arbitrarily shaped planar conducting loop A in the XY plane. Two nonintersecting regions with areas $a_1$ and $a_2$ within the loop are subjected to magnetic fi...

A system is characterized by the following state equation and output equation (U: input, X: state vector, y: output) $\dot{x} = \begin{bmatrix} a & b \\ -a & 0 \end{bmatrix}x + \be...

A system is represented in state-space form as follows: ($u$: input, $\mathbf{x}$: state vector, $y$: output) $\dot{\mathbf{x}}} = \begin{bmatrix} 1 & 2 \\ -3 & 0 \end{bmatrix} \ma...

For the balanced 3-phase transmission line shown, consider the following cases: Case-1: |V1| = 1.1 p. u., |V2| = 0.9 p. u., Z = 0.75∠0° p. u. and θ12 = θ1 - θ2 = 0° Case-2: |V1| =...

Consider an $n \times n$ orthogonal matrix $A$ with real entries and each column having unit Euclidean norm. Which of the following statements is/are correct?

Consider the system of linear equations: $Ax = b$, where $A$ is an $n \times n$ matrix, and $x$ and $b$ are $n$-dimensional column vectors. Suppose this system of equations has a u...

The resistance values of the Wheatstone bridge shown are: P = 2000 Ω, Q = 500 Ω, R = 3000 Ω. The battery voltage E = 50V. The battery has an internal resistance of 1 Ω and the Galv...

A system with two generators G1 and G2 (without generator limits) is shown. The total load on the system is 1184 MW. The expressions for the cost of generation (C1 and C2) and real...

A balanced three-phase supply is given to a 30 kW, 4-pole, 400 V, 50 Hz, wound rotor induction motor with Y-connected stator and rotor windings. The motor is driving a constant tor...

Consider the two-port network shown. For maximum power transfer to the resistive load (RL), the value of RL should be ______ Ω (Round off to two decimal places)

Consider the circuit shown. Assume that the diode (D) is ideal. Given Vs = 100 sin(2π50t) V, Vdc = 50 V, and R = 10 Ω, the average value of the current through the diode is ______...

The magnitude of the contour integral $\oint_C \frac{(z+1)^2}{(z-i)(z-2)} dz$ over the contour C: $|z - 2 - i| = 3/2$ is _______ (Round off to two decimal places) Note: z is a comp...

A uniform spherical volume charge distribution of radius 2 m, centered at the origin, has a strength of $\frac{3}{\pi} \times 10^{-6}$ C/m³. A point charge of strength $\pi \times...

Let X and Y be two real-valued random variables with E(X) = 1, E(Y) = 2, E(X²) = 4, E(Y²) = 9, and E(XY) = 0.9, where E denotes the expectation operator. The value of $\alpha$ that...

The integral $\frac{1}{\pi} \int_0^{\infty} \frac{x^{2026}}{(1 + x^{2026})(1 + x^2)} dx$ evaluates to _______ (Round off to two decimal places)

A coil of inductance 100 mH and series resistance 100 Ω is connected across a 20 V, 1 kHz supply. The quality factor of the coil at 1 kHz is

A device (single-phase, 50 Hz, AC) yields the following measurements at its terminals: voltage = 141.4 ∠ -10° V, current = 0.7 ∠ -55° A. The power factor of the device is ________...

In the circuit, both Q1 and Q2 have a current gain of 100 and VBE = 0.7 V. The small signal voltage gain is ________ (round off to two decimal places).

The current, i, in the circuit shown below, in A, is ________ (round off to the nearest integer).

In a power system, following a sudden increase in load, the frequency will settle at a lower steady-state value if

The bilateral z-transform of a discrete-time signal x[n] is X(z) = (1-a^2) / ((z-a)(z^-1 - a)) with 0 < a < 1, and the region of convergence a < |z| < 1/a. Let u[n] be the discrete...

Given F = a_x(3y + k_1z) + a_y(k_2x - 2z) + a_z(k_3y – x), where a_x, a_y and a_z are the unit vectors in Cartesian coordinate system. The product k_1k_2k_3 for which F is irrotati...

Let $G(s) = \frac{1}{(s+1)(s+2)}$. Then the closed-loop system shown in the figure below, is

Consider the two statements given below, and select the correct option. Statement 1: A directional relay relies on the principle that the CT currents due to fault location on eithe...

In the Wheatstone bridge shown below, the sensitivity of the bridge in terms of change in balancing voltage E for unit change in the resistance R, in mV/Ω, is ____________ (round o...

The relationship between two variables x and y is given by x + py + q = 0 and is shown in the figure. Find the values of p and q. Note: The figure shown is representative.

Spheres of unit diameter are centered at (l, m, n) where l, m, and n take every possible integer values. The distance between two spheres is computed from the center of one sphere...

In the triangle PQR, the lengths of PT and TR are in the ratio of 3:2. ST is parallel to QR. Two semicircles are drawn with PS and PQ as diameters, as shown in the figure. Which on...

Consider the set S of points (x, y) ∈ R² which minimize the real valued function f(x, y) = (x + y − 1)² + (x + y)² Which of the following statements is true about the set S?

Let v₁ and v₂ be the two eigenvectors corresponding to distinct eigenvalues of a 3 × 3 real symmetric matrix. Which one of the following statements is true?

Let $A = \begin{bmatrix} 1 & 1 & 1 \\ -1 & -1 & -1 \\ 0 & 1 & -1 \end{bmatrix}$ and $b = \begin{bmatrix} 1/3 \\ -1/3 \\ 0 \end{bmatrix}$. Then, the system of linear equations $Ax =...

Let $P = \begin{bmatrix} 2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ and let $I$ be the identity matrix. Then $P^2$ is equal to

Consider discrete random variables X and Y with probabilities as follows: P(X = 0 and Y = 0) = $\frac{1}{4}$ P(X = 1 and Y = 0) = $\frac{1}{8}$ P(X = 0 and Y = 1) = $\frac{1}{2}$ P...

Which one of the following statements is true about the small signal voltage gain of a MOSFET based single stage amplifier?

Assuming ideal op-amps, the circuit represents a

The I-V characteristics of the element between the nodes X and Y is best depicted by

A nullator is defined as a circuit element where the voltage across the device and the current through the device are both zero. A series combination of a nullator and a resistor o...

Consider a discrete-time linear time-invariant (LTI) system S, where y[n] = S{x[n]} Let S{δ[n]} = {1, n ∈ {0, 1, 2} {0, otherwise where δ[n] is the discrete-time unit impulse funct...

Consider a continuous-time signal x(t) = -t^2{u(t + 4) - u(t-4)} where u(t) is the continuous-time unit step function. Let δ(t) be the continuous-time unit impulse function. The va...

Selected data points of the step response of a stable first-order linear time-invariant (LTI) system are given below. The closest value of the time-constant, in sec, of the system...