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All questions of Vector Calculus for Mathematics Exam
A vector normal to 
is- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
A vector normal to is
isa)
b)
c)
d)
|
Chirag Verma answered |
We take,
= 1 - 2 + 1 = 0
So, B is normal to A.
The cross-vector product of the vector always equals the vector. Perpendicular is the line and that will make the angle of 900with one another line. Therefore, when two given vectors are perpendicular then their cross product is not zero but the dot product is zero.
at the point (1,-2,-1) is
will be
then div 
is
is
Gauss’s divergence theorem can be written a- a)
- b)
- c)
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
Gauss’s divergence theorem can be written a
a)
b)
c)
d)
None of these
|
|
Chirag Verma answered |
The divergence theorem states that the surface integral of the normal component of a vector point function “F” over a closed surface “S” is equal to the volume integral of the divergence of F⃗ taken over the volume “V” enclosed by the surface S. Thus, the divergence theorem is symbolically denoted as :
A sphere of unit radius is centred at the origin. The unit normal at point (x, y, z) on the surface of the sphere is the vector- a)(x, y, z)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
A sphere of unit radius is centred at the origin. The unit normal at point (x, y, z) on the surface of the sphere is the vector
a)
(x, y, z)
b)
c)
d)
|
Veda Institute answered |
Equation of sphere of unit radius having centre at origin is
f (x, y, z) = x2 + y2 + z2 - 1 = 0
So,
being unit normal at (x,y, z) on the surface.
f (x, y, z) = x2 + y2 + z2 - 1 = 0
So,
being unit normal at (x,y, z) on the surface.
In case of two phase supply the electrical displacement of the winding is –- a)180º
- b)120º
- c)90º
- d)60º
Correct answer is option 'C'. Can you explain this answer?
In case of two phase supply the electrical displacement of the winding is –
a)
180º
b)
120º
c)
90º
d)
60º
|
Ishani Iyer answered |
Explanation:
Two-phase supply is a type of alternating current (AC) electrical power that has two sinusoidal voltages that are 90 degrees out of phase with each other. In two-phase supply, the electrical displacement of the winding is 90 degrees.
What is Electrical Displacement?
Electrical displacement refers to the phase difference between two alternating voltages or currents of the same frequency. It is measured in degrees and determines the overall behavior of the electrical system.
Why is the Electrical Displacement of the Winding 90 Degrees in Two-Phase Supply?
In two-phase supply, the two sinusoidal voltages are displaced from each other by 90 degrees, which means that the electrical displacement of the winding is also 90 degrees. This is because the two-phase system is created by using two separate windings that are displaced by 90 degrees from each other.
What are the Advantages of Two-Phase Supply?
Two-phase supply has some advantages over single-phase supply, including:
1. Improved efficiency: Two-phase supply can support higher power loads with less voltage drop and power loss than single-phase supply.
2. Better motor control: Two-phase supply provides better motor control and reduces the need for starting capacitors.
3. Reduced cost: Two-phase supply can be cheaper to install and maintain than three-phase supply.
Conclusion:
In two-phase supply, the electrical displacement of the winding is 90 degrees. This is because the two sinusoidal voltages that make up the two-phase system are displaced from each other by 90 degrees. Two-phase supply has some advantages over single-phase supply, including improved efficiency, better motor control, and reduced cost.
Two-phase supply is a type of alternating current (AC) electrical power that has two sinusoidal voltages that are 90 degrees out of phase with each other. In two-phase supply, the electrical displacement of the winding is 90 degrees.
What is Electrical Displacement?
Electrical displacement refers to the phase difference between two alternating voltages or currents of the same frequency. It is measured in degrees and determines the overall behavior of the electrical system.
Why is the Electrical Displacement of the Winding 90 Degrees in Two-Phase Supply?
In two-phase supply, the two sinusoidal voltages are displaced from each other by 90 degrees, which means that the electrical displacement of the winding is also 90 degrees. This is because the two-phase system is created by using two separate windings that are displaced by 90 degrees from each other.
What are the Advantages of Two-Phase Supply?
Two-phase supply has some advantages over single-phase supply, including:
1. Improved efficiency: Two-phase supply can support higher power loads with less voltage drop and power loss than single-phase supply.
