A normal random variable $$X$$ has the following probability density function $$${f_x}\left( x \right) = {1 \over {\sqrt {8\pi } }}e{}^{ - \left\{ {{{{{\left( {x - 1} \right)}^2}} \over 8}} \right\}}, - \infty < x < \infty $$$
Then $$\,\int\limits_1^\infty {{f_x}\left( x \right)dx = } $$

To solve the equation $$2sinx=x$$ by Newton- Raphson method, the initial guess was chosen to be $$x=2.0.$$ Consider $$x$$ in radian only. The value of $$x$$ (in radian) obtained after one iteration will be closest to

The function $$f\left( z \right) = {{{z^2} + 1} \over {{z^2} + 4}}$$ is singular at

In abrasive water jet machining, the velocity of water at the exit of the orifice, before mixing with abrasives, is 800 m/s. The mass flow rate of water is 3.4 kg/min. The abrasives are added to the water jet at a rate of 0.6 kg/min with negligible velocity. Assume that at the end of the focusing tube, abrasive particles and water come out with equal velocity. Consider that there is no air in the abrasive water jet. Assuming conservation of momentum, the velocity (in m/s) of the abrasive water jet at the end of the focusing tube is _________.