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GATE ECE 2008 | Question: 1
All the four entries of the $2 \times 2$ matrix $\mathbf{P}=\left[\begin{array}{ll}p_{11} & p_{12} \\ p_{21} & p_{22}\end{array}\right]$ are nonzero, and one of its eigenvalues is zero. Which of the following statements is true? $p_{11} p_{22}-p_{12} p_{21}=1$ $p_{11} p_{22}-p_{12} p_{21}=-1$ $p_{11} p_{22}-p_{12} p_{21}=0$ $p_{11} p_{22}+p_{12} p_{21}=0$
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GATE ECE 2008 | Question: 2
The system of linear equations \[ \begin{array}{l} 4 x+2 y=7 \\ 2 x+y=6 \end{array} \] has a unique solution no solution an infinite number of solutions exactly two distinct solutions
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GATE ECE 2008 | Question: 3
The equation $\sin (z)=10$ has no real or complex solution exactly two distinct complex solutions a unique solution an infinite number of complex solutions
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GATE ECE 2008 | Question: 4
For real values of $x$, the minimum value of the function $f(x)=\exp (x)+\exp (-x)$ is $2$ $1$ $0.5$ $0$
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5
GATE ECE 2008 | Question: 5
Which of the following functions would have only odd powers of $x$ in its Taylor series expansion about the point $x=0$ ? $\sin \left(x^{3}\right)$ $\sin \left(x^{2}\right)$ $\cos \left(x^{3}\right)$ $\cos \left(x^{2}\right)$
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GATE ECE 2008 | Question: 6
Which of the following is a solution to the differential equation $\frac{d x(t)}{d t}+3 x(t)=0$ ? $x(t)=3 e^{-t}$ $x(t)=2 e^{-3 t}$ $x(t)=-\frac{3}{2} t^{2}$ $x(t)=3 t^{2}$
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GATE ECE 2008 | Question: 7
In the following graph, the number of trees $\text{(P)}$ and the number of cut-sets $\text{(Q)}$ are $\mathrm{P}=2, \quad \mathrm{Q}=2$ $\mathrm{P}=2, \mathrm{Q}=6$ $\mathrm{P}=4, \mathrm{Q}=6$ $\mathrm{P}=4, \mathrm{Q}=10$
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GATE ECE 2008 | Question: 8
In the following circuit, the switch $\mathrm{S}$ is closed at $t=0$. The rate of change of current $\frac{d i}{d t}(0+)$ is given by $0$ $\frac{R_{S} I_{S}}{L}$ $\frac{\left(R+R_{s}\right) I_{s}}{L}$ $\infty$
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9
GATE ECE 2008 | Question: 9
The input and output of a continuous time system are respectively denoted by $x(t)$ and $y(t)$. Which of the following descriptions corresponds to a causal system? $y(t)=x(t-2)+x(t+4)$ $y(t)=(t-4) x(t+1)$ $y(t)=(t+4) x(t-1)$ $y(t)=(t+5) x(t+5)$
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10
GATE ECE 2008 | Question: 10
The impulse response $h(t)$ of a linear time-invariant continuous time system is described by $h(t)=\exp (\alpha t) u(t)+\exp (\beta t) u(-t)$, where $u(t)$ denotes the unit step function, and $\alpha$ and $\beta$ are real constants. ... $\beta$ is negative $\alpha$ is positive and $\beta$ is negative $\alpha$ is negative and $\beta$ is positive
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11
GATE ECE 2008 | Question: 11
The pole-zero plot given below corresponds to a Low pass filter High pass filter Band pass filter Notch filter
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GATE ECE 2008 | Question: 12
Step responses of a set of three second-order underdamped systems all have the same percentage overshoot. Which of the following diagrams represents the poles of the three systems?
