Multiple Integrals and Change of Variables - Calculus - Mathematics Previous Year Questions

1

2008 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2008

Let \(E = \{(x, y) \in \mathbb{R}^2 : 0 \leq x \leq 1, 0 \leq y \leq x\}\). Then \(\iint_E (x + y) \, dx \, dy\) is equal to

A

-1

B

0

C

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

D

1

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2

2008 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2008

The possible values of \(\alpha\) for which the variational problem: \[ J[y(x)] = \int_0^1 (3y^2 + 2x^3 y') \, dx, \ y(\alpha) = 1 \] has extremals are

A

-1, 0

B

0, 1

C

-1, 1

D

-1, 0, 1

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3

2008 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2008

The functional \(\int_0^1 (y'^2 + x^2) \, dx\), given \(y(1) = 1\), achieves its

A

weak maximum on all its extremals

B

weak minimum on all its extremals

C

weak maximum on some, but not on all, of its extremals

D

weak minimum on some, but not on all of its extremals

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4

2009 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2009

Let \(E = \{(x, y) \in \mathbb{R}^2 : 0 < x < y\}\). Then \(\iint_E y e^{-(x+y)} dx dy =\)

A

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

B

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

C

\(\frac{4}{3}\)

D

\(\frac{3}{4}\)

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5

2009 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2009

Let \( X \) and \( Y \) be independent and identically distributed \( U(0,1) \) random variables. Then \( P \left( Y < \left( X - \frac{1}{2} \right)^2 \right) = \)

A

\(\frac{1}{12} \)

B

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

C

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

D

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

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6

2009 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2009

The extremal of the functional \( \int_0^1 \left( y + x^2 + \frac{y'^2}{4} \right) dx, \ y(0) = 0, \ y(1) = 0 \) is

A

\(4(x^2 - x) \)

B

\(3(x^2 - x) \)

C

\(2(x^2 - x) \)

D

\(x^2 - x \)

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7

2010 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2010

The Euler's equation for the variational problem: Minimize \( I[y(x)] = \int_{0}^{1} (2x - xy - y') dx \), is

A

\(2y'' - y = 2 \)

B

\(2y'' + y = 2 \)

C

\(y'' + 2y = 0 \)

D

\(2y'' - y = 0 \)

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8

2010 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2010

For a continuous function \( f(t) \), \( 0 \le t \le 1 \), the integral equation \( y(t) = f(t) + 3 \int_0^1 t s y(s) ds \) has

A

a unique solution if \(\int_0^1 s f(s) ds \ne 0 \)

B

no solution if \(\int_0^1 s f(s) ds = 0 \)

C

infinitely many solutions if \(\int_0^1 s f(s) ds = 0 \)

D

infinitely many solutions if \(\int_0^1 s f(s) ds \ne 0 \)

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9

2010 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2010

The values of \( a_1 \) and \( b_1 \) respectively are

A

\(\frac{1}{2} \) and \(-\frac{1}{12} \)

B

\(-\frac{1}{12} \) and \(\frac{1}{2} \)

C

\(\frac{1}{2} \) and \(\frac{1}{12} \)

D

\(\frac{1}{12} \) and \(\frac{1}{2} \)

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10

2011 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2011

A horizontal lever is in static equilibrium under the application of vertical forces \( F_1 \) at a distance \( l_1 \) from the fulcrum and \( F_2 \) at a distance \( l_2 \) from the fulcrum. The equilibrium for the above quantities can be obtained if

A

F_1l_1 = 2F_2l_2

B

2F_1l_1 = F_2l_2

C

F_1l_1 = F_2l_2

D

F_1l_1 < F_2l_2

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11

2011 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2011

Assume \( F \) to be a twice continuously differentiable function. Let \( J(y) \) be a functional of the form \( \int_0^1 F(x, y') dx, 0 \leq x \leq 1 \) defined on the set of all continuously differentiable functions \( y \) on \( [0, 1] \) satisfying \( y(0) = a, y(1) = b. \) For some arbitrary constant \( c, \) a necessary condition for \( y \) to be an extremum of \( J \) is

A

\(\frac{\partial F}{\partial x} = c\)

B

\(\frac{\partial F}{\partial y'} = c\)

C

\(\frac{\partial F}{\partial y} = c\)

D

\(\frac{\partial F}{\partial x} = 0\)

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12

2011 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2011

A massless wire is bent in the form of a parabola \( z = r^2 \) and a bead slides on it smoothly. The wire is rotated about z-axis with a constant angular acceleration \( \alpha \). Assume that \( m \) is the mass of the bead, \( \omega \) is the initial angular velocity and \( g \) is the acceleration due to gravity. Then, the Lagrangian at any time \( t \) is

A

\(\frac{m}{2} \left[ \left( \frac{dr}{dt} \right)^2 (1 + 4 r^2) + r^2 (\omega + \alpha t)^2 + 2 g r^2 \right] \)

