The Laplace transform of f(t) = sin πt is $$F\left( s \right) = \frac{\pi }{{{s^2}\left( {{s^2} + {\pi ^2}} \right)}},\,s > 0.$$ Therefore, the Laplace transform of t sin πt is
A. $$\frac{\pi }{{{s^2}\left( {{s^2} + {\pi ^2}} \right)}}$$
B. $$\frac{{2\pi }}{{{s^2}{{\left( {{s^2} + {\pi ^2}} \right)}^2}}}$$
C. $$\frac{{2\pi s}}{{{{\left( {{s^2} + {\pi ^2}} \right)}^2}}}$$
D. $$\frac{{2\pi }}{{{{\left( {{s^2} + {\pi ^2}} \right)}^2}}}$$
Answer: Option C
A. $$\frac{{1 + i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 + i}}{{\sqrt 2 }}a$$
B. $$ia{\text{ and }} - ia$$
C. $$ia,\, - ia,\,\frac{{1 - i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 - i}}{{\sqrt 2 }}a$$
D. $$\frac{{1 + i}}{{\sqrt 2 }}a,\, - \frac{{1 + i}}{{\sqrt 2 }}a,\,\frac{{1 - i}}{{\sqrt 2 }}a{\text{ and}} - \frac{{1 - i}}{{\sqrt 2 }}a$$
Which of the following functions of the complex variable z is not analytic everywhere?
A. ez
B. $$\sin \frac{{\text{z}}}{{\text{z}}}$$
C. e3
D. |z|3
A. \[\left( {1 - \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + 3{{{\bf{\hat j}}}^{\bf{'}}} + \left( {1 + \sqrt 3 } \right){{{\bf{\hat k}}}^{\bf{'}}}\]
B. \[\left( {1 + \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + 3{{{\bf{\hat j}}}^{\bf{'}}} + \left( {1 - \sqrt 3 } \right){{{\bf{\hat k}}}^{\bf{'}}}\]
C. \[\left( {1 - \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + \left( {3 + \sqrt 3 } \right){{{\bf{\hat j}}}^{\bf{'}}} + 2{{{\bf{\hat k}}}^{\bf{'}}}\]
D. \[\left( {1 - \sqrt 3 } \right){{{\bf{\hat i}}}^{\bf{'}}} + \left( {3 - \sqrt 3 } \right){{{\bf{\hat j}}}^{\bf{'}}} + 2{{{\bf{\hat k}}}^{\bf{'}}}\]
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