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1641
GATE ECE 2022 | Question: 2
Consider a system of linear equations $Ax = b,$ where $A =\begin{bmatrix} 1 & – \sqrt{2} & 3 \\ – 1 & \sqrt{2} & – 3 \end{bmatrix}, \quad b = \begin{bmatrix} 1 \\ 3 \end{bmatrix}.$ This system of equations admits ______________. a unique solution for $x$ infinitely many solutions for $x$ no solutions for $x$ exactly two solutions for $x$
Consider a system of linear equations $Ax = b,$ where$A =\begin{bmatrix} 1 & – \sqrt{2} & 3 \\ – 1 & \sqrt{2} & – 3 \end{bmatrix}, \quad b = \begin{bmatrix} 1 \\ 3 ...
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1642
GATE ECE 2022 | Question: 8
Consider the $\text{CMOS}$ circuit shown in the figure (substrates are connected to their respective sources). The gate width $(W)$ to gate length $(L)$ ratios $\left( \frac{W}{L} \right)$ of the transistors are as shown. Both the transistors have the same gate oxide capacitance ... $2\; \text{V}$ less than $2 \; \text{V}$ equal to $2 \; \text{V}$
Consider the $\text{CMOS}$ circuit shown in the figure (substrates are connected to their respective sources). The gate width $(W)$ to gate length $(L)$ ratios $\left( \f...
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1643
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...
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1644
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 ...
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1645
GATE ECE 2022 | Question: 6
Consider a long rectangular bar of direct bandgap $p-$type semiconductor. The equilibrium hole density is $10^{17} \; \text{cm}^{-3}$ and the intrinsic carrier concentration is $10^{10} \; \text{cm}^{-3}.$ Electron and hole diffusion lengths are $2 \; \mu\text{m}$ ... $3.7 \times 10^{14} \; \text{cm}^{-3}$ $10^{3} \; \text{cm}^{-3}$
Consider a long rectangular bar of direct bandgap $p-$type semiconductor. The equilibrium hole density is $10^{17} \; \text{cm}^{-3}$ and the intrinsic carrier concentrat...
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1646
GATE ECE 2022 | Question: 11
Consider a closed-loop control system with unity negative feedback and $KG(s)$ in the forward path. where the gain $K = 2.$ The complete Nyquist plot of the transfer function $G(s)$ is shown in the figure. Note that the Nyquist contour has been ... of the closed-loop transfer function in the closed right-half of the complex plane is ________________. $0$ $1$ $2$ $3$
Consider a closed-loop control system with unity negative feedback and $KG(s)$ in the forward path. where the gain $K = 2.$ The complete Nyquist plot of the transfer func...
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1647
GATE ECE 2022 | Question: 4
Consider the circuit shown in the figure. The current $I$ flowing through the $10 \; \Omega$ resistor is _____________. $1 \; \text{A}$ $0 \; \text{A}$ $0.1 \; \text{A}$ $ – 0.1 \; \text{A}$
Consider the circuit shown in the figure. The current $I$ flowing through the $10 \; \Omega$ resistor is _____________.$1 \; \text{A}$$0 \; \text{A}$$0.1 \; \text{A}$$ �...
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1648
GATE ECE 2022 | Question: 13
The frequency response $H(f)$ of a linear time-invariant system has magnitude as shown in the figure. Statement $\text{I}:$ The system is necessarily a pure delay system for inputs which are bandlimited to $ - \alpha \leq f \leq \alpha.$ Statement ... $\text{II}$ is correct Statement $\text{I}$ is incorrect, Statement $\text{II}$ is incorrect
The frequency response $H(f)$ of a linear time-invariant system has magnitude as shown in the figure.Statement $\text{I}:$ The system is necessarily a pure delay system f...
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1649
GATE ECE 2022 | Question: 22
A simple closed path $C$ in the complex plane is shown in the figure. If $\oint_{c} \frac{2^{z}}{z^{2} – 1}dz = – i \pi A,$ where $i = \sqrt{-1},$ then the value of $A$ is _________ (rounded off to two decimal places).
A simple closed path $C$ in the complex plane is shown in the figure. If$$\oint_{c} \frac{2^{z}}{z^{2} – 1}dz = – i \pi A,$$where $i = \sqrt{-1},$ then the value of $...
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1650
GATE ECE 2022 | Question: 3
The current $I$ in the circuit shown is ______________. $1.25 \times 10^{-3} \; \text{A}$ $0.75 \times 10^{-3} \; \text{A}$ $ – 0.5 \times 10^{-3} \; \text{A}$ $1.16 \times 10^{-3} \; \text{A}$
The current $I$ in the circuit shown is ______________.$1.25 \times 10^{-3} \; \text{A}$$0.75 \times 10^{-3} \; \text{A}$$ – 0.5 \times 10^{-3} \; \text{A}$$1.16 \times...
