The line integral $$\int\limits_{{P_1}}^{{P_2}} {\left( {ydx + xdy} \right)} $$ from $${P_1}\left( {{x_1},{y_1}} \right)$$ to $${P_2}\left( {{x_2},{y_2}} \right)$$ along the semi-circle $${P_1}$$ $${P_2}$$ shown in the figure is

If $$A$$ $$(0,4,3),$$ $$B(0,0,0)$$ and $$C(3,0,4)$$ are there points defined in $$x, y, z$$ coordinate system, then which one of the following vectors is perpendicular to both the vectors $$\overrightarrow {AB} $$ and $$\overrightarrow {BC} $$.

If $$T(x, y, z)$$ $$ = {x^2} + {y^2} + 2{z^2}$$ defines the temperature at any location $$(x, y, z)$$ then the magnitude of the temperature gradient at point $$P(1,1,1)$$ is _________.

It is estimated that the average number of events during a year is three. What is the probability of occurrence of not more than two events over a two-year duration? Assume that the number of events follow a poisson distribution.