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The vector function expressed byF = ax(5y − k1z) + ay(3z + k2x) + az(k3y−4x)Represents a conservative field, where ax, ay, az are unit vectors along x, y and z directions, respectively. The values of constant k1, k2, k3 are given by: - a)k1 = 3, k2 = 3, k3 = 7
- b)k1 = 3, k2 = 8, k3 = 5
- c)k1 = 4, k2 = 5, k3 = 3
- d)k1 = 0, k2 = 0, k3 = 0
Correct answer is option 'C'. Can you explain this answer?
The vector function expressed by
F = ax(5y − k1z) + ay(3z + k2x) + az(k3y−4x)
Represents a conservative field, where ax, ay, az are unit vectors along x, y and z directions, respectively. The values of constant k1, k2, k3 are given by:
a)
k1 = 3, k2 = 3, k3 = 7
b)
k1 = 3, k2 = 8, k3 = 5
c)
k1 = 4, k2 = 5, k3 = 3
d)
k1 = 0, k2 = 0, k3 = 0
|
Sanchita Pillai answered |
It seems like the expression is incomplete. Please provide the complete expression for the vector function F.
If 2î + 4ĵ - 5k̂ and î + 2ĵ + 3k̂ are two different sides of rhombus. Find the length of diagonals.- a)7, √69
- b)6, √59
- c)5 √65
- d)8, √45
Correct answer is option 'A'. Can you explain this answer?
If 2î + 4ĵ - 5k̂ and î + 2ĵ + 3k̂ are two different sides of rhombus. Find the length of diagonals.
a)
7, √69
b)
6, √59
c)
5 √65
d)
8, √45
|
Sanvi Kapoor answered |
If aî + bĵ + ck̂ and pî + qĵ + rk̂ are 2 different sides of the rhombus.
Suppose 
= aî+bĵ+ck̂ and B = 
= pî+qĵ+rk̂
= aî+bĵ+ck̂ and B = = pî+qĵ+rk̂Then anyone diagonal of the rhombus is given by 
The other diagonal is given by 
The magnitude of the vector 
The magnitude of the vector diagonal D1
D1 = 7
The other diagonal is given by 
D2 = - 1î - 2ĵ + 8k̂
The magnitude of the vector diagonal D2
∴ The length of diagonals of a rhombus is 7 and √69.
The volume determined from ∫∫∫v 8 xyz dv for V = [2, 3] × [1, 2] × [ 0,1 ] will be (in integer) ________.
Correct answer is '15'. Can you explain this answer?
The volume determined from ∫∫∫v 8 xyz dv for V = [2, 3] × [1, 2] × [ 0,1 ] will be (in integer) ________.
|
|
Sanvi Kapoor answered |
Given
Integral
∫∫∫v 8 xyz dv
Limits for x, y and z is given as
[2, 3] × [1, 2] × [0, 1]
Volume of the integral
V = ∫∫∫v 8 xyz dv
i.e. V = ∫ ∫ ∫V 8 xyz dxdydz
V = 5 × 3 × 1
V = 15
∴ Volume is 15
Given the vector A = (cos x)(cos y)âx + (cos x)(cos y) ây, âx & ây denote unit vectors along x,y directions respectively. The curl of A is ________.- a)(cosx cosy - sinx siny) ) âx
- b)(- sinx cosy + cosx siny) âz
- c)(sinx cosy - cosx siny) âz
- d)(- sinx cosy + cosx siny) ây
Correct answer is option 'B'. Can you explain this answer?
Given the vector A = (cos x)(cos y)âx + (cos x)(cos y) ây, âx & ây denote unit vectors along x,y directions respectively. The curl of A is ________.
a)
(cosx cosy - sinx siny) ) âx
b)
(- sinx cosy + cosx siny) âz
c)
(sinx cosy - cosx siny) âz
d)
(- sinx cosy + cosx siny) ây
|
|
Sanchita Pillai answered |
The given vector is A = (cos x)(cos y)ax + (cos x)(cos y)ay.
To find the components of the vector A, we can separate the terms involving ax and ay:
A = (cos x)(cos y)ax + (cos x)(cos y)ay
= (cos x)(cos y)(ax + ay)
Therefore, the x-component of vector A is (cos x)(cos y) and the y-component of vector A is also (cos x)(cos y).
To find the components of the vector A, we can separate the terms involving ax and ay:
A = (cos x)(cos y)ax + (cos x)(cos y)ay
= (cos x)(cos y)(ax + ay)
Therefore, the x-component of vector A is (cos x)(cos y) and the y-component of vector A is also (cos x)(cos y).
Green's theorem is used to-- a)transform the line integral in xy - plane to a surface integral on the same xy - plane.
- b)transform double integrals into triple integral in a region v.
- c)transform surface integral into line integral.
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
Green's theorem is used to-
a)
transform the line integral in xy - plane to a surface integral on the same xy - plane.
b)
transform double integrals into triple integral in a region v.
c)
transform surface integral into line integral.
d)
None of these
|
Engineers Adda answered |
Green's theorem
- It converts the line integral to a double integral.
