The root of the function $$f\left( x \right) = {x^3} + x - 1$$ obtained after first iteration on application of Newton-Raphson scheme using an initial guess of $${x_0} = 1$$ is

Solutions of Laplace's equation having continuous second-order partial derivatives are called

A circular disc of radius 100 mm and mass 1 kg, initially at rest at position A, rolls without slipping down a curved path as shown in figure. The speed v of the disc when it reaches position B is _________ m/s.
Acceleration due to gravity g = 10 m/s²

A rigid rod $$(AB)$$ of length $$L = \sqrt 2 $$ m is undergoing translational as well as rotational motion in the $$ x-y$$ plane (see the figure). The point $$A$$ has the velocity $${V_1} = \widehat i + 2\widehat jm/s.$$ The end $$B$$ is constrained to move only along the $x$$ direction.

The magnitude of the velocity $${V_2}$$ (in $$m/s$$) at the end $$B$ is __________