GATE 2025 Electrical Engineering

The induced emf in a 3.3 kV , 4-pole, 3-phase, star-connected synchronous motor is considered to be equal and in phase with the terminal voltage under no load condition. On applica...

Let $X$ and $Y$ be continuous random variables with probability density functions $P_X(x)$ and $P_Y(y)$, respectively. Further, let $Y=X^2$ and $P_X(x)=\left\{\begin{array}{cc}1, &...

A DC series motor with negligible series resistance is running at a certain speed driving a load, where the load torque varies as cube of the speed. The motor is fed from a 400 V D...

The 3-phase modulating waveforms $\left(v_a(t), v_b(t), v_c(t)\right)$, used in sinusoidal PWM in a voltage source inverter (VSI) are $$ \begin{aligned} & v_a(t)=0.8 \sin (\omega t...

Let $a_R$ be the unit radial vector in the spherical co-ordinate system. For which of the following value(s) of $n$, the divergence of the radial vector field $f(R)=a_R \frac{1}{R^...

Consider two coupled circuits having self inductances $L_1$ and $L_2$ that carry non-zero currents $I_1$ and $I_2$ respectively. The mutual inductance between the circuit is $M$ wi...

Consider ordinary differential equations given by $\dot{x}_1(t)=2 x_2(t), \dot{x}_2(t)=r(t)$ with initial conditions $x_1(0)=1$ and $x_2(0)=0$. If $r(t)=\left\{\begin{array}{ll}1,...

Let $C$ be a clockwise oriented closed curve in the complex plane defined by $|\lambda|=1$. Further, let $f(x)=j z$ be a complex function, where $j=\sqrt{-1}$. Then, $\oint_C f(z)...

Consider the state-space model $$ \begin{aligned} \dot{x}(t) & =A x(t)+B u(t) \\ y(t) & =C x(t) \end{aligned} $$ where $x(t), r(t), y(t)$ are the state, input and output, respectiv...

Let $(x, y) \in \Re^2$. The rate of change of the real valued function, $V(x, y)=x^2+x+y^2+1$ at the origin in the direction of the point $(1,2)$ is _________ (round off to the nea...

A 3-phase, $400 \mathrm{~V}, 4$ pole, 50 Hz star connected induction motor has the following parameters referred to the stator : $$ R_r^{\prime}=1 \Omega, X_S=X_r^{\prime}=2 \Omega...

Which one of the following figures represents the radial electric field distribution $E_R$ caused by a spherical cloud of electrons with a volume charge density. $\rho=-3 \rho_0$ f...

A Boolean function is given as $$ f=(\bar{u}+\bar{v}+\bar{w}+\bar{x}) \cdot(\bar{u}+\bar{v}+\bar{w}+x) \cdot(\bar{u}+v+\bar{w}+\bar{x}) \cdot(\bar{u}+v+\bar{w}+x) $$ The simplified...

Consider a distribution feeder, with $R / X$ ratio of 5 . At the receiving end, a 350 kVA load is connected. The maximum voltage drop will occur from the sending end to the receivi...

Two units, rated at 100 MW and 150 MW , are enabled for economic load dispatch. When the overall incremental cost is $10,000 \mathrm{Rs}$./MWh, the units are dispatched to 50 MW an...

Using shunt capacitors, the power factor of a 3-phase, 4 kV induction motor (drawing 390 kVA at 0.77 pf lag) is to be improved to 0.85 pf lag. The line current of the capacitor ban...

The continuous-time unit impulse signal is applied as an input to a continuous-time linear time-invariant system $S$. The output is observed to be the continuous-time unit step sig...

Consider a discrete-time linear time-invariant (LTI) system, $\boldsymbol{S}$, where $$ y[n]=S\{x(\mathrm{n})\} $$ $$Let\,\,\,\, S\{\delta[n]\}=\left\{\begin{array}{lc} 1, & n \in\...

The maximum percentage error in the equivalent resistance of two parallel connected resistors of $100 \Omega$ and $900 \Omega$ with each having a maximum $5 \%$ error is __________...

