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GATE2019 EC: 16
The value of the contour integral $\frac{1}{2\pi j} \oint(z+\frac{1}{z})^{2}dz$ evaluated over the unit circle $z=1$ is_______.
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2
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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3
GATE2019 EC: 18
If $X$ and $Y$ are random variable such that $E[2X+Y]=0$ and $E[X+2Y]=33$, then $E[X]+E[Y]=$__________________.
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GATE2019 EC: 19
The value of the integral $\int_{0}^{\pi} \int_{y}^{\pi}\frac{sinx}{x}dx dy ,$ is equal to _____________.
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5
GATE2019 EC: 20
Let $Z$ be an exponential random variable with mean $1$. That is, the cumulative distribution function of $Z$is given by $F_{z}(x)=$\begin{Bmatrix} 1e^{x}& if x\geq 0 \\ 0& if x< 0 \end{Bmatrix}$ Then Pr$(z>2z>1)$, rounded off to two decimal places, is equal to ___________.
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GATE2019 EC: 21
Consider the signal $f(t)=1+2 cos(\pi t)+3 sin (\frac{2\pi}{3}t)+4 cos (\frac{\pi}{2}t+\frac{\pi}{4})$, where $t$ is in seconds. Its fundamental time period, in seconds, is _____________________.
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7
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 ... 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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8
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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9
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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10
GATE2019 EC: 25
In the circuit shown, the clock frequency, i.e., the frequency of the ClK signal, is $12kHz$. The frequency of the signal at $Q_{2}$ is _____________ kHz.
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11
GATE2019 EC: 41
The RC circuit shown below has a variable resistance $R(t)$ given by the following expression: $R(t)=R_{0}(1\frac{t}{T}) for \quad 0 \leq t < T$ where $R_{0}=1 \Omega,$ and $C=1F.$ We are also given that $T=3 R_{0}C$ ... 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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12
GATE2019 EC: 42
Consider a unity feedback system, as in the figure shown, with an integral compensator $\frac{K}{s}$ and openloop transfer function $G(s)=\frac{1}{s^{2}+3s+2}$ 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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13
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 $y(1)=1 \quad and \quad y(2)=14$ the value of $y(1.5),$ rounded off to two decimal places, is_______________.
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14
GATE2019 EC: 44
Let $h[n]$ be a length$7$ discretetime finite impluse 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 $n\geq4.$ A length $3$ ... $g[n],$ 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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15
GATE2019 EC: 45
Let a random process $Y(t)$ be described as $Y(t)=h(t)*X(t)+Z(t),$ 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 $H(f)=0.5$ ... $Y(t),$ in watts, is equal to _________ W (rounded off to two decimal places).
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16
GATE2019 EC: 46
A voice signal $m(t)$ is in the frequency range $5kHz$ to $15kHz$. The signal is amplitudemodulated to generated an AM signal $f(t)=A(1+m(t))cos 2\pi f_{c}t,$ where $f_{c}=600 kHz.$ The AM signal $f(t)$ is to be digitized and ... 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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17
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 ... so as to minimize the probability of error $Pr[X \neq X].$ The minimum probability of error, rounded off to $1$ decimal place, is
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18
GATE2019 EC: 48
A Germanium sample of dimensions $1 cm \times 1 cm$ is illuminated with a $20mW,$ $600 nm$ laser light source as shown in the figure. The illuminated sample surface has a $100 nm$ of lossless Silicon dioxide layer that reflects onefourth of the incident ... the bandgap is $0.66eV,$ the thickness of the Germanium layer, rounded off to $3$ decimal places, is _____________ $\mu m.$
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19
GATE2019 EC: 49
In an ideal $pn$ junction with an ideality factor of $1$ at $T=300K,$ the magnitude of the reversebias voltage required to reach $75\%$ of its reverse saturation current, rounded off to $2$ decimal places, is ______ $mV.$ $[k=1.38 \times 10^{23} JK^{1}, h=6.625 \times 10^{34} Js, q=1.602 \times 10^{19}C]$
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20
GATE2019 EC: 50
Consider a longchannel MOSFET with a channel length $1\mu m$ and width $10 \mu m.$ The device parameters are acceptor concentration $N_{A}=5 \times 10^{16} cm^{3},$ electron mobility $\mu_{n}=800 cm^{2}/Vs,$ ... is _______ $mA$ (rounded off to two decimal places.). $[\varepsilon_{0}=8.854 \times 10^{14}F/cm, \varepsilon_{si=11.9}]$
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21
GATE2019 EC: 51
A rectangular waveguide of width $w$ and height $h$ has cutoff frequencies for $TE_{10}$ and $TE_{11}$ modes in the ration $1:2$ . The aspect ratio $w/h$, rounded off to two decimal places , is ____________________.
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22
GATE2019 EC: 52
In the circuit shown. $V_{s}$ is a $10V$ square wave of period, $T=4 ms$ with $R=500 \Omega$ and $C= 10\mu F.$ The capacitor is initially uncharged at $t=0,$ and the diode is assumed to be ideal. The voltage across the capacitor $(V_{c})$ at $3ms$ is equal to _____ volts (rounded off to one decimal place)
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23
GATE2019 EC: 53
A CMOS inverter, designed to have a midpoint voltage $V_{1}$ equal to half of $V_{dd}.$ as shown in the figure, has the following parameters: $V_{dd}=3V$ $\mu_{n} C_{ox}=100 \mu A/V^{2}; V_{tn}=0.7V $ for nMOS $\mu_{p} C_{ox}=40 \mu A/V^{2}; V_{tp}=0.9V $ for pMOS The ration of $(\frac{W}{L})_{n}$ to $(\frac{W}{L})_{p}$ is equal to _______ (rounded off to 3 decimal places).
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24
GATE2019 EC: 54
In the circuit shown, the threshold voltages of the pMOS $(V_{tp})$ and nMOS $(V_{tn})$ transistors are both equal to $1V.$ All the transistors have the same output resistance $r_{ds}$ of $6M\Omega.$ ... . Ignoring the effect of channel length modulation and body bias, the gain of the circuit is ______ (rounded off to $1$ decimal place.)
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25
GATE2019 EC: 55
In the circuit shown, $V_{1}=0$ and $V_{2}=V_{dd}.$ The other relevant parameters are mentioned in the figure. Ignoring the effect of channel length modulation and the body effect, the value of $I_{out}$ is _________ $mA$ (rounded off to $1$ decimal place.)
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