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QUESTION: 1

If φ = xy^{2}z^{3} then match List-I with List-II and select the correct answer using the codes given below the lists:

Solution:

Here

Also,

And,

QUESTION: 2

The unit vector which points from z = h on the z-axis towards (r, φ, 0) in cylindrical co-ordinates as shown below is given by

Solution:

Let the unit vector be given by

Now,

∴ Unit vector,

QUESTION: 3

If the vector V given below is irrotational, then the values of a, b and c will be respectively

Solution:

Since the given vector is irrotational, therefore curl V = 0 or,

Since, therefore, a = 4, b = 2, and c = -1

QUESTION: 4

Match List-I (Vector Identities) with List-ll. (Equivalent expression) and select the correct answer using the codes given below the lists:

Solution:

- is called “ product of four vectors ”.
- is called “vector triple product”.
- is called "vector product of four vectors ”.

QUESTION: 5

The vector differential operator, in spherical co-ordinate system is given by

Solution:

QUESTION: 6

Assertion (A): Divergence of a vector function at each point gives the rate per unit volume at which the physical entity is issuing from that point.

Reason (R): If some physical entity is generated or absorbed within a certain region of the field, then that region is known as source or sink respectively and if there are no sources or sinks in the field, the net outflow of the incompressible physical entity over any part of the region is zero. However, the net outflow is said to be positive, if the total strength of the sources are greater than the total strength of sink and vice-versa.

Solution:

Both assertion and reason are true and reason is the correct explanation of assertion. Reason is the physical interpretation of divergence.

QUESTION: 7

Which of the following identity is not true?

Solution:

- is called “vector triple product" which is a correct expression.

QUESTION: 8

The vector directed from (2, - 4,1) to (0, -2,0) in Cartesian coordinates is given by

Solution:

The vector is given as

QUESTION: 9

What is the value of where

Here, s is the surface bounded by x = 0, x = 1, y = o, y = 1, z = 0, z =1 and are unit vectors along x, yand z axes respectively.

Solution:

By divergence theorem,

Since, the surface s is bounded by x = 0, 1; y = 0, 1 and z = 0, 1 so, putting the limits, we have

QUESTION: 10

The vector field given by

Solution:

The vector field will be irrotational,

Now,

Hence, is irrotationai.

The vector field will be solenoidal,

Here,

Hence, is solenoidal.

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