Vector Derivatives and Line Integrals - Vector Calculus - Engineering Sciences Previous Year Questions

1

2007 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2007

The value of \( \oint_C (xy^2+2x)dx + (x^2y+4x)dy \) along the circle \( C: x^2+y^2=4 \) in the anticlockwise direction is

A

\(-16\pi \)

B

\(-4\pi \)

C

\(4\pi \)

D

\(16\pi \)

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2

2008 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2008

The directional derivative at the point \(P(1,2,3)\) to the surface \(x^2 + \frac{y^2}{4} + \frac{z^2}{9} = 3\) in the direction of the vector \(\overrightarrow{OP}\), where \(O\) denotes the origin, is

A

0

B

\(\frac{2}{\sqrt{14}}\)

C

\(\frac{3}{\sqrt{14}}\)

D

\(\frac{6}{\sqrt{14}}\)

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3

2008 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2008

The absolute value of the integral \( \oint_C (-z dx + x dy + y dz) \), where \( c \) is the curve obtained by the intersection of \( x^2 + y^2 = a^2, a > 0 \) and \( y = z \), is

A

\(\frac{\pi a^2}{\sqrt{2}} \)

B

\(\frac{\pi a^2}{\sqrt{3}} \)

C

\(\pi a^2 \sqrt{2} \)

D

\(2\pi a^2 \)

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4

2009 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2009

Let C be the boundary of the square given by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1. Then \[ \oint_C (x dy - y dx) \] equals

A

-2

B

0

C

1

D

2

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5

2009 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2009

Let \(\vec{u} = -\omega y \hat{i} + \omega x \hat{j}\) and \(\vec{v} = \omega z \hat{j} - \omega y \hat{k}\) be two given vectors, where \(\omega\) is a constant. Then \(div(\vec{u} \times \vec{v})\) equals

A

0

B

\(2\omega^{2} y\)

C

\(4\omega^{2} y\)

D

\(-4\omega^{2} y\)

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6

2011 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2011

A vector field is called solenoidal if its divergence is zero. Consider the vector fields $\vec{P}$ and $\vec{Q}$ given by
$\vec{P}(x, y, z) = (2x^2 + 8xy^2 z)\hat{i} + (3x^3 y - 3xy)\hat{j} - (4y^2 z^2 + 2x^2 z)\hat{k}$ and
$\vec{Q}(x, y, z) = xyz^2 \vec{P}(x, y, z)$.
Then

A

$\vec{P}$ and $\vec{Q}$ are both solenoidal

B

both $\vec{P}$ and $\vec{Q}$ are not solenoidal

C

$\vec{P}$ is solenoidal but not $\vec{Q}$

D

$\vec{Q}$ is solenoidal but not $\vec{P}$

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7

2011 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2011

Consider the function $f(x, y, z) = x^3 e^y \sin z$ and the point $P = \left(1, 0, \frac{\pi}{2}\right)$. The value of $f$ DOES NOT change due to a small displacement of $P$ along the direction of

A

$\left[1, 0, \frac{\pi}{2}\right]$

B

$[1, -1, 1]$

C

$[1, -3, 0]$

D

$[2, 0, -1]$

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8

2012 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2012

For \( f = x^4 - 5xy^2 \), the direction of maximum increase of \( f(x,y) \) at the point \( (2,2) \) is along

A

\(3\hat{i} + 10\hat{j} \)

B

\(-12\hat{i} - 40\hat{j} \)

C

\(3\hat{i} - 10\hat{j} \)

D

\(-12\hat{i} + 40\hat{j} \)

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9

2012 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2012

Evaluation of \(\iint_S (e^x \hat{i} + 3y \hat{j} - z e^x \hat{k}) \cdot \hat{n} \, dA\) over a surface \(S: x^2 + y^2 + z^2 = 1\), using Gauss divergence theorem, gives

A

0

B

\(4\pi\)

C

\(4\pi / 3\)

D

\(12\pi\)

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10

2013 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2013

The work done by the force \(\vec{F}=(x+x^{2}) \hat{i}+(x^{2}+y^{3}) \hat{j}\) in moving a particle once along the triangle with vertices (0,0),(1,0) and (0,1) in the anti-clockwise direction is

A

0

B

1/6

C

1/3

D

5/3

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11

2014 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2014

If the work done in moving a particle once around a circle $x^2 + y^2 = 4$ under the force field $\vec{F}(x,y) = (2x - ay) \hat{i} + (2y + ax) \hat{j}$ is $16\pi$, then $|a|$ is equal to ______.

