Let a given line $$L_1$$ intersects the x and y axes at P and Q, respectively. Let another line $$L_2$$, perpendicular to $$L_1$$, cut the x and y axes at R and S, respectively. Show that the locus of the point of intersection of the lines PS and QR is a circle passing through the origin.

A polygon of nine sides, each of length $$2$$, is inscribed in a circle. The radius of the circle is .................

In a triangle, the lengths of the two larger sides are $$10$$ and $$9$$, respectively. If the angles are in $$AP$$. Then the length of the third side can be

The set of all $$x$$ for which $$in\left( {1 + x} \right) \le x$$ is equal to ..........