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1841

GATE ECE 2022 | Question: 13
The frequency response $H(f)$ of a linear time-invariant system has magnitude as shown in the figure. Statement $\text{I}:$ The system is necessarily a pure delay system for inputs which are bandlimited to $ - \alpha \leq f \leq \alpha.$ Statement ... $\text{II}$ is correct Statement $\text{I}$ is incorrect, Statement $\text{II}$ is incorrect

The frequency response $H(f)$ of a linear time-invariant system has magnitude as shown in the figure.Statement $\text{I}:$ The system is necessarily a pure delay system f...

Arjun

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1842

GATE ECE 2022 | Question: 14
In a circuit, there is a series connection of an ideal resistor and an ideal capacitor. The conduction current $\text{(in Amperes)}$ through the resistor is $2\sin (t+\pi/2).$ The displacement current $\text{(in Amperes)}$ through the capacitor is ______________. $2 \sin (t)$ $2 \sin (t + \pi)$ $2 \sin(t + \pi/2)$ $0$

In a circuit, there is a series connection of an ideal resistor and an ideal capacitor. The conduction current $\text{(in Amperes)}$ through the resistor is $2\sin (t+\pi...

Arjun

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Arjun asked Feb 15, 2022

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1843

GATE ECE 2022 | Question: 15
Consider the following partial differential equation $\text{(PDE)}$ $a \frac{ \partial^{2} f(x ,y)}{\partial x^{2}} + b \frac{ \partial^{2} f(x ,y)}{\partial y^{2}} = f(x, y),$ where $a$ and $b$ are distinct positive real numbers. Select the combination(s) of values of the ... $\xi = 0, \eta = 0$ $\xi = \frac{1}{\sqrt{a}}, \eta = \frac{1}{\sqrt{b}}$

Consider the following partial differential equation $\text{(PDE)}$$$a \frac{ \partial^{2} f(x ,y)}{\partial x^{2}} + b \frac{ \partial^{2} f(x ,y)}{\partial y^{2}} = f(x...

Arjun

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1844

GATE ECE 2022 | Question: 16
An ideal $\text{OPAMP}$ circuit with a sinusoidal input is shown in the figure. The $3 \; \text{dB}$ frequency is the frequency at which the magnitude of the voltage gain decreases by $3 \; \text{dB}$ from the maximum value. Which of the options is/are correct? The circuit ... $1000 \; \text{rad/s}.$ The $3 \; \text{dB}$ frequency is $\frac{1000}{3} \; \text{rad/s}.$

An ideal $\text{OPAMP}$ circuit with a sinusoidal input is shown in the figure. The $3 \; \text{dB}$ frequency is the frequency at which the magnitude of the voltage gain...

Arjun

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1845

GATE ECE 2022 | Question: 18
Select the correct statement(s) regarding $\text{CMOS}$ implementation of $\text{NOT}$ gates. Noise Margin High $(NM_{H})$ is always equal to the Noise Margin Low $(NM_{L}),$ irrespective of the sizing of transistors. Dynamic power consumption ... $\text{NOT}$ gate.

Select the correct statement(s) regarding $\text{CMOS}$ implementation of $\text{NOT}$ gates.Noise Margin High $(NM_{H})$ is always equal to the Noise Margin Low $(NM_{L}...

Arjun

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1846

GATE ECE 2022 | Question: 19
Let $H(X)$ denote the entropy of a discrete random variable $X$ taking $K$ possible distinct real values. Which of the following statements is/are necessarily true? $H(X) \leq \log_{2} K \; \text{bits}$ $H(X) \leq H (2X)$ $H(X) \leq H (X^{2})$ $H(X) \leq H (2^{X})$

Let $H(X)$ denote the entropy of a discrete random variable $X$ taking $K$ possible distinct real values. Which of the following statements is/are necessarily true?$H(X) ...

Arjun

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1847

GATE ECE 2022 | Question: 20
Consider the following wave equation, $\frac{\partial^{2}f(x, t)}{\partial t^{2}} = 10000 \frac{\partial^{2}f(x, t)}{\partial x^{2}} $ Which of the given options is/are solution(s) to the given wave equation? $f(x, t) = e^{ - (x-100t)^{2}} + e^{ - (x+100t)^{2}}$ ... $f(x, t) = e^{ j 100 \pi (- 100x + t)} + e^{ j 100 \pi (100x + t)}$

Consider the following wave equation,$$\frac{\partial^{2}f(x, t)}{\partial t^{2}} = 10000 \frac{\partial^{2}f(x, t)}{\partial x^{2}} $$Which of the given options is/are s...

