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GATE EE 2026 Set 1
All 65 solved GATE EE 2026 Set 1 questions in exam order. Open a question, commit to an answer, and learn from the step-by-step solution. One question at a time.
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General Aptitude (GA)
101
Q1MCQ1 markEasy‘The shopkeeper sells lemons.’ In this sentence, the word ‘lemons’ is the ________.Think it through. Then check your answer.Question
‘The shopkeeper sells lemons.’
In this sentence, the word ‘lemons’ is the ________.Correct answer
(A) object
Step-by-step solution
In the sentence "The shopkeeper sells lemons", "The shopkeeper" is the subject (the doer of the action), "sells" is the transitive verb, and "lemons" is the direct object (the entity receiving the action).2
Q2MCQ1 markEasyThe figure below is supposed to show three non-overlapping shapes – one oval and two triangles. Which one of the following figures P, Q, R, or S fits the missing portion indicated…Think it through. Then check your answer.Question
The figure below is supposed to show three non-overlapping shapes – one oval and two triangles. Which one of the following figures P, Q, R, or S fits the missing portion indicated by ‘?’ and completes the oval and the two triangles?
Correct answer
(A) P [figure]
Step-by-step solution
The missing square must supply three pieces, one for each shape:- Oval (top-left): its lower-right part is cut off, so the square's top-left corner needs a curved segment that is the lower part of an oval.
- Upper triangle (top-right): only the tip of its lower-left portion is missing, so the square's top-right corner needs a thin wedge that slopes down to the right edge.
- Lower shape (bottom): the dark piece below the square is the base of a triangle whose top is missing, so the square's bottom-right part needs a small triangle with its apex pointing up and its base on the bottom edge.
- Q has a right triangle at the bottom-left, which does not continue the lower triangle's sloping sides.
- R has a downward-pointing triangle at the top-right, which does not match the wedge needed for the upper triangle.
- S has a large D-shaped half oval, which is too big for the missing part of the oval.
- P has all three correct pieces: the oval segment, the wedge and the upward triangle.
3
Q3MCQ1 markEasyAt how many points will the curves and intersect in the real plane?Think it through. Then check your answer.Question
At how many points will the curves and intersect in the real plane?Correct answer
(A) 0
Step-by-step solution
To find the intersection points, we equate the two equations:
Now, we check the discriminant () of this quadratic equation :
Since the discriminant is negative (), the equation has no real roots. Therefore, the curves do not intersect at any point in the real plane.4
Q4MCQ1 markEasy‘If Anish had scored hundred runs in today’s match, he would have been made the captain of his team. He would have then become the youngest captain in his team’s history.…Think it through. Then check your answer.Question
‘If Anish had scored hundred runs in today’s match, he would have been made the captain of his team. He would have then become the youngest captain in his team’s history. Unfortunately, he got out without scoring any runs. Hence, there won’t be any change in the captaincy for now.’ Based on the paragraph above, which one of the following statements is true?Correct answer
(C) The current captain is older than Anish.
Step-by-step solution
The passage says that if Anish had become captain, he would have become the youngest captain in his team's history. That can only be true if every captain the team has had so far, including the current one, was older than Anish.Anish scored no runs, so the captaincy does not change and the current captain stays on. Therefore the current captain is older than Anish.The other options fail:- (A) He got out without scoring, so he did not make a hundred.
- (B) If he were already captain, he could not newly "be made the captain".
- (D) The passage only talks about him being the youngest captain, not the youngest player.
5
Q5MCQ1 markEasyWhich one of the following figures P, Q, R, or S, correctly shows the clockwise-rotated version of figure (I)? [figure] [figure]Think it through. Then check your answer.Question
Which one of the following figures P, Q, R, or S, correctly shows the clockwise-rotated version of figure (I)?
Correct answer
(B) Q
Step-by-step solution
Rotating figure (I) by in the clockwise direction results in figure Q. The orientation of the internal blocks and the overall tilt match figure Q.6
Q6MCQ2 marksEasyMatch the words in Column I with their synonyms in Column II. | Column I | Column II | | :--- | :--- | | (i) Lonely | (p) Verbatim | | (ii) Literal | (q) Solitary | | (iii) Lousy…Think it through. Then check your answer.Question
Match the words in Column I with their synonyms in Column II.Column I Column II (i) Lonely (p) Verbatim (ii) Literal (q) Solitary (iii) Lousy (r) Deadly (iv) Lethal (s) Terrible Correct answer
(A) (i)-(q); (ii)-(p); (iii)-(s); (iv)-(r)
Step-by-step solution
The correct matches for the synonyms are:- (i) Lonely: Solitary (q)
- (ii) Literal: Verbatim (p)
- (iii) Lousy: Terrible (s)
- (iv) Lethal: Deadly (r)
7
Q7MCQ2 marksMediumIn the given figure, is the diameter of a circle with center . Two points and are chosen on the circle such that . When…Think it through. Then check your answer.Question
In the given figure, is the diameter of a circle with center . Two points and are chosen on the circle such that . When and are extended, they meet at . The value of is _________________
Correct answer
(B) 50^
Step-by-step solution
1.Since is the diameter of the circle, and (angles in a semicircle).2.Points are collinear and are collinear. Therefore, in , .3.The angle subtended by arc at the center is . The angle subtended by the same arc at the circumference is .4.In the right-angled triangle , the sum of angles is :5.Thus, .8
Q8MCQ2 marksMediumBased on the relationship between each polygon and the number inside it, the value of ‘X’ is ______ [figure]Think it through. Then check your answer.Question
Based on the relationship between each polygon and the number inside it, the value of ‘X’ is ______
Correct answer
(A) 720
Step-by-step solution
The relationship between the number of sides of the polygon () and the number inside it is given by the factorial of the number of sides ():- Triangle ():
- Square ():
- Pentagon ():
- Hexagon ():
9
Q9MCQ2 marksMediumConsider 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…Think it through. Then check your answer.Question
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 of the bulbs are changed based on the following rules:- Any OFF bulb with exactly one ON neighbor at the end of the previous Step is turned ON.
- Any ON bulb with both neighbors ON at the end of the previous Step is turned OFF.
- The state of any bulb not meeting the conditions above is left unchanged.
