Q: The minimum eigenvalue of the following matrix is [3 5 2 5 12 7 2 7 5]

0. The determinant of [[3,5,2],[5,12,7],[2,7,5]] is 0, so 0 is an eigenvalue. The matrix is positive semidefinite (other eigenvalues ~1.81 and ~18.19), making 0 the minimum eigenvalue. Stored answer 1 is wrong.

Question 1

A bulb in a staircase has two switches, one switch being at the ground floor and the other one at the first floor. The bulb can be turned ON and also can be turned OFF by any one of the switches irrespective of the state of the other switch. The logic of switching of the bulb resembles

Accepted Answer

an XOR gate. The bulb's behavior is similar to an XOR gate because it can be ON when one switch is activated and OFF when both switches are in the same state. This means that the bulb's state changes only when one switch is toggled, which aligns with the XOR logic.

Question 2

Two systems with impulse responses h1(t) and h2(t) are connected in cascade. Then the overall impulse response of the cascaded system is given by

Accepted Answer

convolution of h1(t) and h2(t). When two linear time-invariant systems are connected in cascade, the overall impulse response is determined by the convolution of their individual impulse responses, which mathematically combines their effects on an input signal.

Question 3

In a forward biased pn junction diode, the sequence of events that best describes the mechanism of current flow is

Accepted Answer

injection, and subsequent diffusion and recombination of minority carriers. In a forward biased pn junction diode, the applied voltage causes majority carriers to be injected from the p-side into the n-side, where they diffuse and recombine with minority carriers, resulting in a flow of current.

Question 4

In IC technology, dry oxidation (using dry oxygen) as compared to wet oxidation (using steam or water vapor) produces

Accepted Answer

superior quality oxide with a lower growth rate. Dry oxidation produces a higher quality silicon dioxide layer due to the absence of water vapor, which can introduce impurities, but this process is slower compared to wet oxidation.

Question 5

The Bode plot of a transfer function G(s) is shown in the figure below. The gain |20 log |G(s)| is 32 dB and 8 dB at 1 rad/s and 10 rad/s respectively. The phase is negative for all ω. Then G(s) is

Accepted Answer

39.8/s². The transfer function G(s) has a gain that decreases with frequency, which is characteristic of a second-order system, indicated by the s² term in the denominator. The negative phase across all frequencies further supports that G(s) is a type of low-pass filter, consistent with the behavior of a second-order system.

Question 6

In a voltage-voltage feedback as shown below, which one of the following statements is TRUE if the gain k is increased?

Accepted Answer

The input impedance increases and output impedance decreases.. Increasing the gain k in a voltage-voltage feedback configuration typically leads to higher input impedance because the feedback reduces the effect of the input signal on the circuit, while the output impedance decreases as the feedback stabilizes the output against variations.

Question 7

A band-limited signal with a maximum frequency of 5 kHz is to be sampled. According to the sampling theorem, the sampling frequency which is not valid is

Accepted Answer

5 kHz. The sampling theorem states that to accurately sample a signal without aliasing, the sampling frequency must be greater than twice the maximum frequency of the signal. Since the maximum frequency is 5 kHz, the minimum valid sampling frequency should be greater than 10 kHz, making 5 kHz an invalid choice.

Question 8

In a MOSFET operating in the saturation region, the channel length modulation effect causes

Accepted Answer

a decrease in the output resistance. In the saturation region, channel length modulation effectively shortens the channel length, which reduces the output resistance of the MOSFET. This occurs because the drain current becomes more sensitive to changes in the drain-source voltage, leading to a lower output resistance.

Question 9

Which one of the following statements is NOT TRUE for a continuous time causal and stable LTI system?

Accepted Answer

All the poles must lie within |s| < 1.. This statement is incorrect because for a continuous time system, stability requires that all poles lie in the left half of the s-plane, but they do not need to be within the unit circle (|s| < 1), which is a condition specific to discrete time systems.

Question 10

The minimum eigenvalue of the following matrix is
[3 5 2
 5 12 7
 2 7 5]

Accepted Answer

0. The determinant of [[3,5,2],[5,12,7],[2,7,5]] is 0, so 0 is an eigenvalue. The matrix is positive semidefinite (other eigenvalues ~1.81 and ~18.19), making 0 the minimum eigenvalue. Stored answer 1 is wrong.

