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

For a scalar function (x, y, z) = x^{2} + 3y^{2} + 2z^{2}, the directional derivative at the point P( 1, 2, -1) is the direction of a vector is

Solution:

We have,

So, the direction derivative in the direction of

QUESTION: 2

Use the divergence theorem the value of where, S is any closed surface enclosing volume V.

Solution:

where, is an outward drawn unit normal vector to S.

QUESTION: 3

Find the value of

Solution:

So, from vector identity

QUESTION: 4

If then the value of div at the point (1, 1, -1) will be

Solution:

QUESTION: 5

Apply Stoke’s theorem, the value of where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0) and (0, 0, 6) is

Solution:

Taking projection on three planes, we note that the surface S consists of three triangles, Δ OAB in XT- plane, Δ OBC in TZ-plane and Δ OAC in XZ-plane. Using two point formula, the equation of the line AB, BC, CA are respectively 3x + 2y = 6 , 2y + z = 6 , 3x + z = 6

So, by Stake’s theorem

QUESTION: 6

The line integral from the origin to the point P( 1,1,1) is

Solution:

So, potential function of

So, line integral of the vector from point (0, 0, 0) to (1, 1, 1) is

QUESTION: 7

If are to arbitrary vectors with magnitudes a and b respectively, will be equal to

Solution:

Cross checking from option (a),

which is correct answer.

QUESTION: 8

Divergence of the three-dimensional radial vector field is

Solution:

QUESTION: 9

Which of the following holds for any non-zero vector

Solution:

QUESTION: 10

A vector normal to is

Solution:

We take,

= 1 - 2 + 1 = 0

So, B is normal to A.

QUESTION: 11

If and curve C is the arc of the curve y = x^{3} from (0, 0 ) to (2, 8), then the value of

Solution:

Since, C is the curve y = x^{3} from (0, 0) to (2, 8)

So, let x = t ⇒ y = t^{3}

If is the position vector of any point on C, then

or

or

At (0, 0) ⇒ t = x = 0 and at (2, 8) ⇒ t = 2

QUESTION: 12

The value of by Stoke’s theorem, where and C is the boundary of the triangle with vertices at ( 0 ,0 , 0 ) , ( 1 , 0 , 0 ) and ( 1 ,1 , 0 ) is

Solution:

We have, curl

Also we note that z coordinate of each vertex of the triangle is zero.

or The triangle lies in the xy-plane. So,

So, curl

In the figure, we have only considered the xy-plane. So, by Stoke’s Theorem

QUESTION: 13

If is the reciprocal system to the vectors then the value of is

Solution:

Since,

Therefore,

QUESTION: 14

Scalar triple product is equal to

Solution:

QUESTION: 15

R is a closed planar region as shown by the shaded area in the figure below. Its boundary C consists of the circles C_{1} and C_{2}.

If are all continuous everywhere in R, Green’s theorem states that

Which one of the following alternatives correctly depicts the direction of integration along C?

Solution:

The region R is bounded by two closed circles C_{1} and C_{2}, so it is doubly connected. To apply it in Green’s theorem, we need to convert it into simply connected region. For it, we apply cut AD and consider the region R having simple closed curve ABCADEFDA in the anticlockwise direction. So, the directions shown in figure, (c) is correct option.

QUESTION: 16

The value of is

Solution:

Since,

Putting

QUESTION: 17

The vector field are unit vectors) is

Solution:

Given,

Now, for divergence

Hence, vector field is divergence-free.

Now, for irrotational

QUESTION: 18

Use Gauss’s divergence theorem to find where and S is the closed surface in the first octant bounded by y^{2} + z^{2} = 9 and x = 2.

Solution:

Let V be the volume enclosed by the closed surface S, i.e., the volume in the first octant bounded by the cylinder y^{2} I z^{2} = 9 and the planes x = 0, x = 2. Then by Gauss’s divergence theorem, we have

QUESTION: 19

For the scalar field magnitude of the gradient at the point (1, 3) is

Solution:

Since,

So,

At (1, 3),

So,

QUESTION: 20

is equal to

Solution:

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