Correct answer
(B) 4
Step-by-step solution
Write ON as 1 and OFF as 0, with bulbs B1 to B7. The end bulbs B1 and B7 each have only one neighbor, so they can never have "both neighbors ON".- Initial: 0 0 0 1 0 0 0
- Step 1: B3 and B5 are OFF with exactly one ON neighbor (B4), so they turn ON. B4 has both neighbors OFF, so it stays ON.
- Step 2: B2 and B6 have exactly one ON neighbor, so they turn ON. B4 is ON with both neighbors ON, so it turns OFF.
- Step 3: B1 and B7 have exactly one ON neighbor, so they turn ON. B4 is OFF but has two ON neighbors, so it stays OFF. B3 and B5 each have one OFF neighbor (B4), so they stay ON.
- Step 4: B2 and B6 are ON with both neighbors ON, so they turn OFF. B3 and B5 still have B4 OFF beside them, so they stay ON. The end bulbs B1 and B7 stay ON.
- Step 5: Every OFF bulb (B2, B4, B6) has two ON neighbors, not exactly one, so none turns ON. Every ON bulb has no ON neighbor, so none turns OFF.
10
Q10MCQ2 marksEasyand are two positive integers such that . The product of the numbers and is _________Think it through. Then check your answer.Question
and are two positive integers such that . The product of the numbers and is _________Correct answer
(D) 42
Step-by-step solution
Given . Since 13 is a prime number and are positive integers, the only factors of 13 are 1 and 13. Also, for positive integers . Therefore, we must have:1.2.Adding the two equations: .
Subtracting the two equations: .The product of the numbers and is . Thus, option (D) is correct.
Electrical Engineering (EE)
5511
Q11MCQ1 markMediumConsider the infinite-length, discrete-time sequence , where is an integer. The region of convergence of its Z-transform is given by: (Note: is a…Think it through. Then check your answer.Question
Consider the infinite-length, discrete-time sequence , where is an integer. The region of convergence of its Z-transform is given by:
(Note: is a complex variable)Correct answer
(C) 0.9 < z < 1/0.9
Step-by-step solution
The sequence is . This can be split into two parts:
The Z-transform is defined as .1.For the right-sided part , the Z-transform is . This geometric series converges if , which implies .2.For the left-sided part , the Z-transform is . Let , then the sum becomes . This geometric series converges if , which implies .The overall Region of Convergence (ROC) is the intersection of these two individual ROCs: . Thus, option (C) is correct.12
Q12MCQ1 markMediumLet be any continuous-time periodic signal with period . It is sampled uniformly with a sampling period where , resulting in the discrete sequence…Think it through. Then check your answer.Question
Let be any continuous-time periodic signal with period . It is sampled uniformly with a sampling period where , resulting in the discrete sequencewhere is an integer.
Which one of the following statements is correct about ?Correct answer
(D) x[n] will be periodic if and only if T/Tₛ is a rational number
Step-by-step solution
A discrete-time signal is periodic if there exists a positive integer such that for all .
For the sampled signal , the condition for periodicity is:
Since is a continuous-time periodic signal with period , this condition is satisfied if the time interval is an integer multiple of the fundamental period :
for some integer .Rearranging this gives:
Since and are both integers, their ratio is a rational number. Therefore, is periodic if and only if the ratio of the signal period to the sampling period, , is a rational number. Thus, option (D) is correct.13
Q13MCQ1 markMediumThe Laplace transform of the step response of a system is given by The rise time is defined as the time taken for the response to go from 0.1 to…Think it through. Then check your answer.Question
The Laplace transform of the step response of a system is given by The rise time is defined as the time taken for the response to go from 0.1 to 0.9 of its final value. The settling time is defined as the time taken for the response to reach 0.98 of its final value.For this system, the rise time (), settling time (), and time constant (), all expressed in seconds, areCorrect answer
(A) Tᵣ = 0.022, Tₛ = 0.04, T_c = 0.01
Step-by-step solution
Given the step response .
The transfer function is obtained by dividing the step response by the input :
.
This is a first-order system with time constant s.
The time-domain response is .
The final value is .1.Rise time ():
s.2.Settling time ():s.
Thus, .14
Q14MCQ1 markEasyConsider the following differential equation: Which one of the following options is correct?Think it through. Then check your answer.Question
Consider the following differential equation:Which one of the following options is correct?Correct answer
(A) It is a linear differential equation
Step-by-step solution
The given equation is .1.Linearity: A differential equation is linear if the dependent variable and its derivatives appear only to the first power and are not multiplied together. Here, , , and all satisfy this. The coefficients () are functions of the independent variable only. Thus, it is a linear differential equation.2.Time-Invariance: The coefficients depend on , making it a time-varying system.3.Order/Type: It is a second-order ordinary differential equation (ODE), not a partial differential equation (PDE).Therefore, option (A) is correct.15
Q15MCQ1 markEasyA uniform ring charge of radius carries a total charge . Which one of the following options correctly quantifies the magnitude of the force on a point charge of strength…Think it through. Then check your answer.Question
A uniform ring charge of radius carries a total charge . Which one of the following options correctly quantifies the magnitude of the force on a point charge of strength kept at the center of the ring? ( is the permittivity of the medium)Correct answer
(C) 0
Step-by-step solution
By symmetry, the electric field at the center of a uniformly charged ring is zero. For every infinitesimal charge element on the ring, there is a diametrically opposite element that produces an equal and opposite electric field vector at the center. The vector sum of all such fields is zero.
Since the electric field at the center, the force on a point charge placed there is .16
Q16MCQ1 markEasyA positive point charge with velocity enters a region having electric field and magnetic field . Which one of…Think it through. Then check your answer.Question
A positive point charge with velocity enters a region having electric field and magnetic field . Which one of the following statements is correct for the force on the charge as it enters the region?Correct answer
(B) The force will be along the electric field but perpendicular to the magnetic field
Step-by-step solution
The Lorentz force on a charge is given by .
Given:
Calculating the magnetic force component:
Calculating the electric force component:
Total force:
The total force is in the direction of the electric field (along ) and is perpendicular to the magnetic field (along ). Thus, option (B) is correct.17
Q17MCQ1 markMediumA 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?Think it through. Then check your answer.Question
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?Correct answer
(C) 90 kVA
Step-by-step solution
For a two-winding transformer connected as an autotransformer, the power rating increases.
Given:
Two-winding rating kVA
Voltages V, V
Autotransformer voltages kV V, kV VThis configuration implies an additive polarity connection where V.
The rating of the autotransformer is related to the two-winding rating by:
kVA.
