Mathematical Physics — Part 1 MCQs with Explanations

$$ia{\text{ and }} - ia$$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

$$\sin \frac{{\text{z}}}{{\text{z}}}$$ is the correct answer to “Which of the following functions of the complex variable z is not analytic everywhere”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[\left( {1 - \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + 3{{{\bf{\hat j}}}^{\bf{'}}} + \left( {1 + \sqrt 3 } \right){{{\bf{\hat k}}}^{\bf{'}}}\] is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

zero is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[\left[ {\begin{array}{*{20}{c}} 1&1&0 \\ 0&1&{ - 1} \end{array}} \right]\] is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

z = 0 is a removable singularity is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[\left[ {\begin{array}{*{20}{c}} 1&1 \\ 0&1 \end{array}} \right]\] is the correct answer to “Which one of the following matrices is the inverse of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&{ - 1} \\ 0&1 \end{array}} \right]?\]”. This question is standard in competitive examination syllabi for Mathematical Physics.

2a3 is the correct answer to “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 β 3 is equal to”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[\int_{ - 1}^{ + 1} {{x^2}{P_n}\left( x \right){P_{n + 1}}\left( x \right)dx} \] is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[\frac{{6\hat i + 4\hat j - \hat k}}{{\sqrt {53} }}\] is the correct answer to “The unit vector normal to the surface 3x 2 + 4y = z at the point (1, 1, 7) is”. This question is standard in competitive examination syllabi for Mathematical Physics.

5 is the correct answer to “For the function $$\phi = {x^2}y + xy,$$ the value of $$\left| {\overrightarrow \nabla \phi } \right|$$ at x = y = 1 is”. This question is standard in competitive examination syllabi for Mathematical Physics.

zero and +2 is the correct answer to “The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 1&i \\ { - i}&1 \end{array}} \right]\] are”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[\frac{1}{2}\][δ(x - a) + δ(x + a)] is the correct answer to “If the Fourier transform F[δ(x - a)] = exp (-i2πv a), then F -1 (cos 2π av) will correspond to”. This question is standard in competitive examination syllabi for Mathematical Physics.

circles with different radii is the correct answer to “The solutions to the differential equation \[\frac{{dy}}{{dx}} = - \frac{x}{{y + 1}}\] are a family of”. This question is standard in competitive examination syllabi for Mathematical Physics.

$$\frac{{2\pi s}}{{{{\left( {{s^2} + {\pi ^2}} \right)}^2}}}$$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

$$\left( {\frac{{{\partial ^2}}}{{\partial {x^2}}} - \frac{1}{{{v^2}}}\frac{{{\partial ^2}}}{{\partial {t^2}}}} \right)\psi \left( {x,\,t} \right) = 0$$ is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

5, 1, -1 is the correct answer to “The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} 2&3&0 \\ 3&2&0 \\ 0&0&1 \end{array}} \right]\] are”. This question is standard in competitive examination syllabi for Mathematical Physics.

zero is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[ - b\hat i - a\hat j\] is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

related by a rotation is the correct answer to “The two vectors \[\overrightarrow P = i,\,\overrightarrow Q = \frac{{i + j}}{{\sqrt 2 }}\] are”. This question is standard in competitive examination syllabi for Mathematical Physics.

ex2 is the correct answer to “Fourier transform of which of the following functions does not exist”. This question is standard in competitive examination syllabi for Mathematical Physics.

zero is the correct answer to “The value of \[\oint\limits_S {\frac{{\overrightarrow r \cdot d\overrightarrow S }}{{{r^3}}}} ,\] where \[\overrightarrow r \] is the position vector and S is a closed surface enclosing the origin, is”. This question is standard in competitive examination syllabi for Mathematical Physics.

No is the correct answer to “Can the following scalar and vector potentials describe an electromagnetic field? \[\phi \left( {\overrightarrow x ,\,t} \right) = 3xyz - 4t\] \[\overrightarrow A \left( {\overrightarrow x ,\,t} \right) = \left( {2x - \omega t} \right)\hat i + \left( {y - 2z} \right)\hat j + \left( {z - 2{e^{i\omega t}}} \right)\hat k\] where, ω is a constant.”. This question is standard in competitive examination syllabi for Mathematical Physics.

zero for a = $$\frac{1}{2}$$ is the correct answer to “The value of $$\int\limits_C {\frac{{dx}}{{\left( {{z^2} + {a^2}} \right)}},} $$ where C is a unit circle (anticlockwise) centred at the origin in the complex Z-plane is”. This question is standard in competitive examination syllabi for Mathematical Physics.

$$\frac{{\partial u}}{{\partial x}} = \frac{{\partial v}}{{\partial y}}{\text{ and }}\frac{{\partial u}}{{\partial y}} = - \frac{{\partial v}}{{\partial x}}$$ is the correct answer to “If a function f(z) = u (x, y) + iv (x, y) of the complex variable z = x + iy, where x, y, u and v are real, is analytic in a domain D of z, then which of the following is true”. This question is standard in competitive examination syllabi for Mathematical Physics.

anti-Hermitian is the correct answer to “The eigen values of a matrix are i, -2i and 3i. The matrix is”. This question is standard in competitive examination syllabi for Mathematical Physics.

R and S is the correct answer to “Consider the four statements given below about the function f(x) = x 4 - x 2 in the range $$ - \infty < x < + \infty .$$ Which one of the following statement is correct? P. The plot of f(x) versus x has two maxima and two minima. Q. The plot of f(x) versus x cuts the x axis at four points. R. The plot of f(x) versus x has three extrema. S. No part of the plot f(x) versus x lies in the fourth quadrant. Pick the right combination of correct choices from those given below.”. This question is standard in competitive examination syllabi for Mathematical Physics.

$$\frac{1}{a}\overline f \left( {\frac{s}{a}} \right)$$ is the correct answer to “If $$\overline f \left( s \right)$$ is the Laplace transform of f(t) the transform of f(at), where a is a constant is”. This question is standard in competitive examination syllabi for Mathematical Physics.

6 is the correct answer to “The determinant of a 3 × 3 real symmetric matrix is 36. If two of its eigen values are 2 and 3 then the third eigen value is”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[\overrightarrow F = 3z\hat y + 5y\hat z\] is the correct answer to “The curl of a vector field \[\overrightarrow F \] is \[2\hat x.\] Identify the appropriate vector field \[\overrightarrow F \] from the choices given below.”. This question is standard in competitive examination syllabi for Mathematical Physics.

3πa2h is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[{\left( {2\pi } \right)^{ - \frac{1}{2}}}\] is the correct answer to “The k th Fourier component of f(x) = δ(x) is”. This question is standard in competitive examination syllabi for Mathematical Physics.

2πr is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

t sinh(t) is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

3V is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

2πi is the correct answer to “The value of the integral \[\int\limits_C {\frac{{dz}}{{z + 3}},} \] where C is a circle (anticlockwise) with |z| = 4, is”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[\frac{{{\pi ^2}}}{6}\] is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[\left[ {\begin{array}{*{20}{c}} {a{e^{ - i\alpha }}}&{c{e^{ - i\beta }}} \\ b&d \end{array}} \right]\] is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

s-2e-3s is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

S is not a linearly independent set is the correct answer to “Consider the set of vectors in three-dimensional real vector space R 3 , S = {(1, 1, 1), (1, -1, 1), (1, 1, -1)}. Which one of the following statement is true”. This question is standard in competitive examination syllabi for Mathematical Physics.