1
GATE ECE 2024
MCQ (More than One Correct Answer)
+2
-1.33

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 $\int\limits_C (F_1 dx + F_2 dy + F_3 dz)$ evaluates to the same value along any path C that starts at A and ends at B. Then which of the following is/are true?

A

For every closed path Γ, we have $\oint\limits_Γ (F_1 dx + F_2 dy + F_3 dz) = 0$.

B

There exists a differentiable scalar function $f(x, y, z)$ such that $F_1 = \frac{\partial f}{\partial x}$, $F_2 = \frac{\partial f}{\partial y}$, $F_3 = \frac{\partial f}{\partial z}$.

C

$\frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} = 0$.

D

$\frac{\partial F_3}{\partial y} = \frac{\partial F_2}{\partial z}$, $\frac{\partial F_1}{\partial z} = \frac{\partial F_3}{\partial x}$, $\frac{\partial F_2}{\partial x} = \frac{\partial F_1}{\partial y}$.

2
GATE ECE 2024
MCQ (More than One Correct Answer)
+2
-1.33

Consider the matrix $\begin{bmatrix}1 & k \\ 2 & 1\end{bmatrix}$, where $k$ is a positive real number. Which of the following vectors is/are eigenvector(s) of this matrix?

A

$\begin{bmatrix}1 \\ -\sqrt{2/k}\end{bmatrix}$

B

$\begin{bmatrix}1 \\ \sqrt{2/k}\end{bmatrix}$

C

$\begin{bmatrix}\sqrt{2k} \\ 1\end{bmatrix}$

D

$\begin{bmatrix}\sqrt{2k} \\ -1\end{bmatrix}$

3
GATE ECE 2024
Numerical
+1
-0.33

In the given circuit, the current $I_x$ (in mA) is _______ .

4
GATE ECE 2024
Numerical
+1
-0.33

In the circuit given below, the switch $S$ 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 _______ .

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