Thus, option (C) is correct.18
Q18MCQ1 markMediumConsider a power system with buses, of which are generator buses and the remaining are load buses (where there is no generation). Assume that there are no reactive…Think it through. Then check your answer.Question
Consider a power system with buses, of which are generator buses and the remaining are load buses (where there is no generation).
Assume that there are no reactive power-limit violations at the generator buses.
What is the size of the Jacobian matrix in the Newton-Raphson load flow method?Correct answer
(B) (2N - 1 - P) × (2N - 1 - P)
Step-by-step solution
In the Newton-Raphson load flow method, the size of the Jacobian matrix is determined by the number of unknown variables (voltage phase angles and magnitudes).
For a system with total buses:1.One bus is designated as the slack bus. Both and are known. No equations are needed.2. are generator buses. Assuming includes the slack bus, there are PV buses. For PV buses, is known, but is unknown. Number of unknowns = .3. are load buses (PQ buses). Both and are unknown. Number of unknowns = .Total number of unknowns (and thus the size of the Jacobian) is:
Size
Since , we have .
Size .
Thus, the Jacobian matrix is of size . Option (B) is correct.19
Q19MCQ1 markMediumThe initial three-phase voltage phasors (, , and ) at a bus of a power network are as shown in Case-1. Due to a disturbance, the bus voltage…Think it through. Then check your answer.Question
The initial three-phase voltage phasors (, , and ) at a bus of a power network are as shown in Case-1. Due to a disturbance, the bus voltage phasors changed in phase by a small angle , and the magnitudes remained the same as depicted in Case-2.Which one of the following statements is correct about the zero sequence
components?Correct answer
(D) The zero sequence components in Case-1 and Case-2 have the same magnitude but different phase angles
Step-by-step solution
The zero sequence component is defined as . In Case-2, each phasor is rotated by a small angle , which can be represented mathematically as multiplying each phasor by . Thus, the new phasors are , , and . The new zero sequence component is:This shows that the magnitude remains the same, but the phase angle of the zero sequence component changes by . Therefore, the zero sequence components in Case-1 and Case-2 have the same magnitude but different phase angles.20
Q20MCQ1 markEasyA certain application requires power at a frequency of , while the available grid frequency is . A 3-phase synchronous motor connected to the…Think it through. Then check your answer.Question
A certain application requires power at a frequency of , while the available grid frequency is . A 3-phase synchronous motor connected to the , 3-phase grid, driving a synchronous generator, is used for this application.Which one of the following combinations is a suitable choice for the number of poles in the motor and generator, respectively?Correct answer
(B) 6, 2
Step-by-step solution
The synchronous motor and synchronous generator are coupled on the same shaft, meaning they must rotate at the same speed . The synchronous speed is given by . For the motor and generator to have the same speed:Given and :Checking the options, only (B) provides a ratio of 3 ().21
Q21MCQ1 markEasyWhich one of the following options is correct regarding the typical double-squirrel-cage structure used in induction motors?Think it through. Then check your answer.Question
Which one of the following options is correct regarding the typical double-squirrel-cage structure used in induction motors?Correct answer
(A) At starting, a larger portion of the rotor current flows in the outer cage
Step-by-step solution
In a double-squirrel-cage induction motor, the outer cage is designed with high resistance and low reactance, while the inner cage has low resistance and high reactance. At starting, the rotor frequency is high (equal to the supply frequency), which makes the reactance of the inner cage very high. Consequently, the rotor current is forced to flow primarily through the outer cage. This high-resistance path provides the high starting torque required. At rated speed, the slip and rotor frequency are low, making the reactance of the inner cage small, so the current flows through the low-resistance inner cage for efficient operation.22
Q22MCQ1 markMediumThe figure shows the single-line diagram of a synchronous generator delivering of power at unity power factor to an infinite bus. [figure] denotes the…Think it through. Then check your answer.Question
The figure shows the single-line diagram of a synchronous generator delivering of power at unity power factor to an infinite bus.denotes the stator current phasor. If the field excitation is increased, which one of the following options correctly describes its effect on the stator current, power factor, and load angle of the machine?
Correct answer
(C) Stator current increases, power factor becomes lagging, load angle decreases
Step-by-step solution
For a synchronous generator connected to an infinite bus, the real power delivered is given by . Since and are constant, if the field excitation is increased, the load angle must decrease to keep constant. Initially, the generator is at unity power factor. Increasing the excitation makes the generator overexcited, causing it to supply lagging reactive power to the infinite bus, thus the power factor becomes lagging. The stator current is given by . As the power factor decreases from 1, the stator current must increase to maintain the constant real power . Therefore, the stator current increases, the power factor becomes lagging, and the load angle decreases.23
Q23MCQ1 markEasyA circuit with ideal elements is shown. [figure] Which one of the following options correctly identifies all the linear elements in the circuit?Think it through. Then check your answer.Question
A circuit with ideal elements is shown.Which one of the following options correctly identifies all the linear elements in the circuit?
Correct answer
(B) R, L, and C only
Step-by-step solution
Linear elements are those that satisfy the properties of superposition and homogeneity. In electrical circuits, resistors (), inductors (), and capacitors () are considered linear elements because their voltage-current relationships are linear differential or algebraic equations (, , ). A diode () is a non-linear element because its V-I characteristic is exponential and does not follow a linear relationship. Thus, the linear elements are , , and only.24
Q24MCQ1 markMediumFor the circuit shown, which one of the following options correctly identifies the Thevenin’s equivalent parameters between nodes Y and Z? [figure]Think it through. Then check your answer.Question
For the circuit shown, which one of the following options correctly identifies the Thevenin’s equivalent parameters between nodes Y and Z?
Correct answer
(B) V_(TH) = 140 V, R_(TH) = 0 Ω
Step-by-step solution
To find the Thevenin equivalent parameters between nodes Y and Z, we calculate the open-circuit voltage () and the equivalent resistance () with sources deactivated.1. Finding :
We need to find the voltage difference . Let node W be the reference node ().- From the source between X and W, .
- From the source between X and Y (with the positive terminal at Y), .
- From the source between Z and W (with the positive terminal at W), .
- Therefore, .
To find , we deactivate all independent sources. Ideal voltage sources are replaced by short circuits.- Shorting the source between X and Y creates a short between nodes X and Y.
- Shorting the source between X and W creates a short between nodes X and W.
- Shorting the source between Z and W creates a short between nodes Z and W.