Arjun

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1848

GATE ECE 2022 | Question: 22
A simple closed path $C$ in the complex plane is shown in the figure. If $\oint_{c} \frac{2^{z}}{z^{2} – 1}dz = – i \pi A,$ where $i = \sqrt{-1},$ then the value of $A$ is _________ (rounded off to two decimal places).

A simple closed path $C$ in the complex plane is shown in the figure. If$$\oint_{c} \frac{2^{z}}{z^{2} – 1}dz = – i \pi A,$$where $i = \sqrt{-1},$ then the value of $...

Arjun

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1849

GATE ECE 2022 | Question: 23
Let $x_{1} (t) = e^{-t}u(t)$ and $x_{2}(t) = u(t) – u(t – 2),$ where $u(\cdot)$ denotes the unit step function. If $y(t)$ denotes the convolution of $x_{1}(t)$ and $x_{2}(t),$ then $\displaystyle \lim_{t \rightarrow \infty} y(t) =$ ______________ (rounded off to one decimal place).

Let $x_{1} (t) = e^{-t}u(t)$ and $x_{2}(t) = u(t) – u(t – 2),$ where $u(\cdot)$ denotes the unit step function.If $y(t)$ denotes the convolution of $x_{1}(t)$ and $x...

Arjun

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1850

GATE ECE 2022 | Question: 24
An ideal $\text{MOS}$ capacitor (p-type semiconductor) is shown in the figure. The $\text{MOS}$ capacitor is under strong inversion with $V_{G} = 2 \; \text{V}.$ The corresponding inversion charge density $(Q_{IN})$ is $2.2 \; \mu \text{C /cm}^{2}.$ ... $Q_{IN}$ is ______________ $\mu \text{C/cm}^{2}$ (rounded off to one decimal place).

An ideal $\text{MOS}$ capacitor (p-type semiconductor) is shown in the figure. The $\text{MOS}$ capacitor is under strong inversion with $V_{G} = 2 \; \text{V}.$ The corr...

Arjun

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1851

GATE ECE 2022 | Question: 25
A symbol stream contains alternate $\text{QPSK}$ and $\text{16-QAM}$ symbols. If symbols from this stream are transmitted at the rate of $1$ mega-symbols per second, the raw (uncoded) data rate is ____________ mega-bits per second (rounded off to one decimal place).

A symbol stream contains alternate $\text{QPSK}$ and $\text{16-QAM}$ symbols. If symbols from this stream are transmitted at the rate of $1$ mega-symbols per second, the ...

Arjun

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1852

GATE ECE 2022 | Question: 26
The function $f(x) = 8 \log_{e} x – x^{2} + 3$ attains its minimum over the interval $[1, e]$ at $x = $____________. (Here $\log_{e}x$ is the natural logarithm of $x$.) $2$ $1$ $e$ $\frac{1+e}{2}$

The function $f(x) = 8 \log_{e} x – x^{2} + 3$ attains its minimum over the interval $[1, e]$ at $x = $____________.(Here $\log_{e}x$ is the natural logarithm of $x$.)$...

Arjun

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1853

GATE ECE 2022 | Question: 27
Let $\alpha, \beta$ be two non-zero real numbers and $v_{1}, v_{2}$ be two non-zero real vectors of size $3 \times 1.$ Suppose that $v_{1}$ and $v_{2}$ satisfy $v_{1}^{T} v_{2} = 0, v_{1}^{T} v_{1} = 1,$ and $v_{2}^{T} v_{2} = 1.$ Let $A$ be ... $0, \alpha+\beta, \alpha-\beta$ $0, \frac{\alpha+\beta}{2}, \sqrt{\alpha \beta}$ $0, 0, \sqrt{\alpha^{2} + \beta^{2}}$

Let $\alpha, \beta$ be two non-zero real numbers and $v_{1}, v_{2}$ be two non-zero real vectors of size $3 \times 1.$ Suppose that $v_{1}$ and $v_{2}$ satisfy $v_{1}^{T}...

