Given $$f\left( t \right) = {L^{ - 1}}\left[ {{{3s + 1} \over {{s^3} + 4{s^2} + \left( {k - 3} \right)}}} \right].$$
$$\mathop {Lt}\limits_{t \to \propto } \,\,f\left( t \right) = 1$$ then value of $$k$$ is

For the set of equations $$${x_1} + 2{x_2} + {x_3} + 4{x_4} = 2,$$$ $$$3{x_1} + 6{x_2} + 3{x_3} + 12{x_4} = 6.$$$
The following statement is true

An eigen vector of $$p = \left[ {\matrix{ 1 & 1 & 0 \cr 0 & 2 & 2 \cr 0 & 0 & 3 \cr } } \right]$$ is

The value of the quantity, where $$P = \int\limits_0^1 {x{e^x}\,dx\,\,\,} $$ is