MCQs of Vector Calculus (Mathematics - II - 3110015)

Mathematics - II (3110015) MCQs

MCQs of Vector Calculus

Showing 51 to 60 out of 79 Questions

51.

The flow integral along the closed curve

C

is called

(a)	Flow integral
(b)	Circulation
(c)	Flux
(d)	None of above

52.

Let

f (x, y, z) = x - {yz}^{2}

and

C

be the curve

y = x^{2}

joining

(1, 1, 0)

(2, 4, 0)

. Then

\int_{C} f (x, y, z) ds

(a)	$\frac{17^{\frac{5}{2}} - 5^{\frac{3}{2}}}{12}$
(b)	$\frac{5^{\frac{5}{2}} - 17^{\frac{3}{2}}}{12}$
(c)	$\frac{5^{\frac{3}{2}} - 17^{\frac{3}{2}}}{12}$
(d)	$\frac{17^{\frac{3}{2}} - 5^{\frac{3}{2}}}{12}$

53.

Let

F = f_{1} \hat{i} + f_{2} \hat{j} + f_{3} \hat{k}

be the velocity of fluid with continuous components on a smooth curve

C

, then the line integral

\int_{C} F \cdot d r

is called as

(a)	Flow integral
(b)	Circulation
(c)	Flux
(d)	None of above

54.

Let

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

. Then

\int_{C} F (r) \cdot d r

along the curve

x = y^{2}

from the point

A (0, 0)

B (1, 1)

(a)	$\frac{17}{35}$
(b)	$0$
(c)	$- \frac{17}{35}$
(d)	$\frac{24}{35}$

55.

What is the value of

\int_{C} y dx + x dy + z dz

, where

C

is given by

x = \cos (t), y = \sin (t), z = t^{2}, 0 \leq t \leq 2 π ?

(a)	$4 π$
(b)	$4 π^{4}$
(c)	$8 π^{4}$
(d)	$8 π$

56.

Let

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

and

C

be the circle

x^{2} + y^{2} = 1

traversed counterclockwise. Then

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

(a)	$0$
(b)	$- 4 π$
(c)	$- 2 π$
(d)	$8 π$

57.

Let the force field

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

along the straight line joining the points

A (1, - 1, 0)

B (2, 2, 1)

be given. Then the work done is

(a)	$\frac{5}{6}$
(b)	$\frac{6}{5}$
(c)	$\frac{10}{3}$
(d)	$- \frac{5}{6}$

58.

Let

F = (2 x^{2} - 3 y) \hat{i} - 10 xy \hat{j} + 10 {yz}^{2} \hat{k}

be fluid velocity field and curve

C

be the straight line joining the points

A (0, 0, 0)

B (1, 1, 1)

. Then the flow of

F

along

C

(a)	$\frac{5}{3}$
(b)	$\frac{6}{5}$
(c)	$\frac{10}{3}$
(d)	$- \frac{5}{3}$

59.

Let

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

and curve

C

be a circle of radius

3

centered at origin in

XY

plane. Then the circulation of

F

along

C

(a)	$30 π$
(b)	$36 π$
(c)	$- 36 π$
(d)	$18 π$

60.

Which of the following represents the flux of

F = M (x, y) \hat{i} + N (x, y) \hat{j}

(a)	$\int_{C} M dx + N dy$
(b)	$\oint_{C} M dx + N dy$
(c)	$\oint_{C} M dy - N dx$
(d)	$\int_{C} M dx - N dy$

Showing 51 to 60 out of 79 Questions