Arjun

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1854

GATE ECE 2022 | Question: 28
For the circuit shown, the locus of the impedance $Z (j \omega)$ is plotted as $\omega$ increases from zero to infinity. The values of $R_{1}$ and $R_{2}$ are: $R_{1} = 2 \; \text{k} \Omega, R_{2} = 3 \; \text{k} \Omega$ ... $R_{1} = 2 \; \text{k} \Omega, R_{2} = 5 \; \text{k} \Omega$

For the circuit shown, the locus of the impedance $Z (j \omega)$ is plotted as $\omega$ increases from zero to infinity. The values of $R_{1}$ and $R_{2}$ are:$R_{1} = 2 ...

Arjun

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1855

GATE ECE 2022 | Question: 29
Consider the circuit shown in the figure with input $V(t)$ in volts. The sinusoidal steady state current $I(t)$ flowing through the circuit is shown graphically (where $t$ is in seconds). The circuit element $Z$ can be ____________. a capacitor of $1 \; \text{F}$ an inductor of $1 \; \text{H}$ a capacitor of $\sqrt{3} \; \text{F}$ an inductor of $\sqrt{3} \; \text{H}$

Consider the circuit shown in the figure with input $V(t)$ in volts. The sinusoidal steady state current $I(t)$ flowing through the circuit is shown graphically (where $t...

Arjun

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1856

GATE ECE 2022 | Question: 30
Consider an ideal long channel $n \text{MOSFET}$ (enhancement-mode) with gate length $10 \; \mu \text{m}$ and width $100 \; \mu \text{m}.$ The product of electron mobility $(\mu_{n})$ and oxide capacitance per unit area $(C_{\text{ox}})$ ... -to-source current is _____________. $40 \; \text{mA}$ $20 \; \text{mA}$ $15 \; \text{mA}$ $5 \; \text{mA}$

Consider an ideal long channel $n \text{MOSFET}$ (enhancement-mode) with gate length $10 \; \mu \text{m}$ and width $100 \; \mu \text{m}.$ The product of electron mobilit...

Arjun

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1857

GATE ECE 2022 | Question: 31
For the following circuit with an ideal $\text{OPAMP},$ the difference between the maximum and the minimum values of the capacitor voltage $(V_{c})$ is ____________. $15 \; \text{V}$ $27 \; \text{V}$ $13 \; \text{V}$ $14 \; \text{V}$

For the following circuit with an ideal $\text{OPAMP},$ the difference between the maximum and the minimum values of the capacitor voltage $(V_{c})$ is ____________.$15 \...

Arjun

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1858

GATE ECE 2022 | Question: 32
A circuit with an ideal $\text{OPAMP}$ is shown. The Bode plot for the magnitude $\text{(in dB)}$ of the gain transfer function $(A_{V} (j \omega) = V_{\text{out}} (j \omega) / V_{\text{in}} (j \omega))$ of the circuuit is also provided (here, $\omega$ is the angular frequency ... $R = 3 \; \text{k} \Omega, C = 2 \; \mu \text{F}$

A circuit with an ideal $\text{OPAMP}$ is shown. The Bode plot for the magnitude $\text{(in dB)}$ of the gain transfer function $(A_{V} (j \omega) = V_{\text{out}} (j \om...

Arjun

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1859

GATE ECE 2022 | Question: 33
For the circuit shown, the clock frequency is $f_{0}$ and the duty cycle is $25 \%.$ For the signal at the $\text{Q}$ output of the Flip-Flop, _______________. frequency is $f_{0}/4$ and duty cycle is $50 \%$ frequency is $f_{0}/4$ and duty cycle is $25 \%$ frequency is $f_{0}/2$ and duty cycle is $50 \%$ frequency is $f_{0}$ and duty cycle is $25 \%$

For the circuit shown, the clock frequency is $f_{0}$ and the duty cycle is $25 \%.$ For the signal at the $\text{Q}$ output of the Flip-Flop, _______________.frequency i...

