31. 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
32. The kth Fourier component of f(x) = δ(x) is
33. 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
34. 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
35. 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
36. The value of the integral \[\int\limits_C {\frac{{dz}}{{z + 3}},} \] where C is a circle (anticlockwise) with |z| = 4, is
37. 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
Using this, one finds the sum \[\sum\limits_{n = 1}^\infty {{n^{ - 2}}} \] to be
38. 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
39. 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
40. Consider the set of vectors in three-dimensional real vector space
R3, S = {(1, 1, 1), (1, -1, 1), (1, 1, -1)}. Which one of the following statement is true?
R3, S = {(1, 1, 1), (1, -1, 1), (1, 1, -1)}. Which one of the following statement is true?