Mathematical Physics MCQs Question and Answer in Engineering Physics Page-4 section-1

31.
Consider a cylinder of height h and radius a, closed at both ends, centred at the origin. Let \[\hat ix + \hat jy + \hat kz\] be the position vector and \[{\hat n}\] a unit vector normal to the surface. The surface integral \[\int\limits_S {\overrightarrow r \cdot \hat n\,} dS\] over the closed surface-of the cylinder is

A. 2πa²(a + h)

B. 3πa²h

C. 2πa²h

D. zero

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32.
The k^th Fourier component of f(x) = δ(x) is

A. 1

B. zero

C. \[{\left( {2\pi } \right)^{ - \frac{1}{2}}}\]

D. \[{\left( {2\pi } \right)^{ - \frac{3}{2}}}\]

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33.
The value of contour integral, \[\left| {\int\limits_C {\overrightarrow r \times d\overrightarrow \theta } } \right|,\] for a circle C of radius r with centre at the origin is

A. 2πr

B. \[\frac{{{r^2}}}{2}\]

C. πr²

D. r

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34.
The solution of the differential equation for \[y\left( t \right):\frac{{{d^2}y}}{{d{t^2}}} - y = 2\cosh \left( t \right),\] subject to the initial conditions y(0) = 0 and \[{\left. {\frac{{dy}}{{dt}}} \right|_{t = 0}} = 0\] is

A. \[\frac{1}{2}\] cosh(t) + t sinh(t)

B. -sinh(t) + t cosh(t)

C. t cosh(t)

D. t sinh(t)

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35.
If S is the closed surface enclosing a volume V and \[{\hat n}\] is the unit normal vector to the surface and \[\overrightarrow r \] is the positive vector, then the value of the following integral \[\iint\limits_S {\hat n}\,dS\] is

A. V

B. 2V

C. zero

D. 3V

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36.
The value of the integral \[\int\limits_C {\frac{{dz}}{{z + 3}},} \] where C is a circle (anticlockwise) with |z| = 4, is

A. zero

B. πi

C. 2πi

D. 4πi

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37.
A periodic function f(x) = x for -π < x < +π has the Fourier series representation \[f\left( x \right) = \sum\limits_{n = 1}^\infty {\left( { - \frac{2}{n}} \right){{\left( { - 1} \right)}^n}\sin nx.} \]
Using this, one finds the sum \[\sum\limits_{n = 1}^\infty {{n^{ - 2}}} \] to be

A. 2$$l$$n 2

B. \[\frac{{{\pi ^2}}}{3}\]

C. \[\frac{{{\pi ^2}}}{6}\]

D. π$$l$$n 2

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38.
An unitary matrix \[\left[ {\begin{array}{*{20}{c}} {a{e^{i\alpha }}}&b \\ {c{e^{i\beta }}}&d \end{array}} \right]\] is given, were a, b, c, d, α and β are real. The inverse of the matrix is

A. \[\left[ {\begin{array}{*{20}{c}} {a{e^{i\alpha }}}&{ - c{e^{i\beta }}} \\ b&d \end{array}} \right]\]

B. \[\left[ {\begin{array}{*{20}{c}} {a{e^{i\alpha }}}&{c{e^{i\beta }}} \\ b&d \end{array}} \right]\]

C. \[\left[ {\begin{array}{*{20}{c}} {a{e^{ - i\alpha }}}&b \\ {c{e^{ - i\beta }}}&d \end{array}} \right]\]

D. \[\left[ {\begin{array}{*{20}{c}} {a{e^{ - i\alpha }}}&{c{e^{ - i\beta }}} \\ b&d \end{array}} \right]\]

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39.
If \[f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {0,}&{{\text{for }} < 3} \\ {x - 3,}&{{\text{for }} \geqslant 3} \end{array}} \right.\] then, the Laplace transform of f(x) is

A. s^-2e^3s

B. s²e^-3s

C. s^-2

D. s^-2e^-3s

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40.
Consider the set of vectors in three-dimensional real vector space
R³, S = {(1, 1, 1), (1, -1, 1), (1, 1, -1)}. Which one of the following statement is true?