A continuous time periodic signal $x(t)$ is $$ x(t)=1+2 \cos 2 \pi t+2 \cos 4 \pi t+2 \cos 6 \pi t $$ If $T$ is the period of $x(t)$, then $\frac{1}{T} \int_0^T|x(t)|^2 d t=$______...

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 $\delta(t)$ be the continuous-time unit impulse funct...

Let continuous-time signals $x_1(t)$ and $x_2(t)$ be $x_1(t)=\left\{\begin{array}{cc}1, & t \in[0,1] \\ 2-t, & t \in[1,2] \\ 0, & \text { otherwise }\end{array}\right.$ and $x_2(t)...

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...

An ideal low pass filter has frequency response given by $$ H(j \omega)= \begin{cases}1, & |\omega| \leq 200 \pi \\ 0, & \text { otherwise }\end{cases} $$ Let $h(t)$ be its time do...

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

Consider discrete random variable $X$ and $Y$ with probabilities as follows: $$ \begin{aligned} & P(X=0 \text { and } Y=0)=\frac{1}{4} \\ & P(X=1 \text { and } Y=1)=\frac{1}{8} \\...

Let $P=\left[\begin{array}{ccc}2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$ and let $I$ be the identity matrix. Then $P^2$ is equal to

Let $A=\left[\begin{array}{ccc}1 & 1 & 1 \\ -1 & -1 & -1 \\ 0 & 1 & -1\end{array}\right]$ and $b=\left[\begin{array}{c}1 / 3 \\ -1 / 3 \\ 0\end{array}\right]$, then the system of l...

Let $v_1$ and $v_2$ be the two eigen vectors corresponding to distinct eigen values of a $3 \times 3$ real symmetric matrix. Which one of the following statements is true?

What is the value of $\left(\frac{3^{81}}{27^4}\right)^{1 / 3}$ ?

Good : Evil : : Genuine : ? Select the most appropriate option to complete the analogy.

Kavya__________go to work yesterday as she________ feeling well. Select the most approrpriate option to complete the above sentence.

The bus impedance matrix of a 3-bus system (in pu) is $$ Z_{\text {bus }}=\left[\begin{array}{lll} j 0.059 & j 0.061 & j 0.038 \\ j 0.061 & j 0.093 & j 0.066 \\ j 0.038 & j 0.066 &...

Instruments required to synchronize an alternator to the grid is/are

Identify the option that has the most appropriate sequence such that a coherent paragraph is formed : (P) It is because deer, like most of the animals that tigers normally prey on,...

Scores obtained by two students P and Q in seven courses are given in the table below. Based on the information given in the table, which one of the following statements is INCORRE...

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...

Consider the set $S$ of points $(x, y) \in R^2$ 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 $S$...

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...

During a power failure, a domestic household uninterruptible power supply (UPS) supplies AC power to a limited number of lights and fans in various rooms. As per a NewtonRaphson lo...

The input voltage $v(t)$ and current $i(t)$ of a converter are given by, $v(t)=300 \sin (\omega t) \mathrm{V}$ $$ i(t)=10 \sin \left(\omega t-\frac{\pi}{6}\right)+2 \sin \left(3 \o...

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 ________...

Kavya ________ go to work yesterday as she ________ feeling well. Select the most appropriate option to complete the above sentence.

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

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

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).

Good : Evil :: Genuine : ________ Select the most appropriate option to complete the analogy.

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...

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

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...

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...

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...

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.

Each row of Column-I has three items and each item is represented by a circle in Column-II. The arrangement of circles in Column-II represents the relationship among the items in C...

What is the value of $(\frac{3^{81}}{3^{74}})^{\frac{1}{3}}$ ?

Identify the option that has the most appropriate sequence such that a coherent paragraph is formed: P. It is because deer, like most of the animals that tigers normally prey on, r...

In the given figure, EF and HJ are coded as 30 and 80, respectively. Which one among the given options is most appropriate for the entries marked (i) and (ii) ?

