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GATE ECE 2006 | Question: 1
The rank of the matrix $\left[\begin{array}{ccc}1 & 1 & 1 \\ 1 & -1 & 0 \\ 1 & 1 & 1\end{array}\right]$ is $0$ $1$ $2$ $3$
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GATE ECE 2006 | Question: 2
$\nabla \times \nabla \times \mathrm{P}$, where $\mathrm{P}$ is a vector, is equal to $\mathrm{P} \times \nabla \times \mathrm{P}-\nabla^{2} \mathrm{P}$ $\nabla^{2} \text{P} +\nabla(\nabla \cdot \text{P})$ $\nabla^{2} \mathrm{P}+\nabla \times \mathrm{P}$ $\nabla(\nabla \cdot \mathrm{P})-\nabla^{2} \mathrm{P}$
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GATE ECE 2006 | Question: 3
$\iint(\nabla \times \mathrm{P}) \cdot \mathrm{ds}$, where $\mathrm{P}$ is a vector, is equal to $\oint \text{P} \cdot d l$ $\oint \nabla \times \nabla \times \mathrm{P} \cdot d l$ $\oint \nabla \times \mathrm{P} \bullet d l$ $\iiint \nabla \cdot \mathrm{P} d v$
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GATE ECE 2006 | Question: 4
A probability density function is of the form $\qquad p(x)=\mathrm{Ke}^{-\alpha|x|}, x \in(-\infty, \infty)$ The value of $K$ is $0.5$ $1$ $0.5 \alpha$ $\alpha$
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GATE ECE 2006 | Question: 5
A solution for the differential equation $ \qquad \dot{x}(t)+2 x(t)=\delta(t)$ with initial condition $x(0-)=0$ is $e^{-2 t} \; u(t)$ $e^{2 t} \; u(t)$ $e^{-t} \; u(t)$ $e^{t} \; u(t)$
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GATE ECE 2006 | Question: 6
A low-pass filter having a frequency response $\mathrm{H}(\mathrm{j} \omega)=\mathrm{A}(\omega) e^{j^{\phi(\omega)}}$ does not produce any phase distortion, if $\mathrm{A}(\omega)=\mathrm{C}\omega^{2}, \phi(\omega)=k \omega^{3}$ ... $\mathrm{A}(\omega)=\mathrm{C}, \phi(\omega)=k \omega^{-1}$
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GATE ECE 2006 | Question: 7
The values of voltage $\left(\mathrm{V}_{\mathrm{D}}\right)$ across a tunnel-diode corresponding to peak and valley currents are $V_{P}$ and $V_{V}$ respectively. The range of tunnel-diode voltage $V_{D}$ for which the slope of its $I-V_{D}$ characteristics is negative ... $\mathrm{V}_{\mathrm{D}} \geq \mathrm{V}_{\mathrm{V}}$
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GATE ECE 2006 | Question: 8
The concentration of minority carriers in an extrinsic semiconductor under equilibrium is directly proportional to the doping concentration inversely proportional to the doping concentration directly proportional to the intrinsic concentration inversely proportional to the intrinsic concentration
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GATE ECE 2006 | Question: 9
Under low level injection assumption, the injected minority carrier current for an extrinsic semiconductor is essentially the diffusion current drift current recombination current induced current
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GATE ECE 2006 | Question: 10
The phenomenon known as "Early Effect" in a bipolar transistor refers to a reduction of the effective base-width caused by electron-hole recombination at the base the reverse biasing of the base-collector junction the forward biasing of emitter-base junction the early removal of stored base charge during saturation-to-cutoff switching
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GATE ECE 2006 | Question: 11
The input impedance $\left(Z_i\right)$ and the output impedance $\left(Z_0\right)$ of an ideal transconductance (voltage controlled current source) amplifier are $Z_i=0, Z_0=0$ $Z_i=0, Z_0=\infty$ $Z_i=\infty, Z_0=0$ $Z_i=\infty, Z_0=\infty$
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GATE ECE 2006 | Question: 12
An $n$-channel depletion MOSFET has following two points on its $\mathrm{I}_D-\mathrm{V}_{\text {GS}}$ curve $\text{V}_{\text{Gs}}=0$ at $\text{I}_\text{D}=12 \mathrm{~mA}$ and $\mathrm{V}_{\mathrm{GS}}=-6$ Volts at $\mathrm{l}_{\mathrm{D}}=0$ Which of the ... $\text{V}_{\text {Gs }}=0 \; \text{Volts}$ $\mathrm{V}_{\mathrm{Gs}}=3 \; \text{Volts}$
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GATE ECE 2006 | Question: 13
The number of product terms in the minimized sum-of-product expression obtained through the following K-map is (where, " $d$ ... $2$ $3$ $4$ $5$
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GATE ECE 2006 | Question: 14
Let $x(t) \longleftrightarrow \mathrm{X}(j \omega)$ be Fourier Transform pair. The Fourier Transform of the signal $x(5 t-3)$ in terms of $X(j \omega)$ is given as $\frac{1}{5} e^ - \frac{j 3 m}{5} \times\left(\frac{j \omega}{5}\right)$ ... $\frac{1}{5} e^\frac{j 3 m}{5} \times\left(\frac{j \omega}{5}\right)$
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GATE ECE 2006 | Question: 15
The Dirac delta function $\delta(t)$ is defined as $\delta(t)= \begin{cases}1, & t=0 \\ 0, & \text { otherwise }\end{cases}$ $\delta(t)= \begin{cases}\infty, & t=0 \\ 0, & \text { otherwise }\end{cases}$ ...
