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GATE ECE 2007 | Question: 1
If $E$ denotes expectation, the variance of a random variable $X$ is given by $E\left[X^2\right]-E^2[X]$ $E\left[X^2\right]+E^2[X]$ $E\left[X^2\right]$ $E^2[X]$
If $E$ denotes expectation, the variance of a random variable $X$ is given by$E\left[X^2\right]-E^2[X]$$E\left[X^2\right]+E^2[X]$$E\left[X^2\right]$$E^2[X]$
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GATE ECE 2007 | Question: 2
The following plot shows a function $y$ which varies linearly with $x$. The value of the integral $I=\displaystyle{}\int_1^2 y d x$ is $1.0$ $2.5$ $4.0$ $5.0$
The following plot shows a function $y$ which varies linearly with $x$. The value of the integral $I=\displaystyle{}\int_1^2 y d x$ is$1.0$$2.5$$4.0$$5.0$
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GATE ECE 2007 | Question: 3
For $|x| \ll 1$, $\operatorname{coth}(x)$ can be approximated as $x$ $x^2$ $\frac{1}{x}$ $\frac{1}{x^2}$
For $|x| \ll 1$, $\operatorname{coth}(x)$ can be approximated as$x$$x^2$$\frac{1}{x}$$\frac{1}{x^2}$
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GATE ECE 2007 | Question: 4
$\displaystyle{}\lim _{\theta \rightarrow 0} \frac{\sin (\theta / 2)}{\theta}$ is $0.5$ $1$ $2$ not defined
$\displaystyle{}\lim _{\theta \rightarrow 0} \frac{\sin (\theta / 2)}{\theta}$ is$0.5$$1$$2$not defined
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GATE ECE 2007 | Question: 5
Which one of the following functions is strictly bounded? $\frac{1}{x^2}$ $e^x$ $x^2$ $e^{-x^2}$
Which one of the following functions is strictly bounded?$\frac{1}{x^2}$$e^x$$x^2$$e^{-x^2}$
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GATE ECE 2007 | Question: 6
For the function $e^{-x}$, the linear approximation around $x=2$ is $(3-x) e^{-2}$ $1-x$ $[3+2 \sqrt{2}-\left(1+\sqrt{2}\right) x] e^{-2}$ $e^{-2}$
For the function $e^{-x}$, the linear approximation around $x=2$ is$(3-x) e^{-2}$$1-x$$[3+2 \sqrt{2}-\left(1+\sqrt{2}\right) x] e^{-2}$$e^{-2}$
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GATE ECE 2007 | Question: 7
An independent voltage source in series with an impedance $\mathbf{Z}_{\mathrm{s}}=R_{\mathrm{s}}+j X_{\mathrm{s}}$ delivers a maximum average power to a load impedance $\mathbf{Z}_{\mathrm{L}}$ when $\mathbf{Z}_{\mathrm{L}}=R_{\mathrm{S}}+j X_s$ $\mathbf{Z}_{\mathrm{L}}=R_{\mathrm{S}}$ $\mathbf{Z}_{\mathrm{L}}=j X_s$ $\mathbf{Z}_{\mathrm{L}}=R_{\mathrm{S}}-j X_s$
An independent voltage source in series with an impedance $\mathbf{Z}_{\mathrm{s}}=R_{\mathrm{s}}+j X_{\mathrm{s}}$ delivers a maximum average power to a load impedance $...
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GATE ECE 2007 | Question: 8
The $RC$ circuit shown in the figure is a low-pass filter a high-pass filter a band-pass filter a band-reject filter
The $RC$ circuit shown in the figure isa low-pass filtera high-pass filtera band-pass filtera band-reject filter
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GATE ECE 2007 | Question: 9
The electron and hole concentrations in an intrinsic semiconductor are $n_i$ per $\mathrm{cm}^3$ at $300 \mathrm{~K}$. Now, if acceptor impurities are introduced with a concentration of $N_A$ per $\mathrm{cm}^3$ (where $N_A \gg n_i$ ), the electron concentration per $\mathrm{cm}^3$ at $300 \mathrm{~K}$ will be $n_{i}$ $n_i+N_A$ $N_A-n_{i}$ $\frac{n_i^2}{N_A}$
The electron and hole concentrations in an intrinsic semiconductor are $n_i$ per $\mathrm{cm}^3$ at $300 \mathrm{~K}$. Now, if acceptor impurities are introduced with a c...