Arjun

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1860

GATE ECE 2022 | Question: 34
Consider an even polynomial $p(s)$ given by $p(s) = s^{4} + 5s^{2} + 4 + K,$ where $K$ is an unknown real parameter. The complete range of $K$ for which $p(s)$ has all its roots on the imaginary axis is _____________. $ – 4 \leq K \leq \frac{9}{4}$ $ – 3 \leq K \leq \frac{9}{2}$ $ – 6 \leq K \leq \frac{5}{4}$ $ – 5 \leq K \leq 0$

Consider an even polynomial $p(s)$ given by$$p(s) = s^{4} + 5s^{2} + 4 + K,$$where $K$ is an unknown real parameter. The complete range of $K$ for which $p(s)$ has all it...

Arjun

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1861

GATE ECE 2022 | Question: 35
Consider the following series: $ \sum_{n=1}^{\infty} \frac{n^{d}}{c^{n}}$ For which of the following combinations of $c, d$ values does this series converge? $ c= 1, d = – 1$ $ c= 2, d = 1$ $ c= 0.5, d = – 10$ $ c= 1, d = – 2$

Consider the following series:$$ \sum_{n=1}^{\infty} \frac{n^{d}}{c^{n}}$$For which of the following combinations of $c, d$ values does this series converge?$ c= 1, d = �...

Arjun

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1862

GATE ECE 2022 | Question: 36
The outputs of four systems $(S_{1}, S_{2}, S_{3},$ and $\text{S}_{4})$ corresponding to the input signal $\sin(t),$ for all time $t,$ are shown in the figure. Based on the given information, which of the four systems is/are definitely $\text{NOT LTI}$ (linear and time-invariant)? $S_{1}$ $S_{2}$ $S_{3}$ $S_{4}$

The outputs of four systems $(S_{1}, S_{2}, S_{3},$ and $\text{S}_{4})$ corresponding to the input signal $\sin(t),$ for all time $t,$ are shown in the figure.Based on th...

Arjun

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1863

GATE ECE 2022 | Question: 37
Select the $\text{CORRECT}$ statement(s) regarding semiconductor devices. Electrons and holes are of equal density in an intrinsic semiconductor at equilibrium. Collector region is generally more heavily doped than Base region in a $\text{BJT}.$ ... under steady state condition. Mobility of electrons always increases with temperature in Silicon beyond $300 \; \text{K}.$

Select the $\text{CORRECT}$ statement(s) regarding semiconductor devices.Electrons and holes are of equal density in an intrinsic semiconductor at equilibrium.Collector r...

Arjun

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1864

GATE ECE 2022 | Question: 38
A state transition diagram with states $A, B,$ and $C,$ and transition probabilities $p_{1}, p_{2}, \dots, p_{7}$ is shown in the figure (e.g., $\text{p}_{1}$ denotes the probability of transition from state $A$ to $B$). For this state diagram, select the statement(s) which is/are ... $p_{1} + p_{4} + p_{7} = 1$ $p_{2} + p_{5} + p_{7} = 1$

A state transition diagram with states $A, B,$ and $C,$ and transition probabilities $p_{1}, p_{2}, \dots, p_{7}$ is shown in the figure (e.g., $\text{p}_{1}$ denotes the...

Arjun

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1865

GATE ECE 2022 | Question: 40
Two linear time-invariant systems with transfer functions $G_{1}(s) = \frac{10}{s^{2}+s+1} \; \text{and} \; G_{2}(s) = \frac{10}{s^{2}+s\sqrt{10}+10}$ have unit step responses $y_{1}(t)$ and $y_{2}(t),$ respectively. Which of the following ... and $y_{2}(t)$ have the same damped frequency of oscillation. $y_{1}(t)$ and $y_{2}(t)$ have the same $2\%$ settling time.

Two linear time-invariant systems with transfer functions$$G_{1}(s) = \frac{10}{s^{2}+s+1} \; \text{and} \; G_{2}(s) = \frac{10}{s^{2}+s\sqrt{10}+10}$$have unit step resp...

