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41
GATE ECE 2024 | Question: 33
​​​​​Let $z$ be a complex variable. If $f(z)=\frac{\sin (\pi z)}{z^{2}(z-2)}$ and $C$ is the circle in the complex plane with $|z|=3$ then $\oint_{C} f(z) d z$ is $\_\_\_\_\_\_$. $\pi^{2} j$ $j \pi\left(\frac{1}{2}-\pi\right)$ $j \pi\left(\frac{1}{2}+\pi\right)$ $-\pi^{2} j$
​​​​​Let $z$ be a complex variable. If $f(z)=\frac{\sin (\pi z)}{z^{2}(z-2)}$ and $C$ is the circle in the complex plane with $|z|=3$ then $\oint_{C} f(z) d z$ ...
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42
GATE ECE 2024 | Question: 34
Consider two continuous time signals $x(t)$ and $y(t)$ as shown below If $X(f)$ denotes the Fourier transform of $x(t)$, then the Fourier transform of $y(t)$ is $\_\_\_\_\_\_$. $-4 X(4 f) e^{-j \pi f}$ $-4 X(4 f) e^{-j 4 \pi f}$ $-\frac{1}{4} X(f / 4) e^{-j \pi f}$ $-\frac{1}{4} X(f / 4) e^{-j 4 \pi f}$
Consider two continuous time signals $x(t)$ and $y(t)$ as shown belowIf $X(f)$ denotes the Fourier transform of $x(t)$, then the Fourier transform of $y(t)$ is $\_\_\_\_\...
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48
GATE ECE 2024 | Question: 40
In the circuit below, the opamp is ideal. If the circuit is to show sustained oscillations, the respective values of $R_{1}$ and the corresponding frequency of oscillation are $29 R$ and $1 /(2 \pi \sqrt{6} R C)$ $2 R$ and $1 /(2 \pi R C)$ $29 R$ and $1 /(2 \pi R C)$ $2 R$ and $1 /(2 \pi \sqrt{6} R C)$
In the circuit below, the opamp is ideal.If the circuit is to show sustained oscillations, the respective values of $R_{1}$ and the corresponding frequency of oscillation...
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51
GATE ECE 2024 | Question: 43
Consider a system $S$ represented in state space as \[ \frac{d x}{d t}=\left[\begin{array}{ll} 0 & -2 \\ 1 & -3 \end{array}\right] x+\left[\begin{array}{l} 1 \\ 0 \end{array}\right] r, y=\left[\begin{array}{ll} 2 & -5 \end{array}\ ...
Consider a system $S$ represented in state space as\[\frac{d x}{d t}=\left[\begin{array}{ll}0 & -2 \\1 & -3\end{array}\right] x+\left[\begin{array}{l}1 \\0\end{array}\rig...
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52
GATE ECE 2024 | Question: 44
Let $F_{1}, F_{2}$, and $F_{3}$ be functions of $(x, y, z)$. Suppose that for every given pair of points $A$ and $B$ in space, the line integral $\int_{C}\left(F_{1} \mathrm{~d} x+F_{2} \mathrm{~d} y+F_{3} \mathrm{~d} z\right)$ evaluates to ...
Let $F_{1}, F_{2}$, and $F_{3}$ be functions of $(x, y, z)$. Suppose that for every given pair of points $A$ and $B$ in space, the line integral $\int_{C}\left(F_{1} \mat...
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57
GATE ECE 2024 | Question: 50
For the two port network shown below, the value of the $Y_{21}$ parameter (in Siemens) is $\_\_\_\_\_\_$.
For the two port network shown below, the value of the $Y_{21}$ parameter (in Siemens) is $\_\_\_\_\_\_$.
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59
GATE ECE 2024 | Question: 52
In the network shown below, maximum power is to be transferred to the load $R_{L}$. The value of $R_{L}$ (in $\Omega$ ) is $\_\_\_\_\_\_\_$.
In the network shown below, maximum power is to be transferred to the load $R_{L}$.The value of $R_{L}$ (in $\Omega$ ) is $\_\_\_\_\_\_\_$.
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64
GATE ECE 2023 | Question: 2
The rate of increase, of a scalar field $f(x, y, z)=x y z$, in the direction $v=(2,1,2)$ at a point $(0,2,1)$ is $\frac{2}{3}$ $\frac{4}{3}$ $2$ $4$
The rate of increase, of a scalar field $f(x, y, z)=x y z$, in the direction $v=(2,1,2)$ at a point $(0,2,1)$ is$\frac{2}{3}$$\frac{4}{3}$$2$$4$
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65
GATE ECE 2023 | Question: 4
The value of the contour integral, $\oint_C\left(\frac{z+2}{z^2+2 z+2}\right) d z$, where the contour $\mathrm{C}$ is $\left\{z:\left|z+1-\frac{3}{2} j\right|=1\right\}$, taken in the counter clockwise direction, is $-\pi(1+j)$ $\pi(1+j)$ $\pi(1-j)$ $-\pi(1-j)$
The value of the contour integral, $\oint_C\left(\frac{z+2}{z^2+2 z+2}\right) d z$, where the contour $\mathrm{C}$ is $\left\{z:\left|z+1-\frac{3}{2} j\right|=1\right\}$,...
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67
GATE ECE 2023 | Question: 6
In a semiconductor, if the Fermi energy level lies in the conduction band, then the semiconductor is known as degenerate $\text{n}$-type. degenerate $\text{p}$-type. non-degenerate $\text{n}$-type. non-degenerate $\text{p}$-type.
In a semiconductor, if the Fermi energy level lies in the conduction band, then the semiconductor is known asdegenerate $\text{n}$-type.degenerate $\text{p}$-type.non-deg...