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GATE ECE 2007 | Question: 10
In a $p^{+} n$ junction diode under reverse bias, the magnitude of electric field is maximum at the edge of the depletion region on the $p$-side the edge of the depletion region on the $n$-side the $p^{+} n$ junction the centre of the depletion region on the $n$-side
In a $p^{+} n$ junction diode under reverse bias, the magnitude of electric field is maximum atthe edge of the depletion region on the $p$-sidethe edge of the depletion r...
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GATE ECE 2007 | Question: 11
The correct full wave rectifier circuit is
The correct full wave rectifier circuit is
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GATE ECE 2007 | Question: 12
In a transconductance amplifier, it is desirable to have a large input resistance and a large output resistance a large input resistance and a small output resistance a small input resistance and a large output resistance a small input resistance and a small output resistance
In a transconductance amplifier, it is desirable to havea large input resistance and a large output resistancea large input resistance and a small output resistancea smal...
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GATE ECE 2007 | Question: 13
$\mathrm{X}=01110$ and $\mathrm{Y}=11001$ are two $5$-bit binary numbers represented in two's complement format. The sum of $\mathrm{X}$ and $\mathrm{Y}$ represented in two's complement format using $6$ bits is $100111$ $001000$ $000111$ $101001$
$\mathrm{X}=01110$ and $\mathrm{Y}=11001$ are two $5$-bit binary numbers represented in two's complement format. The sum of $\mathrm{X}$ and $\mathrm{Y}$ represented in t...
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GATE ECE 2007 | Question: 14
The Boolean function $Y=A B+C D$ is to be realized using only $2$ -input NAND gates. The minimum number of gates required is $2$ $3$ $4$ $5$
The Boolean function $Y=A B+C D$ is to be realized using only $2$ -input NAND gates. The minimum number of gates required is$2$$3$$4$$5$
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GATE ECE 2007 | Question: 15
If the closed-loop transfer function of a control system is given as $T(s)=\dfrac{s-5}{(s+2)(s+3)},$ then it is an unstable system an uncontrollable system a minimum phase system a non-minimum phase system
If the closed-loop transfer function of a control system is given as $T(s)=\dfrac{s-5}{(s+2)(s+3)},$ then it isan unstable systeman uncontrollable systema minimum phase s...
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GATE ECE 2007 | Question: 16
If the Laplace transform of a signal $y(t)$ is $Y(s)=\dfrac{1}{s(s-1)},$ then its final value is $-1$ $0$ $1$ Unbounded
If the Laplace transform of a signal $y(t)$ is $Y(s)=\dfrac{1}{s(s-1)},$ then its final value is$-1$$0$$1$Unbounded
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GATE ECE 2007 | Question: 17
If $R(\tau)$ is the autocorrelation function of a real, wide-sense stationary random process, then which of the following is $\text{NOT}$ true? $R(\tau)=R(-\tau)$ $|R(\tau)| \leq R(0)$ $R(\tau)=-R(-\tau)$ The mean square value of the process is $R(0)$
If $R(\tau)$ is the autocorrelation function of a real, wide-sense stationary random process, then which of the following is $\text{NOT}$ true?$R(\tau)=R(-\tau)$$|R(\tau)...
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GATE ECE 2007 | Question: 18
If $S(f)$ is the power spectral density of a real, wide-sense stationary random process, then which of the following is $\text{ALWAYS}$ true? $S(0) \geq S(f)$ $S(f) \geq 0$ $S(-f)=-S(f)$ $\displaystyle{}\int_{-\infty}^{\infty} S(f) d f=0$
If $S(f)$ is the power spectral density of a real, wide-sense stationary random process, then which of the following is $\text{ALWAYS}$ true?$S(0) \geq S(f)$$S(f) \geq 0$...
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GATE ECE 2007 | Question: 19
A plane wave of wavelength $\lambda$ is travelling in a direction making an angle $30^{\circ}$ with positive $x$-axis and $90^{\circ}$ with positive $y$-axis. The $\vec{E}$ field of the plane wave can be represented as ( $E_{0}$ ... $\vec{E}=\hat{y} E_{0} e^{j\left(\omega t-\frac{\pi}{\lambda} x+\frac{\sqrt{3} \pi}{\lambda} z\right)}$
A plane wave of wavelength $\lambda$ is travelling in a direction making an angle $30^{\circ}$ with positive $x$-axis and $90^{\circ}$ with positive $y$-axis. The $\vec{E...
