Mathematical Physics MCQs Question and Answer in Engineering Physics Page-5 section-1

41.
The points, where the series solution of the Legendre differential equation \[\left( {1 - {x^2}} \right)\frac{{{d^2}y}}{{d{x^2}}} - 2x\frac{{dy}}{{dx}} + \frac{3}{2}\left( {\frac{3}{2} + 1} \right)y = 0\] will diverge, are located at

A. 0 and 1

B. 0 and -1

C. -1 and 1

D. \[\frac{3}{2}\] and \[\frac{5}{2}\]

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42.
All solutions of the equation e^z = -3 are

A. z = i n π ln 3, n = ±1, ±2, . . . . . . . .

B. z = ln 3 + i(2n + 1)π, n = 0, ±1, ±2, . . . . . . . .

C. z = ln 3 + i 2nπ, n = 0, ±1, ±2, . . . . . . . .

D. z = i 3nπ, n = ±1, ± 2, . . . . . . . .

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43.
The solution of the system of differential equations \[\frac{{dy}}{{dx}} = y - z\] and \[\frac{{dz}}{{dx}} = - 4y + z\] is given by (for A and B are arbitrary constants)

A. y(x) = Ae^3x + Be^-x; z(x) = -2Ae^3x + 2Be^-x

B. y(x) = Ae^3x + Be^-x; z(x) = 2Ae^3x + 2Be^-x

C. y(x) = Ae^3x + Be^-x; z(x) = 2Ae^3x - 2Be^-x

D. y(x) = Ae^3x + Be^-x; z(x) = -2Ae^3x - 2Be^-x

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44.
The value of the integral \[\int\limits_C {\frac{{{e^z}}}{{{z^2} - 3z + 2}}dz,} \] where the contour C is the circle \[\left| z \right| = \frac{3}{2}\] is

A. 2πie

B. πie

C. -2πie

D. -πie

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45.
Inverse Laplace transform of \[\frac{{s + 1}}{{{s^2} - 4}}\] is .

A. cos 2x + \[\frac{1}{2}\] sin 2x

B. cos x + \[\frac{1}{2}\] sin x

C. cosh x + \[\frac{1}{2}\] sinh x

D. cosh 2x + \[\frac{1}{2}\] sinh 2x

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46.
Consider the Bessel equation \[\left( {v = 0} \right),\,\frac{{{d^2}y}}{{d{z^2}}} + \frac{1}{z}\frac{{dy}}{{dz}} + y = 0.\]
Which one of the following statements is correct?

A. Equation has regular singular points at z = 0 and z = \[\infty \]

B. Equation has 2 linearly independent solutions that are entire

C. Equation has an' entire solution and a second linearly independent solution singular at z = 0

D. Limit z → \[\infty \] , taken along X-axis, exists for both the linearly independent solutions

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47.
If \[F\left[ {f\left( x \right)} \right] = \int_{ - \infty }^\infty {f\left( x \right){e^{ - ikx}}dx,} \] then \[{{\hat F}^2}\left[ {f\left( x \right)} \right]\] is equal to

A. f(x)

B. -f(x)

C. f(-x)

D. \[\frac{{\left[ {f\left( x \right) + f\left( { - x} \right)} \right]}}{2}\]

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48.
The value of the integral $$\int\limits_C {{z^{10}}dz,} $$ where C is the unit circle with the origin as the centre is

A. zero

B. $$\frac{{{z^{11}}}}{{11}}$$

C. $$\frac{{2\pi i{z^{11}}}}{{11}}$$

D. $$\frac{1}{{11}}$$

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49.
If u (x, y, z, t) = f(x + iβy - vt) + g(x - iβy - vt), where f and g are arbitrary and twice differentiable functions, is a solution of the wave equation $$\frac{{\partial {u^2}}}{{\partial {x^2}}} = \frac{{{\partial ^2}u}}{{\partial {y^2}}} = \frac{1}{{{c^2}}}\frac{{{\partial ^2}u}}{{\partial {t^2}}}$$ then β is

