Mathematical Physics — Part 2 MCQs with Explanations

-1 and 1 is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

z = ln 3 + i(2n + 1)π, n = 0, ±1, ±2, . . . . . . . . is the correct answer to “All solutions of the equation e z = -3 are”. This question is standard in competitive examination syllabi for Mathematical Physics.

y(x) = Ae3x + Be-x; z(x) = -2Ae3x + 2Be-x is the correct answer to “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)”. This question is standard in competitive examination syllabi for Mathematical Physics.

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

cosh 2x + \[\frac{1}{2}\] sinh 2x is the correct answer to “Inverse Laplace transform of \[\frac{{s + 1}}{{{s^2} - 4}}\] is .”. This question is standard in competitive examination syllabi for Mathematical Physics.

Equation has regular singular points at z = 0 and z = \[\infty \] is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

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

zero is the correct answer to “The value of the integral $$\int\limits_C {{z^{10}}dz,} $$ where C is the unit circle with the origin as the centre is”. This question is standard in competitive examination syllabi for Mathematical Physics.

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

$$\frac{2}{\pi }$$ is the correct answer to “The average of the function f(x) = sin x in the interval (0, π) is”. This question is standard in competitive examination syllabi for Mathematical Physics.

Every eigen state of O1 must necessarily be an eigen state of O2 is the correct answer to “For a physical system, two observables O 1 and O 2 are known to be compatible. Choose the correct implication from amongst those given below.”. This question is standard in competitive examination syllabi for Mathematical Physics.

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

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

Ax + Be-x is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

The determinant of the matrix is zero is the correct answer to “A 3 × 3 matrix has eigen values 0, 2 + i and 2 - i. Which one of the following statement is correct”. This question is standard in competitive examination syllabi for Mathematical Physics.

$$\frac{1}{{5!}}$$ is the correct answer to “The value of the residue of $$\frac{{\sin z}}{{{z^6}}}$$ is”. This question is standard in competitive examination syllabi for Mathematical Physics.

6, 1 and \[\left[ {\begin{array}{*{20}{c}} 4 \\ 1 \end{array}} \right],\,\left[ {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right]\] is the correct answer to “The eigen values and eigen vectors of the matrix \[\left[ {\begin{array}{*{20}{c}} 5&4 \\ 1&2 \end{array}} \right]\] are”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[\overrightarrow \nabla \times \overrightarrow {\mathbf{F}} = 0\] is the correct answer to “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”. This question is standard in competitive examination syllabi for Mathematical Physics.

AB = BA is the correct answer to “If two matrices A and B can be diagonalized simultaneously, which of the following is true”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[m\dot r\] and \[m{r^2}\dot \theta \] is the correct answer to “The Lagrangian of a particle moving in a plane under the influence of a central potential is given by \[L = \frac{1}{2}m\left( {{{\dot r}^2} + {r^2}{{\dot \theta }^2}} \right) - V\left( r \right).\] The generalized momenta corresponding to r and θ are given by”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[\frac{1}{2}\left( {\sqrt 3 \pm i} \right),\] when θ = 30° is the correct answer to “The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} {\cos \theta }&{ - \sin \theta } \\ {\sin \theta }&{\cos \theta } \end{array}} \right]\] are”. This question is standard in competitive examination syllabi for Mathematical Physics.

10 is the correct answer to “Consider an anti-symmetric tensor P ij with the indices i and j running from 1 to 5. The number of independent components of the tensor is”. This question is standard in competitive examination syllabi for Mathematical Physics.

40 is the correct answer to “The average value of the function f(x) = 4x 3 in the interval 1 to 3 is”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[\frac{{8\pi i}}{3}{e^{ - 2}}\] is the correct answer to “The value of \[\oint\limits_C {\frac{{{e^{2z}}}}{{{{\left( {z + 1} \right)}^4}}}dz,} \] where C is a circle defined by |z| = 3, is”. This question is standard in competitive examination syllabi for Mathematical Physics.

6 and 3 respectively is the correct answer to “S ij and A ij represent a symmetric and an anti-symmetric real-valued tensor respectively in three-dimension. The number of independent components of S ij and A ij”. This question is standard in competitive examination syllabi for Mathematical Physics.

