Mathematical Physics MCQs Question and Answer in Engineering Physics

1.
The contour integral $$\oint {\frac{{dz}}{{{z^2} + {a^2}}}} $$ is to be evaluated on a circle of radius 2a centred at the origin. It will have contributions only from the points

A. $$\frac{{1 + i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 + i}}{{\sqrt 2 }}a$$

B. $$ia{\text{ and }} - ia$$

C. $$ia,\, - ia,\,\frac{{1 - i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 - i}}{{\sqrt 2 }}a$$

D. $$\frac{{1 + i}}{{\sqrt 2 }}a,\, - \frac{{1 + i}}{{\sqrt 2 }}a,\,\frac{{1 - i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 - i}}{{\sqrt 2 }}a$$

Answer & Solution Discuss in Board Save for Later

2.
Which of the following functions of the complex variable z is not analytic everywhere?

A. e^z

B. $$\sin \frac{{\text{z}}}{{\text{z}}}$$

C. e³

D. |z|³

Answer & Solution Discuss in Board Save for Later

3.
Consider a vector \[\overrightarrow {\bf{p}} = 2{\bf{\hat i}} + 3{\bf{\hat j}} + 2{\bf{\hat k}}\] in the coordinate system \[\left( {{\bf{\hat i}},\,{\bf{\hat j}},\,{\bf{\hat k}}} \right).\] The axes are rotated anti-clockwise about the Y-axis by an angle of 60°. The vector \[\overrightarrow p \] in the rotate coordinate system \[\left( {{\bf{\hat i}},\,{\bf{\hat j}},\,{\bf{\hat k}}} \right)\] is

A. \[\left( {1 - \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + 3{{{\bf{\hat j}}}^{\bf{'}}} + \left( {1 + \sqrt 3 } \right){{{\bf{\hat k}}}^{\bf{'}}}\]

B. \[\left( {1 + \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + 3{{{\bf{\hat j}}}^{\bf{'}}} + \left( {1 - \sqrt 3 } \right){{{\bf{\hat k}}}^{\bf{'}}}\]

C. \[\left( {1 - \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + \left( {3 + \sqrt 3 } \right){{{\bf{\hat j}}}^{\bf{'}}} + 2{{{\bf{\hat k}}}^{\bf{'}}}\]

D. \[\left( {1 - \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + \left( {3 - \sqrt 3 } \right){{{\bf{\hat j}}}^{\bf{'}}} + 2{{{\bf{\hat k}}}^{\bf{'}}}\]

Answer & Solution Discuss in Board Save for Later

4.
The value of the integral $$\oint\limits_C {\frac{{{e^z}\sin z}}{{{z^2}}}} dz,$$ where the contour C is the unit circle: |z - 2| = 1, is

A. 2πi

B. 4πi

C. πi

D. zero

Answer & Solution Discuss in Board Save for Later

5.
A Iinear transformation T, defined as \[T\left[ {\begin{array}{{20}{c}} {{x_1}} \\ {{x_2}} \\ {{x_3}} \end{array}} \right] = \left[ {\begin{array}{{20}{c}} {{x_1} + {x_2}} \\ {{x_2} - {x_3}} \end{array}} \right],\] transforms a vector \[\overrightarrow x \] for a three-dimensional real space to a two-dimensional real space. The transformation matrix T is

A. \[\left[ {\begin{array}{*{20}{c}} 1&1&0 \\ 0&1&{ - 1} \end{array}} \right]\]

B. \[\left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \end{array}} \right]\]

C. \[\left[ {\begin{array}{*{20}{c}} 1&1&1 \\ { - 1}&1&1 \end{array}} \right]\]

D. \[\left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&0&1 \end{array}} \right]\]

Answer & Solution Discuss in Board Save for Later

6.
For the complex function, $$f\left( z \right) = \frac{{{e^{\sqrt z }} - {e^{ - \sqrt z }}}}{{\sin \left( {\sqrt z } \right)}},$$ which of the following statement is correct?

