Vector Calculus - Engineering Mathematics - Mining Engineering Previous Year Questions

1

2007 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2007

Two sides of a triangle are represented by vectors $\mathbf{a} = \hat{i} + \hat{j} + \hat{k}$ and $\mathbf{b} = -\hat{i} - \hat{j} + \hat{k}$. The area (magnitude) of the triangle is

A

$1/\sqrt{2}$

B

$1$

C

$\sqrt{2}$

D

$2\sqrt{2}$

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2

2008 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2008

The direction of gradient vector at a point (1, 1, 2) on a surface S(x, y, z) = x²+y² - z is

A

\(\frac{1}{3}(2\hat{i}+2\hat{j}+\hat{k})\)

B

\(\frac{1}{3}(-2\hat{i}+2\hat{j}+\hat{k})\)

C

\(\frac{1}{3}(2\hat{i}-2\hat{j}+\hat{k})\)

D

\(\frac{1}{3}(2\hat{i}+2\hat{j}-\hat{k})\)

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3

2008 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2008

A force vector F = (2i+3j-k) in N is acting on a point, whose position vector r = (i-j+6k) in m. The magnitude of the torque about the origin in Nm is

A

20.85

B

21.42

C

21.97

D

22.27

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4

2009 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2009

\( \hat{i}, \hat{j} \) and \( \hat{k} \) represent the unit vectors in the positive \( x, y \) and \( z \) directions of a Cartesian coordinate system. Using the right-hand rule, \( \hat{k} \times \hat{j} \) represents

A

0

B

1

C

\(-\hat{i} \)

D

\(\hat{i} \)

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5

2009 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2009

The value of \(\nabla \cdot \mathbf{F}\) of a vector \(\mathbf{F} = 4x^2\hat{i} + 3xy^2\hat{j} + xyz^2\hat{k}\) at the point (1, 1, 2) is

A

24

B

26

C

30

D

32

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6

2010 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2010

A force of \(50\hat{i} - 50\hat{j}\) N is moved from the origin to a coordinate (4.0m, 2.0m). The work done in the process in J is

A

75.6

B

85.5

C

90.2

D

100.0

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7

2012 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2012

The angle between the tangents to the curve \(\vec{R} = t^2\hat{i} + 2t\hat{j}\) at the point \(t = \pm 1\) is

A

\(\frac{\pi}{2}\)

B

\(\frac{\pi}{3}\)

C

\(\frac{\pi}{4}\)

D

\(\frac{\pi}{6}\)

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8

2014 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2014

The value of k for which the vectors a = 2i - 3j and b = ki + 4j are orthogonal to each other is ___

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9

2014 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2014

The divergence of the vector v = (x + y)(-yi + xj) is

A

y - x

B

x - y

C

x² - y²

D

y² - x²

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10

2016 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2016

Equations of two planes are \( z = 4 \) and \( z = 4 + 3x \). The included angle between the two planes in degrees, is __________

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11

2016 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2016

A force \( \vec{P} = 2\hat{i} - 5\hat{j} + 6\hat{k} \) acts on a particle. The particle is moved from point A to point B, where the position vectors of \( \vec{A} \) and \( \vec{B} \) are \( 6\hat{i} + \hat{j} - 3\hat{k} \) and \( 4\hat{i} - 3\hat{j} - 2\hat{k} \) respectively. The work done is __________

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12

2017 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2017

The position vector of a moving particle is given by $\vec{r}(t) = t^3 \hat{i} + t \hat{j} + t^2 \hat{k}$. The acceleration of the particle in the direction of the motion is

A

0

B

$60 \hat{i} + 2 \hat{k}$

C

$6t \hat{i} + 4 \hat{j} + 2 \hat{k}$

D

$6t \hat{i} + 2 \hat{k}$

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13

2020 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2020

A parallelepiped has edge vectors as shown below.
\(\vec{A} = -4\hat{i} - 10\hat{j} - \hat{k} \)
\(\vec{B} = 7\hat{i} + 9\hat{j} - 2\hat{k} \)
\(\vec{C} = 3\hat{i} + 9\hat{j} + 4\hat{k} \)
The volume of the parallelepiped is ______ (round off to 1 decimal place).

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14

2021 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2021

The vectors \(\vec{a}\) and \(\vec{b}\) act in a plane as shown below. The magnitude of the vector \(\vec{c} = (\vec{a} + \vec{b}) \times (\vec{a} - \vec{b})\) is

A

zero.

B

half to the area bounded by the vectors \(\vec{a}\) and \(\vec{b}\).

C

equal to the area bounded by the vectors \(\vec{a}\) and \(\vec{b}\).

D

twice the area bounded by the vectors \(\vec{a}\) and \(\vec{b}\).

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15

2022 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2022

A velocity field in Cartesian coordinate system is expressed as \(\vec{v} = x\hat{i} + y\hat{j} + p(z)\hat{k}\), where \(p(0) = 0\)
If div \(\vec{v} = 0\), \(p(z)\) is

A

0

B

-2z

C

2

D

2z

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16

2023 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2023

Given two vectors \(\vec{A} = 3\hat{i} + 2\hat{j}\) and \(\vec{B} = \hat{i} + \hat{j}\), the magnitude of projection of \(\vec{A}\) along \(\vec{B}\) is

A

\(\frac{5}{\sqrt{2}}\)

B

\(\frac{5}{\sqrt{13}}\)

C

\(\frac{5}{\sqrt{26}}\)

D

5

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17

2024 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2024

The magnitude of the curl of the vector \(\mathbf{F} = 2x\mathbf{i} + 3y\mathbf{j} + 4z\mathbf{k}\), is

A

0

B

4

C

9

D

25

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18

2024 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2024

Vectors \( \mathbf{a} = 2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k} \) and \( \mathbf{b} = 4\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \) represent the two adjacent sides of a triangle. The magnitude of the area of the triangle and the unit vector perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \) respectively, are

A

28.93 and 0.58\mathbf{i} - 0.76\mathbf{j} - 0.27\mathbf{k}

B

28.93 and 17.0\mathbf{i} - 22.0\mathbf{j} - 8.0\mathbf{k}

C

14.46 and 0.58\mathbf{i} - 0.76\mathbf{j} - 0.27\mathbf{k}

D

14.46 and 17.0\mathbf{i} - 22.0\mathbf{j} - 8.0\mathbf{k}

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19

2025 · Mining Engineering · Engineering Mathematics · Vector Calculus

Mining Engineering (MN) 2025

The directional derivative of a function \(f(x, y, z) = 4x^2 + 8y^2 + 9z^2\) at the point P (3, 4, 5) in the direction vector \(\vec{b} = 2\hat{i} - 3\hat{j} + 4\hat{k}\) is ____ (rounded off to 1 decimal place)

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