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GATE 2025 Electrical Engineering (EE) Question Paper with Detailed Solutions
The Graduate Aptitude Test in Engineering (GATE) is one of India's most prestigious national-level examinations for engineering and science graduates. Every year, lakhs of students appear in the examination to secure admission into M.Tech, MS, PhD programs at IITs, IISc, NITs and other reputed institutes. In addition, a good GATE score opens the door to recruitment in several Public Sector Undertakings (PSUs), research organizations and government departments. Among all engineering branches, the Electrical Engineering (EE) paper is considered one of the most competitive because it covers a wide range of subjects such as Electrical Machines, Power Systems, Power Electronics, Control Systems, Signals & Systems, Analog Electronics, Digital Electronics, Network Theory, Electromagnetic Fields and Engineering Mathematics.
This article provides the GATE 2025 Electrical Engineering (EE) Question Paper with Full Detailed Solutions. Every question has been carefully prepared with a clear explanation so that students can understand not only the correct answer but also the concept behind it. Instead of memorizing answers, aspirants can learn the proper problem-solving techniques that are frequently tested in the GATE examination. Each solution is written step-by-step using standard engineering formulas, logical reasoning and conceptual explanations to make learning easier.
The questions included in this paper cover multiple important topics from the official GATE EE syllabus. Numerical problems, conceptual questions, multiple-select questions (MSQ), multiple-choice questions (MCQ) and NAT (Numerical Answer Type) questions have all been explained in detail. Wherever necessary, circuit diagrams, transformer connections, control system blocks, logic gate diagrams, waveforms and electrical machine figures have been recreated using permanent HTML/SVG graphics so that they remain clear on both desktop and mobile devices. Every diagram is responsive and Blogger-friendly, allowing students to study directly from their smartphones without any loss of quality.
This solved paper is extremely useful for candidates preparing for GATE 2026, GATE 2027 and future examinations because previous year questions remain one of the best sources for understanding the examination pattern, difficulty level and frequently repeated concepts. By solving these questions, students can identify their strengths and weaknesses while improving their speed, accuracy and confidence before the actual examination.
Each question on this page includes interactive answer options. When a student selects the correct option, it is highlighted in green, while an incorrect selection immediately turns red and also reveals the correct answer. In addition, every question contains a Show/Hide Solution button that keeps the page clean and allows learners to attempt the problem first before checking the explanation. This interactive format makes the learning experience similar to an online mock test while also providing complete conceptual understanding.
Whether you are revising important Electrical Engineering subjects, practicing previous year papers or preparing for PSU recruitment through GATE, this complete solved paper will serve as a valuable study resource. We recommend attempting every question on your own before viewing the solution and revisiting difficult concepts regularly for maximum benefit. Consistent practice with authentic previous year questions is one of the most effective strategies for achieving a high GATE score.
Bookmark this page and continue practicing all GATE Electrical Engineering previous year question papers available on our website. Regular practice, detailed analysis and conceptual understanding are the keys to success in the GATE examination. We hope these comprehensive solutions help you strengthen your preparation and move one step closer to achieving your dream IIT, IISc or PSU career.
Question 01
Kavya ______ go to work yesterday as she ______ feeling well.
Select the most appropriate option to complete the sentence.
The word yesterday shows that the sentence is in past tense. In the first blank, we need a modal verb before go. Since Kavya was not feeling well, she was not able to go to work. So, couldn’t is correct.
In the second blank, the past form of is is was. Therefore, wasn’t feeling well is correct.
Question 02
Good : Evil :: Genuine : ________
Select the most appropriate option to complete the analogy.
Correct Answer: (a) Counterfeit
This is an analogy question. The relationship between the first pair is based on opposite meanings.
- Good means morally right.
- Evil means morally wrong.
- Therefore, Good : Evil are opposites.
Similarly,
- Genuine means real, authentic or original.
- Counterfeit means fake, forged or not genuine.
Hence, the opposite of Genuine is Counterfeit.
Why Other Options Are Incorrect?
- Contraband → Illegal goods or prohibited items.
- Counterfoil → The remaining part of a cheque, ticket or receipt.
- Counterpart → A person or thing having the same function as another.
Therefore, the correct answer is (a) Counterfeit.
Question 03
The relationship between two variables x and y is given by x + py + q = 0. The graph of the line is shown below. Find the values of p and q.
Correct Answer: (a) p = -1/2, q = 2
The given equation of the line is
x + py + q = 0
Rearrange the equation in slope-intercept form.
py = -x - q
y = -1/p · x - q/p
Hence,
- Slope (m) = -1/p
- Y-intercept (c) = -q/p
Step 1: Find the Slope
From the graph, the line passes through the points (0, 4) and (2, 8).
