Vector Analysis - Engineering Mathematics - Electronics & Communication Engineering Previous Year Questions

1

2012 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2012

The direction of vector A is radially outward from the origin, with \( |\mathbf{A}| = k r^n \) where \( r^2 = x^2 + y^2 + z^2 \) and k is a constant. The value of n for which \( \nabla \cdot \mathbf{A} = 0 \) is

A

-2

B

2

C

1

D

0

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2

2013 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2013 [Session 1]

Consider a vector field \(\vec{A}(\vec{r})\). The closed loop line integral \(\oint \vec{A} \bullet d\vec{l}\) can be expressed as

A

\(\iint (\nabla \times \vec{A}) \bullet d\vec{s}\) over the closed surface bounded by the loop

B

\(\iiint (\nabla \bullet \vec{A}) dv\) over the closed volume bounded by the loop

C

\(\iiint (\nabla \times \vec{A}) dv\) over the open volume bounded by the loop

D

\(\iint (\nabla \times \vec{A}) \bullet d\vec{s}\) over the open surface bounded by the loop

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3

2013 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2013 [Session 2]

Consider a vector field \(\vec{A}(\vec{r})\). The closed loop line integral \(\oint \vec{A}\cdot d\vec{l}\) can be expressed as

A

\(\iint (\nabla\times\vec{A})\cdot d\vec{s}\) over the closed surface bounded by the loop

B

\(\iiint (\nabla\cdot\vec{A})dv\) over the closed volume bounded by the loop

C

\(\iiint (\nabla\times\vec{A})dv\) over the open volume bounded by the loop

D

\(\iint (\nabla\times\vec{A})\cdot d\vec{s}\) over the open surface bounded by the loop

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4

2013 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2013 [Session 2]

The divergence of the vector field $\vec{A} = x\hat{a}_x + y\hat{a}_y + z\hat{a}_z$ is

A

0

B

1/3

C

1

D

3

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5

2013 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2013 [Session 3]

The divergence of the vector field \( \vec{A} = x \hat{a}_x + y \hat{a}_y + z \hat{a}_z \) is

A

0

B

1/3

C

1

D

3

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6

2013 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2013 [Session 3]

Consider a vector field \( \vec{A}(\vec{r}) \). The closed loop line integral \( \oint \vec{A} \cdot d\vec{l} \) can be expressed as

A

\(\iint (\nabla \times \vec{A}) \bullet d\vec{s} \) over the closed surface bounded by the loop

B

\(\iiint (\nabla \bullet \vec{A}) dv \) over the closed volume bounded by the loop

C

\(\iiint (\nabla \times \vec{A}) dv \) over the open volume bounded by the loop

D

\(\iint (\nabla \times \vec{A}) \bullet d\vec{s} \) over the open surface bounded by the loop

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7

2013 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2013 [Session 4]

The divergence of the vector field \(\vec{A} = x \hat{a}_x + y \hat{a}_y + z \hat{a}_z\) is

A

0

B

1/3

C

1

D

3

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8

2013 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2013 [Session 4]

Consider a vector field \(\vec{A}(\vec{r})\). The closed loop line integral \(\oint \vec{A} \cdot d\vec{l}\) can be expressed as

A

\(\iint (\nabla \times \vec{A}) \cdot d\vec{s}\) over the closed surface bounded by the loop

B

\(\iiint (\nabla \cdot \vec{A}) dv\) over the closed volume bounded by the loop

C

\(\iiint (\nabla \cdot \vec{A}) dv\) over the open volume bounded by the loop

D

\(\iint (\nabla \times \vec{A}) \cdot d\vec{s}\) over the open surface bounded by the loop

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9

2014 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2014 [Session 4]

Given F⃗ = zâ_x + xâ_y + yâ_z. If S represents the portion of the sphere x² + y² + z² = 1 for z ≥ 0, then ∫_S ∇ × F⃗ · dS⃗ is __________

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10

2014 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2014 [Session 4]

The magnitude of the gradient for the function \(f(x,y,z) = x^2 + 3y^2 + z^3\) at the point (1,1,1) is ________.