A. $${\left( {1 - \frac{v}{c}} \right)^{\frac{1}{2}}}$$

B. $$\left( {1 - \frac{v}{c}} \right)$$

C. $${\left( {1 - \frac{{{v^2}}}{{{c^2}}}} \right)^{\frac{1}{2}}}$$

D. $$\left( {1 - \frac{{{v^2}}}{{{c^2}}}} \right)$$

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50.
The average of the function f(x) = sin x in the interval (0, π) is

A. $$\frac{1}{2}$$

B. $$\frac{2}{\pi }$$

C. $$\frac{1}{\pi }$$

D. $$\frac{4}{\pi }$$

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Mathematical Physics MCQs Question and Answer in Engineering Physics

41. The points, where the series solution of the Legendre differential equation \[\left( {1 - {x^2}} \right)\frac{{{d^2}y}}{{d{x^2}}} - 2x\frac{{dy}}{{dx}} + \frac{3}{2}\left( {\frac{3}{2} + 1} \right)y = 0\] will diverge, are located at

Answer & Solution

42. All solutions of the equation ez = -3 are

Answer & Solution

43. The solution of the system of differential equations \[\frac{{dy}}{{dx}} = y - z\] and \[\frac{{dz}}{{dx}} = - 4y + z\] is given by (for A and B are arbitrary constants)

Answer & Solution

44. The value of the integral \[\int\limits_C {\frac{{{e^z}}}{{{z^2} - 3z + 2}}dz,} \] where the contour C is the circle \[\left| z \right| = \frac{3}{2}\] is

Answer & Solution

45. Inverse Laplace transform of \[\frac{{s + 1}}{{{s^2} - 4}}\] is .

Answer & Solution

46. Consider the Bessel equation \[\left( {v = 0} \right),\,\frac{{{d^2}y}}{{d{z^2}}} + \frac{1}{z}\frac{{dy}}{{dz}} + y = 0.\] Which one of the following statements is correct?

Answer & Solution

47. If \[F\left[ {f\left( x \right)} \right] = \int_{ - \infty }^\infty {f\left( x \right){e^{ - ikx}}dx,} \] then \[{{\hat F}^2}\left[ {f\left( x \right)} \right]\] is equal to

Answer & Solution

48. The value of the integral $$\int\limits_C {{z^{10}}dz,} $$ where C is the unit circle with the origin as the centre is

Answer & Solution

Answer & Solution

50. The average of the function f(x) = sin x in the interval (0, π) is

Answer & Solution

41.
The points, where the series solution of the Legendre differential equation \[\left( {1 - {x^2}} \right)\frac{{{d^2}y}}{{d{x^2}}} - 2x\frac{{dy}}{{dx}} + \frac{3}{2}\left( {\frac{3}{2} + 1} \right)y = 0\] will diverge, are located at

42.
All solutions of the equation e^z = -3 are

43.
The solution of the system of differential equations \[\frac{{dy}}{{dx}} = y - z\] and \[\frac{{dz}}{{dx}} = - 4y + z\] is given by (for A and B are arbitrary constants)

44.
The value of the integral \[\int\limits_C {\frac{{{e^z}}}{{{z^2} - 3z + 2}}dz,} \] where the contour C is the circle \[\left| z \right| = \frac{3}{2}\] is

45.
Inverse Laplace transform of \[\frac{{s + 1}}{{{s^2} - 4}}\] is .

46.
Consider the Bessel equation \[\left( {v = 0} \right),\,\frac{{{d^2}y}}{{d{z^2}}} + \frac{1}{z}\frac{{dy}}{{dz}} + y = 0.\]
Which one of the following statements is correct?

47.
If \[F\left[ {f\left( x \right)} \right] = \int_{ - \infty }^\infty {f\left( x \right){e^{ - ikx}}dx,} \] then \[{{\hat F}^2}\left[ {f\left( x \right)} \right]\] is equal to

48.
The value of the integral $$\int\limits_C {{z^{10}}dz,} $$ where C is the unit circle with the origin as the centre is

50.
The average of the function f(x) = sin x in the interval (0, π) is