Vector Calculus - 2


20 Questions MCQ Test Topic-wise Tests & Solved Examples for IIT JAM Mathematics | Vector Calculus - 2


Description
This mock test of Vector Calculus - 2 for Mathematics helps you for every Mathematics entrance exam. This contains 20 Multiple Choice Questions for Mathematics Vector Calculus - 2 (mcq) to study with solutions a complete question bank. The solved questions answers in this Vector Calculus - 2 quiz give you a good mix of easy questions and tough questions. Mathematics students definitely take this Vector Calculus - 2 exercise for a better result in the exam. You can find other Vector Calculus - 2 extra questions, long questions & short questions for Mathematics on EduRev as well by searching above.
QUESTION: 1

Solution:

QUESTION: 2

 then the value of  where C is the curve in the XY-plane, y = 2x2 from (0,0) to (1,2) is

Solution:

Correct Answer :- c

Explanation : y = 2x2

dy/dx = 4x 

So F. dr

∫(x = 0 to 1) [3x(2x2)dx . (2x2)2 d(2x2)]

∫(x = 0 to 1) (6x2 - 16x5)dx

= -7/6

QUESTION: 3

Value of the integral  where C is the square cut from the first quadrant by the lines x = 1 and y = 1 will be (use Green’s theorem to change the line integral into double integral)

Solution:

We know that in Green’s theorem,

On comparing, we get M = -y2 and N = xy

QUESTION: 4

The divergence of the vector field 

Solution:

QUESTION: 5

The directional derivative of f(x, y, z) = x2 + y2 + z2 at the point (1, 1, 1) in the direction

Solution:

We have,



Thus, directional derivative of f in the direction of   the point P(1, 1, 1),

QUESTION: 6

The unit normal vector to the surface of the sphere x2 + y2 + z2 = 1 at the point  and  are unit normal vectors in the Cartesian coordinate system)

Solution:


Problem is to find unit normal vector to the surface of the sphere
f = x2 + y2 + z2 - 1 = 0
We know unit normal vector  is given by



 

QUESTION: 7

If  over the path shown in the figure is 

Solution:


QUESTION: 8

Apply Green’s theorem the value of  where C is the boundary of the area enclosed by the X-axis and the upper half of the circle x2 + y2 = a2 is

Solution:


Since, From Green’s theorem, we have 

(A is the region of the figure)

On changing in polar coordinates, we get
 

QUESTION: 9

A fluid element has a velocity  The motion at (x, y) = 

Solution:


or u = - y2x, v = 2yx2



In two-dim ensional flow, equation of continuity

Fluid is incompressible at this point.


Fluid flow is rotational.
Thus, fluid flow at  is rotational and incompressible.

QUESTION: 10

Unit vectors in X and Z-directions are  respectively. Which one of the following is the directional derivative of the function F(x, z) = In (x2 + z2) at the point P(4, 0), in the direction of  

Solution:

Given, F(x, y) = In(x2 + z2) = log (y + z2)

Coordinate of point p is (4, 0).




or  

QUESTION: 11

For a scalar function f(x, y, z) = x2 + 3y2 + 2z2, the gradient at the point P(1, 2, -1) is

Solution:

QUESTION: 12

 then the value of div Curl is

Solution:



QUESTION: 13

The value of α for which the following three vectors are coplanar is 

Solution:

Given,

Vectors  are coplanar, if

QUESTION: 14

Apply Green’ s theorem the value of where C is the square formed by the lines y = ±1, x = ±1 is

Solution:

Since, On comparing  with  we get M = x2 + xy and 
N = x2 + y2
So, from Green’s theorem


QUESTION: 15

The divergence of the vector field  at a point (1, 1, 1) is equal to

Solution:

Since, We have 
On comparing Eq. (r) with 



QUESTION: 16

 is the reciprocal system to the vectors   then the value of 

Solution:
QUESTION: 17

 where  are constant vectors then  is equal to

Solution:
QUESTION: 18

 , then the value of div curl   is

Solution:
QUESTION: 19

The value of div   will be

Solution:

Correct Answer :- b

Explanation : div r = (d/dx i + d/dy j + d/dz k) * (xi + yj + zk)

= {(i,j,k) (d/dx,d/dy,d/dz) (x,y,z)}

= i(0-0) -j(0-0) +k(0-0)

= 0

QUESTION: 20

Divergence operators is defined for

Solution:

Related tests