2. Better motor control: Two-phase supply provides better motor control and reduces the need for starting capacitors.
3. Reduced cost: Two-phase supply can be cheaper to install and maintain than three-phase supply.
Conclusion:
In two-phase supply, the electrical displacement of the winding is 90 degrees. This is because the two sinusoidal voltages that make up the two-phase system are displaced from each other by 90 degrees. Two-phase supply has some advantages over single-phase supply, including improved efficiency, better motor control, and reduced cost.
Green’s theorem in a plane is a special case of- a)Stoke’s theorem
- b)Gauss divergence theorem
- c)Cauchy’s theorem
- d)None of the above
Correct answer is option 'A'. Can you explain this answer?
Green’s theorem in a plane is a special case of
a)
Stoke’s theorem
b)
Gauss divergence theorem
c)
Cauchy’s theorem
d)
None of the above
|
|
Veda Institute answered |
Green's theorem gives the relationship between a line integral around a simple closed curve, C, in a plane and a double integral over the plane region R bounded by C. It is a special two-dimensional case of the more general Stokes' theorem.
Stokes’ Theorem gives the relationship between a line integral around a simple closed curve, C, in space, and a surface integral over a piecewise, smooth surface.
Green’s theorem in its “curl form”.
where F = P(x,y) i + Q(x,y) j and dr = dx i + dy j
is as follows: (curl form of Green’s Theorem)
c∫ F(x,y) . dr = c∫ F . T ds = r∫ ∫ curl F dA
where curl F is the z-component of curl F = curl F . k
for stoke’s theorem
Use the divergence theorem the value of 
where, S is any closed surface enclosing volume V.- a)V
- b)3V
- c)4V
- d)6V
Correct answer is option 'D'. Can you explain this answer?
Use the divergence theorem the value of where, S is any closed surface enclosing volume V.
where, S is any closed surface enclosing volume V.a)
V
b)
3V
c)
4V
d)
6V
|
|
Chirag Verma answered |
where,
is an outward drawn unit normal vector to S.
Let 
is a solution of the Laplace equation then the vector field 
is - a)neither solenoidal nor irrotational
- b)irrotational but not solenoidal
- c)both solenoidal and irrotational
- d) solenoidal but not irrotational
Correct answer is option 'B'. Can you explain this answer?
Let is a solution of the Laplace equation then the vector field is
a)
neither solenoidal nor irrotational
b)
irrotational but not solenoidal
c)
both solenoidal and irrotational
d)
solenoidal but not irrotational
|
|
Veda Institute answered |
⇒ Divergence is non-zero and curl is zero,
Hence
is not solenoidal but irrotational.
Hence
The work done in moving a particle in the force field F = 3x² i + (2xz - 9y) j+ zk along the line joining (0 , 0, 0) to (2 ,1, 3) is
- a)0
- b)16
- c)24
- d)18
Correct answer is option 'B'. Can you explain this answer?
The work done in moving a particle in the force field F = 3x² i + (2xz - 9y) j+ zk along the line joining (0 , 0, 0) to (2 ,1, 3) is
a)
0
b)
16
c)
24
d)
18
|
|
Chirag Verma answered |
Correct Answer :- B
Explanation : Straight line x = 2t, y = t, z = 3t 0 ≤ t ≤ 1
Work done = ∫F · dr
= ∫(0 to 1) F·dr/dt * dt
= ∫(0 to 1) F·(dr/dt)dt
= ∫(0 to 1)[3(2t)2i + (2.2t.3t − t)j + 3tk] · [2i + j + 3k]dt
= ∫(0 to 1)[24t2 + 12t2 − t + 9t]dt
= [8t3 + 4t3(1/2t2) + 9/2t2](0 to 1)
= 8 + 4 + 4
= 16
The maximum value of the directional derivative of the function φ = 2x2 + 3y2 + 5z2 at a point (1, 1, -1) is- a)10
- b)-4
- c)
- d)152
Correct answer is option 'C'. Can you explain this answer?
The maximum value of the directional derivative of the function φ = 2x2 + 3y2 + 5z2 at a point (1, 1, -1) is
a)
10
b)
-4
c)
d)
152
|
|
Chirag Verma answered |
Hence, vector field is irrotational and divergence-free.
Given,
Directional derivative of φ is
(given)
As x = 1, y = 1, z = - 1
Magnitude of directional derivative is
Given,
Directional derivative of φ is
(given)As x = 1, y = 1, z = - 1
Magnitude of directional derivative is
For a scalar function 
(x, y, z) = x2 + 3y2 + 2z2, the directional derivative at the point P( 1, 2, -1) is the direction of a vector 
is- a)-18
- b)-3√6
- c)3√6
- d)18
Correct answer is option 'B'. Can you explain this answer?
