The value of $$\mathop {\lim }\limits_{x \to \alpha } {\left( {1 + {x^2}} \right)^{{e^{ - x}}}}\,\,$$ is

Choose the most appropriate equation for the function drawn as a thick line, in the plot below.

Suppose $${X_i}$$ for $$i=1,2,3$$ are independent and identically distributed random variables whose probability mass functions are $$\,\,\Pr \left[ {{X_i} = 0} \right] = \Pr \left[ {{X_i} = 1} \right] = 1/2\,\,$$ for $$i=1,2,3.$$ Define another random variable $$\,\,Y = {X_1}{X_2} \oplus {X_3},\,\,$$ where $$ \oplus $$ denotes $$XOR.$$ Then $$\Pr \left[ {Y = 0\left| {{X_3} = 0} \right.} \right]$$ =________.

In the given matrix $$\left[ {\matrix{ 1 & { - 1} & 2 \cr 0 & 1 & 0 \cr 1 & 2 & 1 \cr } } \right],$$ one of the eigenvalues is $$1.$$ The eigen vectors corresponding to the eigen value $$1$$ are