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

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
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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

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?

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

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

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?

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