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1
GATE ECE 2005 | Question: 1
The following differential equation has \[3 \frac{d^{2} y}{d t^{2}}+4\left(\frac{d y}{d t}\right)^{3}+y^{2}+2=x\] degree $=2$, order $=1$ degree $=3$, order $=2$ degree $=4$, order $=3$ degree $=2$, order $=3$
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2
GATE ECE 2005 | Question: 2
Choose the function $f(t); – \infty < 1 < \infty,$ for which a Fourier series cannot be defined. $3 \sin (25 t)$ $4 \cos (20 t+3)+2 \sin (710 t)$ $\exp (-|t|) \sin (25 t)$ $1$
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GATE ECE 2005 | Question: 3
A fair dice is rolled twice. The probability that an odd number will follow an even number is $\frac{1}{2}$ $\frac{1}{6}$ $\frac{1}{3}$ $\frac{1}{4}$
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GATE ECE 2005 | Question: 4
A solution of the following differential equation is given by \[\frac{d^{2} y}{d x^{2}}-5 \frac{d y}{d x}+6 y=0\] $y=e^{2 x}+e^{-3 x}$ $y=e^{2 x}+e^{3 x}$ $y=e^{-2 x}+e^{3 x}$ $y=e^{-2 x}+e^{-3 x}$
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5
GATE ECE 2005 | Question: 5
The function $x(t)$ is shown in the figure. Even and odd parts of a unit-step function $u(t)$ are respectively, $\frac{1}{2}, \frac{1}{2} x(t)$ $-\frac{1}{2}, \frac{1}{2} x(t)$ $\frac{1}{2},-\frac{1}{2} x(t)$ $-\frac{1}{2},-\frac{1}{2} x(t)$
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6
GATE ECE 2005 | Question: 6
The region of convergence of $Z$-transform of the sequence $\left(\frac{5}{6}\right)^{n} u(n)-\left(\frac{6}{5}\right)^{n} u(-n-1)$ must be $|z|<\frac{5}{6}$ $|z|>\frac{6}{5}$ $\frac{5}{6}<|z|<\frac{6}{5}$ $\frac{6}{5}<|z|<\infty$
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7
GATE ECE 2005 | Question: 7
The condition on $\mathrm{R}, \mathrm{L}$ and $\mathrm{C}$ such that the step response $y(t)$ in the figure has no oscillations, is $\mathrm{R} \geq \frac{1}{2} \sqrt{\frac{\mathrm{L}}{\mathrm{C}}}$ $\mathrm{R} \geq \sqrt{\frac{\mathrm{L}}{\mathrm{C}}}$ $\mathrm{R} \geq 2 \sqrt{\frac{\mathrm{L}}{\mathrm{C}}}$ $ \mathrm{R}=\frac{1}{\sqrt{\mathrm{LC}}}$
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8
GATE ECE 2005 | Question: 8
The ABCD parameters of an ideal $n: 1$ transformer shown in the figure are $\left[\begin{array}{ll}n & 0 \\ 0 & \mathrm{X}\end{array}\right]$. The value of $X$ will be $n$ $\frac{1}{n}$ $n^{2}$ $\frac{1}{n^{2}}$
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9
GATE ECE 2005 | Question: 9
In a series $\mathrm{RLC}$ circuit, $\mathrm{R}=2 \; \mathrm{k} \Omega, \mathrm{L}=1 \; \mathrm{H}$, and $\mathrm{C} \frac{1}{400}=\mu \mathrm{F}$. The resonant frequency is $2 \times 10^{4} \mathrm{~Hz}$ $\frac{1}{\pi} \times 10^{4} \mathrm{~Hz}$ $10^{4} \mathrm{~Hz}$ $2 \pi \times 10^{4} \mathrm{~Hz}$
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10
GATE ECE 2005 | Question: 10
The maximum power that can be transferred to the load resistor $R_{L}$ from the voltage source in the figure is $1 \mathrm{~W}$ $10 \mathrm{~W}$ $0.25 \mathrm{~W}$ $0.5 \mathrm{~W}$
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11
GATE ECE 2005 | Question: 11
The bandgap of Silicon at room temperature is $1.3 \; \mathrm{eV}$ $0.7 \; \mathrm{eV}$ $1.1 \; \mathrm{eV}$ $1.4 \; \mathrm{eV}$
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12
GATE ECE 2005 | Question: 12
A Silicon PN junction at a temperature of $20^{\circ} \mathrm{C}$ has a reverse saturation current of $10$ picoAmperes $(\mathrm{pA})$. The reverse saturation current at $40^{\circ} \mathrm{C}$ for the same bias is approximately $30 \; \mathrm{pA}$. $40 \; \mathrm{pA}$. $50 \; \mathrm{pA}$ $60 \; \mathrm{pA}$.
