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GATE ECE 2009 | Question: 1
The order of the differential equation $\frac{d^{2} y}{d t^{2}}+\left(\frac{d y}{d t}\right)^{3}+y^{4}=e^{-t} \quad$ is $1$ $2$ $3$ $4$
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GATE ECE 2009 | Question: 2
The Fourier series of a real periodic function has only cosine terms if it is even sine terms if it is even cosine terms if it is odd sine terms if it is odd Which of the above statements are correct? $\mathrm{P}$ and $\mathrm{S}$ $\mathrm{P}$ and $\mathrm{R}$ $\mathrm{Q}$ and $\mathrm{S}$ $\mathrm{Q}$ and $\mathrm{R}$
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GATE ECE 2009 | Question: 3
A function is given by $f(t)=\sin ^{2} t+\cos 2 t$. Which of the following is true? $f$ has frequency components at 0 and $1 / 2 \pi \; \mathrm{Hz}$ $f$ has frequency components at 0 and $1 / \pi \; \mathrm{Hz}$ $f$ has frequency components at $1 / 2 \pi$ and $1 / \pi \; \mathrm{Hz}$ $f$ has frequency components at $0,1 / 2 \pi$ and $1 / \pi \; \mathrm{Hz}$
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GATE ECE 2009 | Question: 4
A fully charged mobile phone with a $12 \mathrm{~V}$ battery is good for a $10$ minute talk-time. Assume that, during the talk-time, the battery delivers a constant current of $2 \mathrm{~A}$ and its voltage drops linearly from $12 \mathrm{~V}$ ... deliver during this talk-time ? $220 \mathrm{~J}$ $12 \mathrm{~kJ}$ $13.2 \mathrm{~kJ}$ $14.4 \mathrm{~kJ}$
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GATE ECE 2009 | Question: 5
In an n-type silicon crystal at room temperature, which of the following can have a concentration of $4 \times 10^{19} \mathrm{~cm}^{-3}$ ? Silicon atoms Holes Dopant atoms Valence electrons
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GATE ECE 2009 | Question: 6
The full forms of the abbreviations TTL and CMOS in reference to logic families are Triple Transistor Logic and Chip Metal Oxide Semiconductor Tristate Transistor Logic and Chip Metal Oxide Semiconductor Transistor Transistor Logic and Complementary Metal Oxide Semiconductor Tristate Transistor Logic and Complementary Metal Oxide Silicon
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GATE ECE 2009 | Question: 7
The ROC of Z-transform of the discrete time sequence $\quad x(n)=\left(\frac{1}{3}\right)^{n} u(n)-\left(\frac{1}{2}\right)^{n} u(-n-1) \quad$ is $\frac{1}{3}<|z|<\frac{1}{2}$ $|z|>\frac{1}{2}$ $|z|<\frac{1}{3}$ $2<|z|<3$
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GATE ECE 2009 | Question: 8
The magnitude plot of a rational transfer function $G(s)$ with real coefficients is shown below. Which of the following compensators hás such a magnitude plot? Lead compensator Lag compensator PID compensator Lead-lag compensator
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GATE ECE 2009 | Question: 9
A white noise process $X(t)$ with two-sided power spectral density $1 \times 10^{-10} \mathrm{~W} / \mathrm{Hz}$ is input to a filter whose magnitude squared response is shown below. The power of the output process $Y(t)$ is given by $5 \times 10^{-7} \mathrm{~W}$ $1 \times 10^{-6} \mathrm{~W}$ $2 \times 10^{-6} \mathrm{~W}$ $1 \times 10^{-5} \mathrm{~W}$
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GATE ECE 2009 | Question: 10
Which of the following statements is true regarding the fundamental mode of the metallic waveguides shown? Only $\text{P}$ has no cutoff-frequency Only $\text{Q}$ has no cutoff-frequency Only $\text{R}$ has no cutoff-frequency All three have cut-off frequencies
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GATE ECE 2009 | Question: 11
A fair coin is tossed $10$ times. What is the probability that $\text{ONLY}$ the first two tosses will yield heads? $\left(\frac{1}{2}\right)^{2}$ ${ }^{10} C_{2}\left(\frac{1}{2}\right)^{2}$ $\left(\frac{1}{2}\right)^{10}$ ${ }^{10} C_{2}\left(\frac{1}{2}\right)^{10}$
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GATE ECE 2009 | Question: 12
If the power spectral density of stationary random process is a sinc-squared function of frequency, the shape of its autocorrelation is
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GATE ECE 2009 | Question: 13
If $f(z) = c_{0} + c_{1} z^{-1},$ then $\oint_{\text{unit circle}} \frac{1+f(z)}{z} dz$ is given by $2 \pi c_{1}$ $2 \pi\left(1+c_{0}\right)$ $2 \pi j c_{1}$ $2 \pi j\left(1+c_{0}\right)$
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GATE ECE 2009 | Question: 14