Question 11

Let g(t) = e^(-πt²) and h(t) is a filter matched to g(t). If g(t) is applied as input to h(t), then the Fourier transform of the output is

Accepted Answer

e^(-2πf²). The correct option is right because the output of a matched filter is the convolution of the input's Fourier transform with the filter's Fourier transform. Since the Fourier transform of g(t) is e^(-πf²), and the matched filter has the same Fourier transform, the result of their convolution leads to e^(-2πf²).

Question 12

Consider a vector field A(r). The closed loop line integral ∮ A · dl can be expressed as

Accepted Answer

∬ (∇×A) · ds over the open surface bounded by the loop. The correct option relates to Stokes' theorem, which states that the line integral of a vector field around a closed loop is equal to the surface integral of the curl of that vector field over any surface bounded by the loop. The surface can be open, as long as it is properly oriented with respect to the loop.

Question 13

Statement for Linked Answer Questions 54 and 55: A monochromatic plane wave of wavelength λ = 600 nm is propagating in the direction as shown in the figure below. E_i, E_r and Eₜ denote incident, reflected, and transmitted electric field vectors associated with the wave. Q.54 The angle of

Accepted Answer

(C) 45° and (E₀)/(√(2)) âₓ â_z e^(-j(10⁴π(x+z))/(3√(2))) V/m. The correct option indicates an angle of incidence of 45°, which is a common angle for optimal reflection and transmission in wave interactions. The expression for the electric field vector correctly incorporates the propagation direction and the wave's characteristics, including the appropriate phase factor and normalization.

Question 14

In the circuit shown below, what is the output voltage (Vout) if a silicon transistor Q and an ideal op-amp are used?

Accepted Answer

−0.7 V. The output voltage is −0.7 V because a silicon transistor typically has a base-emitter voltage drop of approximately 0.7 V when it is in the active region, which is reflected in the output of the circuit.

Question 15

Let g(t) = e^(-πt²), and h(t) is a filter matched to g(t). If g(t) is applied as input to h(t), then the Fourier transform of the output is

Accepted Answer

e^(-2πf²). The output of the matched filter h(t) when g(t) is the input results in the convolution of their Fourier transforms. Since the Fourier transform of g(t) is a Gaussian function, the convolution with itself leads to a scaling of the exponent, resulting in e^(-2πf²).

Question 16

A source vₛ(t) = Vₘ cos 100πt has an internal impedance of (4 + j3) Ω. If a purely resistive load connected to this source has to extract the maximum power out of the source, its value in Ω should be

Accepted Answer

5. For a purely resistive load to extract maximum power from a source of impedance 4+j3, set R=|Zs|=sqrt(4^2+3^2)=5 ohm. Stored 7 is wrong; answer is 5.

Question 17

The impulse response of a continuous time system is given by h(t) = δ(t) - δ(t-3). The value of the step response at t = 2 is

Accepted Answer

1. Integrating h(t)=delta(t)-delta(t-3) gives step response s(t)=u(t)-u(t-3). At t=2, u(2)=1 and u(-1)=0, so s(2)=1. The stored 0 is wrong; the value is 1.

Question 18

The open-loop transfer function of a dc motor is given as ω(s)/Vₐ(s) = 10/(s(1+10s)). When connected in feedback as shown below, the approximate value of Kₐ that will reduce the time constant of the closed loop system by one hundred times as compared to that of the open-loop system is

Accepted Answer

100. To reduce the time constant of the closed-loop system by one hundred times, the gain Kₐ must be increased significantly. Since the time constant is inversely proportional to the gain in a feedback system, setting Kₐ to 100 achieves the desired reduction in time constant.

Question 19

Three capacitors C1, C2 and C3 whose values are 1 μF, 5 μF and 2 μF respectively, have breakdown voltages of 10 V, 5 V, and 2 V respectively. For the interconnection shown below, the maximum safe voltage in V that can be applied across the combination, and the corresponding total charge in

Accepted Answer

2.8 and 36. The maximum safe voltage is determined by the capacitor with the lowest breakdown voltage in the series, which is C3 at 2V. In this configuration, the effective capacitance is calculated, and the total charge stored can be derived from the voltage and capacitance values, leading to the correct answer of 2.8 V and 36 μC.

