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GATE2020EC: 15
In the given circuit, the twoport network has the impedance matrix $\left [ Z \right ]=\begin{bmatrix} 40 & 60\\ 60& 120 \end{bmatrix}.$ The value of $Z_{L}$ for which maximum power is transferred to the load is _____________$\Omega$.
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GATE2020EC: 16
The current in the $\text{RL}$circuit shown below is $i\left ( t \right )=10\cos\left ( 5t\pi /4 \right )A.$ The value of the inductor $\text{(rounded off to two decimal places)}$ is _______$\text{H}$.
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GATE2020EC: 17
In the circuit shown below, all the components are ideal and the input voltage is sinsoidial. The magnitude of the steadystate output $V_{o}$ ( rounded off to two decimal places) is ______ $V$.
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GATE2020EC: 18
In the circuit shown below, all the components are ideal. If $V_{i}$ is $+2\:V$, the current $I_{o}$ sourced by the opamp is __________ $\text{mA}$.
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GATE2020EC: 19
In an $8085$ microprocessor, the number of address lines required to access a $\text{16 K byte}$ memory bank is ____________ .
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GATE2020EC: 20
A $\text{10bit D/A}$ converter is calibrated over the full range from $\text{0 to 10 V}$. If the input to the $\text{D/A}$ converter is $\text{13A}$ (in hex), the output ( rounded off to three decimal places) is __________ $\text{V}$.
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GATE2020EC: 21
A transmission line of length $3\lambda /4$ and having a characteristic impedance of $50$ $\Omega$ is terminated with a load of $400$ $\Omega$. The impedance (rounded off to two decimal places) seen at the input end of the transmission line is __________ $\Omega$.
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8
GATE2020EC: 22
A binary random variable $\text{X}$ takes the value $+2$ or $2$. The probability $P(X=+2)=\alpha .$ The value of $\alpha$ (rounded off to one decimal place), for which the entropy of $\text{X}$ is maximum, is __________.
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GATE2020EC: 23
The loop transfer function of a negative feedback system is $G\left ( s \right )H\left ( s \right )=\frac{k(s+11)}{s(s+2)(s+8)}.$ The value of $\text{K}$, for which the system is marginally stable, is ___________.
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GATE2020EC: 24
The random variable $Y=\int_{\infty }^{\infty }W\left ( t \right )\phi \left ( t \right )dt,\:\:\:\:\:\:\:where\:\:\:\phi \left ( t \right )=\left\{\begin{matrix} 1;5\leq t\leq 7 &\\ 0;otherwise \end{matrix}\right.$ and $\text{W(t)}$ ... process with twosided power spectral density $S_{W}\left ( f \right )=3 W/Hz,$ for all $\text{f}$. The variance of $\text{Y}$ is ________.
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GATE2020EC: 25
The two sides of a fair coin are labelled as $0$ to $1$. The coin is tossed two times independently. Let $\text{M}$ and $\text{N}$ denote the labels corresponding to the outcomes of those tosses. For a random variable $\text{X}$, defined as $\text{X = min(M, N),}$ the expected value $E(X)$ (rounded off to two decimal places) is ___________.
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12
GATE2020EC: 44
In the voltage regulator shown below, $V_{1}$ is the unregulated imput at $15\:V$. Assume $V_{BE}=0.7\:V$ and the base current is negligible for both the $\text{BJTs}$. If the regulated output $V_{O}$ is $9\:V$, the value of $R_{2}$ is ___________$\Omega$.
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GATE2020EC: 45
The magnetic field of a uniform plane wave in vacuum is given by $\overrightarrow{H}\left ( x,y,z,t \right )=(\hat{a_{x}}+2\hat{a_{y}}+b\hat{a_{z}})\cos\left ( \omega t+3xyz\right ).$ The value of b is ___________.
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14
GATE2020EC: 46
For a $2$port network consisting of an ideal lossless transformer, the parameter $S_{21}$ (rounded off to two decimal places) for a reference impedance of $10$ $\Omega$, is __________.
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15
GATE2020EC: 47
$S_{PM}(t)$ and $S_{FM}(t)$ as defined below, are the phase modulated and the frequency modlated waveforms, respectively, corresponding to the message signal $\text{m(t)}$ ... of $S_{PM}(t)$ and $S_{FM}(t)$ are same, then the value of the ratio $\frac{k_{p}}{k_{f}}$ is ________ seconds.