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GATE ECE 2007 | Question: 21
It is given that $X_{1}, X_{2}, \cdots ,X_{M}$ are $M$ non-zero, orthogonal vectors. The dimension of the vector space spanned by the $2 M$ vectors $X_{1}, X_{2}, \cdots, X_{M},-X_{1},-X_{2}, \cdots,-X_{M}$ is $2 M$ $M+1$ $M$ dependent on the choice of $X_{1}, X_{2}, \cdots, X_{M}$
It is given that $X_{1}, X_{2}, \cdots ,X_{M}$ are $M$ non-zero, orthogonal vectors. The dimension of the vector space spanned by the $2 M$ vectors $X_{1}, X_{2}, \cdots,...
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GATE ECE 2007 | Question: 22
Consider the function $f(x)=x^{2}-x-2$. The maximum value of $f(x)$ in the closed interval $[-4,4]$ is $18$ $10$ $-2.25$ indeterminate
Consider the function $f(x)=x^{2}-x-2$. The maximum value of $f(x)$ in the closed interval $[-4,4]$ is$18$$10$$-2.25$indeterminate
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GATE ECE 2007 | Question: 23
An examination consists of two papers, Paper $1$ and Paper $2.$ The probability of failing in Paper $1$ is $0.3$ and that in Paper $2$ is $0.2$. Given that a student has failed in Paper $2,$ the probability of failing in Paper $1$ is $0.6$. The probability of a student failing in both the papers is $0.5$ $0.18$ $0.12$ $0.06$
An examination consists of two papers, Paper $1$ and Paper $2.$ The probability of failing in Paper $1$ is $0.3$ and that in Paper $2$ is $0.2$. Given that a student has ...
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GATE ECE 2007 | Question: 24
The solution of the differential equation $k^{2} \dfrac{d^{2} y}{d x^{2}}=y-y_{2}$ under the boundary conditions (i) $y=y_{1}$ at $x=0$ and (ii) $y=y_{2}$ at $x=\infty$, where $k, y_{1}$ and $y_{2}$ ... $y=\left(y_{1}-y_{2}\right) \sinh (x / k)+y_{1}$ $y=\left(y_{1}-y_{2}\right) \exp (-x / k)+y_{2}$
The solution of the differential equation $k^{2} \dfrac{d^{2} y}{d x^{2}}=y-y_{2}$ under the boundary conditions (i) $y=y_{1}$ at $x=0$ and (ii) $y=y_{2}$ at $x=\infty$, ...
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GATE ECE 2007 | Question: 25
The equation $x^{3}-x^{2}+4 x-4=0$ is to be solved using the Newton-Raphson method. If $x=2$ is taken as the initial approximation of the solution, then the next approximation using this method will be $\frac{2}{3}$ $\frac{4}{3}$ $1$ $\frac{3}{2}$
The equation $x^{3}-x^{2}+4 x-4=0$ is to be solved using the Newton-Raphson method. If $x=2$ is taken as the initial approximation of the solution, then the next approxim...
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GATE ECE 2007 | Question: 26
Three functions $f_{1}(t), f_{2}(t)$ and $f_{3}(t)$, which are zero outside the interval $[0, T]$, are shown in the figure. Which of the following statements is correct? $f_{1}(t)$ and $f_{2}(t)$ are orthogonal $f_{1}(t)$ and $f_{3}(t)$ are orthogonal $f_{2}(t)$ and $f_{3}(t)$ are orthogonal $f_{1}(t)$ and $f_{2}(t)$ are orthonormal
Three functions $f_{1}(t), f_{2}(t)$ and $f_{3}(t)$, which are zero outside the interval $[0, T]$, are shown in the figure. Which of the following statements is correct?$...
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GATE ECE 2007 | Question: 27
If the semi-circular contour $D$ of radius $2$ is as shown in the figure, then the value of the integral $\displaystyle{}\oint_{D} \dfrac{1}{\left(s^{2}-1\right)} d s$ is $j \pi$ $-j \pi$ $-\pi$ $\pi$
If the semi-circular contour $D$ of radius $2$ is as shown in the figure, then the value of the integral $\displaystyle{}\oint_{D} \dfrac{1}{\left(s^{2}-1\right)} d s$ is...