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69
GATE ECE 2023 | Question: 8
A series $\text{RLC}$ circuit has a quality factor $\text{Q}$ of $1000$ at a center frequency of $10^{6} \; \mathrm{rad} / \mathrm{s}$. The possible values of $\text{R, L}$ and $\text{C}$ are $R=1 \; \Omega, L=1 \; \mu H$ and $C=1 \; \mu F$ ... $C=1 \; \mu F$ $R=0.001 \; \Omega, L=1 \; \mu H$ and $C=1 \; \mu F$
A series $\text{RLC}$ circuit has a quality factor $\text{Q}$ of $1000$ at a center frequency of $10^{6} \; \mathrm{rad} / \mathrm{s}$. The possible values of $\text{R,...
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70
GATE ECE 2023 | Question: 9
For a $\text{MOS}$ capacitor, $\text{V}_{\mathrm{fb}}$ and $\text{V}_{\mathrm{t}}$ are the flat-band voltage and the threshold voltage, respectively. The variation of the depletion width $\left(\mathrm{W}_{\mathrm{dep}}\right)$ for varying gate voltage $\left(\text{V}_g\right)$ is best represented by
For a $\text{MOS}$ capacitor, $\text{V}_{\mathrm{fb}}$ and $\text{V}_{\mathrm{t}}$ are the flat-band voltage and the threshold voltage, respectively. The variation of the...
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71
GATE ECE 2023 | Question: 10
Consider a narrow band signal, propagating in a lossless dielectric medium $\left(\varepsilon_{r}=4, \mu_{r}=1\right)$, with phase velocity $v_{p}$ and group velocity $v_{g}$. Which of the following statement is true? ($c$ ... $v_{p} > c, v_{g} < c$ $v_{p} < c, v_{g} < c$
Consider a narrow band signal, propagating in a lossless dielectric medium $\left(\varepsilon_{r}=4, \mu_{r}=1\right)$, with phase velocity $v_{p}$ and group velocity $v_...
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72
GATE ECE 2023 | Question: 11
In the circuit shown below, $\text{V}_1$ and $\text{V}_2$ are bias voltages. Based on input and output impedances, the circuit behaves as a voltage controlled voltage source. voltage controlled current source. current controlled voltage source. current controlled current source.
In the circuit shown below, $\text{V}_1$ and $\text{V}_2$ are bias voltages. Based on input and output impedances, the circuit behaves as avoltage controlled voltage sour...
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74
GATE ECE 2023 | Question: 14
The synchronous sequential circuit shown below works at a clock frequency $1 \; \mathrm{GHz}$. The throughput, in $\text{Mbits/s,}$ and the latency, in $\text{ns,}$ respectively, are $1000,3$ $333.33,1$ $2000,3$ $333.33,3$
The synchronous sequential circuit shown below works at a clock frequency $1 \; \mathrm{GHz}$. The throughput, in $\text{Mbits/s,}$ and the latency, in $\text{ns,}$ respe...
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76
GATE ECE 2023 | Question: 16
Consider a system with input $x(t)$ and output $y(t)=x\left(e^{t}\right)$. The system is Causal and time invariant. Non-causal and time varying. Causal and time varying. Non-causal and time invariant.
Consider a system with input $x(t)$ and output $y(t)=x\left(e^{t}\right)$. The system isCausal and time invariant.Non-causal and time varying.Causal and time varying.Non-...
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77
GATE ECE 2023 | Question: 17
Let $m(t)$ be a strictly band-limited signal with bandwidth $B$ and energy $E$. Assuming $\omega_{0}=10 B$, the energy in the $m(t) \cos \omega_{0} t$ is $\frac{E}{4}$ $\frac{E}{2}$ $E$ $2E$
Let $m(t)$ be a strictly band-limited signal with bandwidth $B$ and energy $E$. Assuming $\omega_{0}=10 B$, the energy in the $m(t) \cos \omega_{0} t$ is$\frac{E}{4}$$\f...
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78
GATE ECE 2023 | Question: 18
The Fourier transform $X(\omega)$ of $x(t)=e^{-t^{2}}$ is Note: $\int_{-\infty}^{\infty} e^{-y^{2}} d y=\sqrt{\pi}$ $\sqrt{\pi} e^{\frac{\omega^{2}}{2}}$ $\frac{e^{-\frac{\omega^{2}}{4}}}{2 \sqrt{\pi}}$ $\sqrt{\pi} e^{-\frac{\omega^{2}}{4}}$ $\sqrt{\pi} e^{-\frac{\omega^{2}}{2}}$
The Fourier transform $X(\omega)$ of $x(t)=e^{-t^{2}}$ isNote: $\int_{-\infty}^{\infty} e^{-y^{2}} d y=\sqrt{\pi}$$\sqrt{\pi} e^{\frac{\omega^{2}}{2}}$$\frac{e^{-\frac{\o...
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79
GATE ECE 2023 | Question: 19
In the table shown below, match the signal type with its spectral characteristics. Signal type Spectral characteristics (i) Continuous, aperiodic (a) Continuous, aperiodic (ii) Continuous, periodic (b) Continuous, periodic (iii) Discrete, aperiodic (c) Discrete, aperiodic (iv) ...
In the table shown below, match the signal type with its spectral characteristics. Signal type Spectral characteristics (i) Continuous, aperiodic(a) Continuous, aperiod...
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80
GATE ECE 2023 | Question: 20
For a real signal, which of the following is/are valid power spectral density/densities? $\text{S}_X(\omega)=\frac{2}{9+\omega^2}$ $\text{S}_X(\omega)=e^{-\omega^2} \cos ^2 \omega$
For a real signal, which of the following is/are valid power spectral density/densities?$\text{S}_X(\omega)=\frac{2}{9+\omega^2}$$\text{S}_X(\omega)=e^{-\omega^2} \cos ^2...
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