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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
is
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
where, S is any closed surface enclosing volume V.
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
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 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
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 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
- 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 .
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
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
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
Unit vectors in X and Z-directions are 
respectively. Which one of the following is the directional derivative of the function F(x, z) = In (x2 + z2) at the point P(4, 0), in the direction of 
- a)
- b)
- c)1
- d)
Correct answer is option 'D'. Can you explain this answer?
Unit vectors in X and Z-directions are respectively. Which one of the following is the directional derivative of the function F(x, z) = In (x2 + z2) at the point P(4, 0), in the direction of
respectively. Which one of the following is the directional derivative of the function F(x, z) = In (x2 + z2) at the point P(4, 0), in the direction of a)
b)
c)
1
d)
|
|
Chirag Verma answered |
Given, F(x, y) = In(x2 + z2) = log (y + z2)
Coordinate of point p is (4, 0).
or
Coordinate of point p is (4, 0).
or
If 
and curve C is the arc of the curve y = x3 from (0, 0 ) to (2, 8), then the value of 
- a)824
- b)824/21
- c)21/824
- d)0
Correct answer is option 'B'. Can you explain this answer?
If and curve C is the arc of the curve y = x3 from (0, 0 ) to (2, 8), then the value of
and curve C is the arc of the curve y = x3 from (0, 0 ) to (2, 8), then the value of a)
824
b)
824/21
c)
21/824
d)
0
|
|
Chirag Verma answered |
Since, C is the curve y = x3 from (0, 0) to (2, 8)
So, let x = t ⇒ y = t3
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, let x = t ⇒ y = t3
If
is the position vector of any point on C, thenor
or
At (0, 0) ⇒ t = x = 0 and at (2, 8) ⇒ t = 2
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 |
We know that in Green’s theorem,
On comparing, we get M = -y2 and N = xy
On comparing, we get M = -y2 and N = xy
, then the value of div curl 
is- a) 2x + 2y + 2z
- b)zero
- c)6
- d)
Correct answer is option 'B'. Can you explain this answer?
, then the value of div curl isa)
2x + 2y + 2z
b)
zero
c)
6
d)
|
Pooja Choudhury answered |
Since div(curl⇀v)=0, the net rate of flow in vector field curl⇀v\) at any point is zero. Taking the curl of vector field ⇀F eliminates whatever divergence was present in ⇀F
Apply Green’ s theorem the value of 
where C is the square formed by the lines y = ±1, x = ±1 is- a)1
- b)r2
- c) zero
- d)2
Correct answer is option 'C'. Can you explain this answer?
Apply Green’ s theorem the value of where C is the square formed by the lines y = ±1, x = ±1 is
where C is the square formed by the lines y = ±1, x = ±1 isa)
1
b)
r2
c)
zero
d)
2
|
|
Chirag Verma answered |
Since, On comparing 
with 
we get M = x2 + xy and
N = x2 + y2
So, from Green’s theorem
with we get M = x2 + xy and N = x2 + y2
So, from Green’s theorem
then the value of div 
at the point (1, 1, -1) will be
is equal to
Use Gauss’s divergence theorem to find 
where 
and S is the closed surface in the first octant bounded by y2 + z2 = 9 and x = 2.- a)108
- b)810
- c)-180
- d)180
Correct answer is option 'D'. Can you explain this answer?
Use Gauss’s divergence theorem to find where and S is the closed surface in the first octant bounded by y2 + z2 = 9 and x = 2.
where and S is the closed surface in the first octant bounded by y2 + z2 = 9 and x = 2.a)
108
b)
810
c)
-180
d)
180
|
|
Chirag Verma answered |
Let V be the volume enclosed by the closed surface S, i.e., the volume in the first octant bounded by the cylinder y2 I z2 = 9 and the planes x = 0, x = 2. Then by Gauss’s divergence theorem, we have
is the reciprocal system to the vectors 
then the value of 
is
R is a closed planar region as shown by the shaded area in the figure below. Its boundary C consists of the circles C1 and C2.
If 
are all continuous everywhere in R, Green’s theorem states that
Which one of the following alternatives correctly depicts the direction of integration along C?- a)
- b)
- c)
- d)
Correct answer is option 'C'. Can you explain this answer?
R is a closed planar region as shown by the shaded area in the figure below. Its boundary C consists of the circles C1 and C2.
If are all continuous everywhere in R, Green’s theorem states that
Which one of the following alternatives correctly depicts the direction of integration along C?
If
are all continuous everywhere in R, Green’s theorem states thatWhich one of the following alternatives correctly depicts the direction of integration along C?
a)
b)
c)
d)
|
|
Veda Institute answered |
The region R is bounded by two closed circles C1 and C2, so it is doubly connected. To apply it in Green’s theorem, we need to convert it into simply connected region. For it, we apply cut AD and consider the region R having simple closed curve ABCADEFDA in the anticlockwise direction. So, the directions shown in figure, (c) is correct option.