Slope = (8 − 4)/(2 − 0) = 4/2 = 2
Therefore,
-1/p = 2
p = -1/2
Step 2: Find the Value of q
The graph cuts the y-axis at y = 4.
Therefore,
-q/p = 4
Substitute p = -1/2
-q / (-1/2) = 4
2q = 4
q = 2
✔ p = -1/2
✔ q = 2
Shortcut Trick
- Convert the equation into y = mx + c form.
- Find the slope from any two points on the graph.
- Use the y-intercept to calculate the remaining value.
- This method works for every straight-line graph question in JEE Main.
Question 04
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 Column-I. Identify the option that has the most appropriate match between Column-I and Column-II.
Note: The figures shown are representative.
Column-I
1. Animals, Zebra, Giraffe
2. Animals, Zebra, Giraffe
3. Word, Sentence, Novel
4. Pianist, Guitarist, Instrumentalist
Column-II
Correct Answer: (b)
In this question, we have to understand the relationship between the three given words and then match it with the correct circle diagram.
1. Animals, Zebra, Giraffe
Zebra and Giraffe are both animals, but Zebra and Giraffe are different from each other. So, both should be inside the big circle of Animals, but they should not overlap.
So, 1 → Q
2. Animals, Zebra, Giraffe
As per the given options and relationship arrangement, this set is matched with the diagram where three circles overlap with each other.
So, 2 → R
3. Word, Sentence, Novel
A novel is made up of sentences, and sentences are made up of words. Therefore, this relation is like one category inside another category.
Hence, three concentric circles are suitable.
So, 3 → S
4. Pianist, Guitarist, Instrumentalist
Pianist and Guitarist both come under Instrumentalist. Some people may be both pianist and guitarist. Therefore, two small circles overlap inside one big circle.
So, 4 → P
1 → Q
2 → R
3 → S
4 → P
Correct Option: (b)
Question 05
What is the value of
Correct Answer: (c) 323
Step 1: Convert the denominator into the same base.
Step 2: Divide the powers having the same base.
Step 3: Apply the power 1/3.
✔ 323
Question 06
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, run much faster! It simply means, another day of empty stomach for the big cats.
Q. Tigers spend most of their life searching for food.
R. If they trace the scent of deer, tigers follow the trail, chase the deer for a mile or two in the dark, and yet may not catch them.
S. For several nights, they relentlessly prowl through the forest, hunting for a trail that may lead to their prey.
Correct Answer: (c) Q → S → R → P
Step 1:
Sentence Q introduces the main idea: "Tigers spend most of their life searching for food." This is the most suitable opening sentence because it introduces the topic.
Step 2:
Sentence S naturally follows Q by explaining how tigers search for food. It describes their continuous hunting through the forest.
Step 3:
Sentence R continues the story by describing what happens after they find the scent of a deer. Tigers chase the deer but often fail to catch it.
Step 4:
Sentence P gives the reason for their failure. Deer run much faster than tigers, resulting in another day without food.
Q → S → R → P
Therefore, the correct option is (c).
Question 07
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)?
Correct Answer: (c) (i) EG; (ii) 42
This is a coding pattern question. Each pair of letters is converted into a number by multiplying their alphabetical positions.
Step 1: Find the Coding Rule
EF → 30
F = 6
5 × 6 = 30
HJ → 80
J = 10
8 × 10 = 80
Hence, the coding rule is:
Step 2: Find (i)
The number opposite to 35 should correspond to the letter pair whose alphabetical positions multiply to 35.
7 = G
Therefore,
Step 3: Find (ii)
The opposite sector contains the letter pair FG.
G = 7
Therefore,
(i) = EG
(ii) = 42
✔ Correct Option: (c)
Question 08
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 INCORRECT?
| Student | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| P | 22 | 89 | 50 | 45 | 78 | 60 | 39 |
| Q | 35 | 65 | 60 | 56 | 81 | 45 | 50 |
Correct Answer: (b) Median score of P is same as the median score of Q.
To identify the incorrect statement, let's evaluate each option one by one.
Step 1: Calculate the Average Scores
Student P
Student Q
Therefore, Average(P) < Average(Q). Hence, Option (a) is Correct.
Step 2: Find the Median
Arrange the scores in ascending order.
P:
Median = 50
Q:
Median = 56
Since 50 ≠ 56, the statement "Median score of P is same as the median score of Q" is FALSE.
Therefore, Option (b) is the Incorrect Statement.
Step 3: Find the Range (Maximum − Minimum)
Student P
Student Q
Since 67 > 46, Option (c) is also Correct.
Step 4: Compare Median and Average of Q
Average of Q = 56
Median of Q = 56
Both values are equal. Hence, Option (d) is Correct.