B

\(\frac{m}{2} \left[ \left( \frac{dr}{dt} \right)^2 (1 + 4 r^2) - r^2 (\omega + \alpha t)^2 + 2 g r^2 \right] \)

C

\(\frac{m}{2} \left[ \left( \frac{dr}{dt} \right)^2 (1 + 4 r^2) - r^2 (\omega + \alpha t)^2 - 2 g r^2 \right] \)

D

\(\frac{m}{2} \left[ \left( \frac{dr}{dt} \right)^2 (1 + 4 r^2) + r^2 (\omega + \alpha t)^2 - 2 g r^2 \right] \)

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13

2011 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2011

On the interval \( [0, 1] \), let \( y \) be a twice continuously differentiable function which is an extremal of the functional \( J(y) = \int_0^1 \frac{\sqrt{1 + 2 y'^2}}{x} dx \) with \( y(0) = 1 \), \( y(1) = 2 \). Then, for some arbitrary constant \( c \), \( y \) satisfies

A

\(y'^2 (2 - c^2 x^2) = c^2 x^2 \)

B

\(y'^2 (2 + c^2 x^2) = c^2 x^2 \)

C

\(y'^2 (1 - c^2 x^2) = c^2 x^2 \)

D

\(y'^2 (1 + c^2 x^2) = c^2 x^2 \)

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14

2011 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2011

\( P \left( X + Y < \frac{1}{2} \right) \) is

A

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

B

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

C

\(\frac{3}{4} \)

D

1

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15

2012 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2012

A continuous random variable \(X\) has the probability density function
\[ f(x) = \begin{cases} rac{3}{5} e^{-\frac{3x}{5}}, & x > 0 \\ 0, & x \leq 0 \end{cases} \]
The probability density function of \(Y = 3X + 2\) is

A

\(f(y) = \begin{cases} \frac{1}{5} e^{-\frac{1}{5}(y-2)}, & y > 2 \\ 0, & y \leq 2 \end{cases}\)

B

\(f(y) = \begin{cases} \frac{2}{5} e^{-\frac{2}{5}(y-2)}, & y > 2 \\ 0, & y \leq 2 \end{cases}\)

C

\(f(y) = \begin{cases} \frac{3}{5} e^{-\frac{3}{5}(y-2)}, & y > 2 \\ 0, & y \leq 2 \end{cases}\)

D

\(f(y) = \begin{cases} \frac{4}{5} e^{-\frac{4}{5}(y-2)}, & y > 2 \\ 0, & y \leq 2 \end{cases}\)

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16

2012 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2012

The functional \[\int_0^1\left(y''^2+(y+2y')y''+kxyy'+y'^2\right)dx,\quad y(0)=0,\ y(1)=1,\ y'(0)=2,\ y'(1)=3\] is path independent if \(k\) equals

A

1

B

2

C

3

D

4

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17

2012 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2012

The functional \( \int_{0}^{1} (y'^{2} + 4y^{2} + 8ye^{x}) dx, \; y(0) = -\frac{4}{3}, \; y(1) = -\frac{4}{3}e \) possesses :

A

strong minima on \(y = -\frac{1}{3}e^{x} \)

B

strong minima on \(y = -\frac{4}{3}e^{x} \)

C

weak maxima on \(y = -\frac{1}{3}e^{x} \)

D

strong maxima on \(y = -\frac{4}{3}e^{x} \)

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18

2012 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2012

A particle of mass \( m \) is constrained to move on a circle with radius \( a \) which itself is rotating about its vertical diameter with a constant angular velocity \( \omega \). Assume that the initial angular velocity is zero and \( g \) is the acceleration due to gravity. If \( \theta \) be the inclination of the radius vector of the particle with the axis of rotation and \( \dot{\theta} \) denotes the derivative of \( \theta \) with respect to \( t \), then the Lagrangian of this system is

A

\(\frac{1}{2}ma^{2}(\dot{\theta}^{2} + \omega^{2} \sin^{2} \theta) + mga \cos \theta \)

B

\(\frac{1}{2}ma^{2}(\dot{\theta}^{2} + 2\omega \sin \theta) - mga \sin \theta \)

C

\(\frac{1}{2}ma^{2}(\dot{\theta}^{2} + 2\omega^{2} \cos \theta) - mga \sin \theta \)

D

\(\frac{1}{2}ma^{2}(\dot{\theta}^{2} + \omega \sin 2\theta) + mga \sin \theta \)

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19

2012 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2012

The solution of this integral equation is

A

\(\frac{x+1}{1-\lambda} \)

B

\(\frac{x^{2}}{1-\lambda^{2}} \)

C

\(\frac{x}{1+\lambda^{2}} \)

D

\(\frac{x}{1-\lambda} \)

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20

2012 · Mathematics · Calculus · Multiple Integrals and Change of Variables

Mathematics (MA) 2012

A and B are friends. They decide to meet between 1 PM and 2 PM on a given day. There is a condition that whoever arrives first will not wait for the other for more than 15 minutes. The probability that they will meet on that day is

A

1/4

B

1/16

C

7/16

D

9/16

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