For a scalar function (x, y, z) = x2 + 3y2 + 2z2, the directional derivative at the point P( 1, 2, -1) is the direction of a vector is
(x, y, z) = x2 + 3y2 + 2z2, the directional derivative at the point P( 1, 2, -1) is the direction of a vector isa)
-18
b)
-3√6
c)
3√6
d)
18
|
|
Veda Institute answered |
We have, 
So, the direction derivative in the direction of
So, the direction derivative in the direction of
Let S be the oriented surface x2 + y2 + z2 = 1 with the unit normal n pointing outward. For the vector field F(x, y, z) = xi + yj + zk, the value of 
is- a)π/3
- b)2π
- c)4π
- d)4π/3
Correct answer is option 'C'. Can you explain this answer?
Let S be the oriented surface x2 + y2 + z2 = 1 with the unit normal n pointing outward. For the vector field F(x, y, z) = xi + yj + zk, the value of is
a)
π/3
b)
2π
c)
4π
d)
4π/3
|
|
Veda Institute answered |
By Guass is divergence theorem,
,
where V is the volume enclosed by S
since the volume V enclosed by a sphere of unit radius is equal to (4/3)π(13) = (4/3)π .
where V is the volume enclosed by S
since the volume V enclosed by a sphere of unit radius is equal to (4/3)π(13) = (4/3)π .
The directional derivative of f(x, y, z) = 2x2 + 3y2 + z2 at the point P(2, 1, 3) in the direction of the vector 
- a) -2.785
- b)-2.145
- c) -1.789
- d)1.000
Correct answer is option 'C'. Can you explain this answer?
The directional derivative of f(x, y, z) = 2x2 + 3y2 + z2 at the point P(2, 1, 3) in the direction of the vector
a)
-2.785
b)
-2.145
c)
-1.789
d)
1.000
|
|
Veda Institute answered |
So, directional derivative of f indirection of vector
is nothing but the component of grad f in the direction of vector 
The value of 
is- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
The value of is
isa)
b)
c)
d)
|
|
Veda Institute answered |
Since, 
Putting
Putting
Consider a closed surface S surrounding volume of V. if 
is the position vector o f a point inside S, with 
the unit normal on S, the value of the integral 
is- a) 3V
- b)5V
- c)10V
- d)15V
Correct answer is option 'D'. Can you explain this answer?
Consider a closed surface S surrounding volume of V. if is the position vector o f a point inside S, with the unit normal on S, the value of the integral is
is the position vector o f a point inside S, with the unit normal on S, the value of the integral isa)
3V
b)
5V
c)
10V
d)
15V
|
|
Chirag Verma answered |
Apply the divergence theorem
(Since,
is the position vector)
(Since,
is the position vector)
Find the value of 
- a)1
- b)zero
- c)3
- d)-1
Correct answer is option 'B'. Can you explain this answer?
Find the value of
a)
1
b)
zero
c)
3
d)
-1
|
|
Chirag Verma answered |
So, from vector identity
The line integral of 
where, C is the unit circle around the origin traversed once in the counter-clockwise direction, is - a)-2π
- b)zero
- c)2π
- d)π
Correct answer is option 'B'. Can you explain this answer?
The line integral of where, C is the unit circle around the origin traversed once in the counter-clockwise direction, is
where, C is the unit circle around the origin traversed once in the counter-clockwise direction, is a)
-2π
b)
zero
c)
2π
d)
π
|
|
Veda Institute answered |
....(i)C being unit circle around the, origin traversed once in the counter-clockwise direction. We know from Green’s theorem,
Comparing Eq. (i) with LHS of Eq. (ii), we have
So,
A scalar field is given by f = x2/3 + y2/3, where x and y are the Cartesian coordinates. The derivative of f along the line y = x directed away from the origin, at the point (8, 8) is- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
A scalar field is given by f = x2/3 + y2/3, where x and y are the Cartesian coordinates. The derivative of f along the line y = x directed away from the origin, at the point (8, 8) is
a)
b)
c)
d)
|
|
Veda Institute answered |
(Putting y = x)
Derivative of f in the direction of
Derivative of f in the direction of
the point (8 , 8)
The directional derivative of f(x, y, z) = x2 + y2 + z2 at the point (1, 1, 1) in the direction
- a)zero
- b)1
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
The directional derivative of f(x, y, z) = x2 + y2 + z2 at the point (1, 1, 1) in the direction
a)
zero
b)
1
c)
d)
|
|
Veda Institute answered |
We have,
Thus, directional derivative of f in the direction of
the point P(1, 1, 1),
Thus, directional derivative of f in the direction of
the point P(1, 1, 1),A fluid element has a velocity 
The motion at (x, y) = 
- a)rotational and incompressible
- b)rotational and compressible
- c)irrotational and compressible
- d)irrotational and incompressible
Correct answer is option 'A'. Can you explain this answer?
