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QUESTION: 1

If then the value of at (2, 2, 0) will be

Solution:

Given,

QUESTION: 2

If then the value of is

Solution:

Thus,

QUESTION: 3

What is the value of constant b so that the vector

is solenoidal?

Solution:

Since vector is solenoidal, therefore

= 0

or, [1 + 1 + b] = 0 or b = -2

QUESTION: 4

Match List-I with List-ll and select the correct answer using the codes given below the lists:

**List-I**

A. Gauss’s divergence theorem

B. Stroke’s theorem

C. The divergence

D. The curl

**List-ll
**

Solution:

QUESTION: 5

**Assertion (A):** Vector differential operator is a vector quantity and it signifies that certain operations of a differentiation are to be carried out on the scalar function following it.

**Reason (R):** Vector differential operator posses properties similar to ordinary vectors

Solution:

Vector differential operator (‘∇’) is not a vector quantity. Hence, assertion is a false statement.

QUESTION: 6

Consider the following statements:

1. Divergence of a vector function at each point gives the rate per unit volume at which the physical entity is issuing from that point.

2. If a vector function ϕ represents temperature, then grad ϕ or ∇ϕ will represents rate of change of temperature with distance.

3. The curl of a vector function A gives the measure of the angular velocity at every point of the vector field.

Solution:

All the given statements are correct.

QUESTION: 7

**Assertion (A):** The Gauss’s divergence theorem permits us to express certain integrals by means of surface integrals.

**Reason (R):** Gauss’s divergence theorem states that “the surface integral of the curl of a vector field taken over any surface s is equal to the line integral of the vector field around the closed periphery (contour) of the surface.

Solution:

Reason is a statement of stroke’s theorem not that of Gauss's divergence theorem.

QUESTION: 8

If is any vector field in cartesian co-ordinates system, then

Solution:

Let, be any vector field in cartesian co-ordinate system then, we can prove that

Also, Div. Curl

QUESTION: 9

If and , then

Solution:

Given,

∴

Also,

QUESTION: 10

If S is any closed surface enclosing a volume V and then the value of (is a unit vector) will be equal to

Solution:

QUESTION: 11

Assertion (A): The laplacian operator of a scalar function ϕ can be defined as “Gradient of the divergence of the scalar ϕ”

Reason (R): Laplacian operator may be a “scalar laplacian" or a “vector laplacian'’ depending upon whether it is operated with a scalar function or a vector, respectively.

Solution:

Assertion is not true because the laplacian operator (∇^{2})of a scalar function ϕ can be defined as “Divergence of the gradient of the scalar ϕ”. i.e. ∇.∇ϕ

QUESTION: 12

Match List-I (Terms) with List-II (Type) and select the correct answer using the codes given below the lists:

List-I

A. Curl= 0

B. Div = 0

C. Div grad (ϕ) = 0

D. Div div (ϕ) = 0

List-ll

1. Laplace equation

2. Irrotational

3. Solenoidal

4. Not defined

Codes:

A B C D

(a) 2 3 1 4

(b) 4 1 3 2

(c) 2 1 3 4

(d) 4 3 1 2

Solution:

QUESTION: 13

Which of the following relations are not correct?

Solution:

(A x B)^{2} = A^{2}B^{2} - (A·B)^{2}

QUESTION: 14

If uF = ∇_{v}, where u and v are scalar fields and F is a vector field, then F. curl F is equal to

Solution:

Given,

∴

or,

Hence,

QUESTION: 15

A vector field which has a vanishing divergence is called as ____________

Solution:

By the definition: A vector field whose divergence comes out to be zero or Vanishes is called as a Solenoidal Vector Field.

i.e.

is a Solenoidal Vector field.

QUESTION: 16

Which of the following statements is not true of a phasor?

Solution:

A phasor is always a vector quantity.

QUESTION: 17

Match List-I (Physical quantities) with List-ll (Dimensions) and select the correct answer using the codes given below the lists:

List-I

A. Electric potential

B. Magnetic flux

C. Magnetic field intensity

D. Magnetic flux density

List-ll

1. MT^{-2}I^{-1}

2. ML^{2}T^{-3}I^{-1 }

3. IL^{-1}

4. ML^{2}T^{-2}I^{-1}

Codes:

A B C D

(a) 2 4 3 1

(b) 4 2 3 1

(c) 1 2 1 3

(d) 4 2 1 3

Solution:

- Emf induced, V = dφ/dt

or, ϕ = v.t = Magnetic flux

∴ [ϕ] = [ML^{2}T^{-3}I^{-1}[T]

= [ML^{2}T^{-2}I^{-1}] - Magnetic field intensity,

H = NI/I

Magnetic flux density B = ϕ/A

QUESTION: 18

Which of the following statements is not true regarding vector algebra?

Solution:

Option (c) is not correct because cross product of two unlike vectors is a third unit vector having positive sign for normal rotation and negative for reverse rotation while cross product of two like unit vectors is zero.

QUESTION: 19

A rigid body is rotating with an angular velocity of ω where, and v is the line velocity. If is the position vector given by then the value of curl will be equal to

Solution:

Taking the curl, we have:

(Since )

or,

∴

QUESTION: 20

If then which of the following relation will hold true?

Solution:

We have:

= 1.1 + 1.1 + 1.1 = 1 + 1 + 1 = 3

Also,

Thus,

Hence, both (a) and (b) will hold true.

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