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GATE ECE 2006 | Question: 16
If the region of convergence of $x_1[n]+x_2[n]$ is $\frac{1}{3}<|z|<\frac{2}{3}$, then the region of convergence of $x_1[n]-x_2[n]$ includes $\frac{1}{3}<|z|<3$ $\frac{2}{3}<|z|<3$ $\frac{2}{3}<|z|<3$ $\frac{1}{3}<|z|<\frac{2}{3}$
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GATE ECE 2006 | Question: 17
The open-loop transfer function of a unity-gain feedback control system is given by $ G(s)=\frac{K}{(s+1)(s+2)} $ The gain margin of the system in $\text{dB}$ is given by $0$ $1$ $20$ $\propto$
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GATE ECE 2006 | Question: 18
In the system shown below, $x(t)=(\sin t) u(t)$. In steady-steady-state, the response $y(t)$ will be $\frac{1}{\sqrt{2}} \sin \left(t-\frac{\pi}{4}\right)$ $\frac{1}{\sqrt{2}} \sin \left(t+\frac{\pi}{4}\right)$ $\frac{1}{\sqrt{2}} e^{-1} \sin t$ $\sin t-\cos t$
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GATE ECE 2006 | Question: 19
The electric field of an electomagnetic wave propagating in the positive z-direction is given by $ \left.E=a_1, \sin (\omega t-\beta z\right)+\dot{a}_w \sin \left(\omega t-\beta z+\frac{\pi}{2}\right) $ The wave is linearly polarized in the z-direction elliptically polarized left-hand circularly polarized right-hand circularly polarized
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GATE ECE 2006 | Question: 20
A transmission line is feeding $1 \mathrm{Watt}$ of power to a horn antenna having a gain of $10 \mathrm{~dB}$. The antenna is matched to the transmission line. The total power radiated by the horn antenna into the free-space is $10$ Watts $1$ Watt $0.1$ Watt $0.01$ Watt
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GATE ECE 2006 | Question: 21
The eigenvalues and the corresponding eigen vectors of a $2 \times 2$ ... $\left[\begin{array}{ll}4 & 8 \\ 8 & 4\end{array}\right]$
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GATE ECE 2006 | Question: 22
For the function of a complex variable $W=\ln Z$ (where, $\mathrm{W}=u+j \mathrm{v}$ and $\mathrm{Z}=x+j y),$ the $u=$ constant lines get mapped in $Z$-plane as set of radial straight lines set of concentric circles set of confocal hyperbolas set of confocal ellipses
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GATE ECE 2006 | Question: 23
The value of the contour integral $\oint_{\mid=-1=2} \frac{1}{z^2+4} d z$ in positive sense is $\frac{j \pi}{2}$ $-\frac{\pi}{2}$ $-\frac{j \pi}{2}$ $\frac{\pi}{2}$
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GATE ECE 2006 | Question: 24
The integral $\int_0^\pi \sin ^3 \theta d \theta$ is given by $\frac{1}{2}$ $\frac{2}{3}$ $\frac{4}{3}$ $\frac{8}{3}$
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GATE ECE 2006 | Question: 25
Three companies $X, Y$ and $Z$ ... that a computer is defective, the probability that it was supplied by $Y$ is $0.1$ $0.2$ $0.3$ $0.4$
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GATE ECE 2006 | Question: 26
For the matrix $\left[\begin{array}{ll}4 & 2 \\ 2 & 4\end{array}\right]$, the eigen value corresponding to the eigenvector $\left[\begin{array}{l}101 \\ 101\end{array}\right]$ is $2$ $4$ $6$ $8$
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GATE ECE 2006 | Question: 27
For the differential equation $\frac{d^2 y}{d x^2}+k^2 y=0$, the boundary conditions are $y=0$ for $x=0$, and $y=0$ for $x=a$ The form of non-zero solutions of $y$ (where $m$ varies over all integers) are $y=\sum_m \mathrm{~A}_m \sin ^{\frac{m \pi x}{a}}$ ... $y=\sum_m \mathrm{~A}_m x^{\frac{m \pi}{a}}$ $y=\sum_m \mathrm{~A}_{m} e^{-\frac{m \pi x}{a}}$
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GATE ECE 2006 | Question: 28
Consider the function $f(t)$ having Laplace transform $ F(s)=\frac{\omega_0}{s^2+\omega_0^2} \operatorname{Re}[s]>0 $ The final value of $f(t)$ would be $0$ $1$ $-1 \leq f(\infty) \leq 1$ $\infty$
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GATE ECE 2006 | Question: 29