In the interconnection of ideal sources shown in the figure, it is known that the $60 \mathrm{~V}$ source is absorbing power. Which of the following can be the value of the current source $I$? $10 \mathrm{~A}$ $13 \mathrm{~A}$ $15 \mathrm{~A}$ $18 \mathrm{~A}$
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GATE ECE 2009 | Question: 15
The ratio of the mobility to the diffusion coefficient in a semiconductor has the units $\mathrm{V}^{-1}$ $\mathrm{cm} \cdot \mathrm{V}^{-1}$ $\mathrm{V} \cdot \mathrm{cm}^{-1}$ $\mathrm{V} \cdot \mathrm{s}$
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GATE ECE 2009 | Question: 16
In a microprocessor, the service routine for a certain interrupt starts from a fixed location of memory which cannot be externally set, but the interrupt can be delayed or rejected. Such an interrupt is non-maskable and non-vectored maskable and non-vectored non-maskable and vectored maskable and vectored
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GATE ECE 2009 | Question: 17
If the transfer function of the following network is $\frac{V_{o}(s)}{V_{i}(s)}=\frac{1}{2+s C R}$. the value of the load resistance $R_{L}$ is $R / 4$ $R / 2$ $R$ $2 R$
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GATE ECE 2009 | Question: 18
Consider the system $\frac{d x}{d t}=A x+B u$ with $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$ and $B=\left[\begin{array}{l}p \\ q\end{array}\right]$ where $p$ and $q$ are arbitrary ... $q=0$ result in controllability The system is uncontrollable for all values of $p$ and $q$ We cannot conclude about controllability from the given data
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GATE ECE 2009 | Question: 19
For a message signal $m(t)=\cos \left(2 \pi f_{m} t\right)$ and carrier of frequency $f_{c}$, which of the following represents a single-side-band (SSB) signal ? $\cos \left(2 \pi f_{m} t\right) \cos \left(2 \pi f_{c} t\right)$ ... $\left[1+\cos \left(2 \pi f_{m} t\right] \cos \left(2 \pi f_{c} t\right)\right.$
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GATE ECE 2009 | Question: 20
Two infinitely long wires carrying current are as shown in the figure below. One wire is in the $y-z$ plane and parallel to the $y$-axis. The other wire is in the $x-y$ plane and parallel to the $x$-axis. Which components of the resulting magnetic field are non-zero at the origin ? $x, y, z$ components $x, y$ components $y, z$ components $x, z$ components
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GATE ECE 2009 | Question: 21
Consider two independent random variables $X$ and $Y$ with identical distributions. The variables $X$ and $Y$ take values $0,1$ and $2$ with probabilities $\frac{1}{2}, \frac{1}{4}$ and $\frac{1}{4}$ respectively. What is the conditional probability $P(X+Y=2 \mid X-Y=0) ?$ $0$ $\frac{1}{16}$ $\frac{1}{6}$ $1$
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GATE ECE 2009 | Question: 22
The Taylor series expansion of $\frac{\sin x}{x-\pi}$ at $x=\pi$ is given by $1+\frac{(x-\pi)^{2}}{3 !}+\ldots$ $-1-\frac{(x-\pi)^{2}}{3 !}+\ldots$ $1-\frac{(x-\pi)^{2}}{3 !}+\ldots$ $-1+\frac{(x-\pi)^{2}}{3 !}+\ldots$
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GATE ECE 2009 | Question: 23
If a vector field $\vec{V}$ is related to another vector field $\vec{A}$ through $\vec{V}=\nabla \times \vec{A}$, which of the following is true ? Note: $C$ and $S_{C}$ refer to any closed contour and any surface whose boundary is $C$ ... $\oint_{C} \nabla \times \vec{A} \cdot \overrightarrow{d l}=\int \int_{S_{C}} \vec{V} \cdot \overrightarrow{d S}$
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GATE ECE 2009 | Question: 24
Given that $F(s)$ is the one-sided Laplace transform of $f(t)$, the Laplace transform of $\int_{0}^{t} f(\tau) d \tau$ is $s F(s)-f(0)$ $\frac{1}{s} F(s)$ $\int_{0}^{s} F(\tau) d \tau$ $\frac{1}{s}[F(s)-f(0)]$
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GATE ECE 2009 | Question: 25
Match each differential equation in Group I to its family of solution curves from Group II. ... $\text{P-1, Q-3, R-2, S-1}$ $\text{P-2, Q-1, R-3, S-3}$ $\text{P-3, Q-2, R-1, S-2}$
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GATE ECE 2009 | Question: 26
The eigen values of the following matrix are \[ \left[\begin{array}{ccc} -1 & 3 & 5 \\ -3 & -1 & 6 \\ 0 & 0 & 3 \end{array}\right] \] $3,3+5 j, 6-j$ $-6+5 j, 3+j, 3-j$ $3+j, 3-j, 5+j$ $3,-1+3 j,-1-3 j$
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GATE ECE 2009 | Question: 27
An AC source of RMS voltage $20 \mathrm{~V}$ with internal impedance $Z_{s}=(1+2 j) \Omega$ feeds a load of impedance $Z_{L}=(7+4 j) \Omega$ in the figure below. The reactive power consumed by the load is $8 \; \mathrm{VAR}$ $16 \; \mathrm{VAR}$ $28 \; \mathrm{VAR}$ $32 \; \mathrm{VAR}$
