Vector Calculus - Engineering Mathematics - Agricultural Engineering Previous Year Questions

1

2008 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2008

The cross product of \(\vec{x} = 2\mathbf{i} + \mathbf{j}\) and \(\vec{y} = \mathbf{i} - 2\mathbf{j} + \mathbf{k}\) is

A

\(\mathbf{i} - \mathbf{j} + 2\mathbf{k}\)

B

\(\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}\)

C

\(\mathbf{i} - 2\mathbf{j} - 5\mathbf{k}\)

D

\(2\mathbf{i} - 4\mathbf{j}\)

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2

2010 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2010

The curl of the vector A = xyî + yzĵ + zxk̂ (î, ĵ and k̂ represent unit vectors along the three orthogonal axes) is

A

xî + yĵ + zk̂

B

-xî - yĵ - zk̂

C

yî + zĵ + xk̂

D

-yî - zĵ - xk̂

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3

2010 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2010

The angle of intersection between the planes x – 3y + 2z = 10 and 2x + 4y + 5z = 0 is

A

30°

B

60°

C

75°

D

90°

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4

2011 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2011

The divergence of the vector \(F = \sin(xy)\mathbf{i} + y \cos(z)\mathbf{j} + xz \cos(z)\mathbf{k}\) (i, j and k represent unit vectors along the three orthogonal axes) at \(x = y = 0\) is

A

\(\sin(z)\)

B

\(\cos(z)\)

C

\(-\sin(z)\)

D

\(-\cos(z)\)

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5

2011 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2011

A plane contains the following three points: P(2,1,5), Q(-1,3,4) and R(3,0,6). The vector perpendicular to the above plane can be represented as

A

2i - j + k

B

i + 2j + 2k

C

2i + 3j + k

D

i + 2j + k

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6

2016 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2016

Let \(\mathbf{I}, \mathbf{J},\) and \(\mathbf{K}\) are unit vectors along the three mutually perpendicular \(x, y\) and \(z\) axes, respectively. If \(\mathbf{F} = f\mathbf{I} + g\mathbf{J} + h\mathbf{K}\) is a continuously differentiable vector point function, then \(\mathbf{curl\ F}\) is

A

\(\mathbf{I}(\frac{\partial g}{\partial z} - \frac{\partial h}{\partial y}) - \mathbf{J}(\frac{\partial f}{\partial z} - \frac{\partial h}{\partial x}) + \mathbf{K}(\frac{\partial f}{\partial y} - \frac{\partial g}{\partial x})\)

B

\(\mathbf{I}(\frac{\partial h}{\partial y} - \frac{\partial g}{\partial z}) - \mathbf{J}(\frac{\partial h}{\partial x} - \frac{\partial f}{\partial z}) + \mathbf{K}(\frac{\partial g}{\partial x} - \frac{\partial f}{\partial y})\)

C

\(\mathbf{I}(\frac{\partial h}{\partial y} - \frac{\partial g}{\partial z}) + \mathbf{J}(\frac{\partial h}{\partial x} - \frac{\partial f}{\partial z}) + \mathbf{K}(\frac{\partial g}{\partial x} - \frac{\partial f}{\partial y})\)

D

\(\mathbf{I}(\frac{\partial g}{\partial z} - \frac{\partial h}{\partial y}) + \mathbf{J}(\frac{\partial f}{\partial z} - \frac{\partial h}{\partial x}) + \mathbf{K}(\frac{\partial f}{\partial y} - \frac{\partial g}{\partial x})\)

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7

2017 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2017

Direction cosines of the vector \(3\hat{i} - 2\hat{j} + 6\hat{k}\) are

A

[3/7, -2/7, 6/7]

B

[-3/7, 2/7, -6/7]

C

[-7/3, 7/2, -7/6]

D

[7/3, -7/2, 7/6]

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8

2017 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2017

Divergence value of a function (x²y)i - (z³-3x)j + (4y²)k at x = 1, y = 2 and z = 3 is __________.

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9

2019 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2019

For vectors \( \vec{F} = 3xy\hat{i} - y^2\hat{j} \) and \( \vec{R} = x\hat{i} + y\hat{j} \), the value of \( \int_C \vec{F} \cdot d\vec{R} \) on the curve \( C (y = 2x^2) \) in the x-y plane from (0, 0) to (1, 2) is

A

-1.17

B

1.50

C

-2.67

D

2.67

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10

2020 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2020

A particle moves along the curve \(\vec{R} = (2t^3 + 4t^2)\hat{i} + (3t^2 - 5t)\hat{j} + (7t^2 + 6t)\hat{k}\), where t is the time. The velocity component of the particle in the direction \(3\hat{i} + 2\hat{j} + \hat{k}\) at time t = 2 is

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11

2022 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2022

Work done by a moving particle in the force field F⃗ = 6x²î + (3xz + y)ĵ + 4z k̂, moving along the straight line from (0,0,0) to (1,2,3) is __________. [Answer in integer]

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12

2024 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2024

The divergence of the curl of a twice continuously differentiable vector function is ________.

A

0

B

1

C

2

D

∞

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13

2024 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2024

The directional derivative of \(u(x, y, z) = x^2y + y^2z\) at the point (1, 2, 3) in the direction \(\hat{i} + 2\hat{j} + 3\hat{k}\) is __________. (Rounded off to 2 decimal places)

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14

2026 · Agricultural Engineering · Engineering Mathematics · Vector Calculus

Agricultural Engineering (AG) 2026

The following two vectors are adjacent sides of a parallelogram:
\(\vec{A} = 2\hat{i} + 3\hat{j} - \hat{k}\) and \(\vec{B} = 5\hat{i} - 4\hat{k}\)
The magnitude of area of the parallelogram is ______. (Rounded off to two decimal places)

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