✔ Average(P) = 54.71
✔ Average(Q) = 56
✔ Median(P) = 50
✔ Median(Q) = 56
✔ Range(P) = 67
✔ Range(Q) = 46
✅ Correct Answer: (b)
Question 09
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 to the center of the other sphere. For a given sphere, x is the distance to its nearest sphere and y is the distance to its next nearest sphere. The value of y/x is:
Correct Answer: (c) √2
The centers of the spheres are at all possible integer coordinate points (ℓ, m, n). So the centers form a 3D grid.
Step 1: Nearest Sphere Distance
Take one sphere centered at (0, 0, 0). The nearest sphere can be at:
Distance from (0,0,0) to (1,0,0):
Step 2: Next Nearest Sphere Distance
The next nearest sphere lies diagonally, for example at:
Distance from (0,0,0) to (1,1,0):
Step 3: Required Ratio
✔ y/x = √2
Correct Option: (c)
Question 10
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 one of the following statements is true about the shaded area PQS?
Note: The figure shown is representative.
Correct Answer: (a) 16/9 times
Given:
Since ST ∥ QR, triangles PST and PQR are similar.
Now,
So,
Let,
Area Calculation
Area of a semicircle with diameter d is proportional to d².
So,
The shaded area is the difference between the larger semicircle and the smaller semicircle.
Therefore, compared to the semicircle with diameter PS:
✔ The shaded area is 16/9 times the area of the semicircle with diameter PS.
Correct Option: (a)
Question 11
Consider the set S of points (x, y) ∈ R² which minimize the real valued function
Which of the following statements is true about the set S?
Correct Answer: (b) The number of elements in the set S is infinite.
Given function:
Here, the function depends only on x + y. Let
Then,
Now expand:
This quadratic function is minimum at:
So,
This is a straight line in the xy-plane. A straight line contains infinitely many points.
✔ The set S has infinitely many elements.
Correct Option: (b)
Question 12
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?
Correct Answer: (b) v1Tv2 = 0
For a real symmetric matrix, eigenvectors corresponding to distinct eigenvalues are always orthogonal.
Orthogonal vectors have dot product equal to zero.
Therefore, the correct statement is option (b).
✔ v1Tv2 = 0
Question 13
Let
[-1, -1, -1],
[0, 1, -1] ]
and
-1/3,
0 ]
Then, the system of linear equations Ax = b has
Correct Answer: (b) infinitely many solutions
Here, the second row of matrix A is the negative of the first row:
Also, the second entry of b is the negative of the first entry:
So, these two equations are dependent, not contradictory.
After row reduction, the rank of A and the augmented matrix [A|b] is same, but less than the number of variables.
When rank is less than the number of variables and the system is consistent, the system has infinitely many solutions.
✔ Infinitely many solutions
Correct Option: (b)
Question 14
Let
[-1, 0, 0],
[0, 0, 1] ]
and let I be the identity matrix. Then P2 is equal to
Correct Answer: (a) 2P − I
Given matrix:
[-1, 0, 0],
[0, 0, 1] ]
On multiplication,
[-2, -1, 0],
[0, 0, 1] ]
Now calculate 2P − I:
[-2, -1, 0],
[0, 0, 1] ]
Hence,
✔ 2P − I
Correct Option: (a)
Question 15
Consider discrete random variables X and Y with probabilities as follows:
P(X = 1, Y = 0) = 1/8
P(X = 0, Y = 1) = 1/2
P(X = 1, Y = 1) = 1/8
Given X = 1, the expected value of Y is
Correct Answer: (b) 1/2
We need to find:
First find total probability when X = 1:
Now,
Since Y can take values 0 and 1,
✔ 1/2
Correct Option: (b)
Question 16
Which one of the following statements is true about the small signal voltage gain of a MOSFET based single stage amplifier?
Correct Answer: (c) Common source amplifier is inverting and common gate amplifier is non-inverting amplifier.
To determine the correct option, recall the phase relationship between the input and output signals of MOSFET amplifier configurations.
1. Common Source (CS) Amplifier
In a Common Source amplifier, the output voltage is taken from the drain terminal. An increase in the input voltage increases the drain current, causing the drain voltage to decrease. Hence, the output is 180° out of phase with the input.
2. Common Gate (CG) Amplifier
In a Common Gate amplifier, the output voltage changes in the same direction as the input signal. Therefore, there is no phase inversion.
Quick Revision
| Amplifier | Phase Shift | Type |
|---|---|---|
| Common Source (CS) | 180° | Inverting |
| Common Gate (CG) | 0° | Non-Inverting |
| Common Drain (CD) | 0° | Non-Inverting |
✔ Common Source → Inverting
✔ Common Gate → Non-Inverting
Correct Option: (c)
Question 17
Assuming ideal op-amps, the circuit shown above represents a
Correct Answer: (d) Buffer
For an ideal op-amp with negative feedback, the voltage at the inverting input is equal to the voltage at the non-inverting input.