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13
GATE ECE 2008 | Question: 13
Which of the following is NOT associated with a $p-n$ junction? Junction Capacitance Charge Storage Capacitance Depletion Capacitance Channel Length Modulation
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14
GATE ECE 2008 | Question: 14
Which of the following is true? A silicon wafer heavily doped with boron is a $p^{+}$substrate A silicon wafer lightly doped with boron is a ${ }^{+}$substrate A silicon wafer heavily doped with arsenic is a $\mathrm{p}^{+}$substrate A silicon wafer lightly doped with arsenic is a p ${ }^{+}$substrate
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GATE ECE 2008 | Question: 15
For a Hertz dipole antenna, the half power beam width (HPBW) in the E-plane is $360^{\circ}$ $180^{\circ}$ $90^{\circ}$ $45^{\circ}$
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GATE ECE 2008 | Question: 16
For static electric and magnetic fields in an inhomogeneous source-free medium, which of the following represents the correct form of two of Maxwell's equations? $\begin{array}{II}\nabla \cdot \mathbf{E}=0 \\ \nabla \times \textbf{B} = 0 \end{array}$ ... $\begin{array}{II}\nabla \times \mathbf{E}=0 \\ \nabla \cdot \textbf{B} = 0 \end{array}$
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GATE ECE 2008 | Question: 17
In the following limiter circuit, an input voltage $V_{i}=10 \sin 100 \pi t$ is applied. Assume that the diode drop is $0.7 \mathrm{~V}$ when it is forward biased. The Zener breakdown voltage is $6.8 \mathrm{~V}$. The maximum and minimum values of the output voltage respectively ... $7.5 \mathrm{~V},-0.7 \mathrm{~V}$ $7.5 \mathrm{~V},-7.5 \mathrm{~V}$
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18
GATE ECE 2008 | Question: 18
A silicon wafer has $100 \mathrm{~nm}$ of oxide on it and is inserted in a furnace at a temperature above $1000^{\circ} \mathrm{C}$ for further oxidation in dry oxygen. The oxidation rate is independent of current oxide ... of current oxide thickness but depends on temperature slows down as the oxide grows is zero as the existing oxide prevents further oxidation
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19
GATE ECE 2008 | Question: 19
The drain current of a MOSFET in saturation is given by $I_{D}=K\left(V_{G S}-V_{T}\right)^{2}$ where $K$ is a constant. The magnitude of the transconductance $g_{m}$ is $\frac{K\left(V_{G S}-V_{T}\right)^{2}}{V_{D S}}$ $2 K\left(V_{G S}-V_{T}\right)$ $\frac{I_{d}}{V_{G S}-V_{D S}}$ $\frac{K\left(V_{G S}-V_{T}\right)^{2}}{V_{G S}}$
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20
GATE ECE 2008 | Question: 20
Consider the amplitude modulated (AM) signal $A_{c} \cos \omega_{c} t+2 \cos \omega_{m} t \cos \omega_{c} t$. For demodulating the signal using envelope detector, the minimum value of $A_{c}$ should be $2$ $1$ $0.5$ $0$
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21
GATE ECE 2008 | Question: 21
The Thevenin equivalent impedance $Z_{\text {th }}$ between the nodes $P$ and $Q$ in the following circuit is $1$ $1+s+\frac{1}{s}$ $2+s+\frac{1}{s}$ $\frac{s^{2}+s+1}{s^{2}+2 s+1}$
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22
GATE ECE 2008 | Question: 22
The driving point impedance of the following network is given by $Z(s)=\frac{0.2 s}{s^{2}+0.1 s+2}$. The component values are $\begin{array}{lll} \mathrm{L}=5 \mathrm{H}, & \mathrm{R}=0.5 \; \Omega, & \mathrm{C}=0.1 \mathrm{~F} \end{array}$ ... $\begin{array}{lll} L=0.1 \mathrm{H}, & \mathrm{R}=2 \; \Omega, & \mathrm{C}=5 \mathrm{~F} \end{array}$
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23
GATE ECE 2008 | Question: 23
The circuit shown in the figure is used to charge the capacitor $\mathrm{C}$ alternately from two current sources as indicated. The switches $\text{S1}$ and $\text{S2}$ ... $\sum_{n=0}^{\infty}\left[0.5-e^{-(t-2 n T)}+0.5 e^{-(t-2 n T-T)}\right]$
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GATE ECE 2008 | Question: 24
The probability density function (PDF) of a random variable $\mathrm{X}$ is as shown below. The corresponding cumulative distribution function (CDF) has the form
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GATE ECE 2008 | Question: 25
The recursion relation to solve $x=e^{-x}$ using Newton-Raphson method is $x_{n+1}=e^{-x_{n}}$ $x_{n+1}=x_{n}-e^{-x_{n}}$ $x_{n+1}=\left(1+x_{n}\right) \frac{e^{-x_{n}}}{1+e^{-x_{n}}}$ $x_{n+1}=\frac{x_{n}^{2}-e^{-x_{n}}\left(1+x_{n}\right)-1}{x_{n}-e^{-x_{n}}}$
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26
GATE ECE 2008 | Question: 26
The residue of the function $f(z)=\frac{1}{(z+2)^{2}(z-2)^{2}}$ at $z=2$ is $-\frac{1}{32}$ $-\frac{1}{16}$ $\frac{1}{16}$ $\frac{1}{32}$
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GATE ECE 2008 | Question: 27
Consider the matrix $\mathbf{P}=\left[\begin{array}{cc}0 & 1 \\ -2 & -3\end{array}\right]$. The value of $\mathbf{e}^{\mathbf{P}}$ ...