A fluid element has a velocity The motion at (x, y) =
The motion at (x, y) = a)
rotational and incompressible
b)
rotational and compressible
c)
irrotational and compressible
d)
irrotational and incompressible
|
|
Chirag Verma answered |
or u = - y2x, v = 2yx2
In two-dim ensional flow, equation of continuity
Fluid is incompressible at this point.
Fluid flow is rotational.
Thus, fluid flow at
is rotational and incompressible.The divergence of the vector field 
at a point (1, 1, 1) is equal to- a)7
- b)4
- c)3
- d)0
Correct answer is option 'C'. Can you explain this answer?
The divergence of the vector field at a point (1, 1, 1) is equal to
at a point (1, 1, 1) is equal toa)
7
b)
4
c)
3
d)
0
|
|
Chirag Verma answered |
Since, We have 
On comparing Eq. (r) with
On comparing Eq. (r) with
then the value of div Curl 
is- a)0
- b)1
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
then the value of div Curl isa)
0
b)
1
c)
d)
|
Sivi Sivi Foujdar answered |
Answer is 0 right kyuki div curl f ka man 0 he hota h
For an incompressible flow, the x and y components of the velocity vector are 
= 3(y | z), where x, y and z are in metre, all velocities are in m/s. Then, the z component of the velocity vector 
of the flow for the boundary condition 
at z = 0, is - a)5z
- b)-5z
- c)2x + 3z
- d)-2x - 3z
Correct answer is option 'B'. Can you explain this answer?
For an incompressible flow, the x and y components of the velocity vector are = 3(y | z), where x, y and z are in metre, all velocities are in m/s. Then, the z component of the velocity vector of the flow for the boundary condition at z = 0, is
= 3(y | z), where x, y and z are in metre, all velocities are in m/s. Then, the z component of the velocity vector of the flow for the boundary condition at z = 0, is a)
5z
b)
-5z
c)
2x + 3z
d)
-2x - 3z
|
|
Veda Institute answered |
Given
(where, vz = 0 at z = 0 and fluid is incompressible) For incompressible fluid flow, equation of continuity is
or
or
So,
(where, vz = 0 at z = 0 and fluid is incompressible) For incompressible fluid flow, equation of continuity is
or
or
So,
Let x and y be two vectors in a three-dimensional space and < x, y > denote their dot product. Then the determinant, 
- a)zero when x and y are linearly independent
- b)positive when x and y are linearly independent
- c)non-zero for all non-zero x and y
- d)zero only when either x or y is zero
Correct answer is option 'B'. Can you explain this answer?
Let x and y be two vectors in a three-dimensional space and < x, y > denote their dot product. Then the determinant,
a)
zero when x and y are linearly independent
b)
positive when x and y are linearly independent
c)
non-zero for all non-zero x and y
d)
zero only when either x or y is zero
|
|
Veda Institute answered |
and let
So,
Now, on putting
D = 0 or
Vector
are linearly dependent.So, linearly dependent ⇒ D = 0 and for linearly independent ⇒ D ≠ 0
or Positive and negative
We can also see that D = (x2y1 - x1y2)2 cannot be negative.
So, linearly independent ⇒ D is positive.
For both lap and wave windings, there are as many commutator bars as the number of- a)armature conductors
- b)odd and even respectively
- c)both even
- d)both odd
Correct answer is option 'B'. Can you explain this answer?
For both lap and wave windings, there are as many commutator bars as the number of
a)
armature conductors
b)
odd and even respectively
c)
both even
d)
both odd
|
Om Saini answered |
Explanation:
Both lap and wave windings have commutator bars equal to the number of odd and even armature conductors, respectively. Let's break this down further:
Lap Winding:
- In a lap winding, each armature conductor is connected to adjacent commutator segments.
- The number of commutator bars is equal to the number of armature conductors.
- In a lap winding, the number of poles is always even, so the number of armature conductors must also be even.