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16
GATE2020EC: 48
In a digital communication system, a symbol $\text{S}$ randomly chosen from the set $\left \{ s_{1},s_{2},s_{3},s_{4} \right \}$ is transmitted. It is given that $s_{1}=3,s_{2}=1,s_{3}=+1$ and $s_{4}=+2$. The received symbol ... when the transmitted symbol $S=s_{i}$. The index $\text{i}$ for which the conditional symbol error probability $P_{i}$ is the highest is ___________.
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17
GATE2020EC: 49
A system with transfer function $G\left ( s \right )=\frac{1}{\left ( s+1 \right )\left ( s+a \right )},\:\:a> 0$ is subjected to an input $5\cos3t.$ The steady state output of the system is $\frac{1}{\sqrt{10}}\cos\left ( 3t1.892 \right ).$ The value of $\text{a}$ is _______.
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18
GATE2020EC: 50
For the components in the sequential circuit shown below, $t_{pd}$ is the propagation delay, $t_{setup}$ is the setuptime, and $t_{hold}$ is the hold time. The maximum clock frequency (rounded off to the nearest integer), at which the given circuit can operate reliably, is _________$\text{MHz}$.
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19
GATE2020EC: 51
For the solid $\text{S}$ shown below, the value of $\int \int_{S} \int$xdxdydz$ (rounded off to two decimal places) is _______________.
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20
GATE2020EC: 52
$X\left ( \omega \right )$ is the Fourier transform of $\text{x(t)}$ shown below. The value of $\int_{\infty }^{\infty }\left  X\left ( \omega \right ) \right ^{2}d\omega$ (rounded off to two decimal places) is _________________.
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GATE2020EC: 53
The transfer function of a stable discretetime $\text{LTI}$ system is $H\left ( z \right )=\frac{K\left ( z\alpha \right )}{z+0.5},$ where $\text{K}$ and $\alpha$ are real numbers. The value of $\alpha$ (rounded off to one decimal place) with $\left  \alpha \right > 1$ , for which the magnitude response of the system is constant over all frequencies, is ___________.
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22
GATE2020EC: 54
$\text{X}$ is a random variable with uniform probability density function in the interval $[2,\:10]$. For $Y=2X6$, the conditional probability $P\left ( Y\leq 7\mid X\geq 5 \right )$ (rounded off to three decimal places) is __________.
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23
GATE2020EC: 55
Consider the following closed loop control system where $G\left ( s \right )=\frac{1}{s\left ( s+1 \right )}$ and $C\left ( s \right )=K\frac{s+1}{s+3}$. If the steady state error for a unit ramp input is $0.1,$ then the value of $\text{K}$ is ______________.
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24
GATE2019 EC: 16
The value of the contour integral $\frac{1}{2\pi j} \oint\left(z+\frac{1}{z}\right)^{2}dz$ evaluated over the unit circle $\mid z \mid=1$ is_______.
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25
GATE2019 EC: 17
The number of distinct eigenvalues of the matrix $A=\begin{bmatrix} 2&2&3&3\\0&1&1&1\\0&0&3&3\\0&0&0&2 \end{bmatrix}$ is equal to ____________.
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26
GATE2019 EC: 18
If $X$ and $Y$ are random variables such that $E\left[2X+Y\right]=0$ and $E\left[X+2Y\right]=33$, then $E\left[X\right]+E\left[Y\right]=$___________.
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27
GATE2019 EC: 19
The value of the integral $\int_{0}^{\pi} \int_{y}^{\pi}\dfrac{\sin x}{x}dx\: dy ,$ is equal to __________.
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28
GATE2019 EC: 20
Let $Z$ be an exponential random variable with mean $1$. That is, the cumulative distribution function of $Z$ is given by $F_{Z}(x)= \left\{\begin{matrix} 1e^{x}& \text{if}\: x \geq 0 \\ 0& \text{if}\: x< 0 \end{matrix}\right.$ Then $Pr\left(Z>2 \mid Z>1\right),$ rounded off to two decimal places, is equal to ___________.
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29
GATE2019 EC: 21
Consider the signal $f(t)=1+2 \cos(\pi t)+3 \sin \left(\dfrac{2\pi}{3}t\right)+4 \cos \left(\dfrac{\pi}{2}t+\dfrac{\pi}{4}\right)$, where $t$ is in seconds. Its fundamental time period, in seconds, is __________.
