1
GATE ECE 2012
MCQ (Single Correct Answer)
+2
-0.6
The direction of vector $$A$$ is radially outward
from the origin, with $$\left| A \right| = K\,{r^n}$$
where $${r^2} = {x^2} + {y^2} + {z^2}$$ and $$K$$ is constant.
The value of $$n$$ for which $$\nabla .A = 0\,\,$$ is
A
$$-2$$
B
$$2$$
C
$$1$$
D
$$0$$
2
GATE ECE 2010
MCQ (Single Correct Answer)
+2
-0.6
If $$\overrightarrow A = xy\,\widehat a{}_x + {x^2}\widehat a{}_y\,\,$$ then $$\,\,\oint {\overrightarrow A .d\overrightarrow r \,\,}$$ over the path shown in the figure is
A
$$0$$
B
$${2 \over {\sqrt 3 }}$$
C
$$1$$
D
$$2\sqrt 3$$
3
GATE ECE 2009
MCQ (Single Correct Answer)
+2
-0.6
If a vector field$$\overrightarrow V$$ is related to another field $$\overrightarrow A$$ through $$\,\overrightarrow V = \nabla \times \overrightarrow A ,$$ which of the following is true?

Note: $$C$$ and $${S_C}$$ refer to any closed contour and any surface whose boundary is $$C.$$

A
$$\oint\limits_C {\overrightarrow V .\,\overrightarrow {dl} } = \int {\int_{{S_C}} {\overrightarrow A .\,\overrightarrow {ds} } }$$
B
$$\oint\limits_C {\overrightarrow A .\,\overrightarrow {dl} } = \int\limits_{{S_C}} {\int {\overrightarrow \nabla .\,\overrightarrow {ds} } }$$
C
$$\oint\limits_C {\nabla \times \vec V.{\mkern 1mu} \overrightarrow {dl} } = \int\limits_{{S_C}} {\int {\nabla \times \vec A.{\mkern 1mu} \overrightarrow {ds} } }$$
D
$$\oint\limits_C {\nabla \times \vec A.{\mkern 1mu} \overrightarrow {dl} } = \int {\int_{{S_C}} {\vec V.{\mkern 1mu} \overrightarrow {ds} } }$$
4
GATE ECE 2008
MCQ (Single Correct Answer)
+2
-0.6
Consider points $$P$$ and $$Q$$ in $$xy-$$plane with $$P=(1,0)$$ and $$Q=(0,1).$$ The line integral $$2\int\limits_P^Q {\left( {x\,dx + y\,dy} \right)\,\,}$$ along the semicircle with the line segment $$PQ$$ as its diameter
A
is $$-1$$
B
is $$0$$
C
$$1$$
D
depends on the direction (clockwise (or) anti-clockwise) of the semi circle
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