- It transforms the line integral in xy - plane to a surface integral on the same xy - plane.
If M and N are functions of (x, y) defined in an open region then from Green's theorem
The value of the line integral
along a path joining the origin and the point (1,1,1) is- a)0
- b)2
- c)4
- d)6
Correct answer is option 'B'. Can you explain this answer?
The value of the line integral
along a path joining the origin and the point (1,1,1) is
a)
0
b)
2
c)
4
d)
6
|
|
Sanvi Kapoor answered |
Concept:
When two points (x1, y1. z1) and (x1, y1. z2) are mentioned find the relation in terms of the third variable in terms of x,y, and z:
Put the value of z,y, and z and use the end-points of one variable.
Given:
I = ∫(2xy2dx + 2x2ydy + dz), A (0, 0, 0) and B(1, 1, 1).
Equation of line i.e. path
∴ x = y = z = t and t : 0 → 1
∴ 
= 
∴ 
The following surface integral is to be evaluated over a sphere for the given steady vector field, F = xi + yj + zk defined with respect to a Cartesian coordinate system having i, j, and k as unit base vectors.
, Where S is the sphere, x2 + y2 + z2 = 1 and n is the outward unit normal vector to the sphere. The value of the surface integral is- a)π
- b)2π
- c)
- d)4π
Correct answer is option 'A'. Can you explain this answer?
The following surface integral is to be evaluated over a sphere for the given steady vector field, F = xi + yj + zk defined with respect to a Cartesian coordinate system having i, j, and k as unit base vectors.
, Where S is the sphere, x2 + y2 + z2 = 1 and n is the outward unit normal vector to the sphere. The value of the surface integral isa)
π
b)
2π
c)
d)
4π
|
|
Sanya Agarwal answered |
Gauss divergence theorem:
It states that the surface integral of the normal component of a vector function 
taken over a closed surface ‘S’ is equal to the volume integral of the divergence of that vector function 
taken over a volume enclosed by the closed surface ‘S’.
taken over a closed surface ‘S’ is equal to the volume integral of the divergence of that vector function taken over a volume enclosed by the closed surface ‘S’.
Given:
F = xi + yj + zk
The parabolic arc y = √x, 1 ≤ x ≤ 2 is revolved around the x-axis. The volume of the solid of revolution is- a)π/4
- b)π/2
- c)3π/4
- d)3π/2
Correct answer is option 'D'. Can you explain this answer?
The parabolic arc y = √x, 1 ≤ x ≤ 2 is revolved around the x-axis. The volume of the solid of revolution is
a)
π/4
b)
π/2
c)
3π/4
d)
3π/2
|
Partho Choudhary answered |
X^2 is a U-shaped curve that opens upwards. The vertex of the parabola is at the origin (0,0) and the axis of symmetry is the y-axis. The graph of the parabola passes through the point (1,1), (2,4), (-1,1), and (-2,4) as shown below.
For the curve xy3 - yx3 = 6, the slope of the tangent line at the point (1, -1) is:
- a)-1
- b)1/2
- c)2
- d)1
Correct answer is option 'A'. Can you explain this answer?
For the curve xy3 - yx3 = 6, the slope of the tangent line at the point (1, -1) is:
a)
-1
b)
1/2
c)
2
d)
1
|
|
Sanya Agarwal answered |
Concept:
Let y = f(x) be the equation of a curve, then the slope of the tangent at any point say (x1, y1) is given by:
Calculation:
Given curve is xy3 - yx3 = 6
Now by partially differentiating the equation of curve with respect to x we get;
The slope(m) i.e. dy/dx of the tangent at (1, -1) is:
m = -1
The value of 
from A(a, 0, 0) to B(a, 0, 2π b) along r̅ = (a cos t) î + (a sin t) ĵ + b t k̂ is:- a)πa (bĵ + ak̂)
- b)2πa (aĵ + bk̂ + abî)
- c)2πa (aĵ + bk̂)
- d)2πa (bĵ + ak̂)
Correct answer is option 'A'. Can you explain this answer?
The value of from A(a, 0, 0) to B(a, 0, 2π b) along r̅ = (a cos t) î + (a sin t) ĵ + b t k̂ is:
from A(a, 0, 0) to B(a, 0, 2π b) along r̅ = (a cos t) î + (a sin t) ĵ + b t k̂ is:a)
πa (bĵ + ak̂)
b)
2πa (aĵ + bk̂ + abî)
c)
2πa (aĵ + bk̂)
d)
2πa (bĵ + ak̂)
|
|
Sanya Agarwal answered |
The value of 
From A(a, 0, 0) to B(a, 0, 2π b) along r̅ = (a cos t) î + (a sin t) ĵ + b t k̂ is: πa (bĵ + ak̂)
From A(a, 0, 0) to B(a, 0, 2π b) along r̅ = (a cos t) î + (a sin t) ĵ + b t k̂ is: πa (bĵ + ak̂)Chapter doubts & questions for Vector Calculus - Engineering Mathematics 2025 is part of Civil Engineering (CE) exam preparation. The chapters have been prepared according to the Civil Engineering (CE) exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for Civil Engineering (CE) 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.
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