Scores obtained by two students P and Q in seven courses are given in the table below. Based on the information given in the table, which one of the following statements is INCORRE...

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...

The Nyquist plot of a strictly stable G(s) having the numerator polynomial as (s – 3) encircles the critical point –1 once in the anti-clockwise direction. Which one of the followi...

During a power failure, a domestic household uninterruptible power supply (UPS) supplies AC power to a limited number of lights and fans in various rooms. As per a Newton-Raphson l...

Which one of the following figures represents the radial electric field distribution E_R caused by a spherical cloud of electrons with a volume charge density, ρ = -3ρ_0 for 0 ≤ R...

The operating region of the developed torque ($T_{em}$) and speed ($\omega$) of an induction motor drive is given by the shaded region OQRE in the figure. The load torque ($T_L$) c...

The input voltage v(t) and current i(t) of a converter are given by, v(t) = 300 sin(ωt) V i(t) = 10 sin (ωt - π/6) + 2 sin (3ωt + π/6) + sin (5ωt + π/2) A where, ω = 2π × 50 rad/s....

Instrument(s) required to synchronize an alternator to the grid is/are

The open-loop transfer function of the system shown in the figure, is $G(s) = \frac{Ks(s + 2)}{(s + 5)(s + 7)}$ For $K \ge 0$, which of the following real axis point(s) is/are on t...

A continuous time periodic signal x(t) is $x(t) = 1 + 2 \cos 2\pi t + 2 \cos 4\pi t + 2 \cos 6\pi t$ If T is the period of x(t), then $\frac{1}{T} \int_{0}^{T} |x(t)|^2 dt = $ ____...

The maximum percentage error in the equivalent resistance of two parallel-connected resistors of 100 Ω and 900 Ω with each having a maximum 5% error is ________ % (round off to nea...

Consider a distribution feeder, with R/X ratio of 5. At the receiving end, a 350 kVA load is connected. The maximum voltage drop will occur from the sending end to the receiving en...

The bus impedance matrix of a 3-bus system (in pu) is $Z_{bus} = \begin{bmatrix} j0.059 & j0.061 & j0.038 \\ j0.061 & j0.093 & j0.066 \\ j0.038 & j0.066 & j0.110 \end{bmatrix}$ A s...

In the circuit with ideal devices, the power MOSFET is operated with a duty cycle of 0.4 in a switching cycle with I = 10 A and V =15 V. The power delivered by the current source,...

The induced emf in a 3.3 kV, 4 pole, 3-phase star connected synchronous motor is considered to be equal and in phase with the terminal voltage under no load condition. On applicati...

Let X and Y be continuous random variables with probability density functions Px(x) and Py(y), respectively. Further, let Y = X^2 and Px(x) = {1, x ∈ (0,1] {0, otherwise Which one...

A Boolean function is given as $f = (\bar{u} + \bar{v} + \bar{w} + x). (\bar{u} + \bar{v} + w + x). (\bar{u} + v + \bar{w} + x). (\bar{u} + v + w + x)$ The simplified form of this...

In the circuit, I$_{DC}$ is an ideal current source. The transistors M$_{1}$ and M$_{2}$ are assumed to be biased in saturation, wherein V$_{in}$ is the input signal and V$_{DC}$ i...

In the circuit, shown below, if the values of R and C are very large, the form of the output voltage for a very high frequency square wave input, is best represented by

Let continuous-time signals $x_1(t)$ and $x_2(t)$ be $x_1(t) = \begin{cases} 1, & t \in [0,1] \\ 2-t, & t \in [1,2] \\ 0, & \text{otherwise} \end{cases}$ and $x_2(t) = \begin{cases...

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

The continuous-time unit impulse signal is applied as an input to a continuous-time linear time-invariant system S. The output is observed to be the continuous-time unit step signa...

The transformer connection given in the figure is part of a balanced 3-phase circuit where the phase sequence is "abc". The primary to secondary turns ratio is 2:1. If $(I_a + I_b...