- Consequently, there is a continuous short-circuit path from node Y to node Z (Y X W Z). Any resistors in parallel with this path are bypassed.
- Thus, the equivalent resistance between Y and Z is .
25
Q25MCQ1 markMediumRefer to the four circuits shown. Which one of the following options for and makes all of them realizable? [figure]Think it through. Then check your answer.Question
Refer to the four circuits shown. Which one of the following options for and makes all of them realizable?
Correct answer
(A) k₁ = 1, k₄ = -1/3, for all values of k₂ and k₃
Step-by-step solution
Each circuit is a single loop of two sources connected across the same pair of nodes, so the two sources share the same voltage and carry the same loop current. A circuit is realizable only if KVL and KCL can both be satisfied without contradiction.1.Top left (two voltage sources in parallel): Both sources have at the top, so KVL forces . With , this requires .2.Top right (voltage source in parallel with a current source): The voltage source fixes the node voltage and the current source fixes the loop current; these do not conflict, so the circuit is realizable for any .3.Bottom left (current source in parallel with a voltage source): The current source fixes the loop current and the voltage source fixes the voltage across both; again no conflict, so any is allowed.4.Bottom right (two current sources in parallel): Both arrows point up, so both currents enter the top node. KCL at that node gives , i.e. . With , this requires , so .Hence all four circuits are realizable exactly when and , with and arbitrary. Options (B), (C) and (D) all have , so the correct choice is option (A).26
Q26MCQ1 markMediumA single-phase voltage source V delivers a current, A to a load. The load power factor, correct up to two…Think it through. Then check your answer.Question
A single-phase voltage source V delivers a current, A to a load. The load power factor, correct up to two decimal places, is:Correct answer
(B) 0.80
Step-by-step solution
1.Average Power (): Only the fundamental component of the current contributes to real power because the voltage only has a fundamental component.W.2.RMS Voltage ():V.3.RMS Current ():A.4.Apparent Power ():VA.5.Power Factor ():.27
Q27MCQ1 markEasyThe 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. [figure] What…Think it through. Then check your answer.Question
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 power loss in the diode?
Correct answer
(C) 18.6 W
Step-by-step solution
1.Determine the diode model parameters:From the graph, the characteristic is a straight line starting at V (where ).
The slope resistance .
The voltage drop across the diode is .2.Calculate Power Loss:For a continuous DC current A:
W.28
Q28MCQ1 markMediumConsider the circuit shown in Figure (a). A gate pulse is applied between time instants and . After , during the MOSFET turn OFF process, it experiences a…Think it through. Then check your answer.Question
Consider the circuit shown in Figure (a). A gate pulse is applied between time instants and . After , during the MOSFET turn OFF process, it experiences a voltage overshoot.Based on the waveforms shown in Figure (b), which one of the following options is correct?
Correct answer
(B) R₁ R₃ R₂
Step-by-step solution
In the given circuit, when the MOSFET turns off, the inductor current freewheels through the diode and the snubber resistor . The voltage across the MOSFET during this period is given by . Neglecting the diode forward voltage drop, . At the instant of turn-off (), the inductor current is at its peak value. Therefore, the peak voltage overshoot is directly proportional to the resistance . From the waveforms in Figure (b), the peak voltage for is the highest, followed by , and then . This implies that .29
Q29MCQ1 markEasyThe asymptotic Bode magnitude plot of a system is shown. [figure] Which one of the following options best represents the transfer function of the system?Think it through. Then check your answer.Question
The asymptotic Bode magnitude plot of a system is shown.Which one of the following options best represents the transfer function of the system?
Correct answer
(A) G(s) = 1 + (s)/(ω₀)(s)/(ω₀)
Step-by-step solution
1.The initial slope of the Bode magnitude plot is dB/decade, which indicates the presence of a pole at the origin ().2.At the corner frequency , the slope changes from dB/decade to dB/decade. This change in slope is dB/decade, which indicates a simple zero at , represented by the factor .3.The general form of the transfer function is .4.As , the magnitude levels off at dB (which corresponds to a gain of ). In the limit , . Setting this to gives .5.Substituting back into the transfer function: . This matches option (A).30
Q30MSQ1 markMediumTwo matrices and have a common eigenvalue , and the same corresponding nonzero eigenvector. Which of the following options is/are correct? (Note: is…Think it through. Then check your answer.Question
Two matrices and have a common eigenvalue , and the same corresponding nonzero eigenvector.Which of the following options is/are correct?(Note: is the identity matrix.)Correct answer
(A) Determinant (A - 2I) = 0; (B) Determinant (B - 2I) = 0; (D) Determinant (A + B - 4I) = 0
Step-by-step solution
Let be the common eigenvector such that and .- Option (A): Since is an eigenvalue of , the matrix is singular, meaning its determinant is zero. Thus, is correct.
- Option (B): Similarly, since is an eigenvalue of , is correct.
- Option (C): Consider . Since , is not in the null space of . This does not guarantee that the determinant is zero.
- Option (D): Consider . Since there exists a non-zero vector such that , the matrix is singular, and its determinant must be zero. Thus, is correct.
31
Q31MSQ1 markMediumA 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…Think it through. Then check your answer.Question
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 Hz. Which of the following statements is/are correct?Correct answer
(A) No-load current drawn would be smaller for operation in the USA compared to that in India
Step-by-step solution
1.Peak Magnetic Flux Density (): .- In India: .
- In USA: .
2.No-load Current (): The magnetizing component of no-load current . Since is significantly lower in the USA, the no-load current will be smaller. Statement (A) is correct.3.Eddy Current Losses (): (assuming constant core volume and material).- and .
- .
4.Total Losses: Hysteresis losses . Since both is lower and is lower, core losses are significantly reduced. For the same load current, copper losses remain the same. Thus, total losses in the USA will be lower than in India. Statement (B) is incorrect.32
Q32NAT1 markEasyGiven that , the integral over the unit sphere centered at the…Think it through. Then check your answer.Question
Given that , the integralover the unit sphere centered at the origin evaluates to ________ (Round off to one decimal place)Correct answer
0 to 0
Step-by-step solution
Using Gauss's Divergence Theorem:Calculate the divergence of :Since the divergence is zero everywhere, the surface integral over any closed surface (including the unit sphere) is zero.