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13
GATE ECE 2005 | Question: 13
The primary reason for the widespread use of Silicon in semiconductor device technology is aboundance of Silicon on the surface of the Earth. larger bandgap of Silicon in comparison to Germanium. favorable properties of Silicon-dioxide $\left(\mathrm{SiO}_{2}\right)$. lower melting point.
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14
GATE ECE 2005 | Question: 14
The effect of current shunt feedback in an amplifier is to increase the input resistance and decrease the output resistance. increase both input and output resistances. decrease both input and output resistances. decrease the input resistance and increase the output resistance.
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15
GATE ECE 2005 | Question: 15
The input resistance $\text{R}_{\mathrm{i}}$ of the amplifier shown in the figure is $\frac{30}{4} \; \mathrm{k} \Omega$ $10 \; \mathrm{k} \Omega$ $40 \; \mathrm{k} \Omega$ infiinte
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Sep 22
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16
GATE ECE 2005 | Question: 16
The first and the last critical frequency of an RC-driving point impedance function must respectively be a zero and a pole a zero and a zero a pole and a pole a pole and a zero
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Sep 22
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17
GATE ECE 2005 | Question: 17
The cascode amplifier is a multistage configuration of $\mathrm{CC}-\mathrm{CB}$ $\mathrm{CE}-\mathrm{CB}$ $\mathrm{CB}-\mathrm{CC}$ $\mathrm{CE}-\mathrm{CC}$
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Sep 22
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18
GATE ECE 2005 | Question: 18
Decimal $43$ in Hexadecimal and BCD number system is respectively $\text{B2, 01000011}$ $\text{2B, 01000011}$ $\text{2B, 00110100}$ $\text{B2, 0100 0100}$
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Sep 22
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19
GATE ECE 2005 | Question: 19
The Boolean function $f$ implemented in the figure using two input multiplexers is $\mathrm{A} \overline{\mathrm{B}} \text{C}+\mathrm{A} \overline{\mathrm{B}} \overline{\mathrm{C}}$ $\mathrm{ABC}+\mathrm{A} \overline{\mathrm{B}} \overline{\mathrm{C}}$ ... $\overline{\mathrm{A}} \overline{\mathrm{B}} \mathrm{C}+\overline{\mathrm{A}} \mathrm{B} \overline{\mathrm{C}}$
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Sep 22
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20
GATE ECE 2005 | Question: 20
Which of the following can be impulse response of a casual system?
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21
GATE ECE 2005 | Question: 21
Let $x(n)=\left(\frac{1}{2}\right)^{n} u(n), y(n)=x^{2}(n)$ and $\mathrm{Y}\left(e^{j i e}\right)$ be the Fourier transform of $y(n)$. Then $Y\left(e^{j i e}\right)$ is $\frac{1}{4}$ $2$ $4$ $\frac{4}{3}$
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22
GATE ECE 2005 | Question: 22
Find the correct match between group $1$ and group $2.$ ... $\text{P - X, Q - W, R - Z, S - Y,}$ $\text{P - Y, Q - Z, R - W, S - X,}$
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23
GATE ECE 2005 | Question: 23
The power in the signal $s(t)=8 \cos \left(20 \pi t-\frac{\pi}{2}\right)+$ $4 \sin (15 \pi t)$ is $40$ $41$ $42$ $82$
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24
GATE ECE 2005 | Question: 24
Which of the following analog modulation scheme requires the minimum transmitted power and minimum channel bandwidth? VSB DSB-SC SSB AM
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Sep 22
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25
GATE ECE 2005 | Question: 25
A linear system is equivalently represented by two sets of state equations. \[\dot{X}=\mathrm{AX}+\mathrm{BU} \text { and } \dot{W}=\mathrm{CW}+\mathrm{DU} \text {. }\] The eigenvalues of the representations are also computed as $\{\lambda\}$ and $\{\mu\}$. Which one of the ... $X \neq W$ $[\lambda] \neq[\mu]$ and $X=W$ $[\lambda] \neq[\mu]$ and $X \neq W$
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Sep 22
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26
GATE ECE 2005 | Question: 26
Which one of the following polar diagrams corresponds to a lag network?
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Sep 22
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27
GATE ECE 2005 | Question: 27
Despite the presence of negative feedback, control systems still have problems of instability because the components used have nonlinearities. dynamic equations of the subsystems are not known exactly. mathematical analysis involves approximations. system has large negative phase angle at high frequencies.
