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This mock test of Vector Calculus - 2 for IIT JAM helps you for every IIT JAM entrance exam.
This contains 20 Multiple Choice Questions for IIT JAM Vector Calculus - 2 (mcq) to study with solutions a complete question bank.
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QUESTION: 1

Solution:

QUESTION: 2

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

Solution:

**Correct Answer :- c**

**Explanation : **y = 2x^{2}

dy/dx = 4x

So F. dr

∫(x = 0 to 1) [3x(2x^{2})dx . (2x^{2})^{2} d(2x^{2})]

∫(x = 0 to 1) (6x^{2} - 16x^{5})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 = -y^{2} and N = xy

QUESTION: 4

The divergence of the vector field

Solution:

QUESTION: 5

The directional derivative of f(x, y, z) = x^{2} + y^{2} + z^{2} 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 x^{2} + y^{2} + z^{2} = 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 = x^{2} + y^{2} + z^{2} - 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 x^{2} + y^{2} = a^{2} 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 = - y^{2}x, v = 2yx^{2}

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 (x^{2} + z^{2}) at the point P(4, 0), in the direction of

Solution:

Given, F(x, y) = In(x^{2} + z^{2}) = log (y + z^{2})

Coordinate of point p is (4, 0).

or

QUESTION: 11

For a scalar function f(x, y, z) = x^{2} + 3y^{2} + 2z^{2}, 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 = x^{2} + xy and

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

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:

If P (x,y,z) is a variable point in a three- dimensional space, O the origin, i, j and k the unit vectors along the x-axis, y-axis and z-axis respectively, then the vector OP given by x i + y j +z k is called the position vector of the point P and is denoted by r.

Divergence of any vector f = f1i+ f2j +f3k denoted by div f is defined as the scalar ( delta f1)/(delta x) + (delta f2)/(delta y) + (delta f3)/(delta z)

where the delta s denote partial derivatives.

Using this definition we find that div r = delta (x)/ delta x +delta (y)/delta y + delta (z)/ delta z = 1 + 1 +1 = 3.

Hence, divergence of a position vector = div r = 3.

QUESTION: 20

Divergence operators is defined for

Solution:

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