Vector Calculus MCQ Quiz - Objective Question with Answer for Vector Calculus - Download Free PDF
Last updated on Apr 8, 2025
Latest Vector Calculus MCQ Objective Questions
Vector Calculus Question 1:
If f1 and f2 are differentiable scalar functions and v is differentiable vector function such that f1v = ∇f2, then v . curl v is
Answer (Detailed Solution Below)
Vector Calculus Question 1 Detailed Solution
Explanation:
v . curl v = v . ∇ × v = [v ∇ v] = 0
Option (4) is true.
Vector Calculus Question 2:
Consider a cube having the dimension
x, y, z ∈ [1, 3]. If a B̅ =
Divergence of B̅ at the centre of cube is :
Answer (Detailed Solution Below)
Vector Calculus Question 2 Detailed Solution
Concept:
The divergence of a vector field
Given:
Calculation:
Since
Find partial derivatives at the center of the cube:
At
At
Final Result:
Correct Answer: 4) 64
Vector Calculus Question 3:
Three vectors
Which of the following is/are CORRECT?
Answer (Detailed Solution Below)
Vector Calculus Question 3 Detailed Solution
Explanation:
(a)
(This is always true for any three given vectors)
(b) We know that
This can be true only when
So,
This can be true if
(c)
So,
(d)
0 = 0
(Hence this is proved)
Vector Calculus Question 4:
The value of the surface integral
∯s zdxdy
where S is the external surface of the sphere x2 + y2 + z2 = R2 is
Answer (Detailed Solution Below)
Vector Calculus Question 4 Detailed Solution
Explanation:
To evaluate, ∯s z dx dy
where S(sphere) = x2 + y2 + z2 = R2
= ∯z dxdy
Using Gauss Divergence Theorem:
= Volume of sphere =
Vector Calculus Question 5:
Let 𝑆 be the portion of the plane 𝑧 = 5𝑥 + 3𝑦 − 100 which lies inside the cylinder 𝑥2 + 𝑦2 = 3 . If the surface area of 𝑆 is 𝛼𝜋, then the value of 𝛼 is equal to ___________.
Answer (Detailed Solution Below) 17.73 - 17.75
Vector Calculus Question 5 Detailed Solution
Explanation:
𝑆 is the portion of the plane 𝑧 = 5𝑥 + 3𝑦 − 100 which lies inside the cylinder 𝑥2 + 𝑦2 = 3.
zx = 5, zy = 3
then surface area of S
=
=
=
=
Now, the cylinder is 𝑥2 + 𝑦2 = 3
then surface area of S
=
Given area S = απ
Hence α =
17.74 is the answer.
Top Vector Calculus MCQ Objective Questions
The volume determined from ∫∫∫v 8 xyz dv for V = [2, 3] × [1, 2] × [ 0,1 ] will be (in integer) ________.
Answer (Detailed Solution Below) 15
Vector Calculus Question 6 Detailed Solution
Download Solution PDFExplanation
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
Find the area of triangle whose two sides are represented by the vectors 3i + 4j and 5i + 7j + k is
Answer (Detailed Solution Below)
Vector Calculus Question 7 Detailed Solution
Download Solution PDFConcept:
If a triangle is formed by three vectors, then the sum of the vectors should be zero.
AB + BC + CA = 0
Cross product of the vectors:
For two vectors
The magnitude of the cross product is:
Area of a triangle:
If the vectors
Calculation:
Given:
Let, AB = 3i + 4j and CA = 5i +7j + k
If a triangle is formed by three vectors, then the sum of the vectors should be zero.
AB + BC + CA = 0 ⇒ 3i + 4j + BC + 5i +7j + k = 0
BC = - 8i - 11j - k
Let the adjacent vectors be AB (a), AC (b)
First, we will calculate the cross product as follow:
Therefore, the magnitude of the cross product is:
Using the formula for the area of the triangle, the area is given by:
For a position vector
Answer (Detailed Solution Below)
Vector Calculus Question 8 Detailed Solution
Download Solution PDFExplanation:
Position vector
If v = yz î + 3zx ĵ + z k̂, then curl v is
Answer (Detailed Solution Below)
Vector Calculus Question 9 Detailed Solution
Download Solution PDFConcept:
The curl of a vector is given by the expansion of the following matrix
If
Then
Calculation:
Given vector is
Than
The value of the line integral
along a path joining the origin and the point (1,1,1) is
Answer (Detailed Solution Below)
Vector Calculus Question 10 Detailed Solution
Download Solution PDFConcept:
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.
Calculation:
Given:
Equation of line i.e. path
The parabolic arc y = √x, 1 ≤ x ≤ 2 is revolved around the x-axis. The volume of the solid of revolution is
Answer (Detailed Solution Below)
Vector Calculus Question 11 Detailed Solution
Download Solution PDFConcept:
Revolution about x-axis: The volume of the solid generated by the revolution about the x-axis, of the area bounded by the curve y = f(x), the x-axis and the ordinates x = a and x = b is
similarly for revolution about y-axis:
Calculation:
Given:
Hence the required volume will be
The value of
Answer (Detailed Solution Below)
Vector Calculus Question 12 Detailed Solution
Download Solution PDFConcept:
If
Calculation:
Given,
= 0
If
Answer (Detailed Solution Below)
Vector Calculus Question 13 Detailed Solution
Download Solution PDFConcept:
Calculation:
Given:
As
∴
Hence
Let ax and ay be unit vectors along x and y directions, respectively. A vector function is given by
F = ax y - ay x
The line integral of above function
Along the curve C, which follows the parabola y = x2 as shown below is ______ (rounded off to 2 decimal places)
Answer (Detailed Solution Below) -3.05 - -2.95
Vector Calculus Question 14 Detailed Solution
Download Solution PDFGiven that,
F = ax y - ay x
y = x2
By differentiating on both sides,
⇒ dy = 2x dx
In the given graph, the limits of x are: -1 to 2.
The vector function F(r) = -x î + yĵ is defined over a circular are C shown in the figure.
The line integral of ∫C F(r) ⋅ dr is
Answer (Detailed Solution Below)
Vector Calculus Question 15 Detailed Solution
Download Solution PDFConcept:
∫C F(r) dr can be solved by putting:
x = r cos θ
y = r sin θ
dx = -r sin θ dθ
dy = r cos θ dθ
Application:
Given r = 1
θ = 0 to 45°
The required integral can be written as:
With r = 1, the above integral becomes:
r = 1