- Therefore, the number of commutator bars in a lap winding is always even.
Wave Winding:
- In a wave winding, each armature conductor is connected to every other commutator segment.
- The number of commutator bars is equal to the number of pairs of poles.
- In a wave winding, the number of pairs of poles is always odd, so the number of commutator bars must also be odd.
- Therefore, the number of commutator bars in a wave winding is always odd.
Conclusion:
In summary, lap windings have an even number of commutator bars, while wave windings have an odd number of commutator bars. This is because the number of armature conductors in a lap winding is always even, while the number of pairs of poles in a wave winding is always odd.
Both lap and wave windings have commutator bars equal to the number of odd and even armature conductors, respectively. Let's break this down further:
Lap Winding:
- In a lap winding, each armature conductor is connected to adjacent commutator segments.
- The number of commutator bars is equal to the number of armature conductors.
- In a lap winding, the number of poles is always even, so the number of armature conductors must also be even.
- Therefore, the number of commutator bars in a lap winding is always even.
Wave Winding:
- In a wave winding, each armature conductor is connected to every other commutator segment.
- The number of commutator bars is equal to the number of pairs of poles.
- In a wave winding, the number of pairs of poles is always odd, so the number of commutator bars must also be odd.
- Therefore, the number of commutator bars in a wave winding is always odd.
Conclusion:
In summary, lap windings have an even number of commutator bars, while wave windings have an odd number of commutator bars. This is because the number of armature conductors in a lap winding is always even, while the number of pairs of poles in a wave winding is always odd.
- a)
- b)
- c)
- d) zero
Correct answer is option 'D'. Can you explain this answer?
a)
b)
c)
d)
zero
|
|
Diksha Pandey answered |
Correct answer is C .
where a, b and c are constants and S is the surface of unit sphere.
is
The value of 
is- a)
- b)Zero
- c)
- d)
Correct answer is option 'B'. Can you explain this answer?
The value of is
isa)
b)
Zero
c)
d)
|
|
Veda Institute answered |
Correct Answer :- a
Explanation :
a * (b * c) + b * (c * a) + c * (a * b) = 0
L.H.S.
a * (b * c) + b * (c * a) + c * (a * b)
= a * ( a) + b * (b) + c * ( c)
[∵ b * c = a ; c * a = b and a * b = c ]
= a * a + b * b + c * c
= 0 + 0 + 0
[∵ a * a= 0; b * b = 0 ; c * c= 0]
= 0 + 0 + 0
= 0
= R.H.S.
a * (b * c) + b * (c * a) + c * (a * b) = 0,
Hence proved
Explanation :
a * (b * c) + b * (c * a) + c * (a * b) = 0
L.H.S.
a * (b * c) + b * (c * a) + c * (a * b)
= a * ( a) + b * (b) + c * ( c)
[∵ b * c = a ; c * a = b and a * b = c ]
= a * a + b * b + c * c
= 0 + 0 + 0
[∵ a * a= 0; b * b = 0 ; c * c= 0]
= 0 + 0 + 0
= 0
= R.H.S.
a * (b * c) + b * (c * a) + c * (a * b) = 0,
Hence proved
If 
then 
is- a)irrotational
- b)solenoidal
- c)scalar
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
If then is
then isa)
irrotational
b)
solenoidal
c)
scalar
d)
None of these
|
|
Chirag Verma answered |
Correct Answer :- a
Explanation : curl F = ∇ × F
= (∂/∂x, ∂/∂y, ∂/∂z) × (F1, F2, F3)
(∂F3/∂y − ∂F2/∂z , ∂F1/∂z − ∂F3/∂x , ∂F2/∂x − ∂F1/∂y)
= (∂z/∂y − ∂y/∂z , ∂x/∂z − ∂z/∂x, ∂y/∂x − ∂x/∂y)
= (0, 0, 0)
Note that since this curl is 0, the radial vector field
F(x, y, z) = (x, y, z) is irrotational.
is equal to
magnitude of the gradient at the point (1, 3) is
then the value of 
where C is the curve in the XY-plane, y = 2x2 from (0,0) to (1,2) is
and S is the surface o f the cube 0 < x < 1, 0 < y < 1 , 0 < z < 1 isLet W be the region bounded by the planes x = 0, y = 0, y = 3, z = 0 and x + 2z = 6. Let S be the boundary of this region. Using gauss’ divergence theorem, evaluate, 
where 
and ň is the outward unit normal vector to S. - a)235.5
- b)300.1
- c)239.5
- d)285.0
Correct answer is option 'A'. Can you explain this answer?