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30
GATE2019 EC: 22
The baseband signal $m(t)$ shown in the figure is phasemodulated to generate the $PM$ signal $\varphi(t)=\cos(2\pi f_{c}t+ k\:\: m(t)).$ The time $t$ on the $x$ axis in the figure is in milliseconds. If the carrier ... the ratio of the minimum instantaneous frequency (in kHz) to the maximum instantaneous frequency (in kHz) is _________ (rounded off to $2$ decimal places).
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31
GATE2019 EC: 23
Radiation resistance of a small dipole current element of length $l$ at a frequency of $3$ GHz is $3$ ohms. If the length is changed by $1\%$, then the percentage change in the radiation resistance, rounded off to two decimal places, is ________ $\%.$
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32
GATE2019 EC: 24
In the circuit shown, $V_{s}$ is square wave of period $T$ with maximum and minimum values of $8\: V$ and $10\: V$, respectively. Assume that the diode is ideal and $R_{1}=R_{2}=50\: \Omega.$ The average value of $V_{L}$ is _______ volts (rounded off to $1$ decimal place).
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33
GATE2019 EC: 25
In the circuit shown, the clock frequency, i.e., the frequency of the ClK signal, is $12\:kHz$. The frequency of the signal at $Q_{2}$ is _______ kHz.
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34
GATE2019 EC: 41
The $\text{RC}$ circuit shown below has a variable resistance $R(t)$ given by the following expression: $R(t)=R_{0}\left(1\frac{t}{T}\right) \text{for} \:\: 0 \leq t < T$ where $R_{0}=1\: \Omega,$ and $C=1\:F.$ ... at time $t=0$ is $1\: A,$ then the current $I(t)$, in amperes, at time $t=T/2$ is __________ (rounded off to $2$ decimal places).
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35
GATE2019 EC: 42
Consider a unity feedback system, as in the figure shown, with an integral compensator $\dfrac{K}{s}$ and openloop transfer function $G(s)=\dfrac{1}{s^{2}+3s+2}$ where $K>0.$ The positive value of $K$ for which there are exactly two poles of the unity feedback system on the $j\omega$ axis is equal to ________ (rounded off to two decimal places).
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36
GATE2019 EC: 43
Consider the homogenous ordinary differential equation $x^{2}\frac{d^{2}y}{dx^{2}}3x\frac{dy}{dx}+3y=0, \quad x>0$ with $y(x)$ as a general solution. Given that $y(1)=1 \quad \text{and} \quad y(2)=14$ the value of $y(1.5),$ rounded off to two decimal places, is________.
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37
GATE2019 EC: 44
Let $h[n]$ be a length  $7$ discretetime finite impulse response filter, given by $h[0]=4, \quad h[1]=3,\quad h[2]=2,\quad h[3]=1,$ $\quad h[1]=3, \quad h[2]=2, \quad h[3]=1,$ and $h[n]$ is zero for $n\geq4.$ ... $g[n],$ respectively. For the filter that minimizes $E(h,g),$ the value of $10g[1]+g[1],$ rounded off to $2$ decimal places, is __________.
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38
GATE2019 EC: 45
Let a random process $Y(t)$ be described as $Y(t)=h(t) \ast X(t)+Z(t),$ where $X(t)$ is a white noise process with power spectral density $S_{x}(f)=5$W/Hz. The filter $h(t)$ has a magnitude response given by $ \mid H(f) \mid =0.5$ ... , with power spectral density as shown in the figure. The power in $Y(t),$ in watts, is equal to _________ $W$ (rounded off to two decimal places).
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39
GATE2019 EC: 46
A voice signal $m(t)$ is in the frequency range $5\:kHz$ to $15\:kHz$. The signal is amplitudemodulated to generated an AM signal $f(t)=A\left(1+m(t)\right)\cos 2\pi f_{c}t,$ where $f_{c}=600\: kHz.$ The AM signal $f(t)$ ... bits required for the encoding. The rate, in Megabits per second (rounded off to $2$ decimal places), of the resulting stream of coded bits is ________ Mbps.
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40
GATE2019 EC: 47
A random variable $X$ takes values $1$ and $+1$ with probabilities $0.2$ and $0.8$, respectively. It is transmitted across a channel which adds noise $N,$ so that the random variable at the channel output is $Y=X+N$. The noise $N$ is independent of ... the probability of error $Pr[ \hat{X} \neq X].$ The minimum probability of error, rounded off to $1$ decimal place, is _________.
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