Result = 0.033
Q33NAT1 markMediumIn the linear regulator circuit shown, the base to emitter voltage of the BJT is . The Zener diode clamps the base voltage to . Ignore the…Think it through. Then check your answer.Question
In the linear regulator circuit shown, the base to emitter voltage of the BJT is . The Zener diode clamps the base voltage to . Ignore the biasing current of the Zener diode and BJT.The maximum possible efficiency of the regulator circuit is ___________ % (Round off to one decimal place)
Correct answer
47.5 to 48
Step-by-step solution
1.Output voltage: The Zener diode holds the base at , so the emitter (output) voltage is2.Output power: With ,3.Input power: Ignoring the base current, the transistor's collector current equals its emitter current, , drawn from the source. Since the Zener and base bias currents are to be ignored, the input current is just :This is the maximum possible efficiency, since any bias current only adds to the input power.4.Effect of the bias resistor: Strictly, the resistor carries . Including it gives and . Because the question says to ignore the bias currents, both readings are accepted, so any value from to is correct.Answer:34
Q34NAT1 markMediumConsider the circuit shown. Assume that the diode is ideal. The supply voltage V, , and . The peak diode current…Think it through. Then check your answer.Question
Consider the circuit shown. Assume that the diode is ideal. The supply voltage V, , and . The peak diode current (in amperes) is ______________ (Round off to one decimal place)
Correct answer
32.3 to 32.5
Step-by-step solution
The given circuit is a half-wave rectifier with an load. The current during the conduction period of the diode is governed by the differential equation:where:- V
- Hz rad/s
- H
35
Q35NAT1 markEasyis an skew-symmetric matrix with real-valued entries, and is an -dimensional column vector with real-valued entries such that . The quantity…Think it through. Then check your answer.Question
is an skew-symmetric matrix with real-valued entries, and is an -dimensional column vector with real-valued entries such that . The quantity evaluates to _________
(Answer in integer)Correct answer
0 to 0
Step-by-step solution
For a skew-symmetric matrix , we have .
Let . Since is a scalar ( matrix), .Thus, , which implies , so .36
Q36MCQ2 marksMediumA time-limited waveform is specified as follows: A new waveform…Think it through. Then check your answer.Question
A time-limited waveform is specified as follows:A new waveform is constructed from as follows:The sum of the coefficients of the third harmonics of the sine and cosine terms in the trigonometric Fourier series expansion of is .
What is the value of ?Correct answer
(B) (1)/(2)
Step-by-step solution
The waveform is a periodic square wave with period . Since is an odd function, all cosine coefficients are zero ().
The sine coefficients are given by:For the third harmonic ():The sum of the coefficients of the third harmonic is .
Given that this sum is :37
Q37MCQ2 marksMediumIn the circuit shown, the open loop gain of the operational amplifier is . What is the voltage gain of the circuit? (Round off to two decimal places) [figure]Think it through. Then check your answer.Question
In the circuit shown, the open loop gain of the operational amplifier is . What is the voltage gain of the circuit?(Round off to two decimal places)
Correct answer
(A) -16.67
Step-by-step solution
The given circuit is an inverting amplifier with a finite open-loop gain . The resistors are and .The exact voltage gain for an inverting amplifier with finite open-loop gain is given by:Substituting the given values:Rounding off to two decimal places, we get . Thus, the correct option is (A).38
Q38MCQ2 marksHardThree single-phase transformers are connected to form a three-phase transformer bank with connections as shown. [figure] Considering ABC phase…Think it through. Then check your answer.Question
Three single-phase transformers are connected to form a three-phase transformer bank with connections as shown.Considering ABC phase sequence, the vector group of the transformer is:
Correct answer
(D) Dd10
Step-by-step solution
1.HV Side (Delta): Phase A is connected to the dot of the first winding, and Phase B is connected to the non-dot. Thus, the voltage across the first HV winding is .2.LV Side (Delta): Terminal 'a' is connected to the dot of the first winding, and terminal 'c' is connected to the non-dot. Thus, the voltage across the first LV winding is .3.Phase Relationship: Since the windings are on the same core, (LV) is in phase with (HV). Let (Reference at 12 o'clock). Then .4.Line Voltage Calculation: In a balanced delta, . For the second winding, terminal 'b' is at the dot and 'a' is at the non-dot, so is in phase with . Since , we have , which implies .5.Vector Group: The LV line voltage leads the HV line voltage by . In clock notation, a lead corresponds to 10 o'clock. Therefore, the vector group is Dd10.39
Q39MCQ2 marksHardIn the circuit shown, the phase currents are A A A [figure] Given that the CTs are ideal with no…Think it through. Then check your answer.Question
In the circuit shown, the phase currents are
A
A
AGiven that the CTs are ideal with no saturation, and the turns ratio of the Main CT is 300:5 and that of the Auxiliary Transformer (Yn) is 2:1 on every phase, the value of , rounded off to three decimal places, is:
Correct answer
(D) 8.954 4.105^o A
Step-by-step solution
The main CTs have a turns ratio of , i.e. a current ratio of , so each CT secondary carries its line current divided by 60.Step 1 - zero-sequence component of the line currentsStep 2 - what the auxiliary transformer doesThe three relay burdens are star-connected with an isolated star point, so the three relay currents must sum to zero: no zero-sequence current can flow through the relays. The auxiliary transformer is connected from the three CT secondary leads to earth, its primary star point being solidly earthed and its secondary forming a closed delta. For zero-sequence currents this is a complete path (equal currents in the three star windings return through the earthed neutral and are balanced by a current circulating in the closed delta), so the zero-sequence part of the CT secondary currents is diverted into the auxiliary transformer. For positive- and negative-sequence currents the closed delta offers no balancing path, so (with an ideal, unmagnetised core) the auxiliary transformer draws no such current and they pass on unchanged to the relays. The relay in phase A therefore carries the phase-A CT secondary current with its zero-sequence part removed:(The ratio only sets how large the circulating delta current is; it does not change .)Step 3 - evaluateCarried to three decimals, the given data produce A, which differs from option (D), A, only in the last digit, owing to rounding in the listed currents. Option (D) is the only choice with this magnitude and angle.The other options: (C), A, is (approximately) this current before the CT ratio is applied; (B), A, is close to the zero-sequence part A, which goes into the auxiliary transformer and not the relay; and (A), A, would require the relay current to vanish, which happens only if were purely zero-sequence.Answer: (D) A40
Q40MCQ2 marksMediumThe operating characteristic of a reactance relay is given by , where is the reactance calculated by the relay. Its operating characteristic in the admittance…Think it through. Then check your answer.Question
The operating characteristic of a reactance relay is given by , where is the reactance calculated by the relay. Its operating characteristic in the admittance plane (- plane, where and denote conductance and susceptance, respectively, expressed in ) is given by:Correct answer
(A) G² + (B + 0.5)² ≥ (1)/(4)
Step-by-step solution
The reactance of a component with admittance is given by the imaginary part of the impedance .Thus, the reactance is .
