Vector Calculus MCQ Quiz in मराठी - Objective Question with Answer for Vector Calculus - मोफत PDF डाउनलोड करा

Last updated on Mar 11, 2025

पाईये Vector Calculus उत्तरे आणि तपशीलवार उपायांसह एकाधिक निवड प्रश्न (MCQ क्विझ). हे मोफत डाउनलोड करा Vector Calculus एमसीक्यू क्विझ पीडीएफ आणि बँकिंग, एसएससी, रेल्वे, यूपीएससी, स्टेट पीएससी यासारख्या तुमच्या आगामी परीक्षांची तयारी करा.

Latest Vector Calculus MCQ Objective Questions

Top Vector Calculus MCQ Objective Questions

Vector Calculus Question 1:

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 1 Detailed Solution

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

Vector Calculus Question 2:

Using Evaluate  where C is the boundary of the triangle with vertices (2, 0, 0) (0, 3, 0) and (0, 0, 6).

Answer (Detailed Solution Below) 20.5 - 21.5

Vector Calculus Question 2 Detailed Solution

By stokes theorem,

Here F = (x + y) I + (2x - z) J + (y + z) K

Equation of the plane through A, B, C is

Vector N normal to this plane is

∇(3x + 2y + z - 6) = 3I + 2J + K

 where S is the triangle ABC

Vector Calculus Question 3:

Which of the following represent the Green's theorem 

  1.  = 

Answer (Detailed Solution Below)

Option 3 :

Vector Calculus Question 3 Detailed Solution

Green's theorem:

Stokes theorem:

 = 

Gauss Divergence theorem:

Vector Calculus Question 4:

Divergence of the vector field  at (1, -1, 1) is

  1. 0
  2. 3
  3. 5
  4. 6

Answer (Detailed Solution Below)

Option 3 : 5

Vector Calculus Question 4 Detailed Solution

Concept:

The divergence of the given vector field,

Calculation:

Given:

Using equation (1),

Vector Calculus Question 5:

Divergence of the curl of a twice differentiable continuous vector function is

  1. Unity
  2. Infinity
  3. Zero
  4. A unit vector

Answer (Detailed Solution Below)

Option 3 : Zero

Vector Calculus Question 5 Detailed Solution

Explanation:

If  is twice continuously differentiable, then its second derivatives are independent of the order in which the derivatives are applied. All the terms cancel in the expression for curl , and we conclude that curlf=0." id="MathJax-Element-12-Frame" role="presentation" style=" word-spacing: 0px; position: relative;" tabindex="0">

  • curlf=0." role="presentation" style=" word-spacing: 0px; position: relative;" tabindex="0">Divergence operates on a vector field but results in a scalar.
  • Curl operates on a vector field and results in a vector field.
  • Gradient operates on a scalar but results in a vector field.
  • Divergence of curl, Curl of the gradient is always zero.
  • Thus, the gradient of curl gives the result of curl (which is a vector field) to the gradient to operate upon, which is a mathematically invalid expression.curlf=0." role="presentation" style=" word-spacing: 0px; position: relative;" tabindex="0">..curlf=0.

Vector Calculus Question 6:

The line integral of the vector function  along the straight line from (0, 0) to (2, 4) is ________

  1. 10
  2. 11
  3. 12
  4. 0

Answer (Detailed Solution Below)

Option 3 : 12

Vector Calculus Question 6 Detailed Solution

Straight line from (0, 0) to (2, 4)

x = 2t, y = 4t

dx = 2dt

dy = 4dt

Since x (2t) is from 0 to 2 and y(4t) is from 0 to 4, the limits of t are: 0 to 1

Now, the line integral becomes:

Vector Calculus Question 7:

The Cartesian coordinates of a point P in a right-handed coordinate system are (1, 1, 1). The transformed coordinates of P due to a 45° clockwise rotation of the coordinate system about the positive x-axis are

  1. (1, 0, √2)
  2. (1, 0, -√2)
  3. (-1, 0, √2)
  4. (-1, 0, -√2)

Answer (Detailed Solution Below)

Option 1 : (1, 0, √2)

Vector Calculus Question 7 Detailed Solution

Concept:

In 3D, rotations can also be defined as linear transformations, a rotation in 3D can be represented by a matrix equation P'=RP where R is a rotation matrix.

R =  

where θ is the angle of rotation with the positive axis in the clockwise direction.

Given Data and Calculation:

Given that, Coordinate of point P = (1,1,1)

The angle of rotation = 45°

New Coordinate will be,

So the solution will be 1.3×3" id="MathJax-Element-163-Frame" role="presentation" style=" position: relative;" tabindex="0">rotation matrix.(16)p′=Rp" id="MathJax-Element-161-Frame" role="presentation" style=" position: relative;" tabindex="0">p=Rp

Vector Calculus Question 8:

If 2î + 4ĵ - 5k̂ and î + 2ĵ + 3k̂ are two different sides of rhombus. Find the length of diagonals.

  1. 7, 
  2. 6, 
  3. 5, 
  4. 8, 

Answer (Detailed Solution Below)

Option 1 : 7, 

Vector Calculus Question 8 Detailed Solution

Concept:

If aî + bĵ + ck̂ and pî + qĵ + rk̂ are 2 different sides of the rhombus.

Suppose  and B = 

Then anyone diagonal of the rhombus is given by 

The other diagonal is given by 

The magnitude of the vector  

Calculation:

Given:

2î + 4ĵ - 5k̂ and î + 2ĵ + 3k̂ are 2 different sides of rhombus.

Suppose,  and B = 

Then anyone diagonal of the rhombus is given by 

D1 = (2î + 4ĵ - 5k̂) + (1î + 2ĵ + 3k̂)

D1 = 3î + 6ĵ - 2k̂

The magnitude of the vector diagonal D1

D1 = 7 

The other diagonal is given by 

D2 = (1î + 2ĵ + 3k̂) - (2î + 4ĵ - 5k̂)

D2 = - 1î - 2ĵ + 8k̂

The magnitude of the vector diagonal D2

∴ The length of diagonals of a rhombus is 7 and .

Vector Calculus Question 9:

If the directional derivative of the function z = y2e2x at (2, -1) along the unit vector  is zero, then |α + β| equals

  1. √2
  2. 2√2

Answer (Detailed Solution Below)

Option 3 : √2

Vector Calculus Question 9 Detailed Solution

Concept:

Directional derivative of a function f along the vector  is given by:

where 

grad f or ∇ f is defined by the equation,

Calculation:

Given:

z = y2e2x

At (2, -1), 

Unit vector 

Directional derivative  = 2e4α – 2e4β

Given that, the directional derivative is zero.

⇒ 2e4α – 2e4β = 0 ⇒ α = β

As b is a unit vector, 

Vector Calculus Question 10:

Given a vector  and n̂ as the unit normal vector to the surface of the hemisphere (x2 + y2 + z2 = 1; z ≥ 0), the value of integral  evaluated on the curved surface of the hemisphere S is

  1. – π/2
  2. π/3
  3. π/2
  4. π

Answer (Detailed Solution Below)

Option 3 : π/2

Vector Calculus Question 10 Detailed Solution

Given vector  

Bounded by open surface of hemisphere x2 + y2 + z2 = 1

Such that z ≥ 0, and closed curve (x2 + y2 = 1)

We have to find

⇒ from stokes theorem, we have

Now, changing the integral with substitution

x = cos θ ⇒ dx = -sin θ dθ

y = sin θ ⇒ dy = cos θ dθ

Hot Links: teen patti bodhi teen patti master downloadable content teen patti apk