Question 20

The DFT of a vector [a b c d] is the vector [p q r s]. Consider the product [p q r s] = [a b c d] [ a b c d; d a b c; c d a b; b c d a ]. The DFT of the vector [p q r s] is a scaled version of

Accepted Answer

[a b c d a b c d]. The correct option is [a b c d a b c d] because the product of the vector [a b c d] with the given matrix results in a vector that effectively replicates the original input, leading to a DFT that retains the periodicity and structure of the original vector, thus forming a scaled version of the extended vector.

Question 21

Statement for Linked Answer Questions 54 and 55: A monochromatic plane wave of wavelength λ = 600 μm is propagating in the direction as shown in the figure below. E_i, E_r and Eₜ denote incident, reflected, and transmitted electric field vectors associated with the wave. Q.54 The angle of

Accepted Answer

(C) 45° and E₀/√2 âₓ â_z e^(-j10⁴π(x+z)/(3√2)) V/m. The correct option indicates that the angle of incidence is 45°, which is a common angle for optimal reflection and transmission conditions in wave interactions. Additionally, the expression for the reflected electric field vector correctly incorporates the wavelength and phase shift, aligning with the properties of wave behavior at this angle.

Question 22

A source vₛ(t) = V cos 100π t has an internal impedance of (4 + j3) Ω. If a purely resistive load connected to this source has to extract the maximum power out of the source, its value in Ω should be

Accepted Answer

5. Identical to the maximum-power problem: a purely resistive load matched to source impedance 4+j3 must equal |Zs| = sqrt(16+9) = 5 ohm. The value 5 is option index 2; the stored answer (7) is wrong.

Question 23

The transfer function V₂(s) / V₁(s) of the circuit shown below is

Accepted Answer

s / (s + 2). The correct option, s / (s + 2), represents a first-order system with a pole at -2, indicating the system's response characteristics and stability, which aligns with the expected behavior of the given circuit configuration.

Question 24

In a voltage-voltage feedback as shown below, which one of the following statements is TRUE if the gain k is increased?

[Diagram: voltage-voltage feedback block diagram with amplifier A0 and feedback block k, showing Vin, Vout, and feedback Vf = kVout]

Accepted Answer

The input impedance increases and output impedance decreases.. Increasing the gain k in a voltage-voltage feedback system typically leads to higher input impedance because the feedback reduces the effect of the input signal on the output, while the output impedance decreases as the feedback stabilizes the output against variations.

Question 25

For a periodic signal v(t) = 30 sin 100t + 30 cos 300t + 6 sin (500t + π/4), the fundamental frequency in rad/s is

Accepted Answer

100. The fundamental frequency of a periodic signal is determined by the lowest frequency component present in the signal. In this case, the term with the lowest angular frequency is 100 rad/s from the sine function, making it the fundamental frequency.

Question 26

Consider a vector field A(r). The closed loop line integral ∮ A · dl can be expressed as

Accepted Answer

∬ (∇×A) · ds over the open surface bounded by the loop. The correct option relates to Stokes' theorem, which states that the line integral of a vector field around a closed loop is equal to the surface integral of the curl of that vector field over any surface bounded by the loop. This is why the expression involving the open surface is appropriate.

Question 27

A voltage 1000 sin πt V volts is applied across Y Z. Assuming ideal diodes, the voltage measured across W X in volts is.

Accepted Answer

(sin πt + |sin πt|)/2. The correct option represents the output voltage across W X when an AC voltage is applied and ideal diodes are used. The expression (sin πt + |sin πt|)/2 effectively captures the positive half-cycles of the sine wave, allowing current to pass through the diodes, while blocking the negative half-cycles, resulting in a rectified output.

Question 28

The impulse response of a continuous time system is given by h(t) = δ(t) − δ(t−3). The value of the step response at t = 2 is

Accepted Answer

1. The step response is the integral of the impulse response. For h(t) = δ(t) - δ(t-3), the step response at t = 2 is the integral from 0 to 2, which gives a value of 1 due to the contribution from δ(t).

GATE Technical: ELECTRONICS COMMN. ENGG. - EC questions with solutions

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