In the given circuit, the output is fed back to the inverting terminal. This makes the op-amp work as a voltage follower.
A circuit in which output voltage follows the input voltage is called a buffer.
✔ Buffer
Correct Option: (d)
Question 18
The I-V characteristics of the element between the nodes X and Y is best depicted by:
Correct Answer: (b)
In the circuit, a 1 A current source is connected in parallel with a 1 kΩ resistor.
Let the current entering from node X be Ixy and voltage across X-Y be Vxy.
Current through the resistor is:
Using KCL:
So,
This is a straight line equation:
where:
- y = Ixy
- x = Vxy
- slope m = 1/1000 = 1/1k
- intercept c = 1 A
✔ Correct graph is option (b)
Question 19
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 of value R will behave as a
Correct Answer: (b) Nullator
A nullator is an ideal circuit element for which:
When a resistor is connected in series with a nullator, the same current must flow through both elements.
Since the nullator current is zero:
Voltage across the resistor:
Also, the voltage across the nullator is already zero. Therefore, the total voltage and current of the series combination are both zero.
✔ The combination behaves as a nullator.
Correct Option: (b)
Question 20
Consider a discrete-time linear time-invariant (LTI) system S, where y[n] = S{x[n]}.
Let S{δ[n]} =
0, otherwise
Where δ[n] is the discrete-time unit impulse function. For an input signal x[n], the output y[n] is
Correct Answer: (a) x[n] + x[n − 1] + x[n − 2]
For an LTI system, the response to δ[n] is the impulse response h[n].
Output is convolution of input and impulse response:
Using impulse shifting property:
Therefore,
✔ x[n] + x[n − 1] + x[n − 2]
Correct Option: (a)
Question 21
Consider a continuous-time signal
Where u(t) is the continuous-time unit step function. Let δ(t) be the continuous-time unit impulse function. The value of
is
Correct Answer: (a) −9
Using the sifting property of impulse:
Now,
At t = −3:
So,
✔ −9
Correct Option: (a)
Question 22
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 seconds, of the system is
| Time (sec) | 0.6 | 1.6 | 2.6 | 10 | ∞ |
|---|---|---|---|---|---|
| Output | 0.78 | 1.65 | 2.18 | 2.98 | 3 |
Correct Answer: (b) 2
For a first-order system step response:
From the table:
Using t = 0.6 sec and C(t) = 0.78:
Taking natural log:
✔ τ ≈ 2 seconds
Correct Option: (b)
Question 23
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 following statements on the closed-loop system shown in figure is correct?
Correct Answer: (d) The system is unstable.
Given that G(s) is strictly stable, the number of open-loop poles in the right half plane is:
The Nyquist plot encircles the critical point −1 once in the anti-clockwise direction. So,
Using Nyquist stability criterion:
Here, Z = 1 means the closed-loop system has one pole in the right half plane. Therefore, the closed-loop system is unstable.
✔ The system is unstable.
Correct Option: (d)
Question 24
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 load-flow formulation, the UPS would be represented as a
Correct Answer: (a) Slack bus
In load-flow analysis, the bus which balances the active and reactive power demand is called the slack bus.
During a power failure, the UPS acts as the main source of power for the connected household loads.
Since it supplies the required power to lights and fans and balances the load demand, it is represented as a slack bus.
✔ Slack bus
Correct Option: (a)
Question 25
Which one of the following figures represents the radial electric field distribution ER caused by a spherical cloud of electrons with a volume charge density, ρ = −3ρ0 for 0 ≤ R ≤ a, and ρ = 0 for R > a?
Correct Answer: (c) Fig. (iii)
For a spherical charge distribution, electric field is found using Gauss law.
Given:
Since charge density is negative, the radial electric field will also be negative. Therefore, the graph must lie below the R-axis.
For 0 ≤ R ≤ a
Inside a uniformly charged sphere, electric field is directly proportional to R.
So, from R = 0 to R = a, the field decreases linearly from 0 to a negative value.
For R > a
Outside the sphere, the entire charge behaves like a point charge at the center. Hence field magnitude decreases as:
So, after R = a, the curve approaches zero from the negative side.
✔ The correct graph is Fig. (iii)
Correct Option: (c)
Question 26
The operating region of the developed torque (Tem) and speed (ω) of an induction motor drive is given by the shaded region OQRE in the figure. The load torque (TL) characteristic is also shown. The motor drive moves from the initial operating point O to the final operating point S. Which one of the following trajectories will take the shortest time?