Arjun

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1866

GATE ECE 2022 | Question: 41
Consider an $\text{FM}$ broadcast that employs the pre-emphasis filter with frequency response $H_{pe}(\omega) = 1 + \frac{j \omega}{\omega_{0}},$ where $\omega_{0} = 10^{4} \; \text{rad/sec}.$ For the network shown in the figure to act as a corresponding de-emphasis filter, the appropriate ... $R = 1 \; k \Omega, C = 2 \; \mu F$ $R = 2 \; k \Omega, C = 0.5 \; \mu F$

Consider an $\text{FM}$ broadcast that employs the pre-emphasis filter with frequency response$$H_{pe}(\omega) = 1 + \frac{j \omega}{\omega_{0}},$$where $\omega_{0} = 10^...

Arjun

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1867

GATE ECE 2022 | Question: 42
A waveguide consists of two infinite parallel plates (perfect conductors) at a separation of $10^{-4} \; \text{cm},$ with air as the dielectric. Assume the speed of light in air to be $3 \times 10^{8} \; \text{m/s}.$ The frequency/frequencies of $\text{TM}$ waves which can ... $8 \times 10^{14} \; \text{Hz}$ $1 \times 10^{13} \; \text{Hz}$

A waveguide consists of two infinite parallel plates (perfect conductors) at a separation of $10^{-4} \; \text{cm},$ with air as the dielectric. Assume the speed of light...

Arjun

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1868

GATE ECE 2022 | Question: 43
The value of the integral $ \iint_{D} 3(x^{2} + y^{2})dxdy,$ where $D$ is the shaded triangular region shown in the diagram, is ___________ (rounded off to the nearest integer).

The value of the integral$$ \iint_{D} 3(x^{2} + y^{2})dxdy,$$where $D$ is the shaded triangular region shown in the diagram, is ___________ (rounded off to the nearest in...

Arjun

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1869

GATE ECE 2022 | Question: 44
A linear $\text{2-port}$ network is shown in $\text{Fig.(a)}.$ An ideal $\text{DC}$ voltage source of $10 \; \text{V}$ is connected across Port $1.$ A variable resistance $R$ is connected across Port $2.$ As $R$ is varied, the measured voltage ... by the load shown in $\text{Fig.(c)},$ the current $I_{2}$ is ___________ $\text{mA}$ (rounded off to one decimal place).

A linear $\text{2-port}$ network is shown in $\text{Fig.(a)}.$ An ideal $\text{DC}$ voltage source of $10 \; \text{V}$ is connected across Port $1.$ A variable resistance...

Arjun

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1870

GATE ECE 2022 | Question: 45
For a vector $\overline{x} = \left[ x [0], x[1], \dots, x[7] \right],$ the $\text{8-point}$ discrete Fourier transform $\text{(DFT)}$ is denoted by $\overline{X} = DFT (\overline{x}) = \left[ X [0], X[1], \dots, X[7] \right],$ ... $\overline{y} = DFT (DFT(\overline{x})),$ then the value of $y[0]$ is _____________ (rounded off to one decimal place).

For a vector $\overline{x} = \left[ x [0], x , \dots, x[7] \right],$ the $\text{8-point}$ discrete Fourier transform $\text{(DFT)}$ is denoted by $\overline{X} = DFT (\ov...

Arjun

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1871

GATE ECE 2022 | Question: 46
A $p-$type semiconductor with zero electric field is under illumination (low level injection) in steady state condition. Excess minority carrier density is zero at $x = \pm 2l_{n},$ where $l_{n} = 10 ^{-4} \; \text{cm}$ is the diffusion length of ... photo-generated electrons at $x = + 2l_{n}$ is ______________ $\text{mA/cm}^{2}$ (rounded off to two decimal places).

A $p-$type semiconductor with zero electric field is under illumination (low level injection) in steady state condition. Excess minority carrier density is zero at $x = \...

Arjun

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1872

GATE ECE 2022 | Question: 47
A circuit and the characteristics of the diode $\text{(D)}$ in it are shown. The ratio of the minimum to the maximum small signal voltage gain $\frac{\partial V_{\text{out}}}{\partial V_{\text{in}}}$ is ____________ (rounded off to two decimal places).

A circuit and the characteristics of the diode $\text{(D)}$ in it are shown. The ratio of the minimum to the maximum small signal voltage gain $\frac{\partial V_{\text{ou...