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Sep 22
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28
GATE ECE 2005 | Question: 28
The magnetic field intensity vector of a plane wave is given by $\overline{\mathrm{H}}(x, y, z, t)=10 \sin \left(50000 t+0.004 x+30 \; \hat{a}_{y}\right)$, where $\hat{a}_{y}$ denotes the unit vector in $y$ ... $-1.25 \times 10^{7} \mathrm{~m} / \mathrm{s}$. $3 \times 10^{8} \mathrm{~m} / \mathrm{s}$.
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Sep 22
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29
GATE ECE 2005 | Question: 29
Many circles are drawn in a Smith chart used for transmission line calculations. The circles shown in the figure represent unit circles. constant resistance circles. constant reactance circles. constant reflection coefficient circles.
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30
GATE ECE 2005 | Question: 30
Refractive index of glass is $1.5.$ Find the wavelength of a beam of light with a frequency of $10^{14} \mathrm{~Hz}$ in glass. Assume velocity of light is $3 \times 10^{8} \mathrm{~m} / \mathrm{s}$ in vacuum. $3 \; \mu \mathrm{m}$ $3 \mathrm{~mm}$ $2 \; \mu \mathrm{m}$ $1 \; \mu \mathrm{m}$
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Sep 22
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31
GATE ECE 2005 | Question: 31
In what range should $\operatorname{Re}(s)$ remain so that the Laplace transform of the function $e^{(n+2)t+5}$ exits? $\operatorname{Re}(s)>a+2$ $\operatorname{Re}(\mathrm{s})>a+7$ $\operatorname{Re}(s)<2$ $\operatorname{Re}(s)>a+5$
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32
GATE ECE 2005 | Question: 32
Given the matrix $\left[\begin{array}{cc}-4 & 2 \\ 4 & 3\end{array}\right]$, the eigenvector is $\left[\begin{array}{l}3 \\ 2\end{array}\right]$ $\left[\begin{array}{l}4 \\ 3\end{array}\right]$ $\left[\begin{array}{c}2 \\ -1\end{array}\right]$ $\left[\begin{array}{c}-2 \\ 1\end{array}\right]$
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33
GATE ECE 2005 | Question: 33
Let, $\mathrm{A}=\left[\begin{array}{cc}2 & -0.1 \\ 0 & 3\end{array}\right]$ and $\mathrm{A}^{-1}=\left[\begin{array}{ll}\frac{1}{2} & \mathrm{a} \\ 0 & \mathrm{~b}\end{array}\right]$ Then $(a+b)=$ $\frac{7}{20}$ $\frac{3}{20}$ $\frac{19}{60}$ $\frac{11}{20}$
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34
GATE ECE 2005 | Question: 34
The value of the integral $\text{I}=\frac{1}{\sqrt{2 \pi}} \int_{0}^{\infty} \exp \left(-\frac{x^{2}}{8}\right)$ $d x$ is $1$ $\pi$ $2$ $2 \pi$
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35
GATE ECE 2005 | Question: 35
The derivative of the symmetric function drawn in given figure will look like
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36
GATE ECE 2005 | Question: 36
Match the following and choose the correct combination: ... $\text{E-1, F-3, G-4, H-2}$ $\text{E-5, F-3, G-4, H-1}$
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37
GATE ECE 2005 | Question: 37
Given an orthogonal matrix $A=\left[\begin{array}{cccc}1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1\end{array}\right]$ $\left[\mathrm{AA}^{\mathrm{T}}\right]^{-1}$ ...
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38
GATE ECE 2005 | Question: 38
For the circuit show in the figure, the instantaneous current $i_{i}(t)$ is $\frac{10 \sqrt{3}}{2} \angle 90^{\circ} \mathrm{Amps}$. $\frac{10 \sqrt{3}}{2} \angle-90^{\circ}$ Amps. $5 \angle 60^{\circ}$ Amps. $5 \angle-60^{\circ}$ Amps.
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39
GATE ECE 2005 | Question: 39
Impedance $\mathrm{Z}$ as shown in the given figure is $j 29 \Omega$ $j9 \Omega$ $j 19 \Omega$ $j 39 \Omega$
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40
GATE ECE 2005 | Question: 40
For the circuit shown in the figure, Thevenin's voltage and Thevenin's equivalent resistance atterminals $a-b$ is $5 \mathrm{~V}$ and $2 \; \Omega$. $7.5 \mathrm{~V}$ and $2.5 \; \Omega$. $4 \mathrm{~V}$ and $2 \; \Omega$. $3 \mathrm{~V}$ and $2.5 \; \Omega$.
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