Vector Calculus - Calculus - Mathematics Previous Year Questions

1

2008 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2008

In any system of particles, suppose we do not assume that the internal forces come in pairs. Then the fact that the sum of internal forces is zero follows from

A

Newton's second law

B

conservation of angular momentum

C

conservation of energy

D

principle of virtual displacement

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2

2008 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2008

Let \(W = \{(x, y, z) \in \mathbb{R}^3 : 1 \le x^2 + y^2 + z^2 \le 4\}\) and \(F: W \to \mathbb{R}^3\) be defined by \(F(x, y, z) = \frac{(x, y, z)}{[x^2 + y^2 + z^2]^{3/2}}\) for \((x, y, z) \in W\). If \(\partial W\) denotes the boundary of \(W\) oriented by the outward normal \(n\) to \(W\), then \(\iint_{\partial W} F \cdot n dS\) is equal to

A

0

B

\(4\pi\)

C

\(8\pi\)

D

\(12\pi\)

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3

2010 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2010

Let \( I = \oint_C \frac{e^x}{x} dx + (e^x \ln x + x) dy \), where \( C \) is the positively oriented boundary of the region enclosed by \( y = 1 + x^2 \), \( y = 2 \), \( x = \frac{1}{2} \). Then the value of \( I \) is

A

\(\frac{1}{8} \)

B

\(\frac{5}{24} \)

C

\(\frac{7}{24} \)

D

\(\frac{3}{8} \)

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4

2011 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2011

Let \(I = \oint_C (2x^2 + y^2) dx + e^y dy\), where \(C\) is the boundary (oriented anticlockwise) of the region in the first quadrant bounded by \(y=0\), \(x^2+y^2=1\) and \(x=0\). The value of \(I\) is

A

-1

B

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

C

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

D

1

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5

2012 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2012

The flux of the vector field \(\vec{u}=x\vec{i}+y\vec{j}+z\vec{k}\) flowing out through the surface of the ellipsoid \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1,\ a>b>c>0,\) is

A

\(\pi abc\)

B

\(2\pi abc\)

C

\(3\pi abc\)

D

\(4\pi abc\)

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6

2014 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2014

A particle P of mass \(m\) moves along the cycloid \(x=(\theta-\sin \theta)\) and \(y=(1+\cos \theta)\), \(0 \leq \theta \leq 2 \pi\). Let \(g\) denote the acceleration due to gravity. Neglecting the frictional force, the Lagrangian associated with the motion of the particle P is:

A

\(m(1-\cos \theta) \dot{\theta}^{2}-m g(1+\cos \theta)\)

B

\(m(1+\cos \theta) \dot{\theta}^{2}+m g(1+\cos \theta)\)

C

\(m(1+\cos \theta) \dot{\theta}^{2}+m g(1-\cos \theta)\)

D

\(m(\theta-\sin \theta) \dot{\theta}^{2}-m g(1+\cos \theta)\)

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7

2014 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2014

Let \(\vec{F}\) be a vector field defined on \(\mathbb{R}^{2} \backslash\{(0,0)\}\) by \(\vec{F}(x, y)=\frac{-y}{x^{2}+y^{2}} \hat{i}+\frac{x}{x^{2}+y^{2}} \hat{j}\). Let \(\gamma, \alpha:[0,1] \rightarrow \mathbb{R}^{2}\) be defined by
\(\gamma(t)=(8 \cos 2 \pi t, 17 \sin 2 \pi t)\) and \(\alpha(t)=(26 \cos 2 \pi t,-10 \sin 2 \pi t)\).
If \(3 \int_{\alpha} \vec{F} \cdot d \vec{r}-4 \int_{\gamma} \vec{F} \cdot d \vec{r}=2 m \pi\), then \(m\) is ______________

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8

2015 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2015

Consider the unit sphere \(S = \{(x,y,z) \in \mathbb{R}^3: x^2 + y^2 + z^2 = 1\}\) and the unit normal vector \(\hat{n} = (x,y,z)\) at each point \((x,y,z)\) on \(S\). The value of the surface integral
\(\iint_S \left[ \left( \frac{2x}{\pi} + \sin(y^2) \right) x + \left( e^z - \frac{y}{\pi} \right) y + \left( \frac{2z}{\pi} + \sin^2 y \right) z \right] d\sigma\)
is equal to ______

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9

2016 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2016

Let \(\gamma\) be the triangular path connecting the points \((0,0), (2,2)\) and \((0,2)\) in the counter-clockwise direction in \(\mathbb{R}^2\). Then \[ I = \oint_\gamma \sin(x^3) dx + 6xy dy \] is equal to __________

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10

2017 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2017

Let \(C: x^2 + y^2 = 9\) be the circle in \(\mathbb{R}^2\) oriented positively. Then \(\frac{1}{\pi} \oint_C \left(3y - e^{\cos x^2}\right) dx + \left(7x + \sqrt{y^4 + 11}\right) dy\) equals ________.

