Mathematical Physics MCQ
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A real traceless 4 × 4 matrix has to eigen values -1 and +1. The other eigen values are
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The eigen values of the matrix \[A = \left[ {\begin{array}{*{20}{c}} 0&i \\ i&0 \end{array}} \right
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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
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Given the four vectors \[{u_1} = \left[ {\begin{array}{*{20}{c}} 1 \\ 2 \\ 3 \end{array}} \right
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Consider the differential equation $$\frac{{{d^2}x}}{{d{t^2}}} + 2\frac{{dx}}{{dt}} + x = 0$$<br>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
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For arbitrary matrices E, F, G and H, if EF - FE = 0 then Trace (EFGH) is equal to
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The value of $$\int_{ - i}^i {\pi \left( {z + 1} \right)dz} $$ is
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The unit vector normal to the surface x<sup>2</sup> + y<sup>2</sup> - z = 1 at the point P (1, 1, 1) is
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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
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The inverse of the complex number $$\frac{{3 + 4i}}{{3 - 4i}}$$ is
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Two matrices A and B are said to be similar, if B = P<sup>-1</sup> AP for some invertible matrix P. Which of the following statements is not true?
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For any opeartor A, i(A<sup>+</sup> - A) is
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S<sub>ij</sub> and A<sub>ij</sub> represent a symmetric and an anti-symmetric real-valued tensor respectively in three-dimension. The number of independent components of S<sub>ij</sub> and A<sub>ij</sub>
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The average value of the function f(x) = 4x<sup>3</sup> in the interval 1 to 3 is
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Consider an anti-symmetric tensor P<sub>ij</sub> with the indices i and j running from 1 to 5. The number of independent components of the tensor is
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The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}} {\cos \theta }&{ - \sin \theta } \\ {\sin \theta }&{\cos \theta } \end{array}} \right
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If two matrices A and B can be diagonalized simultaneously, which of the following is true?
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Which one of the following curves gives the solution of the differential equation $${k_1}\frac{{dx}}{{dt}} + {k_2}x = {k_3},$$ where k<sub>1</sub>, k<sub>2</sub> and k<sub>3</sub> are positive constant with initial conditions x = 0 and t = 0?
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The value of the residue of $$\frac{{\sin z}}{{{z^6}}}$$ is
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A 3 × 3 matrix has eigen values 0, 2 + i and 2 - i. Which one of the following statement is correct?
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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<br> where A and B are constants
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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
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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
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For a physical system, two observables O<sub>1</sub> and O<sub>2</sub> are known to be compatible. Choose the correct implication from amongst those given below.
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The average of the function f(x) = sin x in the interval (0, π) is
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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
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The value of the integral $$\int\limits_C {{z^{10}}dz,} $$ where C is the unit circle with the origin as the centre is
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All solutions of the equation e<sup>z</sup> = -3 are
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Consider the set of vectors in three-dimensional real vector space<br>R<sup>3</sup>, S = {(1, 1, 1), (1, -1, 1), (1, 1, -1)}. Which one of the following statement is true?
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If \[f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {0,}&{{\text{for }}