Consider a $$3 \times 3$$ real symmetric matrix $$S$$ such that two of its eigen values are $$a \ne 0,$$ $$b\,\, \ne 0$$ with respective eigen vectors $$\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr {{x_3}} \cr } } \right],\left[ {\matrix{ {{y_1}} \cr {{y_2}} \cr {{y_3}} \cr } } \right].$$ If $$a\, \ne b$$ then $${x_1}{y_1} + {x_2}{y_2} + {x_3}{y_3}$$ equals

If a function is continuous at a point,

Divergence of the vector field $${x^2}z\widehat i + xy\widehat j - y{z^2}\widehat k\,\,$$ at $$(1, -1, 1)$$ is

A group consists of equal number of men and women. Of this group $$20$$% of the men and $$50$$% of the women are unemployed. If a person is selected at random from this group, the probability of the selected person being employed is