Vector Calculus MCQ Quiz in मराठी - Objective Question with Answer for Vector Calculus - मोफत PDF डाउनलोड करा
Last updated on Mar 11, 2025
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)
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
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
Vector Calculus Question 3:
Which of the following represent the Green's theorem
Answer (Detailed Solution Below)
Vector Calculus Question 3 Detailed Solution
Green's theorem:
Stokes theorem:
Gauss Divergence theorem:
Vector Calculus Question 4:
Divergence of the vector field
Answer (Detailed Solution Below)
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
Answer (Detailed Solution Below)
Vector Calculus Question 5 Detailed Solution
Explanation:
If
- curl∇f=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.curl∇f=0." role="presentation" style=" word-spacing: 0px; position: relative;" tabindex="0">
..curl∇f=0.
Vector Calculus Question 6:
The line integral of the vector function
Answer (Detailed Solution Below)
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
Answer (Detailed Solution Below)
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">
Vector Calculus Question 8:
If 2î + 4ĵ - 5k̂ and î + 2ĵ + 3k̂ are two different sides of rhombus. Find the length of diagonals.
Answer (Detailed Solution Below)
Vector Calculus Question 8 Detailed Solution
Concept:
If aî + bĵ + ck̂ and pî + qĵ + rk̂ are 2 different sides of the rhombus.
Suppose
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,
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
Answer (Detailed Solution Below)
Vector Calculus Question 9 Detailed Solution
Concept:
Directional derivative of a function f along the vector
where
grad f or ∇ f is defined by the equation,
Calculation:
Given:
z = y2e2x
At (2, -1),
Unit vector
Directional derivative
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
Answer (Detailed Solution Below)
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