The given operating characteristic is :Completing the square for the terms:This represents the region outside a circle centered at with radius in the - plane.41
Q41MCQ2 marksHardA three-phase two-winding transformer has a voltage transformation ratio , where is the primary side voltage in p.u., and is the…Think it through. Then check your answer.Question
A three-phase two-winding transformer has a voltage transformation ratio , where is the primary side voltage in p.u., and is the secondary side voltage in p.u. and represent the currents injected into the primary and secondary sides of the transformer, respectively. The admittance corresponding to the leakage impedance of the transformer referred to the secondary is p.u. Neglect the magnetizing branch.The bus representation of this transformer is:
Correct answer
(D) bmatrix I_P \ I_S bmatrix = bmatrix yₜ & -(yₜ)/(0.866 - j0.5) \ -(yₜ)/(0.866 + j0.5) & yₜ bmatrix bmatrix V_P \ V_S bmatrix
Step-by-step solution
For a phase-shifting transformer with a complex turns ratio and series admittance referred to the secondary side, the -bus matrix is given by:Given , we find its magnitude and conjugate:Substituting these into the matrix:This matches Option D.42
Q42MCQ2 marksMediumAn electrical component has voltage drop , when the current through it is . What is the average power dissipated over a…Think it through. Then check your answer.Question
An electrical component has voltage drop , when the current through it is . What is the average power dissipated over a half cycle corresponding to ?Correct answer
(C) (Vₘ Iₘ)/(2) cos θ
Step-by-step solution
The instantaneous power is the product of instantaneous voltage and current:Using the trigonometric identity , we get:The average power over a half cycle is:Since and , the sine terms cancel out:This is the same as the average power over a full cycle.43
Q43MCQ2 marksMediumThe electrical network shown has an independent voltage source () and a current source (). The voltage across the capacitor at time…Think it through. Then check your answer.Question
The electrical network shown has an independent voltage source () and a current source (). The voltage across the capacitor at time instants (in seconds) , , and , respectively, is:
Correct answer
(B) 8.00 V, 26.36 V, 28.00 V
Step-by-step solution
The circuit consists of a voltage source, a current source, a resistor, a capacitor, and a resistor.1. Find :
At , the current source is (since for ). The capacitor acts as an open circuit. The circuit consists of the source, resistor, and resistor. The capacitor voltage is the voltage across the resistor.
.
Since the voltage across a capacitor cannot change instantaneously, .2. Find :
At , the current source is (since for ). The capacitor acts as an open circuit (steady state for DC). We can use superposition to find .- Due to source (current source open):
- Due to source (voltage source short):
.
The voltage across this parallel combination is .- Total voltage:
For , the capacitor voltage follows the transient response: .
First, calculate the Thevenin resistance seen by the capacitor. Turn off all independent sources (voltage source short, current source open).
.
The time constant .
Now, substitute the values into the transient response equation:
.
At :
.
Using ,
.Thus, the voltages are , , and .The final answer is44
Q44MCQ2 marksMediumThe terminal voltage and current of a linear electrical network shown in Figure (a) are given in the table. | Terminal voltage () | Terminal current () |…Think it through. Then check your answer.Question
The terminal voltage and current of a linear electrical network shown in Figure (a) are given in the table.Terminal voltage () Terminal current () 18 V A 30 V 0.5 A 36 V 1.0 A The correct choice for the parameters (, ) of the Norton equivalent circuit shown in Figure (b) is:
Correct answer
(C) I_N = 2.0 A, R_N = 12.0 Ω
Step-by-step solution
In Figure (a) the terminal current is marked into the network at terminal X, and is the terminal voltage with its sign at X. A linear network therefore obeys a straight-line terminal relationStep 1 - find and from two data pointsSubtracting the first from the second:Step 2 - check with the third data pointwhich is exactly the measured value, so all three readings lie on the same line.Step 3 - convert to the Norton form of Figure (b)The other options are inconsistent with the slope of the measured - line (which fixes ) or with its intercept (which fixes V).Answer: (C) A,45
Q45MCQ2 marksMediumThe digital circuit shown has 3 inputs (, , and ). [figure] The simplified logical expression for the output (OUT) is:Think it through. Then check your answer.Question
The digital circuit shown has 3 inputs (, , and ).The simplified logical expression for the output (OUT) is:
Correct answer
(B) 0
Step-by-step solution
Read the circuit from the figure:- The 3-input NOR gate takes , , and gives .
- The NOT gate inverts , giving ; its output feeds both AND gates (note the junction dot after the inverter).
- The upper AND gate takes and : output .
- The lower AND gate takes and : output .
- The OR gate combines them: .
- The final AND gate multiplies the NOR output by the OR output.
46
Q46MCQ2 marksMediumThe MOSFET switches shown in the circuit are ideal. [figure] Which of the following is the correct option for Boolean logical expression of the output (OUT), and the maximum…Think it through. Then check your answer.Question
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 (P) consumed by the circuit?
