The equation of the common tangent touching the circle $${\left( {x - 3} \right)^2} + {y^2} = 9$$ and the parabola $${y^2} = 4x$$ above the $$x$$-axis is

Let $$f:\left( {0,\infty } \right) \to R$$ and $$F\left( x \right) = \int\limits_0^x {f\left( t \right)dt.} $$ If $$F\left( {{x^2}} \right) = {x^2}\left( {1 + x} \right)$$, then $$f(4)$$ equals

A quantity X is given by $${\varepsilon _0}L{{\Delta V} \over {\Delta t}}$$ where $${\varepsilon _0}$$ is the permittivity of the free space, L is a length, $${\Delta V}$$ is a potential difference and $${\Delta t}$$ is a time interval. The dimentional formula for X is the same as that of

The pulleys and strings shown in the figure are smooth and of negligible mass. For the system to remain in equilibrium, the angle $$\theta $$ should be