Hermitian is the correct answer to “For any opeartor A, i(A + - A) is”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[{\bf{\hat i}} + {\bf{\hat j}} + {\bf{\hat k}}\] is the correct answer to “The curl of the vector \[\overrightarrow {\bf{A}} = z{\bf{\hat i}} + x{\bf{\hat j}} + y{\bf{\hat k}}\] is given by”. This question is standard in competitive examination syllabi for Mathematical Physics.

$$\sqrt \pi \exp \left( {\frac{{ - {k^2}}}{4}} \right)$$ is the correct answer to “The Fourier transform F(k) of a function f(x) is defined as $$F\left( k \right)\int_{ - \infty }^\infty {dxf\left( x \right)\exp \left( {ikx} \right).} $$ Then F(k) for f(x) = exp(-x 2 ) is $$\left[ {{\text{Given: }}\int_{ - \infty }^\infty {\exp \left( { - {x^2}} \right)dx = \sqrt \pi } } \right]$$”. This question is standard in competitive examination syllabi for Mathematical Physics.

A and B have the same eigen vectors is the correct answer to “Two matrices A and B are said to be similar, if B = P -1 AP for some invertible matrix P. Which of the following statements is not true”. This question is standard in competitive examination syllabi for Mathematical Physics.

$$ - \frac{7}{{25}} - i\frac{{24}}{{25}}$$ is the correct answer to “The inverse of the complex number $$\frac{{3 + 4i}}{{3 - 4i}}$$ is”. This question is standard in competitive examination syllabi for Mathematical Physics.

9 is the correct answer to “A 3 × 3 matrix has elements such that its trace is 11 and its determinant is 36. The eigen values of the matrix are all known to be positive integers. The largest eigen value of the matrix is”. This question is standard in competitive examination syllabi for Mathematical Physics.

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

(-1 + 2i)π is the correct answer to “The value of $$\int_{ - i}^i {\pi \left( {z + 1} \right)dz} $$ is”. This question is standard in competitive examination syllabi for Mathematical Physics.

Trace (HGFE) is the correct answer to “For arbitrary matrices E, F, G and H, if EF - FE = 0 then Trace (EFGH) is equal to”. This question is standard in competitive examination syllabi for Mathematical Physics.

$$\frac{2}{e}$$ is the correct answer to “Consider the differential equation $$\frac{{{d^2}x}}{{d{t^2}}} + 2\frac{{dx}}{{dt}} + x = 0$$ At time t = 0, it is given that x = 1 and $$\frac{{dx}}{{dt}} = 0.$$ At t = 1, the value of x is given by”. This question is standard in competitive examination syllabi for Mathematical Physics.

u3, u4 is the correct answer to “Given the four vectors \[{u_1} = \left[ {\begin{array}{*{20}{c}} 1 \\ 2 \\ 3 \end{array}} \right],\,{u_2} = \left[ {\begin{array}{*{20}{c}} 3 \\ { - 5} \\ 1 \end{array}} \right],\,{u_3} = \left[ {\begin{array}{*{20}{c}} 2 \\ 4 \\ { - 8} \end{array}} \right].\,{u_4} = \left[ {\begin{array}{*{20}{c}} 3 \\ 6 \\ { - 12} \end{array}} \right]\] The linearly dependent pair is”. This question is standard in competitive examination syllabi for Mathematical Physics.

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

complex and distinct is the correct answer to “The eigen values of the matrix \[A = \left[ {\begin{array}{*{20}{c}} 0&i \\ i&0 \end{array}} \right]\] are”. This question is standard in competitive examination syllabi for Mathematical Physics.

\[\overrightarrow \nabla \cdot \overrightarrow {\mathbf{r}} = 2{\text{ and }}\overrightarrow \nabla \left| {\overrightarrow {\mathbf{r}} } \right| = \frac{{{\mathbf{\hat r}}}}{r}\] is the correct answer to “If \[\overrightarrow {\mathbf{r}} = x{\mathbf{\hat i}} + y{\mathbf{\hat j}},\] then”. This question is standard in competitive examination syllabi for Mathematical Physics.

-1 and +1 is the correct answer to “A real traceless 4 × 4 matrix has to eigen values -1 and +1. The other eigen values are”. This question is standard in competitive examination syllabi for Mathematical Physics.