MCQs of Vector Calculus (Mathematics - II - 3110015)

Mathematics - II (3110015) MCQs

MCQs of Vector Calculus

Showing 61 to 70 out of 79 Questions

61.

The flux of the field

F = x^{2} \hat{i} + y \hat{j}

around and across the closed curve

r = \cos (t) \hat{i} + \sin (t) \hat{j}, 0 \leq t \leq 2 π

, is

(a)	$π$
(b)	$- π$
(c)	$2 π$
(d)	$0$

62.

For any two paths

C_{1}

and

C_{2}

in some domain

D

with same initial and end points, the line integral is independent of path, then

(a)	$\int_{C_{1}} F \cdot d r \neq \int_{C_{2}} F \cdot d r$
(b)	$\int_{C_{1}} F \cdot d r = \int_{C_{2}} F \cdot d r$
(c)	$\int_{C_{1}} F \cdot d r < \int_{C_{2}} F \cdot d r$
(d)	$\int_{C_{1}} F \cdot d r > \int_{C_{2}} F \cdot d r$

63.

Let

ϕ

be a scalar potential function of a conservative field

F

. Then the work done in moving a particle along a curve

C

from point

A

B

(a)	$ϕ (A) - ϕ (B)$
(b)	$0$
(c)	$ϕ (B) + ϕ (A)$
(d)	$ϕ (B) - ϕ (A)$

64.

If a vector field

F

is conservative and curve

C

is closed, then

\oint_{C} F \cdot d r

(a)	$ϕ (A) - ϕ (B)$
(b)	$ϕ (B) - ϕ (A)$
(c)	$0$
(d)	$ϕ (A) + ϕ (B)$

65.

Let

F = (y + z^{3}) \hat{i} + (x + 2 y) \hat{j} + (3 {xz}^{2}) \hat{k}

be a conservative field. Then the integral when

C

is any path joining

A (1, 1, 1)

and

B (- 1, 2, 0)

(a)	$- 11$
(b)	$11$
(c)	$5$
(d)	$0$

66.

Let

\int_{(1, 2)}^{(3, 4)} ({xy}^{2} + y^{3}) dx + (x^{2} y + 3 {xy}^{2}) dy

be independent of path. Then the value of integral is

(a)	$202$
(b)	$254$
(c)	$- 202$
(d)	$- 254$

67.

The necessary and sufficient condition that

\int_{A}^{B} F \cdot d r

be independent of path is

(a)	$div (F) = 0$
(b)	$curl (F) \neq 0$
(c)	$curl (F) = (0, 0, 0)$
(d)	$curl (F) = (1, 1, 1)$

68.

\oint_{C} F (r) \cdot d r = 0

, then

F

(a)	Irrotational
(b)	Solenoidal
(c)	Incompressible
(d)	None of these

69.

By Green’s theorem

\oint_{c} (M dx + N dy) =

(a)	$\iint_{R} \{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}\} dx dy$
(b)	$\iint_{R} \{\frac{\partial N}{\partial y} - \frac{\partial M}{\partial x}\} dx dy$
(c)	$\iint_{R} \{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\} dx dy$
(d)	$\iint_{R} \{\frac{\partial M}{\partial y} + \frac{\partial N}{\partial x}\} dx dy$

70.

Green’s theorem is useful for changing a line integral around a closed curve

C

into ___________ over the region

R

enclosed by

C

(a)	Triple integral
(b)	Double integral
(c)	Volume integral
(d)	Line integral

Showing 61 to 70 out of 79 Questions