Mathematical Physics MCQs Question and Answer in Engineering Physics Page-6 section-1

51.
For a physical system, two observables O₁ and O₂ are known to be compatible. Choose the correct implication from amongst those given below.

A. Every eigen state of O₁ must necessarily be an eigen state of O₂

B. Every non-degenerate eigen state of O₁ must necessarily be an eigen state of O₂

C. When an observation of O₁ is carried out on an arbitrary state $$\left| \psi \right\rangle $$ of the physical system, a subsequent observation of O₂ leads to an unambiguous result

D. Observation of O₁ and O₂, carried out on an arbitrary state $$\left| \psi \right\rangle $$ of the physical system, lead to the identical results irrespective of the order in which the observations are made

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52.
A vector $$\overrightarrow A = \left( {5x + 2y} \right)\hat i + \left( {3y - z} \right)\hat j + \left( {2x - az} \right)\hat k$$ is solenoidal, if the constant a has a value

A. 4

B. -4

C. 8

D. -8

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53.
The Fourier transform of the function f(x) is $$F\left( k \right) = \int {{e^{ikx}}f\left( x \right)dx.} $$ The Fourier transform of $$\frac{{df\left( x \right)}}{{dx}}$$ is

A. $$\frac{{dF\left( k \right)}}{{dk}}$$

B. $$\int {\frac{{F\left( k \right)}}{{dk}}} $$

C. -ikF(k)

D. ikF(k)

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54.
The solution of the differential equation $$\left( {1 + x} \right)\frac{{{d^2}y\left( x \right)}}{{d{x^2}}} + x\frac{{dy\left( x \right)}}{{dx}} - y\left( x \right) = 0$$ is
where A and B are constants

A. Ax² + B

B. Ax + Be^-x

C. Ax + Be^x

D. Ax + Bx²

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55.
A 3 × 3 matrix has eigen values 0, 2 + i and 2 - i. Which one of the following statement is correct?

A. The matrix, is Hermitian

B. The matrix is unitary

C. The inverse of the matrix exists

D. The determinant of the matrix is zero

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56.
The value of the residue of $$\frac{{\sin z}}{{{z^6}}}$$ is

A. $$ - \frac{1}{{5!}}$$

B. $$\frac{1}{{5!}}$$

C. $$\frac{{2\pi i}}{{5!}}$$

D. $$ - \frac{{2\pi i}}{{5!}}$$

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57.
Which one of the following curves gives the solution of the differential equation $${k_1}\frac{{dx}}{{dt}} + {k_2}x = {k_3},$$ where k₁, k₂ and k₃ are positive constant with initial conditions x = 0 and t = 0?

A. Mathematical Physics mcq question image

B. Mathematical Physics mcq question image

C. Mathematical Physics mcq question image

D. Mathematical Physics mcq question image

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58.
The eigen values and eigen vectors of the matrix \[\left[ {\begin{array}{*{20}{c}} 5&4 \\ 1&2 \end{array}} \right]\] are

A. 6, 1 and \[\left[ {\begin{array}{*{20}{c}} 4 \\ 1 \end{array}} \right],\,\left[ {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right]\]

B. 2, 5 and \[\left[ {\begin{array}{*{20}{c}} 4 \\ 1 \end{array}} \right],\,\left[ {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right]\]

C. 6 , 1 and \[\left[ {\begin{array}{*{20}{c}} 1 \\ 4 \end{array}} \right],\,\left[ {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right]\]

D. 2, 5 and \[\left[ {\begin{array}{*{20}{c}} 1 \\ 4 \end{array}} \right],\,\left[ {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right]\]

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59.
If a force \[\overrightarrow {\mathbf{F}} \] is derivable from a potential function V(r), where r is the distance from the origin of the coordinate system, it follows that

A. \[\overrightarrow \nabla \times \overrightarrow {\mathbf{F}} = 0\]

B. \[\overrightarrow \nabla \cdot \overrightarrow {\mathbf{F}} = 0\]

C. \[\overrightarrow {\nabla \,} V = 0\]

D. \[{\nabla ^2}\,V = 0\]

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60.
If two matrices A and B can be diagonalized simultaneously, which of the following is true?