Correct answer
(D) OUT = AB C, P = 7.5 mW
Step-by-step solution
Step 1 - the right-hand branch fixes node The resistor pulls node up to V and the NMOS gated by pulls it down to 0. This is a resistive-load NMOS inverter, soStep 2 - the output branch is pulled up through the resistor. Its pull-down network is the series pair of NMOS devices driven by and , in parallel with a third NMOS whose gate is node . is pulled low when either path conducts, i.e. when or when :Step 3 - maximum powerA branch draws current from the supply only while its pull-down path conducts:Both branches conduct together for : turns on the device of the second branch, and turns on the series pair of the first branch. HenceStep 4 - pick the optionOptions (A) and (D) carry the same logic function, since , but (A) quotes mW, which ignores the current drawn by the second branch. Options (B) and (C) do not match the derived function.Answer: (D) , mW47
Q47MCQ2 marksMediumConsider the single-phase voltage source inverter circuit feeding an inductive load (). Assume that the power MOSFET switches are ideal. and are switched on during…Think it through. Then check your answer.Question
Consider the single-phase voltage source inverter circuit feeding an inductive load (). Assume that the power MOSFET switches are ideal. and are switched on during the first , and and are switched on during the next in a switching cycle. The switches in the same leg are thus switched in a complementary fashion. Neglect the dead time. The waveform of the inductor current () in the steady state is triangular with a peak value of as shown.The rms value of the current through the switch is:
Correct answer
(B) 2.04 A
Step-by-step solution
When and are on, the inductor current flows through ; when and are on, it flows through them and .Step 1 - read the waveformThe switching period is , and conducts during , in which interval rises linearly from A to A:Step 2 - set up the rms integral over the full periodStep 3 - evaluateSubstituting (so and runs from to ):Equivalently: the rms of the full triangle of peak 5 A is A, and conducting for only half the period divides the mean square by 2, giving A - so option (A) is the value for continuous conduction and (C), (D) do not arise.Answer: (B) A48
Q48MCQ2 marksHardConsider the boost converter circuit shown. Assume that the semiconductor devices are ideal. In steady state, the inductor current rises linearly from to…Think it through. Then check your answer.Question
Consider the boost converter circuit shown. Assume that the semiconductor devices are ideal. In steady state, the inductor current rises linearly from to in the first and then falls linearly from to in the next of every switching cycle as shown. The load resistance is and the capacitance is .Neglect the ripple in the output voltage. What is the input voltage ?
Correct answer
(C) 7.5 V
Step-by-step solution
Step 1 - identify the modeThe inductor current starts from A, peaks at A at and returns to A at , so the converter works at the boundary of continuous conduction withStep 2 - average load currentIn a boost converter the diode carries the inductor current only during , and (the output ripple being negligible) all of its average flows into the load:Step 3 - output voltageStep 4 - volt-second balance on the inductor(The same result follows from with : V.)Answer: (C) V49
Q49MCQ2 marksEasyWhich one of the following statements is ALWAYS correct about a collection of column vectors, each having real-valued entries?Think it through. Then check your answer.Question
Which one of the following statements is ALWAYS correct about a collection of column vectors, each having real-valued entries?Correct answer
(A) If p n, then the column vectors must be linearly dependent
Step-by-step solution
In a vector space of dimension (like ), the maximum number of linearly independent vectors is . Therefore, any set containing more than vectors () must be linearly dependent.50
Q50MCQ2 marksEasyConsider the second-order differential equation with initial conditions …Think it through. Then check your answer.Question
Consider the second-order differential equation with initial conditionsThe solution is given byCorrect answer
(A) y(x) = exp(-x/2) (cos (√(3)x2) + √(3) sin (√(3)x2))
Step-by-step solution
The characteristic equation is . The roots are .
The general solution is .
Using : .
Differentiating: .
Using : .
Thus, .51
Q51MCQ2 marksHardThe figure shows an arbitrarily shaped planar conducting loop A in the XY plane. Two nonintersecting regions with areas and within the loop are subjected to magnetic…Think it through. Then check your answer.Question
The figure shows an arbitrarily shaped planar conducting loop A in the XY plane. Two nonintersecting regions with areas and within the loop are subjected to magnetic fields , and , respectively.What is the expression for the induced rms voltage in loop A?
Correct answer
(A) √((a₁² ω² m² + 4 a₂² ω² n²)/(4))
Step-by-step solution
The loop is in the XY plane, so the area vector is .
Flux .
.
Induced voltage .
Since the frequencies and are different, the total RMS voltage is .
.
.
.52
Q52MCQ2 marksMediumA system is characterized by the following state equation and output equation (: input, : state vector, : output)…Think it through. Then check your answer.Question
A system is characterized by the following state equation and output equation (: input, : state vector, : output)What are the values of and for which the poles of the transfer function are at and ?Correct answer
(D) a = -4, b = -3.25
Step-by-step solution
The poles of the transfer function are the eigenvalues of the state matrix .
The characteristic equation is given by :The desired poles are at . The corresponding characteristic equation is:Comparing the coefficients of the two equations:
1)
2)
Thus, and .53
Q53MCQ2 marksMediumA system is represented in state-space form as follows: (: input, : state vector, : output)…Think it through. Then check your answer.Question
A system is represented in state-space form as follows:
(: input, : state vector, : output)Consider the new state vector
What is the state-space representation of the system in terms of the new state vector ?Correct answer
(C) z = bmatrix 4 & 9 \ -2 & -3 bmatrix z + bmatrix 4 \ -1 bmatrix u y = bmatrix 2 & 3 bmatrix z
Step-by-step solution
Given the transformation where .
The inverse transformation is .
. Thus, .
The new state-space matrices are:
So the representation is and .54
Q54MSQ2 marksMediumFor the balanced 3-phase transmission line shown, consider the following cases: [figure] **Case-1:**…Think it through. Then check your answer.Question
For the balanced 3-phase transmission line shown, consider the following cases:Case-1: and
Case-2: and Which of the following statements is/are correct about real power loss and reactive power loss in the line?Correct answer
(A) Real power loss in Case-1 is more than that in Case-2; (D) Reactive power loss in Case-2 is more than that in Case-1
Step-by-step solution
The line current is .
Real power loss and reactive power loss .Case-1: . .
Case-2: . .
Comparing the results:
Statement (A) is correct.
Statement (D) is correct.55
Q55MSQ2 marksMediumConsider an orthogonal matrix with real entries and each column having unit Euclidean norm. Which of the following statements is/are correct?Think it through. Then check your answer.Question
Consider an orthogonal matrix with real entries and each column having unit Euclidean norm.Which of the following statements is/are correct?Correct answer
(A) The value of the determinant of A is either +1 or -1; (B) The eigenvalues of A have modulus 1; (D) \ Ax \ = \ x \, for all x ∈ Rⁿ, where \ x \ denotes the Euclidean norm of x, and (Ax)^T(Ay) = x^Ty, for all x, y ∈ Rⁿ
Step-by-step solution
For a real orthogonal matrix , the columns are orthonormal, so (equivalently ).(A) . Correct.(B) An eigenvalue of a real matrix may be complex, with a complex eigenvector : . Because is real, , so andBut , so . Correct.(D) For real :and with this gives . Both parts hold for all . Correct.(C) Its first part is true, but its second part claims for distinct , which contradicts the identity just shown. Incorrect.Answer: (A), (B), (D)56
Q56MSQ2 marksMediumConsider the system of linear equations: , where is an matrix, and and are -dimensional column vectors.…Think it through. Then check your answer.Question
Consider the system of linear equations: , where is an matrix, and and are -dimensional column vectors.Suppose this system of equations has a unique solution. Which of the following statements is/are correct?Correct answer
(A) A⁻¹ exists; (B) The system of equations A^mx = b also has a unique solution for m = 1, 2, 3, …; (C) rank(A) = rank(A^m), for m = 1, 2, 3, …
Step-by-step solution
A unique solution for an system implies that is non-singular (invertible).