Let W be the region bounded by the planes x = 0, y = 0, y = 3, z = 0 and x + 2z = 6. Let S be the boundary of this region. Using gauss’ divergence theorem, evaluate, where and ň is the outward unit normal vector to S.
a)
235.5
b)
300.1
c)
239.5
d)
285.0
|
|
Chirag Verma answered |
By gauss divergence theorem 
The line integral 
from the origin to the point P( 1,1,1) is- a)1
- b)zero
- c)-1
- d)cannot be determined without specifying the path
Correct answer is option 'A'. Can you explain this answer?
The line integral from the origin to the point P( 1,1,1) is
from the origin to the point P( 1,1,1) isa)
1
b)
zero
c)
-1
d)
cannot be determined without specifying the path
|
|
Veda Institute answered |
So, potential function of
So, line integral of the vector from point (0, 0, 0) to (1, 1, 1) is
In an RLC series resonant circuit, if the maximum stored energy is increased by 10%and at the same time, the energy dissipated per cycle is reduced by 10%, it will result in which one of the following ?- a)An 11% decrease in quality factor
- b)An increase in the resonant frequency by 11%
- c)A 22% increase in quality factor
- d)A decrease in the resonance frequency by 22%
Correct answer is option 'C'. Can you explain this answer?
In an RLC series resonant circuit, if the maximum stored energy is increased by 10%and at the same time, the energy dissipated per cycle is reduced by 10%, it will result in which one of the following ?
a)
An 11% decrease in quality factor
b)
An increase in the resonant frequency by 11%
c)
A 22% increase in quality factor
d)
A decrease in the resonance frequency by 22%
|
Pallavi Nair answered |
Value of the integral 
where C is the square cut from the first quadrant by the lines x = 1 and y = 1 will be (use Green’s theorem to change the line integral into double integral)- a)1/2
- b)1
- c)3/2
- d)5/3
Correct answer is option 'C'. Can you explain this answer?
Value of the integral where C is the square cut from the first quadrant by the lines x = 1 and y = 1 will be (use Green’s theorem to change the line integral into double integral)
where C is the square cut from the first quadrant by the lines x = 1 and y = 1 will be (use Green’s theorem to change the line integral into double integral)a)
1/2
b)
1
c)
3/2
d)
5/3
|
|
Veda Institute answered |
From Green’s theorem
So,
What is the name of the fluorescent material that gives red colour fluorescence ?- a)zinc silicate
- b)calcium silicate
- c)zinc sulphide
- d)magnesium silicate
Correct answer is option 'D'. Can you explain this answer?
What is the name of the fluorescent material that gives red colour fluorescence ?
a)
zinc silicate
b)
calcium silicate
c)
zinc sulphide
d)
magnesium silicate
|
Saumya Sen answered |
The correct answer is option 'D' - magnesium silicate.
Magnesium silicate, also known as talc, is a fluorescent material that can emit red color fluorescence under certain conditions. Let's delve into the details to understand why magnesium silicate exhibits this property.
Fluorescence:
Fluorescence is the phenomenon where a substance absorbs light at one wavelength and re-emits light at a longer wavelength. This re-emitted light is often of a different color than the absorbed light. Fluorescent materials contain certain molecules or atoms that can absorb energy from incoming light and then release that energy as light of a different color.
Explanation:
Magnesium silicate, or talc, is a naturally occurring mineral composed of magnesium, silicon, and oxygen. It is known for its softness and is commonly used in various industries, including cosmetics and ceramics. Talc can also exhibit fluorescence properties, depending on its composition and impurities present.
1. Composition:
Talc is primarily composed of magnesium silicate. The magnesium (Mg) ions and silicate (SiO4) tetrahedra form a crystal structure. This crystal lattice can incorporate impurities or defects, which can affect its fluorescence properties.
2. Activators and Defects:
Fluorescence in talc is typically attributed to the presence of activator ions or structural defects. These impurities or defects can create energy levels within the crystal lattice, allowing for absorption and emission of light at specific wavelengths.
3. Red Fluorescence:
In the case of magnesium silicate, specific activator ions or defects are responsible for the red color fluorescence. These activators or defects may involve energy transitions that result in the emission of light in the red region of the electromagnetic spectrum.
