The real numbers $${x_1}$$, $${x_2}$$, $${x_3}$$ satisfying the equation $${x^3} - {x^2} + \beta x + \gamma = 0$$ are in AP. Find the intervals in which $$\beta \,\,and\,\gamma $$ lie.

A rectangle $$PQRS$$ has its side $$PQ$$ parallel to the line $$y = mx$$ and vertices $$P, Q$$ and $$S$$ on the lines $$y = a, x = b$$ and $$x = -b,$$ respectively. Find the locus of the vertex $$R$$.

The intercept on the line y = x by the circle $${x^2} + {y^2} - 2x = 0$$ is AB. Equation of the circle with AB as a diameter is................................

General value of $$\theta $$ satisfying the equation $${\tan ^2}\theta + \sec \,2\,\theta = 1$$ is _________.