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11

2016 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2016 [Session 1]

M and N start from the same location. M travels 10 km East and then 10 km North-East. N travels 5 km South and then 4 km South-East. What is the shortest distance (in km) between M and N at the end of their travel?

A

18.60

B

22.50

C

20.61

D

25.00

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12

2016 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2016 [Session 2]

Suppose C is the closed curve defined as the circle \( x^2 + y^2 = 1 \) with C oriented anti-clockwise. The value of \( \oint_C (xy^2 dx + x^2 y dy) \) over the curve C equals ______

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13

2017 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2017

The smaller angle (in degrees) between the planes x + y + z = 1 and 2x - y + 2z = 0 is _______.

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14

2017 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2017

If the vector function F̅ = a̅x(3y − k1z) + a̅γ(k2x − 2z) − a̅z(k3y + z) is irrotational, then the values of the constants k1, k2 and k3, respectively, are

A

0.3, −2.5, 0.5

B

0.0, 3.0, 2.0

C

0.3, 0.33, 0.5

D

4.0, 3.0, 2.0

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15

2017 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2017

Correct : 1 Wrong : -0.33

Fatima starts from point P, goes North for 3 km, and then East for 4 km to reach point Q. She then turns to face point P and goes 15 km in that direction. She then goes North for 6 km. How far is she from point P, and in which direction should she go to reach point P?

A

8 km, East

B

12 km, North

C

6 km, East

D

10 km, North

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16

2019 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2019

Consider the line integral
\[ \int_C (x \, dy - y \, dx) \]
the integral being taken in a counterclockwise direction over the closed curve \( C \) that forms the boundary of the region \( R \) shown in the figure below. The region \( R \) is the area enclosed by the union of a \( 2 \times 3 \) rectangle and a semi-circle of radius 1. The line integral evaluates to

A

\(6 + \pi/2 \)

B

\(8 + \pi \)

C

\(12 + \pi \)

D

\(16 + 2\pi \)

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17

2020 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2020

For a vector field \(\vec{A}\), which one of the following is FALSE?

A

\(\vec{A}\) is solenoidal if \(\nabla \cdot \vec{A} = 0\).

B

\(\nabla \times \vec{A}\) is another vector field.

C

\(\vec{A}\) is irrotational if \(\nabla^2 \vec{A} = 0\).

D

\(\nabla \times (\nabla \times \vec{A}) = \nabla(\nabla \cdot \vec{A}) - \nabla^2 \vec{A}\)

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18

2021 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2021

The vector function \mathbf{F}(r) = -x\hat{i} + y\hat{j} is defined over a circular arc C shown in the figure.
The line integral of \int_C \mathbf{F}(r) \cdot d\mathbf{r} is

A

\(\frac{1}{2}\)

B

\(\frac{1}{4}\)

C

\(\frac{1}{6}\)

D

\(\frac{1}{3}\)

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19

2021 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2021

Consider the vector field \(\mathbf{F} = \mathbf{a}_x(4y - c_1 z) + \mathbf{a}_y(4x + 2z) + \mathbf{a}_z(2y + z)\) in a rectangular coordinate system (x, y, z) with unit vectors \(\mathbf{a}_x\), \(\mathbf{a}_y\), and \(\mathbf{a}_z\). If the field \(\mathbf{F}\) is irrotational (conservative), then the constant \(c_1\) (in integer) is __________

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20

2022 · Electronics & Communication Engineering · Engineering Mathematics · Vector Analysis

Electronics & Communication Engineering (EC) 2022

Consider the two-dimensional vector field \( \vec{F}(x, y) = x \hat{i} + y \hat{j} \), where \( \hat{i} \) and \( \hat{j} \) denote the unit vectors along the \( x \)-axis and the \( y \)-axis, respectively. A contour \( C \) in the \( x \)-\( y \) plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral \[ \oint_C \vec{F}(x, y) \cdot (dx \hat{i} + dy \hat{j}) \] is ________.

A

0

B

1

C

8 + 2\pi

D

-1

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