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IIT JAM Mathematics Practice Test- 9 - Mathematics MCQ


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30 Questions MCQ Test - IIT JAM Mathematics Practice Test- 9

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IIT JAM Mathematics Practice Test- 9 - Question 1

Force 3i + 2j + 5k and 2i + j - 3k are acting on a particle and displace it from the point 2i - 2j + 10k, then the work done by the force is

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 1

Resultant force F = 5i + 3j + 2k and displacement d = 2i - 2j + 10k, then Work done W = F.cl = 10 - 6 + 20 = 24 unit.

IIT JAM Mathematics Practice Test- 9 - Question 2

Let  be a vector point function and C be the an Arc from (1,0, 1) to (3, 4, 5) of the curve, which is the intersection of the surface Z2 = x2 + y2 and Z = y + 1, then the value of  is

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 2

Take


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IIT JAM Mathematics Practice Test- 9 - Question 3

Force of magnitude 5 units acting along the vector 2i - 2j + k displaces the point of application from (1, 2, 3) to (5, 3, 7), then the work done is

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 3

Required work done = (Force vector). (Displacement vector)

Force vector 

∴ Required work done

IIT JAM Mathematics Practice Test- 9 - Question 4

The value of line integral  , where F = (x- 3y, 2x- y, 0) and C is the ellipse  x2/9 + y2/4 = 1 in the XY plane described in the positive sense.

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 4

The parametric equation of the ellipse is 

x = 3cosθ, y = 2sinθ, z = 0

∴ r = (3cosθ, 2sinθ, 0), dr/dθ = (-3sinθ, 2cos0, 0), and

F = (x - 3y, 2x - y, 0) = (3cosθ - 6sinθ, 6cosθ - 2sinθ, 0).

Now,

IIT JAM Mathematics Practice Test- 9 - Question 5

The value of line integral  where F = xzi + yj +x2 k and C is the curve x = sinθ cosθ, y = sin2 θ, z = cos θ with θ increasing from 0 to π/2, is

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 5

F .dr = ( xzi + yj + x2k ).( idx + jdy + kdz) = x z dx + y dy + x2 dz 

∴ Along the given curve, we have

Putting values,

IIT JAM Mathematics Practice Test- 9 - Question 6

If  then the value of  is

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 6

IIT JAM Mathematics Practice Test- 9 - Question 7

If u = 3x2y and v = xz2 - 2y. then grad [(gradu). (gradv)] is

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 7

IIT JAM Mathematics Practice Test- 9 - Question 8

Determine if the following vector field is 

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 8

Here is P and Q as well as the appropriate derivatives.


Let  be a vector field on an open and simply-connected region D

and  

The vector field f is conservative.
Here the two partial derivatives are equal and so this is a conservative vector field.

 

IIT JAM Mathematics Practice Test- 9 - Question 9

The values of a, b, c so that the function 

is irrotational.

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 9

For an irrotational vector function


As  are independent vectors, so their coefficients must be zero separately, i.e.,

c + 1 = 0, 4 - a = 0, b - 2 = 0

These equations give

a = 4 ,b = 2, c = - 1

IIT JAM Mathematics Practice Test- 9 - Question 10

The value of  where  and S is the surface of the paraboloid Z = x2 + y2 bounded by Z = 4 and C is the boundary cuive described in the positive sense.

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 10

Here S is an open surface so we can apply stokes Theorem , 

= 3y dx - 4 x dy    ( ∵ z = 4 ⇒ dz = 0 )

Let x = 2 cos θ , y = 2 sin θ

⇒ dx = -2 sin θ dθ , dy = 2 cosθ dθ

IIT JAM Mathematics Practice Test- 9 - Question 11

If F = xy2 i + 2x2 yz j - 3yz2 k , then curl F at (1 ,-1,1) is

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 11

= i [ - 3z2 - 2x2 y ] - j [0] + k [ 4 xyz - 2xy ]

IIT JAM Mathematics Practice Test- 9 - Question 12

Find grad 

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 12

we have

IIT JAM Mathematics Practice Test- 9 - Question 13

If f = (x + y + 1) i + j + (-x -y) k , then f. curl f = 

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 13

= i[- 1 ]- j[- 1 ] + k[-1]

= H + j - k )

IIT JAM Mathematics Practice Test- 9 - Question 14

Let  and S is the surface of a rectangle bounded 0  Then the value of 

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 14

Take o.p. on xy - plane that be a rectangle bounded by 0 ≤ x ≤ n , 0 ≤ y ≤ 2

IIT JAM Mathematics Practice Test- 9 - Question 15

Let   for a square surface shown in the figure, then 

 

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 15

Since x = 0 for this surface

*Multiple options can be correct
IIT JAM Mathematics Practice Test- 9 - Question 16

If  then which one of the following is /are  true?