A. A²B = B²A

B. A²B² = B²A

C. AB = BA

D. AB²AB = BABA²

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Mathematical Physics MCQs Question and Answer in Engineering Physics

51. For a physical system, two observables O1 and O2 are known to be compatible. Choose the correct implication from amongst those given below.

Answer & Solution

52. A vector $$\overrightarrow A = \left( {5x + 2y} \right)\hat i + \left( {3y - z} \right)\hat j + \left( {2x - az} \right)\hat k$$ is solenoidal, if the constant a has a value

Answer & Solution

53. The Fourier transform of the function f(x) is $$F\left( k \right) = \int {{e^{ikx}}f\left( x \right)dx.} $$ The Fourier transform of $$\frac{{df\left( x \right)}}{{dx}}$$ is

Answer & Solution

54. The solution of the differential equation $$\left( {1 + x} \right)\frac{{{d^2}y\left( x \right)}}{{d{x^2}}} + x\frac{{dy\left( x \right)}}{{dx}} - y\left( x \right) = 0$$ is where A and B are constants

Answer & Solution

55. A 3 × 3 matrix has eigen values 0, 2 + i and 2 - i. Which one of the following statement is correct?

Answer & Solution

56. The value of the residue of $$\frac{{\sin z}}{{{z^6}}}$$ is

Answer & Solution

57. Which one of the following curves gives the solution of the differential equation $${k_1}\frac{{dx}}{{dt}} + {k_2}x = {k_3},$$ where k1, k2 and k3 are positive constant with initial conditions x = 0 and t = 0?

Answer & Solution

58. The eigen values and eigen vectors of the matrix \[\left[ {\begin{array}{*{20}{c}} 5&4 \\ 1&2 \end{array}} \right]\] are

Answer & Solution

59. If a force \[\overrightarrow {\mathbf{F}} \] is derivable from a potential function V(r), where r is the distance from the origin of the coordinate system, it follows that

Answer & Solution

60. If two matrices A and B can be diagonalized simultaneously, which of the following is true?

Answer & Solution

51.
For a physical system, two observables O₁ and O₂ are known to be compatible. Choose the correct implication from amongst those given below.

52.
A vector $$\overrightarrow A = \left( {5x + 2y} \right)\hat i + \left( {3y - z} \right)\hat j + \left( {2x - az} \right)\hat k$$ is solenoidal, if the constant a has a value

53.
The Fourier transform of the function f(x) is $$F\left( k \right) = \int {{e^{ikx}}f\left( x \right)dx.} $$ The Fourier transform of $$\frac{{df\left( x \right)}}{{dx}}$$ is

54.
The solution of the differential equation $$\left( {1 + x} \right)\frac{{{d^2}y\left( x \right)}}{{d{x^2}}} + x\frac{{dy\left( x \right)}}{{dx}} - y\left( x \right) = 0$$ is
where A and B are constants

55.
A 3 × 3 matrix has eigen values 0, 2 + i and 2 - i. Which one of the following statement is correct?

56.
The value of the residue of $$\frac{{\sin z}}{{{z^6}}}$$ is

57.
Which one of the following curves gives the solution of the differential equation $${k_1}\frac{{dx}}{{dt}} + {k_2}x = {k_3},$$ where k₁, k₂ and k₃ are positive constant with initial conditions x = 0 and t = 0?

58.
The eigen values and eigen vectors of the matrix \[\left[ {\begin{array}{*{20}{c}} 5&4 \\ 1&2 \end{array}} \right]\] are

59.
If a force \[\overrightarrow {\mathbf{F}} \] is derivable from a potential function V(r), where r is the distance from the origin of the coordinate system, it follows that

60.
If two matrices A and B can be diagonalized simultaneously, which of the following is true?