The value of 
by Stoke’s theorem, where 
and C is the boundary of the triangle with vertices at ( 0 ,0 , 0 ) , ( 1 , 0 , 0 ) and ( 1 ,1 , 0 ) is- a)1/2
- b)1/3
- c)1/4
- d)1/5
Correct answer is option 'B'. Can you explain this answer?
The value of by Stoke’s theorem, where and C is the boundary of the triangle with vertices at ( 0 ,0 , 0 ) , ( 1 , 0 , 0 ) and ( 1 ,1 , 0 ) is
by Stoke’s theorem, where and C is the boundary of the triangle with vertices at ( 0 ,0 , 0 ) , ( 1 , 0 , 0 ) and ( 1 ,1 , 0 ) isa)
1/2
b)
1/3
c)
1/4
d)
1/5
|
|
Chirag Verma answered |
We have, curl
Also we note that z coordinate of each vertex of the triangle is zero.
or The triangle lies in the xy-plane. So,
So, curl
In the figure, we have only considered the xy-plane. So, by Stoke’s Theorem
The unit normal vector to the surface of the sphere x2 + y2 + z2 = 1 at the point 
and 
are unit normal vectors in the Cartesian coordinate system)- a)
- b)
- c)
- d)
Correct answer is option 'B'. Can you explain this answer?
The unit normal vector to the surface of the sphere x2 + y2 + z2 = 1 at the point and are unit normal vectors in the Cartesian coordinate system)
and are unit normal vectors in the Cartesian coordinate system)a)
b)
c)
d)
|
|
Chirag Verma answered |
Problem is to find unit normal vector to the surface of the sphere
f = x2 + y2 + z2 - 1 = 0
We know unit normal vector
is given by
Apply Stoke’s theorem, the value of 
where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0) and (0, 0, 6) is- a)12
- b)21
- c)11
- d)0
Correct answer is option 'B'. Can you explain this answer?
Apply Stoke’s theorem, the value of where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0) and (0, 0, 6) is
where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0) and (0, 0, 6) isa)
12
b)
21
c)
11
d)
0
|
|
Veda Institute answered |
Taking projection on three planes, we note that the surface S consists of three triangles, Δ OAB in XT- plane, Δ OBC in TZ-plane and Δ OAC in XZ-plane. Using two point formula, the equation of the line AB, BC, CA are respectively 3x + 2y = 6 , 2y + z = 6 , 3x + z = 6
So, by Stake’s theorem
So, by Stake’s theorem
The vector field 
are unit vectors) is- a)divergence-free but not irrotational
- b) irrotational but not divergence-free
- c)divergence-free and irrotational
- d)neither divergence-free nor irrotational
Correct answer is option 'C'. Can you explain this answer?
The vector field are unit vectors) is
are unit vectors) isa)
divergence-free but not irrotational
b)
irrotational but not divergence-free
c)
divergence-free and irrotational
d)
neither divergence-free nor irrotational
|
|
Chirag Verma answered |
Given, 
Now, for divergence
Hence, vector field is divergence-free.
Now, for irrotational
Now, for divergence
Hence, vector field is divergence-free.
Now, for irrotational
over the path shown in the figure is 
If 
are to arbitrary vectors with magnitudes a and b respectively, 
will be equal to- a)
- b)
- c)
- d)
Correct answer is option 'A'. Can you explain this answer?
If are to arbitrary vectors with magnitudes a and b respectively, will be equal to
are to arbitrary vectors with magnitudes a and b respectively, will be equal toa)
b)
c)
d)
|
|
Veda Institute answered |
Cross checking from option (a),
which is correct answer.
which is correct answer.
Consider the vector field 
, where a is a constant. If 
, then the value of a is- a)1
- b)-1
- c)0
- d)3/2
Correct answer is option 'A'. Can you explain this answer?
Consider the vector field , where a is a constant. If , then the value of a is
a)
1
b)
-1
c)
0
d)
3/2
|
|
Veda Institute answered |
⇒ – (ax + y + a) + 1 + x + y = 0
⇒ (– a + 1)x + (– a + 1) = 0
⇒ (– a + 1) (1 + x) = 0
⇒ a = 1 (because x is not constant).
The value of α for which the following three vectors are coplanar is 
- a)4
- b)zero
- c)- 2
- d)-10
Correct answer is option 'C'. Can you explain this answer?
The value of α for which the following three vectors are coplanar is
a)
4
b)
zero
c)
- 2
d)
-10
|
|
Veda Institute answered |
Given,
Vectors
are coplanar, if
Vectors
are coplanar, if
Apply Green’s theorem the value of 
where C is the boundary of the area enclosed by the X-axis and the upper half of the circle x2 + y2 = a2 is- a)
- b)
- c)
- d)1/3
Correct answer is option 'B'. Can you explain this answer?