A. z = 0 is a branch point

B. z = 0 is a pole of order one

C. z = 0 is a removable singularity

D. z = 0 is an essential singularity

Answer & Solution Discuss in Board Save for Later

7.
Which one of the following matrices is the inverse of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&{ - 1} \\ 0&1 \end{array}} \right]?\]

A. \[\left[ {\begin{array}{*{20}{c}} 1&1 \\ { - 1}&1 \end{array}} \right]\]

B. \[\left[ {\begin{array}{*{20}{c}} 1&0 \\ 1&1 \end{array}} \right]\]

C. \[\left[ {\begin{array}{*{20}{c}} 1&1 \\ 0&1 \end{array}} \right]\]

D. \[\left[ {\begin{array}{*{20}{c}} { - 1}&1 \\ 0&{ - 1} \end{array}} \right]\]

Answer & Solution Discuss in Board Save for Later

8.
Let \[{T_{ij}} = \sum\limits_K {{\varepsilon _{ijk}}{a_k}} \] and \[{\beta _k} = \sum\limits_{i,\,j} {{\varepsilon _{ijk}}{T_{ij}}} ,\] where \[{\varepsilon _{ijk}}\] is the Levi-Civita density, defined to be zero, if two 'of the indices. coincide and +1 and -1 depending on whether ijk is even or odd permutation of 1, 2, 3. Then β₃ is equal to

A. 2a₃

B. -2a₃

C. a₃

D. -a₃

Answer & Solution Discuss in Board Save for Later

9.
Given the recurrence relation for the Legendre polynomials (2n + 1) xP_n(x) = (n + 1) P_{n + 1}(x) + P_{n - 1}(x), which of the following integrals has a non-zero value?

A. \[\int_{ - 1}^{ + 1} {{x^2}{P_n}\left( x \right){P_{n + 1}}\left( x \right)dx} \]

B. \[\int_{ - 1}^{ + 1} {x{P_n}\left( x \right){P_{n + 2}}\left( x \right)dx} \]

C. \[\int_{ - 1}^{ + 1} {x{{\left[ {{P_n}\left( x \right)} \right]}^2}dx} \]

D. \[\int_{ - 1}^{ + 1} {{x^2}{P_n}\left( x \right){P_{n + 2}}\left( x \right)dx} \]

Answer & Solution Discuss in Board Save for Later

10.
The unit vector normal to the surface 3x² + 4y = z at the point (1, 1, 7) is

A. \[\frac{{ - 6\hat i + 4\hat j + \hat k}}{{\sqrt {53} }}\]

B. \[\frac{{4\hat i + 6\hat j - \hat k}}{{\sqrt {53} }}\]

C. \[\frac{{6\hat i + 4\hat j - \hat k}}{{\sqrt {53} }}\]

D. \[\frac{{4\hat i + 6\hat j + \hat k}}{{\sqrt {53} }}\]

Answer & Solution Discuss in Board Save for Later

Mathematical Physics MCQs Question and Answer in Engineering Physics

1. The contour integral $$\oint {\frac{{dz}}{{{z^2} + {a^2}}}} $$ is to be evaluated on a circle of radius 2a centred at the origin. It will have contributions only from the points

Answer & Solution

2. Which of the following functions of the complex variable z is not analytic everywhere?

Answer & Solution

Answer & Solution

4. The value of the integral $$\oint\limits_C {\frac{{{e^z}\sin z}}{{{z^2}}}} dz,$$ where the contour C is the unit circle: |z - 2| = 1, is

Answer & Solution

Answer & Solution

6. For the complex function, $$f\left( z \right) = \frac{{{e^{\sqrt z }} - {e^{ - \sqrt z }}}}{{\sin \left( {\sqrt z } \right)}},$$ which of the following statement is correct?

Answer & Solution

7. Which one of the following matrices is the inverse of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&{ - 1} \\ 0&1 \end{array}} \right]?\]

Answer & Solution

Answer & Solution

9. Given the recurrence relation for the Legendre polynomials (2n + 1) xPn(x) = (n + 1) Pn + 1(x) + Pn - 1(x), which of the following integrals has a non-zero value?

Answer & Solution

10. The unit vector normal to the surface 3x2 + 4y = z at the point (1, 1, 7) is

Answer & Solution

1.
The contour integral $$\oint {\frac{{dz}}{{{z^2} + {a^2}}}} $$ is to be evaluated on a circle of radius 2a centred at the origin. It will have contributions only from the points

2.
Which of the following functions of the complex variable z is not analytic everywhere?

4.
The value of the integral $$\oint\limits_C {\frac{{{e^z}\sin z}}{{{z^2}}}} dz,$$ where the contour C is the unit circle: |z - 2| = 1, is

6.
For the complex function, $$f\left( z \right) = \frac{{{e^{\sqrt z }} - {e^{ - \sqrt z }}}}{{\sin \left( {\sqrt z } \right)}},$$ which of the following statement is correct?

7.
Which one of the following matrices is the inverse of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&{ - 1} \\ 0&1 \end{array}} \right]?\]

9.
Given the recurrence relation for the Legendre polynomials (2n + 1) xP_n(x) = (n + 1) P_{n + 1}(x) + P_{n - 1}(x), which of the following integrals has a non-zero value?

10.
The unit vector normal to the surface 3x² + 4y = z at the point (1, 1, 7) is