(A) Since is non-singular, exists. Correct.
(B) If is non-singular, then is also non-singular for any positive integer . Thus, has a unique solution. Correct.
(C) . Since is also non-singular, . Correct.
(D) For a unique solution, . The statement says , which is incorrect.57
Q57NAT2 marksEasyThe resistance values of the Wheatstone bridge shown are: , , . The battery voltage . The battery has an…Think it through. Then check your answer.Question
The resistance values of the Wheatstone bridge shown are:
, , . The battery voltage .
The battery has an internal resistance of and the Galvanometer (G) has a resistance of .The value of the resistance for balanced condition is _____________
(Answer in integer)Correct answer
750 to 750
Step-by-step solution
For a Wheatstone bridge to be balanced, the ratio of the resistances in the arms must satisfy:Given , , and , we can solve for :The balance condition is independent of the battery voltage, internal resistance, and galvanometer resistance.58
Q58NAT2 marksHardA system with two generators G1 and G2 (without generator limits) is shown. [figure] The total load on the system is 1184 MW. The expressions for the cost of generation ( and…Think it through. Then check your answer.Question
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 ( and ) and real power loss () are as follows: Rs/MWh
Rs/MWh
MWWhen the generators are operating at their optimal generation, meeting the total load requirement, the real power loss in the system is ________ MW (Round off to one decimal place)Consider the Lagrange multiplier for optimal generation.Answer checking is unavailable for this question. You can review the published solution without a score.
Step-by-step solution
With transmission losses, the optimal (coordination) condition for each generator isGenerator G1Since depends only on , :Generator G2LossCheck of the power balance: MW, which agrees with the 1184 MW load only approximately. The problem fixes and also demands that the load be met exactly, so its data are over-determined and not perfectly consistent; the cost functions are also stated in Rs/MWh where Rs/h is meant. Because of these defects, this question was awarded full marks to every candidate. Using the given , the intended loss is MW59
Q59NAT2 marksMediumA 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…Think it through. Then check your answer.Question
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 torque load. With shorted sliprings, the machine runs at 1476 rpm.When an external non-inductive resistance of 0.27 per phase is connected in series in the rotor circuit, the steady-state speed drops to 1404 rpm.Neglecting rotational losses, the actual per phase rotor winding resistance is ____ (Round off to two decimal places)Correct answer
0.09 to 0.09
Step-by-step solution
Synchronous speed .Case 1 (shorted sliprings):
Slip .Case 2 (external resistance ):
Slip .For an induction motor driving a constant torque load, the torque is constant. Neglecting stator impedance and for small slips, . Thus:60
Q60NAT2 marksMediumConsider the two-port network shown. For maximum power transfer to the resistive load (), the value of should be__________ (Round off to two decimal places)…Think it through. Then check your answer.Question
Consider the two-port network shown. For maximum power transfer to the resistive load (), the value of should be__________
(Round off to two decimal places)
Correct answer
2.81 to 2.91
Step-by-step solution
To achieve maximum power transfer to the resistive load , the load resistance must be equal to the Thevenin resistance () of the network as seen from the load terminals.Step 1: Deactivate independent sources.
Replace the independent voltage source with a short circuit.Step 2: Analyze the simplified network.- When the source is short-circuited, the resistor is in parallel with a short circuit, effectively removing it from the circuit analysis for .
- The remaining network consists of the resistor and the resistor connected in parallel with respect to the load terminals.
61
Q61NAT2 marksMediumConsider the circuit shown. Assume that the diode () is ideal. Given V, V, and , the average value of the current…Think it through. Then check your answer.Question
Consider the circuit shown. Assume that the diode () is ideal.Given V, V, and , the average value of the current through the diode is ___________ A(Round off to two decimal places)
Correct answer
1.04 to 1.14
Step-by-step solution
The average current is given by . The diode conducts when , i.e., . This occurs for . The current is . Thus, A.62
Q62NAT2 marksMediumThe magnitude of the contour integral over the contour is ______ (Round off to two decimal places)…Think it through. Then check your answer.Question
The magnitude of the contour integralover the contour is ______(Round off to two decimal places)Note: is a complex variable andCorrect answer
25 to 25.6
Step-by-step solution
The integrand has poles at and . The contour is a circle . The distance from the center to is (outside). The distance to is (inside). By Cauchy's Residue Theorem, the integral is . . The integral is . The magnitude is .63
Q63NAT2 marksHardA uniform spherical volume charge distribution of radius m, centered at the origin, has a strength of C/m. A point charge of strength…Think it through. Then check your answer.Question
A uniform spherical volume charge distribution of radius m, centered at the origin, has a strength of C/m. A point charge of strength C is moved from to in Cartesian coordinate system. The relative permittivity of the medium is and the coordinate values are in meters.The work done during the process is ______ J(Round off to two decimal places)Correct answer
0.38 to 0.42
Step-by-step solution
Total charge C. The point charge is C. The work done is . Both points are outside the sphere (, ). J = J.64
Q64NAT2 marksMediumLet and be two real-valued random variables with , and , where denotes the expectation operator. The value of…Think it through. Then check your answer.Question
Let and be two real-valued random variables with , and , where denotes the expectation operator. The value of that minimizes is __________(Round off to one decimal place)Correct answer
0.1 to 0.1
Step-by-step solution
To minimize the expectation , we expand the expression:This is a quadratic function of of the form , where , , and .
The minimum occurs at :Rounding to one decimal place, the value is 0.1.65
Q65NAT2 marksHardThe integral evaluates to _________ (Round off to two decimal places)Think it through. Then check your answer.Question
The integral evaluates to _________(Round off to two decimal places)Correct answer
0.25 to 0.25
Step-by-step solution
Let the integral be .
Using the substitution , then . The limits change from to :Since the variable of integration is dummy, we can write:Adding the two expressions for :The question asks for the value of :Rounding to two decimal places, the value is 0.25.