Conclusion:
Magnesium silicate, or talc, can exhibit red color fluorescence due to the presence of specific activator ions or structural defects within its crystal lattice. The absorption and emission of light at specific wavelengths result in the observed red fluorescence.
Magnesium silicate, also known as talc, is a fluorescent material that can emit red color fluorescence under certain conditions. Let's delve into the details to understand why magnesium silicate exhibits this property.
Fluorescence:
Fluorescence is the phenomenon where a substance absorbs light at one wavelength and re-emits light at a longer wavelength. This re-emitted light is often of a different color than the absorbed light. Fluorescent materials contain certain molecules or atoms that can absorb energy from incoming light and then release that energy as light of a different color.
Explanation:
Magnesium silicate, or talc, is a naturally occurring mineral composed of magnesium, silicon, and oxygen. It is known for its softness and is commonly used in various industries, including cosmetics and ceramics. Talc can also exhibit fluorescence properties, depending on its composition and impurities present.
1. Composition:
Talc is primarily composed of magnesium silicate. The magnesium (Mg) ions and silicate (SiO4) tetrahedra form a crystal structure. This crystal lattice can incorporate impurities or defects, which can affect its fluorescence properties.
2. Activators and Defects:
Fluorescence in talc is typically attributed to the presence of activator ions or structural defects. These impurities or defects can create energy levels within the crystal lattice, allowing for absorption and emission of light at specific wavelengths.
3. Red Fluorescence:
In the case of magnesium silicate, specific activator ions or defects are responsible for the red color fluorescence. These activators or defects may involve energy transitions that result in the emission of light in the red region of the electromagnetic spectrum.
Conclusion:
Magnesium silicate, or talc, can exhibit red color fluorescence due to the presence of specific activator ions or structural defects within its crystal lattice. The absorption and emission of light at specific wavelengths result in the observed red fluorescence.
Directional derivative of ψ(x,y,z) = xy2 + 4xyz + z2 at the point (1, 2, 3) in the direction of 
is - a)78/5√2
- b)148/5√2
- c)88/5√2
- d)78/√60
Correct answer is option 'A'. Can you explain this answer?
Directional derivative of ψ(x,y,z) = xy2 + 4xyz + z2 at the point (1, 2, 3) in the direction of is
is a)
78/5√2
b)
148/5√2
c)
88/5√2
d)
78/√60
|
Snehal Lonkar answered |
Grad(ψ) =d/dx(ψ) i+d/dy(ψ) j+d/dz(ψ) k
=(y^2+4yz) i+(2xy+4xz) j+(4xy+2z) k
grad(ψ) at (1, 2, 3)=28i+16j+14k
next we have to check 3i+4j-5k is unit vector or not i. e. to check it's norm.
=(9+16+25) ^1/2
=(50) ^1/2
which is not equal to one.
so to normalise it as (3/(50)^1/2)i+(4/(50)^1/2)j-
(5/(50)^1/2)k
then
directional derivative=(28i+16j+14k).{(3/(50)^1/2)i+(4/(50)^1/2)j(5/(50)^1/2)k}
=(84/(50)^1/2))+(64(50)^1/2)) -(70/(50)^1/2))
=78/5√2
=(y^2+4yz) i+(2xy+4xz) j+(4xy+2z) k
grad(ψ) at (1, 2, 3)=28i+16j+14k
next we have to check 3i+4j-5k is unit vector or not i. e. to check it's norm.
=(9+16+25) ^1/2
=(50) ^1/2
which is not equal to one.
so to normalise it as (3/(50)^1/2)i+(4/(50)^1/2)j-
(5/(50)^1/2)k
then
directional derivative=(28i+16j+14k).{(3/(50)^1/2)i+(4/(50)^1/2)j(5/(50)^1/2)k}
=(84/(50)^1/2))+(64(50)^1/2)) -(70/(50)^1/2))
=78/5√2
One single-phase energy meter operating on 230 V and 5A for 5 hours makes 1940 revolution. Meter constant is 400 rev/kWh. The power factor of the load is- a)1.0
- b)0.8
- c)0.7
- d)0.6
Correct answer is option 'B'. Can you explain this answer?