Correct Answer: (a) O − Q − R − S
The motor drive has to move from the initial operating point O to the final operating point S. To reach the final point in the shortest time, the motor should accelerate with the maximum possible developed torque.
In the torque-speed plane, the maximum available torque corresponds to the upper boundary of the shaded operating region.
So, the motor should first move vertically from O to Q, then along the top boundary from Q to R, and finally come down to the operating point S.
✔ O − Q − R − S
Correct Option: (a)
Question 27
The input voltage v(t) and current i(t) of a converter are given by,
i(t) = 10 sin(ωt − π/6) + 2 sin(3ωt + π/6) + sin(5ωt + π/2) A
Where, ω = 2π × 50 rad/sec. The input power factor of the converter is closest to
Correct Answer: (a) 0.845
Supply voltage is purely sinusoidal:
Only the fundamental component of current contributes to real power.
Displacement power factor:
RMS value of total current:
RMS value of fundamental current:
True power factor:
✔ Input Power Factor ≈ 0.845
Correct Option: (a)
Question 28
Instrument(s) required to synchronize an alternator to the grid is/are
Correct Answer: (a) and (c)
To synchronize an alternator with the grid, its voltage, frequency, phase sequence, and phase angle must match the grid conditions.
A voltmeter is used to compare the alternator voltage with the grid voltage.
A synchroscope shows the phase relationship and indicates the correct instant for synchronization.
A wattmeter measures power, and a stroboscope is not used for grid synchronization.
✔ Voltmeter and Synchroscope
Correct Options: (a), (c)
Question 29
The open-loop transfer function of the system shown in the figure is
For K ≥ 0, which of the following real axis point(s) is/are on the root locus?
Correct Answer: (a) −1 and (c) −6
Given open-loop transfer function:
Open-loop zeros are:
Open-loop poles are:
Root Locus Rule
A point on the real axis lies on the root locus if the total number of open-loop poles and zeros to the right of that point is odd.
Now check the given points:
- s = −1: Points to the right are 0 only → 1 point → odd → on root locus.
- s = −4: Points to the right are 0 and −2 → 2 points → even → not on root locus.
- s = −6: Points to the right are −5, −2 and 0 → 3 points → odd → on root locus.
- s = −10: Points to the right are −7, −5, −2 and 0 → 4 points → even → not on root locus.
✔ −1 and −6 are on the root locus.
Correct Options: (a), (c)
Question 30
A continuous-time periodic signal x(t) is given by
If T is the period of x(t), then
(round off to the nearest integer)
Correct Answer: 7
Using Parseval's theorem for continuous-time periodic signals:
Here,
and cosine coefficients are:
There are no sine terms, so all bn = 0.
✔ 7
Question 31
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 nearest integer value)
Correct Answer: 5%
Given:
Nominal equivalent resistance:
For maximum positive error:
Percentage error:
✔ 5%
Question 32
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 end, when the power factor of the load is ______.
(Round off to three decimal places)
Correct Answer: 0.981
Given:
Therefore:
For maximum voltage drop in a distribution feeder:
Power factor:
✔ 0.981
Question 33
The bus impedance matrix of a 3-bus system (in pu) is
[j0.061, j0.093, j0.066],
[j0.038, j0.066, j0.110] ]
A symmetrical fault through a fault impedance of j0.007 pu occurs at bus 2. Neglecting pre-fault loading conditions, the voltage at bus 1 during the fault is ______ pu.
(Round off to three decimal places)
Correct Answer: 0.390 pu
Given:
Fault occurs at bus 2, so:
Pre-fault voltage is assumed as:
Fault current:
Voltage at bus 1 during fault:
Here,
✔ 0.390 pu
Question 34
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, in W, is ______.
(Round off to the nearest integer)
Correct Answer: 90 W
Given:
When the MOSFET is ON, the ideal MOSFET shorts the current source, so power delivered to the voltage source is zero.
When the MOSFET is OFF, the current flows through the diode and charges the voltage source.
Average power delivered by the current source:
✔ 90 W
Question 35
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 application of a mechanical load, the induced emf phasor is deflected by an angle of 2° mechanical with respect to the terminal voltage phasor. If the synchronous reactance is 2 Ω, and stator resistance is negligible, then the motor armature current magnitude, in ampere, during loaded condition is closest to ______.
(Round off to two decimal places)
Correct Answer: 66.49 A
Phase voltage:
At no load:
For a 4-pole machine, electrical angle is:
For motor operation, induced emf lags terminal voltage:
Armature current:
After calculation:
✔ 66.49 A
Question 36
Let X and Y be continuous random variables with probability density function PX(x) given by:
PX(x) = 0, otherwise
Further, let Y = X². Which one of the following is correct?
