1
GATE ECE 2022
+1
-0.33

Consider the two-dimensional vector field $$\overrightarrow F (x,y) - x\overrightarrow i + y\overrightarrow j$$, where $$\overrightarrow i$$ and $$\widehat j$$ denote the unit vectors along the x-axis and the y-axis, respectively. A contour C in the x-y plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral

$$\oint\limits_C {\overrightarrow F (x,y)\,.\,(dx\overrightarrow i + dy\overrightarrow j )}$$

is ___________.

A
0
B
1
C
8 + 2$$\pi$$
D
$$-$$1
2
GATE ECE 2022
+1
-0.33

The function f(x) = 8loge x $$-$$ x2 + 3 attains its minimum over the interval [1, e] at x = __________.

(Here loge x is the natural logarithm of x.)

A
2
B
1
C
e
D
$${{1 + e} \over 2}$$
3
GATE ECE 2022
Numerical
+1
-0.33

The value of the integral

$$\int\!\!\!\int\limits_D {3({x^2} + {y^2})dx\,dy}$$,

where D is the shaded triangular region shown in the diagram, is ___________ (rounded off to the nearest integer).

4
GATE ECE 2019
+1
-0.33
The families of curves represented by the solution of the equation

$${{dy} \over {dx}} = - {\left( {{x \over y}} \right)^n}$$

for n = –1 and n = 1 respectively, are
A
Circles and Hyperbolas
B
Hyperbolas and Circles
C
Parabolas and Circles
D
Hyperbolas and Parabolas
EXAM MAP
Medical
NEET