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GATE ECE 2007 | Question: 28
Two series resonant filters are as shown in the figure. Let the $\text{3-dB}$ bandwidth of Filter $1$ be $B_{1}$ and that of Filter $2$ be $B_{2}$. The value of $\dfrac{B_{1}}{B_{2}}$ is $4$ $1$ $\frac{1}{2}$ $\frac{1}{4}$
Two series resonant filters are as shown in the figure. Let the $\text{3-dB}$ bandwidth of Filter $1$ be $B_{1}$ and that of Filter $2$ be $B_{2}$. The value of $\dfrac{B...
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GATE ECE 2007 | Question: 29
For the circuit shown in the figure, the Thevenin voltage and resistance looking into $\mathrm{X}-\mathrm{Y}$ are $4 / 3 \mathrm{~V}, 2 \; \Omega$ $4 \mathrm{~V}, 2 / 3 \; \Omega$ $4 / 3 \mathrm{~V}, 2 / 3 \; \Omega$ $4 \mathrm{~V}, 2 \; \Omega$
For the circuit shown in the figure, the Thevenin voltage and resistance looking into $\mathrm{X}-\mathrm{Y}$ are$4 / 3 \mathrm{~V}, 2 \; \Omega$$4 \mathrm{~V}, 2 / 3 \; ...
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GATE ECE 2007 | Question: 30
In the circuit shown, $V_{C}$ is $0$ volts at $t=0 \; \mathrm{sec}$. For $t>0$, the capacitor current $i_{c}(t)$, where $t$ is in seconds, is given by $0.50 \exp (-25 t) \; \mathrm{mA}$ $0.25 \exp (-25 t) \; \mathrm{mA}$ $0.50 \exp (-12.5 t) \; \mathrm{mA}$ $0.25 \exp (-6.25 t) \; \mathrm{mA}$
In the circuit shown, $V_{C}$ is $0$ volts at $t=0 \; \mathrm{sec}$. For $t>0$, the capacitor current $i_{c}(t)$, where $t$ is in seconds, is given by$0.50 \exp (-25 t) \...
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GATE ECE 2007 | Question: 31
In the $\text{AC}$ network shown in the figure, the phasor voltage $\text{V}_{\text{AB}}$ (in Volts) is $0$ $5 \angle 30^{\circ}$ $12.5 \angle 30^{\circ}$ $17 \angle 30^{\circ}$
In the $\text{AC}$ network shown in the figure, the phasor voltage $\text{V}_{\text{AB}}$ (in Volts) is$0$$5 \angle 30^{\circ}$$12.5 \angle 30^{\circ}$$17 \angle 30^{\cir...
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GATE ECE 2007 | Question: 32
A $p^{+} n$ junction has a built-in potential of $0.8 \mathrm{~V}$. The depletion layer width at a reverse bias of $1.2 \mathrm{~V}$ is $2 \; \mu \mathrm{m}$. For a reverse bias of $7.2 \mathrm{~V},$ the depletion layer width will be $4 \; \mu \mathrm{m}$ $4.9 \; \mu \mathrm{m}$ $8 \; \mu \mathrm{m}$ $12 \; \mu \mathrm{m}$
A $p^{+} n$ junction has a built-in potential of $0.8 \mathrm{~V}$. The depletion layer width at a reverse bias of $1.2 \mathrm{~V}$ is $2 \; \mu \mathrm{m}$. For a rever...
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GATE ECE 2007 | Question: 33
Group I lists four types of $p-n$ ... $\mathrm{P}-2, \mathrm{Q}-1, \mathrm{R}-2, \mathrm{S}-2$
Group I lists four types of $p-n$ junction diodes. Match each device in Group I with one of the options in Group II to indicate the bias condition of that device in its n...
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GATE ECE 2007 | Question: 34
The $\text{DC}$ current gain ($\beta$ ) of a $\text{BJT}$ is $50.$ Assuming that the emitter injection efficiency is $0.995$, the base transport factor is $0.980$ $0.985$ $0.990$ $0.995$
The $\text{DC}$ current gain ($\beta$ ) of a $\text{BJT}$ is $50.$ Assuming that the emitter injection efficiency is $0.995$, the base transport factor is$0.980$$0.985$$0...