Correct Answer: (a)
Given:
For transformation of variables:
Jacobian:
Since X is uniform in (0,1), fX(x) = 1. So,
Hence,
✔ 1/(2√y)
Correct Option: (a)
Question 37
A Boolean function is given as
The simplified form of this function is represented by
Correct Answer: (a)
The given Boolean expression is in Product of Sums (POS) form.
Step 1: Write Maxterms
The given POS expression corresponds to:
Step 2: Karnaugh Map
| uv \ wx | 00 | 01 | 11 | 10 |
|---|---|---|---|---|
| 00 | 1 | 1 | 1 | 1 |
| 01 | 1 | 1 | 1 | 1 |
| 11 | 1 | 1 | 0 | 0 |
| 10 | 1 | 1 | 0 | 0 |
The four adjacent 0's are grouped together.
Step 3: Circuit Representation
The simplified Boolean function is implemented using:
- NOT gate on u
- NOT gate on w
- 2-input OR gate
✔ Simplified Function = u̅ + w̅
Correct Option: (a)
Question 38
In the circuit, IDC is an ideal current source. The transistors M1 and M2 are assumed to be biased in saturation, wherein Vin is the input signal and VDC is fixed DC voltage. Both transistors have a small signal resistance of rds and trans-conductance of gm. The small signal output impedance of this circuit is
Correct Answer: (c) gmrds2 + 2rds
For small signal output impedance, the ideal current source is treated as open circuit and VDC is AC ground.
Given:
This is a cascode structure. Using resistance reflection rule:
Substitute ro1 = ro2 = rds:
✔ gmrds2 + 2rds
Correct Option: (c)
Question 39
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
Correct Answer: (a) Triangular Waveform
For the given RC network,
Since both R and C are very large,
where T is the period of the input square wave. Therefore, the capacitor voltage cannot change instantaneously.
High Frequency Condition
For a very high-frequency square-wave input, the RC circuit behaves as an integrator.
Waveform
Hence the output voltage is a triangular waveform.
✔ Triangular Waveform
Correct Option: (a)
Question 40
Let continuous-time signals x1(t) and x2(t) be
x1(t) = 2 − t, t ∈ [1,2]
x1(t) = 0, otherwise
x2(t) = t, t ∈ [0,1]
x2(t) = 2 − t, t ∈ [1,2]
x2(t) = 0, otherwise
Consider the convolution y(t) = x1(t) * x2(t). Then
Correct Answer: (a) 1.5
Using the area property of convolution:
Area of x1(t):
Area of x2(t):
Therefore:
✔ 1.5
Correct Option: (a)
Question 41
Let
Then the closed-loop system shown in the figure below is
Correct Answer: (b)
Open-loop transfer function:
Given:
So,
Characteristic equation:
For stability of a second-order system, all coefficients must be positive.
So, system is stable only when:
Therefore, for K > 2, the system becomes unstable.
✔ Unstable for all K > 2
Correct Option: (b)
Question 42
The continuous-time unit impulse signal is applied as an input to a continuous-time linear time-invariant (LTI) system S. The output is observed to be the continuous-time unit step signal u(t). Which one of the following statements is true?
Correct Answer: (b)
For an LTI system,
Given:
Hence, the impulse response is
The output of an LTI system is
Since convolution with the unit-step is equivalent to integration,
Now choose a bounded input,
Its output becomes
which is a ramp signal. A ramp grows without bound, therefore the output is unbounded even though the input is bounded.
Hence the system is not BIBO stable.
✔ A bounded input can produce an unbounded output.
Correct Option: (b)
Question 43
A balanced three-phase transformer has a primary star connection and a secondary delta connection. The phase sequence is abc. The primary to secondary turns ratio is 2 : 1. If
then the relationship between IA and Iad is to be determined.
Correct Answer: (a)
Primary is star connected and secondary is delta connected. Turns ratio is:
For an ideal transformer, current ratio is inverse of turns ratio:
In delta connection:
Therefore:
For the given Y–Δ connection with phase sequence abc, secondary line current Iad lags primary current IA by 30°.
✔ |IA| / |Iad| = 1/(2√3)
✔ Iad lags IA by 30°
Correct Option: (a)
Question 44
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 DC source and draws 40 A armature current. Assume linear magnetic circuit. The external resistance, in Ω, that must be connected in series with the armature to reduce the speed of the motor by half, is closest to
Correct Answer: (a) 23.28 Ω
Given:
For a DC series motor with a linear magnetic circuit,
Load torque varies as cube of speed:
Since motor speed is reduced to half,
Back emf at the new operating point:
For a DC series motor:
✔ 23.28 Ω
Correct Option: (a)
Question 45
A 3-phase, 400 V, 4-pole, 50 Hz star-connected induction motor has the following parameters referred to the stator:
Xs = X′r = 2 Ω
The stator resistance, magnetizing reactance and core loss of the motor are neglected. The motor is operated with constant V/f control from a drive.
