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

  1. Zero
  2. 1

Answer (Detailed Solution Below)

Option 4 : Zero

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 :

  1. 60
  2. 48
  3. 72
  4. 64

Answer (Detailed Solution Below)

Option 4 : 64

Vector Calculus Question 2 Detailed Solution

Concept:

The divergence of a vector field is given by:

Given:

, and the cube lies in

Calculation:

Since has no component,

Find partial derivatives at the center of the cube:

At

At

Final Result:

Correct Answer: 4) 64

Vector Calculus Question 3:

Three vectors  and  are given as

Which of the following is/are CORRECT?

Answer (Detailed Solution Below)

Option :

Vector Calculus Question 3 Detailed Solution

Explanation:

(a) 

(This is always true for any three given vectors)

(b) We know that  is always true but  because 

This can be true only when 

So, 

This can be true if

 this is true in this case

(c)  can't be equal to  because  and 

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

  1. 4πR
  2. 0
  3. πR3

Answer (Detailed Solution Below)

Option 3 :

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

=  = √35 × π ×  = 3√35 π

Given area S = απ

Hence α =   = 17.74

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

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Explanation

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 

  1. 13

Answer (Detailed Solution Below)

Option 1 :

Vector Calculus Question 7 Detailed Solution

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Concept:

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  and  the cross, the product is given by: 

The magnitude of the cross product is:

Area of a triangle:

If the vectors  form adjacent sides of the triangle then the area of the triangle is given by: 

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)   , and .

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  the norm of the vector can be defined as . Given a function , its gradient ∇ϕ is

Answer (Detailed Solution Below)

Option 3 :

Vector Calculus Question 8 Detailed Solution

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Explanation:

Position vector 

If v = yz î + 3zx ĵ + z k̂, then curl v is

  1. -3xî + yĵ + 2zk̂
  2. 3xî – yĵ + 2zk̂
  3. -3xî – yĵ – 2xk̂
  4. 3xî + yĵ – 2zk̂

Answer (Detailed Solution Below)

Option 1 : -3xî + yĵ + 2zk̂

Vector Calculus Question 9 Detailed Solution

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Concept:

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

  1. 0
  2. 2
  3. 4
  4. 6

Answer (Detailed Solution Below)

Option 2 : 2

Vector Calculus Question 10 Detailed Solution

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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.

Calculation:

Given:

, A (0, 0, 0) and B(1, 1, 1).

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

  1. π/4
  2. π/2
  3. 3π/4
  4. 3π/2

Answer (Detailed Solution Below)

Option 4 : 3π/2

Vector Calculus Question 11 Detailed Solution

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Concept:

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 .

Answer (Detailed Solution Below)

Option 1 : 0

Vector Calculus Question 12 Detailed Solution

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Concept:

If  are any non-zero vector then

Calculation: 

Given, 

= 0

If  are three vectors such that  and  then the value of  is:

  1. 0
  2. 1
  3. -19
  4. 38

Answer (Detailed Solution Below)

Option 3 : -19

Vector Calculus Question 13 Detailed Solution

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Concept:

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

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Given 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)

Option 3 :

Vector Calculus Question 15 Detailed Solution

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Concept:

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

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