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GATE ECE 2007 | Question: 35
Group I lists four different semiconductor devices. Match each device in Group I with its characteristic property in Group II. ... $\text{P-3, Q-4, R-1, S-2}$ $\text{P-3, Q-2, R-1, S-4}$
Group I lists four different semiconductor devices. Match each device in Group I with its characteristic property in Group II.$$\begin{array}{ll} \textbf{Group I} & \qqua...
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GATE ECE 2007 | Question: 36
For the $\text{Op-Amp}$ circuit shown in the figure, $V_{o}$ is $-2 \mathrm{~V}$ $-1 \mathrm{~V}$ $-0.5 \mathrm{~V}$ $0.5 \mathrm{~V}$
For the $\text{Op-Amp}$ circuit shown in the figure, $V_{o}$ is$-2 \mathrm{~V}$$-1 \mathrm{~V}$$-0.5 \mathrm{~V}$$0.5 \mathrm{~V}$
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GATE ECE 2007 | Question: 37
For the $\text{BJT}$ circuit shown, assume that the $\beta$ of the transistor is very large and $V_{B E}=0.7 \mathrm{~V}$. The mode of operation of the $\text{BJT}$ is cut-off saturation normal active reverse active
For the $\text{BJT}$ circuit shown, assume that the $\beta$ of the transistor is very large and $V_{B E}=0.7 \mathrm{~V}$. The mode of operation of the $\text{BJT}$ iscut...
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GATE ECE 2007 | Question: 38
In the $\text{Op-Amp}$ circuit shown, assume that the diode current follows the equation $I=I_{S} \exp \left(V / V_{T}\right)$. For $V_{i}=2 \mathrm{~V}, V_{o}=V_{o 1},$ and for $V_{i}=4 \mathrm{~V}, V_{o}=V_{o 2}$. The relationship between $V_{o 1}$ and $V_{o 2}$ ... $V_{o 2}=e^{2} V_{o 1}$ $V_{o 2}=V_{o 1} \ln 2$ $V_{o 1}-V_{o 2}=V_{T} \ln 2$
In the $\text{Op-Amp}$ circuit shown, assume that the diode current follows the equation $I=I_{S} \exp \left(V / V_{T}\right)$. For $V_{i}=2 \mathrm{~V}, V_{o}=V_{o 1},$ ...
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GATE ECE 2007 | Question: 39
In the CMOS inverter circuit shown, if the transconductance parameters of the NMOS and PMOS transistors are $k_{n}=k_{p}=\mu_{n} C_{o x} \dfrac{W_{n}}{L_{n}}=\mu_{p} C_{o x} \dfrac{W_{p}}{L_{p}}=40 \; \mu \mathrm{A} / \mathrm{V}^{2}$ and their threshold voltages ... the current $I$ is $0 \mathrm{~A}$ $25 \; \mu \mathrm{A}$ $45 \; \mu \mathrm{A}$ $90 \; \mu \mathrm{A}$
In the CMOS inverter circuit shown, if the transconductance parameters of the NMOS and PMOS transistors are $k_{n}=k_{p}=\mu_{n} C_{o x} \dfrac{W_{n}}{L_{n}}=\mu_{p} C_{o...
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GATE ECE 2007 | Question: 40
For the Zener diode shown in the figure, the Zener voltage at knee is $7 \mathrm{~V}$, the knee current is negligible and the Zener dynamic resistance is $10 \; \Omega$. If the input voltage $\left(V_{i}\right)$ range is from $10$ to $16 \; \mathrm{V}$ ... $7.14$ to $7.29 \mathrm{~V}$ $7.14$ to $7.43 \mathrm{~V}$ $7.29$ to $7.43 \mathrm{~V}$
For the Zener diode shown in the figure, the Zener voltage at knee is $7 \mathrm{~V}$, the knee current is negligible and the Zener dynamic resistance is $10 \; \Omega$. ...
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GATE ECE 2007 | Question: 41
The Boolean expression $Y=\overline{A} \;\overline{B}\; \overline{C} D+\overline{A} B C \overline{D}+A \overline{B}\; \overline{C} D+A B \overline{C}\; \overline{D}$ ... $Y=\overline{A} B C \overline{D}+\overline{B}\; \overline{C} D+A B \overline{C} \;\overline{D}$
The Boolean expression $Y=\overline{A} \;\overline{B}\; \overline{C} D+\overline{A} B C \overline{D}+A \overline{B}\; \overline{C} D+A B \overline{C}\; \overline{D}$ can ...
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