For maximum starting torque, the voltage and frequency output respectively from the drive are closest to
Correct Answer: (c) 100 V and 12.5 Hz
For an induction motor, maximum starting torque occurs when:
At 50 Hz:
But given:
Reactance is directly proportional to frequency. So, to make total leakage reactance equal to 1 Ω:
Since the drive uses constant V/f control:
✔ 100 V and 12.5 Hz
Correct Option: (c)
Question 46
The three-phase modulating waveforms va(t), vb(t) and vc(t), used in sinusoidal PWM in a Voltage Source Inverter (VSI), are
vb(t)=0.8 sin(ωt−2π/3) V
vc(t)=0.8 sin(ωt+2π/3) V
Where ω = 2π × 40 rad/s is the fundamental frequency. The modulating waveforms are compared with a 10 kHz triangular carrier whose magnitude varies between +1 and −1. The VSI has a DC link voltage = 600 V and feeds a star connected motor. The per-phase fundamental RMS motor voltage is closest to
Correct Answer: (a) 169.71 V
Carrier amplitude:
Reference signal amplitude:
Modulation index:
DC-link voltage:
For sinusoidal PWM,
✔ 169.71 V
Correct Option: (a)
Question 47
An ideal sinusoidal voltage source
feeds an ideal inductor L through an ideal SCR with firing angle α = 0°. If L = 100 mH, then the peak of the inductor current, in ampere, is closest to
Correct Answer: (a) 20.71 A
Given:
Peak Source Voltage
Current through Ideal Inductor
For an ideal inductor,
Hence,
The current becomes maximum when
Calculation
✔ Peak Inductor Current = 20.71 A
Correct Option: (a)
Question 48
In the following circuit, the average voltage
where α is the firing angle. If the power dissipated in the resistor is 64 W, then the closest value of α, in degrees, is
Correct Answer: (a) 35.9°
Given average output voltage:
Power dissipated in resistor:
Resistor value:
So,
For the load side,
Now substitute in the given equation:
✔ α ≈ 35.9°
Correct Option: (a)
Question 50
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-sectional view is shown in Fig. (i). In order to increase the capacitance, a dielectric material of relative permittivity εr is inserted inside 50% of the annular region as shown in Fig. (ii). The value of εr for which the capacitance of the capacitor in Fig. (ii) becomes 5C0 is
Correct Answer: (c) 9
The capacitance of the original air-filled coaxial capacitor is
Since the dielectric occupies exactly 50% of the annular region, the capacitor is equivalent to two capacitors connected in parallel.
Capacitance of Each Half
Total Capacitance
Given:
✔ Relative Permittivity, εr = 9
Correct Option: (c)
Question 51
Let aR be the unit radial vector in the spherical coordinate system. For which of the following value(s) of n, the divergence of the radial vector field
is independent of R ?
Correct Answer: (b) and (d)
For a purely radial vector field in spherical coordinates,
Given
Therefore,
For divergence to be independent of R,
- Case 1: Exponent of R must be zero.
- Case 2: Coefficient becomes zero.
✔ n = −1 and n = 2
Correct Options: (b) and (d)
Question 52
Consider two coupled circuits, having self-inductances L1 and L2, that carry non-zero currents I1 and I2, respectively. The mutual inductance between the circuits is M with unity coupling coefficient. The stored magnetic energy of the coupled circuits is minimum at which of the following value(s) of I1/I2?
Correct Answer: (a) and (d)
For two coupled inductors, stored magnetic energy is:
For minimum energy:
Also,
Since the coupling coefficient is unity:
So both expressions are equivalent.
✔ −M/L1 and −L2/M
Correct Options: (a), (d)
Question 53
Let V(x,y) ∈ ℝ². The rate of change of the real-valued function
at the origin in the direction of the point (1,2) is ______ (round off to the nearest integer).
Correct Answer: 0
Given,
Gradient of V:
At the origin (0,0):
Direction from (0,0) to (1,2):
Directional derivative:
Nearest integer:
✔ 0
Question 54
Consider the ordinary differential equations
ẋ2(t)=r(t)
with initial conditions
If
0, t<0 }
then at t = 1, x1(t)= ______ (round off to the nearest integer)
Correct Answer: 2
Given,
For t ≥ 0,
Using x₂(0)=0,
Now,
Using x₁(0)=1,
At t=1,
✔ 2
Question 55
Let C be a clockwise oriented closed curve in the complex plane defined by |z| = 1. Further, let f(z) = iz be a complex function, where i = √−1. Then,
(round off to the nearest integer)
Correct Answer: 0
Given:
The function f(z) = iz is a polynomial function in z. Therefore, it is analytic everywhere in the complex plane.