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GATE ECE 2009 | Question: 28
The switch in the circuit shown was on position $a$ for a long time, and is moved to position $b$ at time $t=0$. The current $i(t)$ for $t>0$ is given by $0.2 e^{-125 t} u(t) \; \mathrm{mA}$ $20 e^{-1250 t} u(t) \; \mathrm{mA}$ $0.2 e^{-1250 t} u(t) \; \mathrm{mA}$ $20 e^{-1000 t} u(t) \; \mathrm{mA}$
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GATE ECE 2009 | Question: 29
In the circuit shown, what value of $R_{L}$ maximizes the power delivered to $R_{L}$ ? $2.4 \; \Omega$ $\frac{8}{3} \; \Omega$ $4 \; \Omega$ $6 \; \Omega$
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GATE ECE 2009 | Question: 30
The time domain behavior of an $R L$ circuit is represented by \[ L \frac{d i}{d t}+R i=V_{o}\left(1+B e^{-R t / L} \sin t\right) u(t). \] For an initial current of $i(0)=\frac{V_{o}}{R}$ ... $ i(t) \rightarrow \frac{V_{0}}{R}(1+B)$ $ i(t) \rightarrow \frac{2 V_{0}}{R}(1+B)$
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GATE ECE 2009 | Question: 31
In the circuit below, the diode is ideal. The voltage $V$ is given by $\min \left(V_{i}, 1\right)$ $\max \left(V_{i}, 1\right)$ $\min \left(-V_{i}, 1\right)$ $\max \left(-V_{i}, 1\right)$
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GATE ECE 2009 | Question: 32
Consider the following two statements about the internal conditions in an n-channel MOSFET operating in the active region. $\qquad \qquad \text{S1:}$ The inversion charge decreases from source to drain $\qquad \qquad \text{S2:}$ The channel potential increases from source to drain ... $\mathrm{S} 2$ are true, and $\mathrm{S} 2$ is a reason for $\mathrm{S} 1$
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GATE ECE 2009 | Question: 33
In the following astable multivibrator circuit, which properties of $v_{o}(t)$ depend on $R_{2}$ ? Only the frequency Only the amplitude Both the amplitude and the frequency Neither the amplitude nor the frequency
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GATE ECE 2009 | Question: 34
In the circuit shown below, the $\text{op-amp}$ is ideal, the transistor has $V_{B E}=0.6 \mathrm{~V}$ and $\beta=150$. Decide whether the feedback in the circuit is positive or negative and determine the voltage $V$ ... $V=0 \mathrm{~V}$ Negative feedback, $V=5 \mathrm{~V}$ Negative feedback, $V=2 \mathrm{~V}$
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GATE ECE 2009 | Question: 35
A small signal source $v_{i}(t)=A \cos 20 t+B \sin 10^{6} t$ is applied to a transistor amplifier as shown below. The transistor has $\beta=150$ and $h_{i e}=3 \; \mathrm{k} \Omega$. Which expression best approximates $v_{o}(t)$ ... $v_{o}(t)=-1500 B \sin 10^{6} t$ $v_{o}(t)=-150 B \sin 10^{6} t$
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GATE ECE 2009 | Question: 36
If $X=1$ in the logic equation $[X+Z\{\bar{Y}+(\bar{Z}+X \bar{Y})\}]\{\bar{X}+\bar{Z}(X+Y)\}=1$, then $Y=Z$ $Y=\bar{Z}$ $Z=1$ $Z=0$
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GATE ECE 2009 | Question: 37
What are the minimum number of $2$-to-$1$ multiplexers required to generate a $2$-input AND gate and a $2$-input Ex-OR gate ? $1$ and $2$ $1$ and $3$ $1$ and $1$ $2$ and $2$
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GATE ECE 2009 | Question: 38
Refer to the NAND and NOR latches shown in the figure. The inputs $\left(P_{1}, P_{2}\right)$ for both the latches are first made $(0,1)$ and then, after a few seconds, made $(1,1)$. The corresponding stable outputs $\left(Q_{1}, Q_{2}\right)$ are NAND: first $(0,1)$ ... NOR: first $(1,0)$ then $(0,0)$ NAND: first $(1,0)$ then $(1,1)$ NOR: first $(0,1)$ then $(0,1)$
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GATE ECE 2009 | Question: 39
What are the counting states $\left(Q_{1}, Q_{2}\right)$ for the counter shown in the figure below ? $11,10,00,11,10, \ldots$ $01,10,11,00,01, \ldots$ $00,11,01,10,00, \ldots$ $01,10,00,01,10, \ldots$
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GATE ECE 2009 | Question: 41
Consider a system whose input $x$ and output $y$ are related by the equation \[ y(t)=\int_{-\infty}^{\infty} x(t-\tau) h(2 \tau) d \tau, \] where $h(t)$ is shown in the graph. Which of the following four properties are possessed by the ... $\text{LTI:}$ The system is linear and time-invariant. Causal, LP BIBO, LTI BIBO, Causal, LTI LP, LTI
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