The integral $$\,\,{1 \over {\sqrt {2\pi } }}\int\limits_{ - \infty }^\infty {{e^{{{ - {x^2}} \over 2}}}} dx\,\,$$ is equal to

If $$f\left( x \right) = \sin \left| x \right|\,\,$$ then the value of $${{df} \over {dx}}\,\,$$ at $$\,\,x = {{ - \pi } \over 4}\,\,$$ is

If $$(1, 0, -1)$$$${}^T$$ is an eigen vector of the following matrix $$\left[ {\matrix{ 1 & { - 1} & 0 \cr { - 1} & 2 & { - 1} \cr 0 & { - 1} & 1 \cr } } \right]$$ then the corresponding eigen value is

An industrial gas $$\left( {{C_p} = 1kJ/kgK} \right)$$ enters a parallel flow heat exchanger at $${250^ \circ }C$$ with a flow rate of $$2$$ $$kg/s$$ to heat a water stream.
The water stream $$\left( {{C_p} = 4kJ/kgK} \right)$$ enters the heat exchanger at $${50^ \circ }C$$ with a flow rate of $$1kg/s.$$ The heat exchanger has an effectiveness of $$0.75.$$ The gas stream exit temperature will be