Since f(z) is analytic inside and on the closed curve |z| = 1, by Cauchy's theorem:
The direction of the curve is clockwise, but the integral is still zero because there is no singularity inside the contour.
✔ 0
Question 56
The op-amps in the following circuit are ideal. The voltage gain of the circuit is (round off to the nearest integer).
Correct Answer: 2
Let the differential input voltage of the first op-amp be
Since the first op-amp is an ideal non-inverting amplifier with equal resistors,
For the second op-amp,
Applying KCL at the inverting terminal,
Since
Therefore,
✔ Voltage Gain = 2
Nearest Integer = 2
Question 57
The switch (S) closes at t = 0 s. The time, in seconds, that the capacitor takes to charge to 50 V is ______ (round off to one decimal place).
Correct Answer: 4.1 s
For a first-order RC circuit,
Step 1: Initial Capacitor Voltage
Before the switch closes (t = 0⁻), the capacitor is in open-circuit condition. Applying KVL,
Step 2: Final Capacitor Voltage
After a long time (t → ∞), the capacitor again behaves as an open circuit. Applying KVL,
Step 3: Time Constant
Deactivate independent sources and find equivalent resistance seen by the capacitor.
Step 4: Capacitor Voltage
Given,
✔ 4.1 s
Question 58
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 below.
The voltmeter readings are:
VR = 9 V
VLoad = 199 V
Assuming perfect resistors and ideal voltmeters, the load-active power measured in this experiment, in W, is ______ (round off to one decimal place).
Correct Answer: 79.0 W
Given:
Step 1: Load Current
The resistor is connected in series with the load. Hence,
Step 2: Power Factor
Using the voltage triangle,
Step 3: Load Active Power
✔ Load Active Power = 79.0 W
Correct Answer: 79.0 W
Question 59
In the Wheatstone bridge shown below, the sensitivity of the bridge in terms of change in balancing voltage E for unit change in resistance R, in mV/Ω, is ______.
(Round off to two decimal places)
Correct Answer: 1.96 mV/Ω
From the bridge, the output voltage is:
Using voltage division:
Therefore,
✔ 1.96 mV/Ω
Question 60
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 used, in mH, is ______ (round off to one decimal place).
Correct Answer: 1.8 mH
For a buck converter operating in CCM:
From the waveform:
The capacitor current ripple is approximately:
During ON time:
Considering the closest rounded range from the graph:
✔ 1.8 mH
Question 61
An ideal low pass filter has frequency response given by
0, otherwise }
Let h(t) be its time-domain representation. Then h(0)= ______ (round off to the nearest integer).
Correct Answer: 200
Using inverse Fourier transform:
Since H(jω)=1 only from −200π to 200π:
✔ 200
Question 62
Consider the state-space model
y(t) = Cx(t)
where x(t), r(t), y(t) are the state, input and output, respectively. The matrices A, B, C are given below:
[-2, -3] ]
B = [ 0,
1 ]
C = [1 0]
The sum of the magnitudes of the poles is ______ (round off to the nearest integer).
Correct Answer: 3
The poles of the state-space system are the eigenvalues of matrix A.
Given:
Characteristic equation:
Therefore, the poles are:
Sum of magnitudes:
✔ 3
Question 63
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 bank, in ampere, is ______ (round off to one decimal place).
Correct Answer: 9.1 A
Given:
Active power:
Power-factor angles:
Required capacitor reactive power:
Capacitor line current:
✔ 9.1 A
Question 64
Two units, rated at 100 MW and 150 MW, are enabled for economic load dispatch. The overall incremental cost is 10,000 Rs./MWh. So, the units are dispatched to 50 MW and 80 MW respectively. At an overall incremental cost of 10,600 Rs./MWh, the power output of the units are 80 MW and 92 MW respectively. The total plant output, without overloading any unit, at an overall incremental cost of 11,800 Rs./MWh is ______ MW.
(Round off to the nearest integer)
Correct Answer: 216 MW
For unit 1, incremental cost changes from 10000 to 10600 when power changes from 50 MW to 80 MW.
At IC = 11800:
But unit 1 rating is 100 MW, so:
For unit 2:
At IC = 11800:
Total plant output:
✔ 216 MW
Question 65
A controller D(s) of the form (1 + KDs) is to be designed for the plant
as shown in the figure. The value of KD that yields a phase margin of 45° at the gain crossover frequency of 10 rad/sec is ______ (rounds)
Correct Answer: 0.1
Open-loop transfer function:
At gain crossover frequency:
Magnitude of plant at ω = 10:
For gain crossover:
Now,
✔ KD = 0.1
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