Vector Calculus - Engineering Mathematics - Metallurgical Engineering Previous Year Questions

1

2007 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2007

If \( \mathbf{V} = (4xy - 3z^2)\mathbf{i} + 2x^2\mathbf{j} - 9xz^2\mathbf{k} \), the divergence of \( \mathbf{V} \) is

A

\(4xy - 18z \)

B

\(4y^2 - 9xz \)

C

\(4y - 18xz \)

D

\(2xy + 18z^2 \)

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2

2010 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2010

A vector makes angles α, β and γ with the three axes x, y and z, respectively. The value of cos²α + cos²β + cos²γ is

A

−1

B

0

C

1

D

Not determinable

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3

2012 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2012

Given that v is a vector field and f is a scalar field, match the equations in Group I with their physical meaning in Group II

A

P-1, Q-2, R-3, S-4

B

P-2, Q-1, R-4, S-3

C

P-1, Q-3, R-2, S-4

D

P-2, Q-1, R-3, S-4

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4

2013 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2013

For scalar fields φ and ψ, the value of \( \nabla \cdot (\phi \nabla \psi) \) is __________

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5

2014 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2014

If a vector \(\vec{a}\) is defined as gradient of a scalar field \(\varphi\) such that \(\vec{a} = \vec{\nabla}\varphi\), then which one of the following is equal to the curl of \(\vec{a}\)?

A

\(\nabla^2\varphi\)

B

0

C

\(\vec{\nabla}\varphi\)

D

\(\varphi\)

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6

2014 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2014

If \(\vec{r}\) is a position vector relative to the origin in three dimensional space, then \(\vec{\nabla} \cdot \vec{r}\) is

A

0

B

1

C

2

D

3

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7

2016 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2016

If \( \vec{V} = x^2 y \, \hat{i} + y^2 x \, \hat{j} + xyz \, \hat{k} \), the divergence of \( \vec{V} \) is

A

x^3 y + y^3 x + xyz^2

B

x^2 y + y^2 x + xyz

C

5xy

D

0

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8

2016 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2016

The vector parallel to the plane \(3x - 2y + z = -1\) is

A

\(\hat{i} + \hat{j} - \hat{k} \)

B

\(3\hat{i} - 2\hat{j} + \hat{k} \)

C

\(-\hat{i} + \hat{j} - \hat{k} \)

D

\(3\hat{i} - 2\hat{j} + 2\hat{k} \)

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9

2018 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2018

A vector field is given as follows:
\(\vec{f} = (-x^2 y + xy^2) \hat{i} + (x^2 y) \hat{j}\)

The divergence of this field evaluated at (x, y) = (2, 2) is ________.

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10

2018 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2018

Two equal and opposite point charges + Q and − Q are located as shown in the figure above. A surface integral, F_i is defined on surface S_i of a sphere of radius r_i as follows:
F_i = ∮_Si (E⃗ · n̂)dS_i
where E⃗ is the electric field, and n̂ is the unit normal to the surface of integration.
If r₁:r₂:r₃ are in the ratio 1: 2: 5, use the Gauss divergence theorem to determine the ratio F₁:F₂:F₃ .

A

1 : 2 : 5

B

5 : 2 : 1

C

1 : −1 : 0

D

1 : 1 : 0

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11

2019 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2019

The curl of vector fields shown below is not zero for ______.

A

B

C

D

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12

2019 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2019

Numerical value of work done (rounded off to the nearest integer) by a position-dependent force \(\vec{F} = x\hat{i} + 5xy\hat{j}\) (where \(\hat{i}\) and \(\hat{j}\) are unit vectors) along the path, \(y = \frac{x^2}{2}\), from (0,0) to (2,2) in the xy plane is ______.

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13

2020 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2020

Given the three vectors X = −i − j + k, Y = −i + 2j + k and Z = i + k, which one of the following statements is TRUE?

A

X, Y and Z are mutually perpendicular.

B

X, Y and Z are coplanar.

C

X makes an angle of 30° with the normal to the plane containing Y and Z.

D

Z makes an angle of 60° with the normal to the plane containing X and Y.

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14

2020 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2020

The divergence of the vector field (x³ + y³)i + 3xy²j + 3zy²k is:

A

3y² + 6xy + 6x²

B

3x² + 6y² + 9xy + 6yz

C

12xyz

D

3(x + y)²

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15

2021 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2021

The divergence of a vector field \(\vec{V}(x,y,z)\), where its three components \(V_x, V_y, V_z\) are functions of x,y,z, is:

A

\(\frac{\partial V_x}{\partial x} + \frac{\partial V_y}{\partial y} + \frac{\partial V_z}{\partial z}\)

B

\(\left(\frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z}\right)\mathbf{i} + \left(\frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x}\right)\mathbf{j} + \left(\frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\right)\mathbf{k}\)

C

\(\frac{\partial V_x}{\partial x}\mathbf{i} + \frac{\partial V_y}{\partial y}\mathbf{j} + \frac{\partial V_z}{\partial z}\mathbf{k}\)

D

\(\frac{\partial^2 V_x}{\partial x^2} + \frac{\partial^2 V_y}{\partial y^2} + \frac{\partial^2 V_z}{\partial z^2}\)

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16

2021 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2021

The work done by a force \( \vec{F} = 2x\hat{i} + 3y\hat{j} \) along a straight line from point (0, 0) to (1, 2) is: ______ (round off to nearest integer).

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17

2022 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2022

Given that \( V \) is a closed volume in space bounded by the surface \( S \) with unit normal \( \vec{n} \). If \( \vec{f} \) is any non-zero vector and \( \vec{\nabla} \) is the gradient operator, then the volume integral \( \int_V (\vec{\nabla} \cdot \vec{f}) \, dV \) is equal to the surface integral \( \int_S (\vec{n} \cdot \vec{f}) \, dS \) by virtue of ______

A

Stokes Curl theorem

B

Reynolds transport theorem

C

Buckingham Pi theorem

D

Gauss divergence theorem

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18

2023 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2023

A fluid flow field is given by the velocity vector \(\vec{V} = e^{xyz}(x\hat{i} + z\hat{k})\). The curl of velocity at (1, 2, 3) is

A

\(e^6(9\hat{i} - 16\hat{j} - 3\hat{k})\)

B

\(e^6(9\hat{i} - 3\hat{k})\)

C

\(e^6(9\hat{i} + 16\hat{j} - 3\hat{k})\)

D

\(e^6(-16\hat{i} + 9\hat{j} - 3\hat{k})\)

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19

2023 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2023

Given, \(\vec{\varphi} = xy\hat{i} + yz\hat{j} + xz\hat{k}\). S is a surface bounded by the planes x = 0, y = 0, z = 0, x = 3, y = 2, and z = 1. If \(\hat{n}\) is the unit vector normal to S, then \(\iint_S \vec{\varphi} \cdot \hat{n} dS\) is

A

18

B

9

C

36

D

3

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20

2024 · Metallurgical Engineering · Engineering Mathematics · Vector Calculus

Metallurgical Engineering (MT) 2024

The divergence of the vector field \[\vec{V} = x^2 y\ \hat{i} + y^3 z\ \hat{j} + z^4\ \hat{k}\] at the point (1,1,1) is __________. (Round off to the nearest integer)

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