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162
GATE ECE 2024 | Question: 19
In a number system of base $r$, the equation $x^{2}-12 x+37=0$ has $x=8$ as one of its solutions. The value of $r$ is $\_\_\_\_\_\_\_$.
In a number system of base $r$, the equation $x^{2}-12 x+37=0$ has $x=8$ as one of its solutions. The value of $r$ is $\_\_\_\_\_\_\_$.
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164
GATE ECE 2024 | Question: 21
In the given circuit, the current $I_{x}$ (in $m A$ ) is $\_\_\_\_\_\_\_\_$.
In the given circuit, the current $I_{x}$ (in $m A$ ) is $\_\_\_\_\_\_\_\_$.
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165
GATE ECE 2024 | Question: 22
In the circuit given below, the switch $\text{S}$ was kept open for a sufficiently long time and is closed at time $t=0$. The time constant (in seconds) of the circuit for $t>0$ is $\_\_\_\_\_\_$.
In the circuit given below, the switch $\text{S}$ was kept open for a sufficiently long time and is closed at time $t=0$. The time constant (in seconds) of the circuit fo...
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166
GATE ECE 2024 | Question: 23
Suppose $\text{X}$ and $\text{Y}$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0,1]$. The probability that $\text{X} \geq \text{Y}$ is $\_\_\_\_\_\_$.
Suppose $\text{X}$ and $\text{Y}$ are independent and identically distributed random variables that are distributed uniformly in the interval $[0,1]$. The probability tha...
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167
GATE ECE 2024 | Question: 24
A source transmits symbols from an alphabet of size $16$. The value of maximum achievable entropy (in bits) is $\_\_\_\_\_\_$.
A source transmits symbols from an alphabet of size $16$. The value of maximum achievable entropy (in bits) is $\_\_\_\_\_\_$.
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176
GATE ECE 2024 | Question: 33
​​​​​Let $z$ be a complex variable. If $f(z)=\frac{\sin (\pi z)}{z^{2}(z-2)}$ and $C$ is the circle in the complex plane with $|z|=3$ then $\oint_{C} f(z) d z$ is $\_\_\_\_\_\_$. $\pi^{2} j$ $j \pi\left(\frac{1}{2}-\pi\right)$ $j \pi\left(\frac{1}{2}+\pi\right)$ $-\pi^{2} j$
​​​​​Let $z$ be a complex variable. If $f(z)=\frac{\sin (\pi z)}{z^{2}(z-2)}$ and $C$ is the circle in the complex plane with $|z|=3$ then $\oint_{C} f(z) d z$ ...
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177
GATE ECE 2024 | Question: 34
Consider two continuous time signals $x(t)$ and $y(t)$ as shown below If $X(f)$ denotes the Fourier transform of $x(t)$, then the Fourier transform of $y(t)$ is $\_\_\_\_\_\_$. $-4 X(4 f) e^{-j \pi f}$ $-4 X(4 f) e^{-j 4 \pi f}$ $-\frac{1}{4} X(f / 4) e^{-j \pi f}$ $-\frac{1}{4} X(f / 4) e^{-j 4 \pi f}$
Consider two continuous time signals $x(t)$ and $y(t)$ as shown belowIf $X(f)$ denotes the Fourier transform of $x(t)$, then the Fourier transform of $y(t)$ is $\_\_\_\_\...
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183
GATE ECE 2024 | Question: 40
In the circuit below, the opamp is ideal. If the circuit is to show sustained oscillations, the respective values of $R_{1}$ and the corresponding frequency of oscillation are $29 R$ and $1 /(2 \pi \sqrt{6} R C)$ $2 R$ and $1 /(2 \pi R C)$ $29 R$ and $1 /(2 \pi R C)$ $2 R$ and $1 /(2 \pi \sqrt{6} R C)$
In the circuit below, the opamp is ideal.If the circuit is to show sustained oscillations, the respective values of $R_{1}$ and the corresponding frequency of oscillation...
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186
GATE ECE 2024 | Question: 43
Consider a system $S$ represented in state space as \[ \frac{d x}{d t}=\left[\begin{array}{ll} 0 & -2 \\ 1 & -3 \end{array}\right] x+\left[\begin{array}{l} 1 \\ 0 \end{array}\right] r, y=\left[\begin{array}{ll} 2 & -5 \end{array}\ ...
Consider a system $S$ represented in state space as\[\frac{d x}{d t}=\left[\begin{array}{ll}0 & -2 \\1 & -3\end{array}\right] x+\left[\begin{array}{l}1 \\0\end{array}\rig...
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187
GATE ECE 2024 | Question: 44
Let $F_{1}, F_{2}$, and $F_{3}$ be functions of $(x, y, z)$. Suppose that for every given pair of points $A$ and $B$ in space, the line integral $\int_{C}\left(F_{1} \mathrm{~d} x+F_{2} \mathrm{~d} y+F_{3} \mathrm{~d} z\right)$ evaluates to ...
Let $F_{1}, F_{2}$, and $F_{3}$ be functions of $(x, y, z)$. Suppose that for every given pair of points $A$ and $B$ in space, the line integral $\int_{C}\left(F_{1} \mat...
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188
GATE ECE 2024 | Question: 45
Consider the matrix $\left[\begin{array}{ll}1 & k \\ 2 & 1\end{array}\right]$, where $k$ is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix? $\left[\begin{array}{c}1 \\ -\sqrt{2 / k}\end{array}\right]$ ... $\left[\begin{array}{c}\sqrt{2 k} \\ -1\end{array}\right]$
​​​​​Consider the matrix $\left[\begin{array}{ll}1 & k \\ 2 & 1\end{array}\right]$, where $k$ is a positive real number. Which of the following vectors is/are e...
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192
GATE ECE 2024 | Question: 50
For the two port network shown below, the value of the $Y_{21}$ parameter (in Siemens) is $\_\_\_\_\_\_$.
For the two port network shown below, the value of the $Y_{21}$ parameter (in Siemens) is $\_\_\_\_\_\_$.
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194
GATE ECE 2024 | Question: 52
In the network shown below, maximum power is to be transferred to the load $R_{L}$. The value of $R_{L}$ (in $\Omega$ ) is $\_\_\_\_\_\_\_$.
In the network shown below, maximum power is to be transferred to the load $R_{L}$.The value of $R_{L}$ (in $\Omega$ ) is $\_\_\_\_\_\_\_$.
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199
GATE ECE 2023 | Question: 2
The rate of increase, of a scalar field $f(x, y, z)=x y z$, in the direction $v=(2,1,2)$ at a point $(0,2,1)$ is $\frac{2}{3}$ $\frac{4}{3}$ $2$ $4$
The rate of increase, of a scalar field $f(x, y, z)=x y z$, in the direction $v=(2,1,2)$ at a point $(0,2,1)$ is$\frac{2}{3}$$\frac{4}{3}$$2$$4$
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200
GATE ECE 2023 | Question: 4
The value of the contour integral, $\oint_C\left(\frac{z+2}{z^2+2 z+2}\right) d z$, where the contour $\mathrm{C}$ is $\left\{z:\left|z+1-\frac{3}{2} j\right|=1\right\}$, taken in the counter clockwise direction, is $-\pi(1+j)$ $\pi(1+j)$ $\pi(1-j)$ $-\pi(1-j)$
The value of the contour integral, $\oint_C\left(\frac{z+2}{z^2+2 z+2}\right) d z$, where the contour $\mathrm{C}$ is $\left\{z:\left|z+1-\frac{3}{2} j\right|=1\right\}$,...
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