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12

2015 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2015

The divergence of a vector field \(\vec{v}(x, y, z) = 2x^2 \hat{i} + 2y^2 \hat{j} + 2z^2 \hat{k}\) at a point \((1,1,1)\) is

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13

2016 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2016

The value of the surface integral \( \iint_{\Gamma} \vec{F} \cdot \mathbf{n} \, dS \) over the sphere \( \Gamma \) given by \( x^2 + y^2 + z^2 = 1 \), where \( \vec{F} = 4x \hat{i} - z \hat{k} \), and \( \mathbf{n} \) denotes the outward unit normal, is

A

\(\pi \)

B

\(2\pi \)

C

\(3\pi \)

D

\(4\pi \)

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14

2017 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2017

Let $C$ be a simple smooth closed curve enclosing the region $R$ in the $xy$-plane. Let $C$ be oriented counterclockwise. If the value of the integral \(\oint_{C} (y + e^{\sqrt{x}}) dx + (3x + \cos y) dy\) is 16, then the area of $R$ is ________.

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15

2018 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2018

Let \( \mathbb{R}^3 \) denote the three dimensional Euclidean space and \( \mathbf{F}(x, y, z) = -y \hat{i} + x \hat{j} + z \hat{k} \) for all \( (x, y, z) \in \mathbb{R}^3 \). If \( C \) is the curve described by the parametric equation \( \mathbf{r}(t) = \cos t \ \hat{i} + \sin t \ \hat{j} + 2t^2 \ \hat{k}, \ 0 \le t \le 1 \), then the value of the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) is __________

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16

2019 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2019

The value of the line integral \( \frac{2}{\pi} \oint_{\gamma} (-y^3 dx + x^3 dy) \), where \( \gamma \) is the circle \( x^2 + y^2 = 1 \) oriented counter clockwise, is ______.

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17

2020 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2020

Let \( \vec{V}(x, y, z) = ax \hat{i} - hz \hat{j} + cy \hat{k} \) be a vector field whose curl is zero. Then necessarily

A

\(a = h = c \).

B

\(a = -h = c \).

C

\(h = c \).

D

\(h = -c \).

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18

2021 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2021

Let C be the boundary of the region R : \( 0 \le x \le \pi, 0 \le y \le \sin x \) in the xy-plane and \( \alpha \) be the area of the region R. If C traverses once in the counter clockwise direction, then the value of the line integral \( \oint_C (2y\,dx + 5x\,dy) \) is equal to

A

\(\alpha \)

B

\(2\alpha \)

C

\(3\alpha \)

D

\(4\alpha \)

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19

2021 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2021

If \(\theta\) is the angle, in degrees, between the longest diagonal of the cube and any one of the edges of the cube, then, cos \(\theta\)=

A

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

B

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

C

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

D

\(\frac{\sqrt{3}}{2}\)

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20

2021 · Engineering Sciences · Vector Calculus · Vector Derivatives and Line Integrals

Engineering Sciences (XE) 2021

Let \(C_1\) be the line segment from \((0, 1)\) to \((\frac{4}{5}, \frac{3}{5})\), and let \(C_2\) be the arc of the circle \(x^2 + y^2 = 1\) from \((0, 1)\) to \((\frac{4}{5}, \frac{3}{5})\). If \(\alpha = \int_{C_1} \left( \frac{2x}{y} \, i + \frac{1 - x^2}{y^2} \, j \right) \cdot d\vec{r}\) and \(\beta = \int_{C_2} \left( \frac{2x}{y} \, i + \frac{1 - x^2}{y^2} \, j \right) \cdot d\vec{r}\), where \(\vec{r} = x \, i + y \, j\), then the value of \(\alpha^2 + \beta^2\) is ______________________ (round off to two decimal places).

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