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1
GATE ECE 2024
MCQ (More than One Correct Answer)
+2
-0

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 2017 Set 1
MCQ (Single Correct Answer)
+2
-0.6
If the vector function
$$\,\,\overrightarrow F = \widehat a{}_x\left( {3y - k{}_1z} \right) + \widehat a{}_y\left( {k{}_2x - 2z} \right) - \widehat a{}_z\left( {k{}_3y + z} \right)\,\,\,$$
is irrotational, then the values of the constants $$\,{k_1},\,{k_2}\,\,$$ and $$\,{k_3}$$ respectively, are
A
$$0.3, -2.5, 0.5$$
B
$$0.0, 3.0, 2.0$$
C
$$0.3, 0.33, 0.5$$
D
$$4.0, 3.0, 2.0$$
3
GATE ECE 2017 Set 1
Numerical
+2
-0
Let $$\,\,\,{\rm I} = \int_c {\left( {2z\,dx + 2y\,dy + 2x\,dz} \right)} \,\,\,\,$$ where $$x, y, z$$ are real, and let $$C$$ be the straight line segment from point $$A: (0, 2, 1)$$ to point $$B: (4,1,-1).$$ The value of $${\rm I}$$ is ___________.
Your input ____
4
GATE ECE 2016 Set 2
Numerical
+2
-0
Suppose $$C$$ is the closed curve defined as the circle $$\,\,{x^2} + {y^2} = 1\,\,$$ with $$C$$ oriented anti-clockwise. The value of $$\,\,\oint {\left( {x{y^2}dx + {x^2}ydx} \right)\,\,} $$ over the curve $$C$$ equals _______.
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Questions Asked from Marks 2
GATE ECE 2024 (1) GATE ECE 2017 Set 1 (2) GATE ECE 2016 Set 2 (1) GATE ECE 2014 Set 4 (1) GATE ECE 2013 (1) GATE ECE 2012 (1) GATE ECE 2010 (1) GATE ECE 2009 (1) GATE ECE 2008 (1)
GATE ECE Subjects
Signals and Systems
Network Theory
Control Systems
Digital Circuits
General Aptitude
Electronic Devices and VLSI
Analog Circuits
Engineering Mathematics
Microprocessors
Communications
Electromagnetics