$$f\left( z \right) = u\left( {x,y} \right) + i\,\,\,\,v\left( {x,y} \right)$$ is an analytic function of complex variable $$z=x+iy$$ , where $$i = \sqrt { - 1} $$ If $$u(x,y)=2xy,$$ then $$v(x,y)$$ may be expressed as

A rigid ball of weight $$100$$ $$N$$ is suspended with the help of a string. The ball is pulled by a horizontal force $$F$$ such that the string makes an angle of $${30^0}$$ with the vertical. The magnitude of force $$F$$ (in N) is __________

A point mass $$M$$ is released from rest and slides down a spherical bowl (of radius $$R$$) from a height $$H$$ as shown in the figure below. The surface of the bowl is smooth (no friction). The velocity of the mass at the bottom of the bowl is

A block of mass m rests on an inclined plane and is attached by a string to the wall as shown in the figure. The coefficient of static friction between the plane and the block is $$0.25.$$ The string can withstand a maximum force of $$20$$ N. The maximum value of the mass (m) for which the string will not break and the block will be in static equilibrium is ____________ kg.
Take $$\cos \theta = 0.8$$ and $$\sin \theta = 0.6$$. Acceleration due to gravity g $$=$$ $$10$$ m/s²