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28
GATE ECE 2008 | Question: 28
In the Taylor series expansion of $\exp (x)+\sin (x)$ about the point $x=\pi$, the coefficient of $(x-\pi)^{2}$ is $\exp (\pi)$ $0.5 \exp (\pi)$ $\exp (\pi)+1$ $\exp (\pi)-1$
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GATE ECE 2008 | Question: 29
$\quad P_{X}(x)=M \exp (-2|x|)+N \exp (-3|x|)$ is the probability density function for the real random variable $X$, over the entire $x$ axis. $M$ and $N$ are both positive real numbers. The equation relating $M$ and $N$ is $M+\frac{2}{3} N=1$ $2 M+\frac{1}{3} N=1$ $M+N=1$ $M+N=3$
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GATE ECE 2008 | Question: 30
The value of the integral of the function $g(x, y)=4 x^{3}+10 y^{4}$ along the straight line segment from the point $(0,0)$ to the point $(1,2)$ in the $x-y$ plane is $33$ $35$ $40$ $56$
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31
GATE ECE 2008 | Question: 31
A linear, time-invariant, causal continuous time system has a rational transfer function with simple poles at $s=-2$ and $s=-4$, and one simple zero at $s=-1$. A unit step $u(t)$ is applied at the input of the system. At steady state, the output has constant value of $1.$ The impulse response ... $[-0.5 \exp (-2 t)+1.5 \exp (-4 t)] u(t)$
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GATE ECE 2008 | Question: 32
The signal $x(t)$ is described by $x(t)=\left\{\begin{array}{ll} 1 & \text { for }-1 \leq t \leq+1 \\ 0 & \text { otherwise } \end{array}\right.$ Two of the angular frequencies at which its Fourier transform becomes zero are $\pi, 2 \pi$ $0.5 \pi, 1.5 \pi$ $0, \pi$ $2 \pi, 2.5 \pi$
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GATE ECE 2008 | Question: 33
A discrete time linear shift-invariant system has an impulse response $h[n]$ with $h[0]=1, h[1]=-1$, $h[2]=2$, and zero otherwise. The system is given an input sequence $x[n]$ with $x[0]=x[2]=1$, and zero otherwise. The number of nonzero samples in the output sequence $y[n]$, and the value of $y[2]$ are, respectively $5,2$ $6,2$ $6,1$ $5,3$
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GATE ECE 2008 | Question: 34
Consider points $\mathrm{P}$ and $\mathrm{Q}$ in the $x-y$ plane, with $P=(1,0)$ and $Q=(0,1)$. The line integral $2 \int_{P}^{Q}(x d x+y d y)$ along the semicircle with the line segment $P Q$ as its diameter is $-1$ is $0$ is $1$ depends on the direction (clockwise or anti-clockwise) of the semicircle
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GATE ECE 2008 | Question: 35
Let $x(t)$ be the input and $y(t)$ be the output of a continuous time system. Match the system properties $\mathrm{P} 1, \mathrm{P} 2$ and $\mathrm{P} 3$ with system relations $\mathrm{R} 1, \mathrm{R} 2, \mathrm{R} 3, \mathrm{R} 4 .$ ... $\text{(P1, R3), (P2, R1), (P3, R2)}$ $\text{(P1, R1), (P2, R2), (P3, R3)}$
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GATE ECE 2008 | Question: 36
A memoryless source emits $n$ symbols each with a probability $p$. The entropy of the source as a function of $n$ increases as $\log n$ decreases as $\log (1 / n)$ increases as $n$ increases as $n \log n$
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GATE ECE 2008 | Question: 37
$\{x(n)\}$ is a real-valued periodic sequence with a period N. $x(n)$ and $X(k)$ form $\mathrm{N}$-point Discrete Fourier Transform (DFT) pairs. The DFT $Y(k)$ of the sequence $y(n)=\frac{1}{N} \sum_{r=0}^{N-1} x(r) x(n+r)$ is $|X(k)|^{2}$ $\frac{1}{N} \sum_{r=0}^{N-1} X(r) X^{*}(k+r)$ $\frac{1}{N} \sum_{r=0}^{N-1} X(r) X(k+r)$ $0$
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GATE ECE 2008 | Question: 38
$\text{Group I}$ lists a set of four transfer functions. $\text{Group II}$ gives a list of possible step responses $y(t)$. Match the step responses with the corresponding transfer functions. $\textbf{Group I}$ \[ P=\frac{25}{s^{2}+25} \quad Q=\frac{36}{s^{2}+20 s+36} \quad R=\frac{36}{s^ ... $\text{P-2, Q-1, R-4, S-3}$ $\text{P-3, Q-4, R-1, S-2}$
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GATE ECE 2008 | Question: 39
A certain system has transfer function $G(s)=\frac{s+8}{s^{2}+\alpha s-4}$, where $\alpha$ is a parameter. Consider the standard negative unity feedback configuration as shown below. Which of the following statements is true? The closed loop ... , the closed loop system is stable. The closed loop system is stable for all values of $\alpha$, both positive and negative.
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GATE ECE 2008 | Question: 40
A signal flow graph of a system is given below. The set of equations that correspond to this signal flow graph is ...
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