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1651
GATE ECE 2022 | Question: 5
The Fourier transform $X(j \omega)$ of the signal $x(t) = \frac{t}{(1+t^{2})^{2}}$ is ______________. $\frac{\pi}{2j} \omega e^{– |\omega|}$ $\frac{\pi}{2} \omega e^{– |\omega|}$ $\frac{\pi}{2j}e^{– |\omega|}$ $\frac{\pi}{2}e^{– |\omega|}$
The Fourier transform $X(j \omega)$ of the signal$$x(t) = \frac{t}{(1+t^{2})^{2}}$$is ______________.$\frac{\pi}{2j} \omega e^{– |\omega|}$$\frac{\pi}{2} \omega e^{– ...
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1652
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...
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1653
GATE ECE 2022 | Question: 12
The root-locus plot of a closed-loop system with unity negative feedback and transfer function $KG(s)$ in the forward path is shown in the figure. Note that $K$ is varied from $0$ to $\infty.$ Select the transfer function $G(s)$ that results in the root-locus plot of the closed-loop system as ... $G(s) = \frac{s - 1}{(s+1)^{6}}$ $G(s) = \frac{s + 1}{s^{6}+1}$
The root-locus plot of a closed-loop system with unity negative feedback and transfer function $KG(s)$ in the forward path is shown in the figure. Note that $K$ is varied...
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1654
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) ...
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1655
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...
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1656
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...
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1657
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...
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1658
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...
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1659
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 ...
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1660
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...
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1661
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}...
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1662
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}...
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1663
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...
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1664
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 $x$.)$...
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1665
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...
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1666
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...
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1667
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...
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1668
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...
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1669
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} \...
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1670
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)}$...
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1671
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...
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1672
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...
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1673
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}$$have unit step resp...
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1674
GATE ECE 2022 | Question: 34
Consider an even polynomial $p(s)$ given by $p(s) = s^{4} + 5s^{2} + 4 + K,$ where $K$ is an unknown real parameter. The complete range of $K$ for which $p(s)$ has all its roots on the imaginary axis is _____________. $ – 4 \leq K \leq \frac{9}{4}$ $ – 3 \leq K \leq \frac{9}{2}$ $ – 6 \leq K \leq \frac{5}{4}$ $ – 5 \leq K \leq 0$
Consider an even polynomial $p(s)$ given by$$p(s) = s^{4} + 5s^{2} + 4 + K,$$where $K$ is an unknown real parameter. The complete range of $K$ for which $p(s)$ has all it...
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GATE ECE 2022 | Question: 36
The outputs of four systems $(S_{1}, S_{2}, S_{3},$ and $\text{S}_{4})$ corresponding to the input signal $\sin(t),$ for all time $t,$ are shown in the figure. Based on the given information, which of the four systems is/are definitely $\text{NOT LTI}$ (linear and time-invariant)? $S_{1}$ $S_{2}$ $S_{3}$ $S_{4}$
The outputs of four systems $(S_{1}, S_{2}, S_{3},$ and $\text{S}_{4})$ corresponding to the input signal $\sin(t),$ for all time $t,$ are shown in the figure.Based on th...
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GATE ECE 2022 | Question: 41
Consider an $\text{FM}$ broadcast that employs the pre-emphasis filter with frequency response $H_{pe}(\omega) = 1 + \frac{j \omega}{\omega_{0}},$ where $\omega_{0} = 10^{4} \; \text{rad/sec}.$ For the network shown in the figure to act as a corresponding de-emphasis filter, the appropriate ... $R = 1 \; k \Omega, C = 2 \; \mu F$ $R = 2 \; k \Omega, C = 0.5 \; \mu F$
Consider an $\text{FM}$ broadcast that employs the pre-emphasis filter with frequency response$$H_{pe}(\omega) = 1 + \frac{j \omega}{\omega_{0}},$$where $\omega_{0} = 10^...
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1677
GATE ECE 2022 | Question: 37
Select the $\text{CORRECT}$ statement(s) regarding semiconductor devices. Electrons and holes are of equal density in an intrinsic semiconductor at equilibrium. Collector region is generally more heavily doped than Base region in a $\text{BJT}.$ ... under steady state condition. Mobility of electrons always increases with temperature in Silicon beyond $300 \; \text{K}.$
Select the $\text{CORRECT}$ statement(s) regarding semiconductor devices.Electrons and holes are of equal density in an intrinsic semiconductor at equilibrium.Collector r...
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1678
GATE ECE 2022 | Question: 35
Consider the following series: $ \sum_{n=1}^{\infty} \frac{n^{d}}{c^{n}}$ For which of the following combinations of $c, d$ values does this series converge? $ c= 1, d = – 1$ $ c= 2, d = 1$ $ c= 0.5, d = – 10$ $ c= 1, d = – 2$
Consider the following series:$$ \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 = �...
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1679
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 \...
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1680
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),...
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