A DC series motor with negligible series resistance is running at a certain speed driving a load, where the load torque varies as cube of the speed. The motor is fed from a 400 V D...

A 3-phase, 400 V, 4 pole, 50 Hz star connected induction motor has the following parameters referred to the stator: R = 1Ω, X_s = X'_r = 2Ω Stator resistance, magnetizing reactance...

The 3-phase modulating waveforms ($v_a(t)$, $v_b(t)$ and $v_c(t)$), used in sinusoidal PWM in a Voltage Source Inverter (VSI) are $v_a(t) = 0.8 \sin (\omega t)$ V $v_b(t) = 0.8 \si...

An ideal sinusoidal voltage source $v(t) = 230\sqrt{2} \sin(2\pi \times 50t)$ V feeds an ideal inductor L through an ideal SCR with firing angle $\alpha = 0^{\circ}$. If $L = 100$...

In the following circuit, the average voltage $V_o = 400 \left(1 + \frac{\cos \alpha}{3}\right)$ V, where $\alpha$ is the firing angle. If the power dissipated in the resistor is 6...

In the system shown below, the generator was initially supplying power to the grid. A temporary LLLG bolted fault occurs at F very close to circuit breaker 1. The circuit breakers...

An air filled cylindrical capacitor (capacitance C₀) of length L, with a and b as its inner and outer radii, respectively, consists of two coaxial conducting surfaces. Its cross-se...

Let $\mathbf{a}_R$ be the unit radial vector in the spherical co-ordinate system. For which of the following value(s) of $n$, the divergence of the radial vector field $\mathbf{f}(...

Consider two coupled circuits, having self-inductances $L_1$ and $L_2$, that carry non-zero currents $I_1$ and $I_2$, respectively. The mutual inductance between the circuits is $M...

Let (x, y) ∈ R². The rate of change of the real valued function, V(x, y) = x² + x + y² + 1 at the origin in the direction of the point (1,2) is ________ (round off to the nearest i...

Consider ordinary differential equations given by ẋ₁(t) = 2x₂(t) ẋ₂(t) = r(t) with initial conditions x₁(0) = 1 and x₂(0) = 0. If r(t) = {1, t ≥ 0; 0, t < 0}, then at t = 1, x₁(t)...

Let C be a clockwise oriented closed curve in the complex plane defined by |z| = 1. Further, let f(z) = jz be a complex function, where j = √-1. Then, ∮_C f(z)dz = ________ (round...

The op-amps in the following circuit are ideal. The voltage gain of the circuit is ________ (round off to the nearest integer).

The switch (S) closes at t = 0 sec. The time, in sec, the capacitor takes to charge to 50 V is ________ (round off to one decimal place).

In an experiment to measure the active power drawn by a single-phase RL Load connected to an AC source through a 2 Ω resistor, three voltmeters are connected as shown in the figure...

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 off t...

The steady state capacitor current of a conventional DC-DC buck converter, working in CCM, is shown in one switching cycle. If the input voltage is 30 V, the value of the inductor...

An ideal low pass filter has frequency response given by H(jω) = {1, |ω| ≤ 200π {0, otherwise Let h(t) be its time domain representation. Then h(0) = ________ (round off to the nea...

Consider the state-space model $\dot{x}(t) = Ax(t) + Br(t)$, $y(t) = Cx(t)$ where $x(t)$, $r(t)$, $y(t)$ are the state, input and output, respectively. The matrices A, B, C are giv...

Using shunt capacitors, the power factor of a 3-phase, 4 kV induction motor (drawing 390 kVA at 0.77 pf lag) is to be corrected to 0.85 pf lag. The line current of the capacitor ba...

Two units, rated at 100 MW and 150 MW, are enabled for economic load dispatch. When the overall incremental cost is 10,000 Rs./MWh, the units are dispatched to 50 MW and 80 MW resp...

A controller $D(s)$ of the form $(1+K_D s)$ is to be designed for the plant $G(s) = \frac{1000\sqrt{2}}{s(s+10)^2}$ as shown in the figure. The value of $K_D$ that yields a a phase...