A. S is not a linearly independent set

B. S is a basis for R³

C. The vectors in S are orthogonal

D. An orthogonal set of vectors cannot be generated from S

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Mathematical Physics MCQs Question and Answer in Engineering Physics

Answer & Solution

32. The kth Fourier component of f(x) = δ(x) is

Answer & Solution

33. The value of contour integral, \[\left| {\int\limits_C {\overrightarrow r \times d\overrightarrow \theta } } \right|,\] for a circle C of radius r with centre at the origin is

Answer & Solution

34. The solution of the differential equation for \[y\left( t \right):\frac{{{d^2}y}}{{d{t^2}}} - y = 2\cosh \left( t \right),\] subject to the initial conditions y(0) = 0 and \[{\left. {\frac{{dy}}{{dt}}} \right|_{t = 0}} = 0\] is

Answer & Solution

35. If S is the closed surface enclosing a volume V and \[{\hat n}\] is the unit normal vector to the surface and \[\overrightarrow r \] is the positive vector, then the value of the following integral \[\iint\limits_S {\hat n}\,dS\] is

Answer & Solution

36. The value of the integral \[\int\limits_C {\frac{{dz}}{{z + 3}},} \] where C is a circle (anticlockwise) with |z| = 4, is

Answer & Solution

37. A periodic function f(x) = x for -π < x < +π has the Fourier series representation \[f\left( x \right) = \sum\limits_{n = 1}^\infty {\left( { - \frac{2}{n}} \right){{\left( { - 1} \right)}^n}\sin nx.} \] Using this, one finds the sum \[\sum\limits_{n = 1}^\infty {{n^{ - 2}}} \] to be

Answer & Solution

38. An unitary matrix \[\left[ {\begin{array}{*{20}{c}} {a{e^{i\alpha }}}&b \\ {c{e^{i\beta }}}&d \end{array}} \right]\] is given, were a, b, c, d, α and β are real. The inverse of the matrix is

Answer & Solution

39. If \[f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {0,}&{{\text{for }} < 3} \\ {x - 3,}&{{\text{for }} \geqslant 3} \end{array}} \right.\] then, the Laplace transform of f(x) is

Answer & Solution

40. Consider the set of vectors in three-dimensional real vector space R3, S = {(1, 1, 1), (1, -1, 1), (1, 1, -1)}. Which one of the following statement is true?

Answer & Solution

32.
The k^th Fourier component of f(x) = δ(x) is

33.
The value of contour integral, \[\left| {\int\limits_C {\overrightarrow r \times d\overrightarrow \theta } } \right|,\] for a circle C of radius r with centre at the origin is

34.
The solution of the differential equation for \[y\left( t \right):\frac{{{d^2}y}}{{d{t^2}}} - y = 2\cosh \left( t \right),\] subject to the initial conditions y(0) = 0 and \[{\left. {\frac{{dy}}{{dt}}} \right|_{t = 0}} = 0\] is

35.
If S is the closed surface enclosing a volume V and \[{\hat n}\] is the unit normal vector to the surface and \[\overrightarrow r \] is the positive vector, then the value of the following integral \[\iint\limits_S {\hat n}\,dS\] is

36.
The value of the integral \[\int\limits_C {\frac{{dz}}{{z + 3}},} \] where C is a circle (anticlockwise) with |z| = 4, is

37.
A periodic function f(x) = x for -π < x < +π has the Fourier series representation \[f\left( x \right) = \sum\limits_{n = 1}^\infty {\left( { - \frac{2}{n}} \right){{\left( { - 1} \right)}^n}\sin nx.} \]
Using this, one finds the sum \[\sum\limits_{n = 1}^\infty {{n^{ - 2}}} \] to be

38.
An unitary matrix \[\left[ {\begin{array}{*{20}{c}} {a{e^{i\alpha }}}&b \\ {c{e^{i\beta }}}&d \end{array}} \right]\] is given, were a, b, c, d, α and β are real. The inverse of the matrix is

39.
If \[f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {0,}&{{\text{for }} < 3} \\ {x - 3,}&{{\text{for }} \geqslant 3} \end{array}} \right.\] then, the Laplace transform of f(x) is

40.
Consider the set of vectors in three-dimensional real vector space
R³, S = {(1, 1, 1), (1, -1, 1), (1, 1, -1)}. Which one of the following statement is true?