As $x$ is increased from $-\infty$ to $\infty$, the function $ f(x)=\frac{e^x}{1+e^x} $ monotonically increases monotonically decreases increases to a maximum value and then decreases decreases to a minimum value and then increases
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GATE ECE 2006 | Question: 30
A two-port network is represented by $\text{ABCD}$ ... $\frac{\mathrm{B}+\mathrm{AR}_{\mathrm{L}}}{\mathrm{D}+\mathrm{CR}_{\mathrm{L}}}$
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GATE ECE 2006 | Question: 31
In the two port network shown in the figure below, $z_{12}$ and $z_{21}$ are, respectively $r_e$ and $\beta r_0$ $0$ and $-\beta r_0$ $0,$ and $\beta r_0$ $r_e,$ and $-\beta r_0$
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GATE ECE 2006 | Question: 32
The first and the last critical frequencies (singularities) of a driving point impedance function of a passive network having two kinds of elements, are a pole and a zero respectively. The above property will be satisfied by $\text{RL}$ network only $\text{RC}$ network only $\text{LC}$ network only $\mathrm{RC}$ as well as $\mathrm{RL}$ networks
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GATE ECE 2006 | Question: 33
A $2 \; \mathrm{mH}$ inductor with some initial current can be represented as shown below, where $s$ is the Laplace Transform variable. The value of initial current is $0.5 \mathrm{~A}$ $2.0 \mathrm{~A}$ $1.0 \mathrm{~A}$ $0.0 \mathrm{~A}$
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GATE ECE 2006 | Question: 34
In the figure shown, assume that all the capacitors are initially uncharged. If $V_i(t)=10 u(t)$ Volts, then $V_0(t)$ is given by $8 e^{-0.004 t}$ Volts $8\left(1-e^{-0.004 t}\right)$ Volts $8 u(t)$ Volts $8$ Volts
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GATE ECE 2006 | Question: 35
Consider two transfer functions $ \mathrm{G}_1(s)=\frac{1}{s^2+a s+b} \text { and } \mathrm{G}_2(s)=\frac{s}{s^2+a s+b} $ The $3-d B$ bandwidths of their frequency responses are, respectively $\sqrt{a^2-4 b}, \sqrt{a^2+4 b}$ $\sqrt{a^2+4 b}, \sqrt{a^2-4 b}$ $\sqrt{a^2-4 b}, \sqrt{a^2-4 b}$ $\sqrt{a^2+4 b}, \sqrt{a^2+4 b}$
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GATE ECE 2006 | Question: 36
A negative resistance $R_{\text {neg }}$ is connected to a passive network $N$ having driving point impedance $Z_{1}$ (s) as shown below. For $Z_{2}(s)$ ... $\left|\text{R}_{\text {neg }}\right| \leq \angle Z_{1}(j \omega), \forall \omega$
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GATE ECE 2006 | Question: 37
In the circuit shown below, the switch was connected to position $1$ at $t<0$ and at $t=0$, it is changed to position $2$. Assume that $y$ the diode has zero voltage drop and a storage time $t_{s}$. For $0 < t \leq t_{s}, \text{V}_{\text{R}}$ is given by ... $0 \leq \mathrm{V}_{\mathrm{R}}<5$ $-5 < \mathrm{V}_{\mathrm{R}} < 0$
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GATE ECE 2006 | Question: 38
The majority carriers in an $n$-type semiconductor have an average drift velocity $\mathrm{V}$ in a direction perpendicular to a uniform magnetic field $\text{B}.$ The electric field $\mathrm{E}$ induced due to Hall effect acts in the direction $\mathrm{V} \times \mathrm{B}$ $\mathrm{B} \times \mathrm{V}$ along $\text{V}$ opposite to $\mathrm{V}$
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GATE ECE 2006 | Question: 39
Find the correct match between Group $1$ and Group $2$ ... $\text{E - 3, F - 4, G - 1, H - 2}$ $\text{E - 1, F - 3, G - 2, H - 4}$
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GATE ECE 2006 | Question: 40
A heavily doped n-type semiconductor has the following data Hole-electron mobility ratio : $0.4$ Doping concentration : $4.2 \times 10^{8} \; \mathrm{atoms/m}^{3}$ Intrinsic concentration : $1.5 \times 10^{4} \; \mathrm{atoms/m}^{3}$ The ... that of the intrinsic semiconductor of same material and at the same temperature is given by $0.00005$ $2,000$ $10,000$ $20,000$
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