Apply Green’s theorem the value of where C is the boundary of the area enclosed by the X-axis and the upper half of the circle x2 + y2 = a2 is
where C is the boundary of the area enclosed by the X-axis and the upper half of the circle x2 + y2 = a2 isa)
b)
c)
d)
1/3
|
|
Veda Institute answered |
Since, From Green’s theorem, we have
(A is the region of the figure)
On changing in polar coordinates, we get
Let T (x, y, z) = xy2 + 2z – x2z2 be the temperature at the point (x, y, z). The unit vector in the direction in which the temperature decrease most rapidly at (1, 0, – 1) is- a)
- b)
- c)
- d)
Correct answer is option 'D'. Can you explain this answer?
Let T (x, y, z) = xy2 + 2z – x2z2 be the temperature at the point (x, y, z). The unit vector in the direction in which the temperature decrease most rapidly at (1, 0, – 1) is
a)
b)
c)
d)
|
|
Chirag Verma answered |
Let T(x, y), z) = xy2 + 2z – x2z2 be the temperature at a point (x, y, z). Temperature increase most rapidly in the direction of gradient i.e., ∇ T
So, temperature decreases most rapidly in the direction of
.
So, temperature decreases most rapidly in the direction of
The value of 
- a)zero
- b)1
- c)-1
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
The value of
a)
zero
b)
1
c)
-1
d)
None of these
|
Puneet Sharma answered |
Change this in determinant form you will get two rows are equivalent...
when two rows are equivalent in a determinant then it's value is zero..
when two rows are equivalent in a determinant then it's value is zero..
is the unit outward drawn normal to any dosed surface S, then 
Using the Stoke’s theorem, evaluate 
[(x + 2y) dx + (x-2) dy+ (y - z)dz], where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0) and (0, 0, 6) oriented in the anti-clockwise direction.- a)12
- b)15
- c)9
- d)zero
Correct answer is option 'B'. Can you explain this answer?
Using the Stoke’s theorem, evaluate [(x + 2y) dx + (x-2) dy+ (y - z)dz], where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0) and (0, 0, 6) oriented in the anti-clockwise direction.
[(x + 2y) dx + (x-2) dy+ (y - z)dz], where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0) and (0, 0, 6) oriented in the anti-clockwise direction.a)
12
b)
15
c)
9
d)
zero
|
Umang Kumawat answered |
A
Potential function φ is given as φ = x2 - y2. What will be the stream function (ψ) with the condition ψ = 0 at x = y = 0?- a)2xy
- b)x2 + y2
- c)x2 - y2
- d)2x2y2
Correct answer is option 'A'. Can you explain this answer?
Potential function φ is given as φ = x2 - y2. What will be the stream function (ψ) with the condition ψ = 0 at x = y = 0?
a)
2xy
b)
x2 + y2
c)
x2 - y2
d)
2x2y2
|
Aarav Singh answered |
A potential function, also known as a scalar potential or simply a potential, is a function that describes the energy of a system in terms of its configuration. It is commonly used in physics to analyze conservative force fields, such as gravitational or electrostatic fields.
In the context of classical mechanics, a potential function is defined as a scalar function of the position coordinates of a particle or a system of particles. The gradient of the potential function gives the force acting on the particle(s) at any given point in space. This allows for a convenient mathematical representation of the force field, simplifying calculations and analysis.
In electromagnetism, the electrostatic potential is a potential function that describes the electric field in terms of the distribution of charges in space. Similarly, the gravitational potential is a potential function that describes the gravitational field in terms of the mass distribution in space.
Potential functions have many applications in various fields of science and engineering, such as in fluid dynamics, quantum mechanics, and thermodynamics. They provide a powerful tool for analyzing and understanding the behavior of physical systems in terms of their energy.
In the context of classical mechanics, a potential function is defined as a scalar function of the position coordinates of a particle or a system of particles. The gradient of the potential function gives the force acting on the particle(s) at any given point in space. This allows for a convenient mathematical representation of the force field, simplifying calculations and analysis.
In electromagnetism, the electrostatic potential is a potential function that describes the electric field in terms of the distribution of charges in space. Similarly, the gravitational potential is a potential function that describes the gravitational field in terms of the mass distribution in space.
Potential functions have many applications in various fields of science and engineering, such as in fluid dynamics, quantum mechanics, and thermodynamics. They provide a powerful tool for analyzing and understanding the behavior of physical systems in terms of their energy.
Chapter doubts & questions for Vector Calculus - Mathematics for Competitive Exams 2025 is part of Mathematics exam preparation. The chapters have been prepared according to the Mathematics exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for Mathematics 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.
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Mathematics for Competitive Exams
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