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This mock test of Vector Calculus - 8 for Mathematics helps you for every Mathematics entrance exam.
This contains 20 Multiple Choice Questions for Mathematics Vector Calculus - 8 (mcq) to study with solutions a complete question bank.
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QUESTION: 1

If then div curl is equal to

Solution:

QUESTION: 2

If Φ is a differentiable scalar function, then div grad Φ is equal to

Solution:

QUESTION: 3

If then is

Solution:

QUESTION: 4

The value of along the circle x^{2} + y^{2} = 1 is

Solution:

QUESTION: 5

The value of is

Solution:

QUESTION: 6

Directional derivative of ψ(x,y,z) = xy^{2} + 4xyz + z^{2} at the point (1, 2, 3) in the direction of is

Solution:

QUESTION: 7

If and are two vectors then the value of will be

Solution:

QUESTION: 8

The value of div grad Φ is

Solution:

QUESTION: 9

The value of is

Solution:

QUESTION: 10

Gauss’s divergence theorem is

Solution:

QUESTION: 11

If then the value of div will be

Solution:

QUESTION: 12

If S denotes the surface of the cube bounded by the planes x = 0, x = a, y = 0, y = a, z = 0, z = a then by Gauss divergence theorem the value of is

Solution:

QUESTION: 13

If is a differentiable vector point function, then the value of div curl is

Solution:

QUESTION: 14

Let and

Q. The unit vector perpendicular to the plane containing and is

Solution:

Given that

and

Therefore, The unit vector perpendicular to the plane containing vector and is

QUESTION: 15

Let us consider the scalar point function f(x y, z) =x^{2} + y^{2} + z^{2}

Q. The grad of f(x, y, z) is

Solution:

Grad f=

QUESTION: 16

Let us consider the scalar point function f(x y, z) =x^{2} + y^{2} + z^{2}

Q. The directional derivative of f(x, y, z) at the point P(1, 1, 1) along is

Solution:

(Grad f)_{(1,1,1)}=

Now, the directional derivative o f f at P(1,1,1) along is

QUESTION: 17

Let where a, b and c are constants and S is the surface of unit sphere.

Q. The value of is

Solution:

By Gauss divergence theorem,

QUESTION: 18

Let where a, b and c are constants and S is the surface of unit sphere.

Q. The value of is

Solution:

By Gauss divergence theorem,

Since, V is enclosed by a sphere of unit radius. Thereofore

a + b + c = 1

and

QUESTION: 19

A vector normal to is

Solution:

We take,

= 1 - 2 + 1 = 0

So, B is normal to A.

QUESTION: 20

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

So,

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