The Lagrangian of a particle moving in a plane under the influence of a central potential is given by The generalized momenta corresponding to r and θ are given by
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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] are
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Consider an anti-symmetric tensor Pij 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 average value of the function f(x) = 4x3 in the interval 1 to 3 is
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The value of where C is a circle defined by |z| = 3, is
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Sij and Aij represent a symmetric and an anti-symmetric real-valued tensor respectively in three-dimension. The number of independent components of Sij and Aij
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For any opeartor A, i(A+ - A) is
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The curl of the vector is given by
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The Fourier transform F(k) of a function f(x) is defined as Then F(k) for f(x) = exp(-x2) is
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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?
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The inverse of the complex number 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 unit vector normal to the surface x2 + y2 - z = 1 at the point P (1, 1, 1) is
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The value of is
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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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Consider the differential equation
At time t = 0, it is given that x = 1 and At t = 1, the value of x is given by
At time t = 0, it is given that x = 1 and At t = 1, the value of x is given by
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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
The linearly dependent pair is
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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] are
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The eigen values of the matrix A = \left[ {\begin{array}{*{20}{c}}
0&i \\
i&0
\end{array}} \right] are
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If then
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