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 16

given

*Multiple options can be correct
IIT JAM Mathematics Practice Test- 9 - Question 17

if  are three non - zero vector’s, then which of the following(s) is/are false statements ?

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 17

we know 

*Multiple options can be correct
IIT JAM Mathematics Practice Test- 9 - Question 18

if   and  with r =  then which of the followings is /are true ? (where f be a SPF).

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 18

Take

=> (A) is correct

Now

*Multiple options can be correct
IIT JAM Mathematics Practice Test- 9 - Question 19

if  then

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 19


again

Hence

*Multiple options can be correct
IIT JAM Mathematics Practice Test- 9 - Question 20

If the scalar function is f = x2yz3 then at that point (2,1,-1)

Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 20

Hence the maximum directional derivative at the point (2,1 ,-1 )=  

Again the magnitude of the directional derivative at the point (2,1 ,-1)   

*Answer can only contain numeric values
IIT JAM Mathematics Practice Test- 9 - Question 21

A particle acted on by constant forces 4i + j - 3k and 3i + j - k is displaced from the point i + 2j + 3k to the point 5i + 4j + k. The total work done by the force is _____.


Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 21

*Answer can only contain numeric values
IIT JAM Mathematics Practice Test- 9 - Question 22

The value of line integral  where  F = x2y2 i + yj and cuive C is y2 = 4x in the xy-plane from (0, 0) to (4, 4), where r = xi + yj, is _____


Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 22

*Answer can only contain numeric values
IIT JAM Mathematics Practice Test- 9 - Question 23

The value of   and C is given by  is __________.


Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 23

Now, we could use the techniques we discussed when we first looked at line integrals of vector fields. However that would be particularly unpleasant solution.
This vector field is conseivative and a potential function for the vector field is,

Using this, we know that integral must be independent of path and so all we need to do is, to use the theorem from the previous section to do the evaluation.

where

*Answer can only contain numeric values
IIT JAM Mathematics Practice Test- 9 - Question 24

The directional derivative of f = xyz2 at (1,0, 3) in the direction of the vector i - j + k is _________.


Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 24

since

Now the unit vector t in the direction of the vector i - j + k is given by t  = 

∴ The required directional derivative is

*Answer can only contain numeric values
IIT JAM Mathematics Practice Test- 9 - Question 25

The angle between the surface x2 + y2 + z2 = 9 and x2 + y2 - z = 3 at the point (2, -1, 2), i s ________.


Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 25


Now angle between at ( 2 , -1 , 2 ) be 

*Answer can only contain numeric values
IIT JAM Mathematics Practice Test- 9 - Question 26

The directional derivatives of f = x2 + y2 + z2 at (1,2, 3) in the direction of the line x/3 = y/4 = z/5, i s ________.


Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 26

The directional derivative is

*Answer can only contain numeric values
IIT JAM Mathematics Practice Test- 9 - Question 27

If vector A = ( ax + 3y + 4 z ) i + ( x - 2y + 3z) j + ( 3x + 2 y - z) k is solenoidal, then a is ______ .


Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 27

If the vector A is solenoidal, then div A= 0.

*Answer can only contain numeric values
IIT JAM Mathematics Practice Test- 9 - Question 28

If the vector  has its curl identically equal to zero,then the value of a is _________ .


Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 28

Given

if is identically equal to zero , then all three components of  are zero

*Answer can only contain numeric values
IIT JAM Mathematics Practice Test- 9 - Question 29

A particle moves anticlock wise along the curve x2 + y2 = 9 from (3,0) to a point  Q under the force action  Then the total no. of possible locations of Q so that the work done is equal to 3/2  is ______.


Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 29

We have x2+ y2 = 9 point on the circ le is represented as ( 3cos θ , 3 sin θ)

Given W = 3/2

Here both the values are possible .

here are two possible locations for Q so that the work done by  is equal to 3/2

*Answer can only contain numeric values
IIT JAM Mathematics Practice Test- 9 - Question 30

The directional derivative of f(x, y) = x2 - 6y2 at the point (7, 2) in the direction  is _____________.


Detailed Solution for IIT JAM Mathematics Practice Test- 9 - Question 30

since

  

so

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