Arjun

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1873

GATE ECE 2022 | Question: 48
Consider the circuit shown with an ideal $\text{OPAMP}.$ The output voltage $V_{0}$ is ____________ $V$ (rounded off to two decimal places).

Consider the circuit shown with an ideal $\text{OPAMP}.$ The output voltage $V_{0}$ is ____________ $V$ (rounded off to two decimal places).

Arjun

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1874

GATE ECE 2022 | Question: 49
Consider the circuit shown with an ideal long channel $n \text{MOSFET}$ (enhancement-mode, substrate is connected to the source). The transistor is appropriately biased in the saturation region with $V_{GG}$ and $V_{DD}$ such that it acts as a linear amplifier ... $\dfrac{v_{A}}{v_{B}}$ is _____________ (rounded off to one decimal place).

Consider the circuit shown with an ideal long channel $n \text{MOSFET}$ (enhancement-mode, substrate is connected to the source). The transistor is appropriately biased i...

Arjun

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1875

GATE ECE 2022 | Question: 50
The block diagram of a closed-loop control system is shown in the figure. $R(s), Y(s),$ and $D(s)$ are the Laplace transforms of the time-domain signals $r(t), y(t),$ and $d(t),$ ... $e (\infty),$ due to a unit step disturbance $d(t),$ is __________ (rounded off to two decimal places).

The block diagram of a closed-loop control system is shown in the figure. $R(s), Y(s),$ and $D(s)$ are the Laplace transforms of the time-domain signals $r(t), y(t),$ and...

Arjun

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1876

GATE ECE 2022 | Question: 51
The transition diagram of a discrete memoryless channel with three input symbols and three output symbols is shown in the figure. The transition probabilities are as marked. The parameter $\alpha$ lies in the interval $[0.25, 1].$ The value of $\alpha$ for which the capacity of this channel is maximized, is ___________ (rounded off to two decimal places).

The transition diagram of a discrete memoryless channel with three input symbols and three output symbols is shown in the figure. The transition probabilities are as mark...

Arjun

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1877

GATE ECE 2022 | Question: 52
Consider communication over a memoryless binary symmetric channel using a $(7, 4)$ Hamming code. Each transmitted bit is received correctly with probability $(1 - \in),$ and flipped with probability $\in.$ For each codeword transmission, the ... the probability that a transmitted codeword is decoded correctly is ___________(rounded off to two decimal places).

Consider communication over a memoryless binary symmetric channel using a $(7, 4)$ Hamming code. Each transmitted bit is received correctly with probability $(1 – \in),...

Arjun

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1878

GATE ECE 2022 | Question: 53
Consider a channel over which either symbol $x_{A}$ or symbol $x_{B}$ is transmitted. Let the output of the channel $Y$ be the input to a maximum likelihood $\text{(ML)}$ detector at the receiver. The conditional probability density ... the standard unit step function. The probability of symbol error for this system is _____________ (rounded off to two decimal places).

Consider a channel over which either symbol $x_{A}$ or symbol $x_{B}$ is transmitted. Let the output of the channel $Y$ be the input to a maximum likelihood $\text{(ML)}$...

Arjun

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1879

GATE ECE 2022 | Question: 54
Consider a real valued source whose samples are independent and identically distributed random variables with the probability density function, $f(x),$ as shown in the figure. Consider a $1$ bit quantizer that maps positive samples to value $\alpha$ and others to ... error, then $(\alpha^{\ast} - \beta^{\ast}) = $____________ (rounded off to two decimal places).

Consider a real valued source whose samples are independent and identically distributed random variables with the probability density function, $f(x),$ as shown in the fi...

Arjun

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1880

GATE ECE 2022 | Question: 55
In an electrostatic field, the electric displacement density vector, $\overrightarrow{D},$ is given by $\overrightarrow{D} (x, y, z) = (x^{3} \overrightarrow{i} + y^{3} \overrightarrow{j} + xy^{2} \overrightarrow{k}) \; \text{C/m}^{2},$ where ... $R$ is _____________ $\text{C}$ (rounded off to two decimal places).

In an electrostatic field, the electric displacement density vector, $\overrightarrow{D},$ is given by$$\overrightarrow{D} (x, y, z) = (x^{3} \overrightarrow{i} + y^{3} \...

Arjun

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Arjun asked Feb 15, 2022