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11

2018 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2018

Let \(S\) be the surface of the solid \[V = \{(x,y,z) : 0 \leq x \leq 1, \ 0 \leq y \leq 2, \ 0 \leq z \leq 3\}.\] Let \(\hat{n}\) denote the unit outward normal to \(S\) and let \[\vec{F}(x,y,z) = x\hat{i} + y\hat{j} + z\hat{k}, \quad (x,y,z) \in V.\] Then the surface integral \(\iint_S \vec{F} \cdot \hat{n} \, dS\) equals ________.

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12

2019 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2019

Let \(\Gamma = \{(x,y,z) \in \mathbb{R}^3 : -1 < x < 1, -1 < y < 1, -1 < z < 1\}\) and \(\phi : \Gamma \to \mathbb{R}\) be a function whose all second order partial derivatives exist and are continuous. If \(\phi\) satisfies the Laplace equation \(\nabla^2 \phi = 0\) for all \((x,y,z) \in \Gamma\), then which one of the following statements is TRUE in \(\Gamma\)? ( \(\mathbb{R}\) is the set of all real numbers, and \(\mathbb{R}^3 = \{(x,y,z): x,y,z \in \mathbb{R}\}\) )

A

\(\vec{\nabla} \phi\) is solenoidal but not irrotational

B

\(\vec{\nabla} \phi\) is irrotational but not solenoidal

C

\(\vec{\nabla} \phi\) is both solenoidal and irrotational

D

\(\vec{\nabla} \phi\) is neither solenoidal nor irrotational

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13

2019 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2019

Let \(L\) denote the value of the line integral \(\oint_C (3x-4x^3y)dx+(4xy^2+2y)dy\), where \(C\), a circle of radius 2 with centre at origin of the xy-plane, is traversed once in the anti-clockwise direction. Then \(\frac{L}{\pi}\) is equal to ______.

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14

2020 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2020

Let \(\vec{F}(x,y,z) = (2x - 2y \cos x) \hat{i} + (2y - y^2 \sin x) \hat{j} + 4z \hat{k}\) and let \(S\) be the surface of the tetrahedron bounded by the planes \[ x = 0, y = 0, z = 0 \text{ and } x + y + z = 1. \] If \(\hat{n}\) is the unit outward normal to the tetrahedron, then the value of \[ \iint_S \vec{F} \cdot \hat{n} \, dS \] is ________ (rounded off to two decimal places)

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15

2020 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2020

Let \(\vec{F} = (x + 2y) e^z \hat{i} + (y e^z + x^2) \hat{j} + y^2 z \hat{k}\) and let \(S\) be the surface \[ x^2 + y^2 + z = 1, \, z \ge 0. \] If \(\hat{n}\) is a unit normal to \(S\) and \[ \left| \iint_S (\nabla \times \vec{F}) \cdot \hat{n} \, dS \right| = a\pi. \] Then \(a\) is equal to ________

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16

2021 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2021

Let \( f: \mathbb{R}^3 \to \mathbb{R} \) be a twice continuously differentiable scalar field such that \( div(\nabla f) = 6 \). Let \( S \) be the surface \( x^2 + y^2 + z^2 = 1 \) and \( \hat{n} \) be unit outward normal to \( S \). Then the value of \( \iint_S (\nabla f \cdot \hat{n}) \, dS \) is

A

\(2\pi \)

B

\(4\pi \)

C

\(6\pi \)

D

\(8\pi \)

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17

2021 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2021

Let \( \Gamma \) denote the boundary of the square region \( R \) with vertices \( (0, 0), (2, 0), (2, 2) \) and \( (0, 2) \) oriented in the counter-clockwise direction. Then \( \oint_{\Gamma} (1 - y^2) \, dx + x \, dy = \) ________.

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18

2022 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2022

The work done by the force \( F = (x + y)\hat{i} - (x^2 + y^2)\hat{j} \), where \( \hat{i} \) and \( \hat{j} \) are unit vectors in \( \overrightarrow{OX} \) and \( \overrightarrow{OY} \) directions, respectively, along the upper half of the circle \( x^2 + y^2 = 1 \) from \( (1, 0) \) to \( (-1, 0) \) in the \( xy \)-plane is

A

\(-\pi \)

B

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

C

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

D

\(\pi \)

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19

2023 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2023

Let \( C \) be the curve of intersection of the cylinder \( x^2 + y^2 = 4 \) and the plane \( z - 2 = 0 \). Suppose \( C \) is oriented in the counterclockwise direction around the z-axis, when viewed from above. If
\[ \left| \int_C (\sin x + e^x) \, dx + 4x \, dy + e^x \cos^2 z \, dz \right| = a\pi, \]
then the value of \( a \) equals ________.

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20

2024 · Mathematics · Calculus · Vector Calculus

Mathematics (MA) 2024

If the outward flux of \( \mathbf{F}(x, y, z) = (x^3, y^3, z^3) \) through the unit sphere \( x^2 + y^2 + z^2 = 1 \) is \( \alpha \pi \), then \( \alpha \) is equal to ________ (round off to TWO decimal places)

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