Mathematical Physics MCQs Question and Answer in Engineering Physics Page-2 section-1

11.
For the function $$\phi = {x^2}y + xy,$$ the value of $$\left| {\overrightarrow \nabla \phi } \right|$$ at x = y = 1 is

A. 5

B. $$\sqrt 5 $$

C. 13

D. $$\sqrt {13} $$

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12.
The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&i \\ { - i}&1 \end{array}} \right]\] are

A. +1 and +1

B. zero and +1

C. zero and +2

D. -1 and +1

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13.
If the Fourier transform F[δ(x - a)] = exp (-i2πv a), then F^-1(cos 2π av) will correspond to

A. δ(x - a) - δ(x + a)

B. a constant

C. \[\frac{1}{2}\][δ(x - a) + iδ(x + a)]

D. \[\frac{1}{2}\][δ(x - a) + δ(x + a)]

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14.
The solutions to the differential equation \[\frac{{dy}}{{dx}} = - \frac{x}{{y + 1}}\] are a family of

A. circles with different radii

B. circles with different centres

C. straight lines with different slopes

D. straight lines with different intercepts on the Y-axis

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15.
The Laplace transform of f(t) = sin πt is $$F\left( s \right) = \frac{\pi }{{{s^2}\left( {{s^2} + {\pi ^2}} \right)}},\,s > 0.$$ Therefore, the Laplace transform of t sin πt is

A. $$\frac{\pi }{{{s^2}\left( {{s^2} + {\pi ^2}} \right)}}$$

B. $$\frac{{2\pi }}{{{s^2}{{\left( {{s^2} + {\pi ^2}} \right)}^2}}}$$

C. $$\frac{{2\pi s}}{{{{\left( {{s^2} + {\pi ^2}} \right)}^2}}}$$

D. $$\frac{{2\pi }}{{{{\left( {{s^2} + {\pi ^2}} \right)}^2}}}$$

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16.
A finite wave train, of an unspecified nature, propagates along the positive X-axis with a constant speed v and without any change of shape. The differential equation among the four listed below, whose solution it must be, is

A. $$\left( {\frac{{{\partial ^2}}}{{\partial {x^2}}} - \frac{1}{{{v^2}}}\frac{{{\partial ^2}}}{{\partial {t^2}}}} \right)\psi \left( {x,\,t} \right) = 0$$

B. $$\left( {{\nabla ^2} - \frac{1}{{{v^2}}}\frac{{{\partial ^2}}}{{\partial {t^2}}}} \right)\psi \left( {\overrightarrow r ,\,t} \right) = 0$$

C. $$\left( { - \frac{h}{{2m}}\frac{{{\partial ^2}}}{{\partial {x^2}}} - ih\frac{\partial }{{\partial t}}} \right)\psi \left( {x,\,t} \right) = 0$$

D. $$\left( {{\nabla ^2} + a\frac{\partial }{{\partial t}}} \right)\psi \left( {\overrightarrow r ,\,t} \right) = 0$$

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17.
The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 2&3&0 \\ 3&2&0 \\ 0&0&1 \end{array}} \right]\] are

A. 5, 2, -2

B. -5, -1, 1

C. 5, 1, -1

D. -5, 1, 1

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18.
The value of the integral \[\int_C {\frac{{dz}}{{{z^2}}},} \] where z is a complex variable and C is the unit circle with the origin as its centre, is

A. zero

B. 2πi

C. 4πi

D. -4πi

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19.
Which of the following vectors is orthogonal to the vector \[\left( {a\hat i + b\hat j} \right),\] where a and b (a ≠ b) are constants, and \[{\hat i}\] and \[{\hat j}\] are unit orthogonal vectors?

A. \[ - b\hat i + a\hat j\]

B. \[ - a\hat i + b\hat j\]

C. \[ - a\hat i - b\hat j\]

D. \[ - b\hat i - a\hat j\]

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20.
The two vectors \[\overrightarrow P = i,\,\overrightarrow Q = \frac{{i + j}}{{\sqrt 2 }}\] are

A. related by a rotation

B. related by a reflection through the XY-plane

C. related by an inversion

D. not linearly independent

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Mathematical Physics MCQs Question and Answer in Engineering Physics

11. For the function $$\phi = {x^2}y + xy,$$ the value of $$\left| {\overrightarrow \nabla \phi } \right|$$ at x = y = 1 is

Answer & Solution

12. The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&i \\ { - i}&1 \end{array}} \right]\] are

Answer & Solution

13. If the Fourier transform F[δ(x - a)] = exp (-i2πv a), then F-1(cos 2π av) will correspond to

Answer & Solution

14. The solutions to the differential equation \[\frac{{dy}}{{dx}} = - \frac{x}{{y + 1}}\] are a family of

Answer & Solution

15. The Laplace transform of f(t) = sin πt is $$F\left( s \right) = \frac{\pi }{{{s^2}\left( {{s^2} + {\pi ^2}} \right)}},\,s > 0.$$ Therefore, the Laplace transform of t sin πt is

Answer & Solution

16. A finite wave train, of an unspecified nature, propagates along the positive X-axis with a constant speed v and without any change of shape. The differential equation among the four listed below, whose solution it must be, is

Answer & Solution

17. The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 2&3&0 \\ 3&2&0 \\ 0&0&1 \end{array}} \right]\] are

Answer & Solution

18. The value of the integral \[\int_C {\frac{{dz}}{{{z^2}}},} \] where z is a complex variable and C is the unit circle with the origin as its centre, is

Answer & Solution

19. Which of the following vectors is orthogonal to the vector \[\left( {a\hat i + b\hat j} \right),\] where a and b (a ≠ b) are constants, and \[{\hat i}\] and \[{\hat j}\] are unit orthogonal vectors?

Answer & Solution

20. The two vectors \[\overrightarrow P = i,\,\overrightarrow Q = \frac{{i + j}}{{\sqrt 2 }}\] are

Answer & Solution

11.
For the function $$\phi = {x^2}y + xy,$$ the value of $$\left| {\overrightarrow \nabla \phi } \right|$$ at x = y = 1 is

12.
The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&i \\ { - i}&1 \end{array}} \right]\] are

13.
If the Fourier transform F[δ(x - a)] = exp (-i2πv a), then F^-1(cos 2π av) will correspond to

14.
The solutions to the differential equation \[\frac{{dy}}{{dx}} = - \frac{x}{{y + 1}}\] are a family of

15.
The Laplace transform of f(t) = sin πt is $$F\left( s \right) = \frac{\pi }{{{s^2}\left( {{s^2} + {\pi ^2}} \right)}},\,s > 0.$$ Therefore, the Laplace transform of t sin πt is

16.
A finite wave train, of an unspecified nature, propagates along the positive X-axis with a constant speed v and without any change of shape. The differential equation among the four listed below, whose solution it must be, is

17.
The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 2&3&0 \\ 3&2&0 \\ 0&0&1 \end{array}} \right]\] are

18.
The value of the integral \[\int_C {\frac{{dz}}{{{z^2}}},} \] where z is a complex variable and C is the unit circle with the origin as its centre, is

19.
Which of the following vectors is orthogonal to the vector \[\left( {a\hat i + b\hat j} \right),\] where a and b (a ≠ b) are constants, and \[{\hat i}\] and \[{\hat j}\] are unit orthogonal vectors?

20.
The two vectors \[\overrightarrow P = i,\,\overrightarrow Q = \frac{{i + j}}{{\sqrt 2 }}\] are