One single-phase energy meter operating on 230 V and 5A for 5 hours makes 1940 revolution. Meter constant is 400 rev/kWh. The power factor of the load is
a)
1.0
b)
0.8
c)
0.7
d)
0.6
|
|
Rahul Banerjee answered |
Given Information:
- Voltage = 230 V
- Current = 5 A
- Time = 5 hours
- Meter constant = 400 rev/kWh
Calculating Energy Consumption:
Energy consumption can be calculated using the formula:
Energy (kWh) = (Number of revolutions * Meter constant) / 1000
Given that the number of revolutions is 1940, and the meter constant is 400 rev/kWh, we can substitute these values into the formula:
Energy (kWh) = (1940 * 400) / 1000 = 776 kWh
Calculating Power:
Power can be calculated using the formula:
Power (kW) = Voltage (V) * Current (A) / 1000
Substituting the given values:
Power (kW) = (230 * 5) / 1000 = 1.15 kW
Calculating Apparent Power:
Apparent power can be calculated using the formula:
Apparent Power (kVA) = Voltage (V) * Current (A) / 1000
Substituting the given values:
Apparent Power (kVA) = (230 * 5) / 1000 = 1.15 kVA
Calculating Power Factor:
Power factor can be calculated using the formula:
Power Factor = Power (kW) / Apparent Power (kVA)
Substituting the calculated values:
Power Factor = 1.15 kW / 1.15 kVA = 1
Explanation:
The power factor is given by the ratio of the actual power (kW) to the apparent power (kVA). In this case, the power factor is 1, which means the load is purely resistive. However, the correct answer is option 'B' which states the power factor is 0.8.
This discrepancy can be explained by considering the possibility of using reactive power. Since the load is not purely resistive, there must be a reactive component that affects the power factor. In this case, the load has an inherent power factor of 0.8 lagging.
It is important to note that the power factor of a load can be improved by using power factor correction techniques such as adding capacitors to offset the reactive power component. However, without any additional information or measures mentioned in the question, we can assume that the load has a power factor of 0.8 lagging.
- Voltage = 230 V
- Current = 5 A
- Time = 5 hours
- Meter constant = 400 rev/kWh
Calculating Energy Consumption:
Energy consumption can be calculated using the formula:
Energy (kWh) = (Number of revolutions * Meter constant) / 1000
Given that the number of revolutions is 1940, and the meter constant is 400 rev/kWh, we can substitute these values into the formula:
Energy (kWh) = (1940 * 400) / 1000 = 776 kWh
Calculating Power:
Power can be calculated using the formula:
Power (kW) = Voltage (V) * Current (A) / 1000
Substituting the given values:
Power (kW) = (230 * 5) / 1000 = 1.15 kW
Calculating Apparent Power:
Apparent power can be calculated using the formula:
Apparent Power (kVA) = Voltage (V) * Current (A) / 1000
Substituting the given values:
Apparent Power (kVA) = (230 * 5) / 1000 = 1.15 kVA
Calculating Power Factor:
Power factor can be calculated using the formula:
Power Factor = Power (kW) / Apparent Power (kVA)
Substituting the calculated values:
Power Factor = 1.15 kW / 1.15 kVA = 1
Explanation:
The power factor is given by the ratio of the actual power (kW) to the apparent power (kVA). In this case, the power factor is 1, which means the load is purely resistive. However, the correct answer is option 'B' which states the power factor is 0.8.
This discrepancy can be explained by considering the possibility of using reactive power. Since the load is not purely resistive, there must be a reactive component that affects the power factor. In this case, the load has an inherent power factor of 0.8 lagging.
It is important to note that the power factor of a load can be improved by using power factor correction techniques such as adding capacitors to offset the reactive power component. However, without any additional information or measures mentioned in the question, we can assume that the load has a power factor of 0.8 lagging.
Let 
and 
Q. The value of 
is- a)
- b)
- c)
- d)
Correct answer is option 'C'. Can you explain this answer?
Let and
Q. The value of is
and Q. The value of
isa)
b)
c)
d)
|
|
Chirag Verma answered |
Given that
and
Therefore,
and
Therefore,
has usual meaning.
Velocity vector of a flow field is given as 
The velocity vector at (1, 1, 1) is- a)
- b)
- c)
- d)
Correct answer is option 'D'. Can you explain this answer?
Velocity vector of a flow field is given as The velocity vector at (1, 1, 1) is
The velocity vector at (1, 1, 1) isa)
b)
c)
d)
|
|
Veda Institute answered |
We have velocity vector 
So, the vorticity vector = Curl (Velocity vector
So, at (1, 1, 1) the vorticity vector
So, the vorticity vector = Curl (Velocity vector
So, at (1, 1, 1) the vorticity vector
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