A vertical tower $$PQ$$ stands at a point $$P$$. Points $$A$$ and $$B$$ are located to the South and East of $$P$$ respectively. $$M$$ is the mid point of $$AB$$. $$PAM$$ is an equilateral triangle; and $$N$$ is the foot of the perpendicular from $$P$$ and $$AB$$. Let $$AN$$$$=20$$ mrtres and the angle of elevation of the top of the tower at $$N$$ is $${\tan ^{ - 1}}\left( 2 \right)$$. Determine the height of the tower and the angles of elevation of the top of the tower at $$A$$ and $$B$$.

Show that $$2\sin x + \tan x \ge 3x$$ where $$0 \le x < {\pi \over 2}$$.

A point $$P$$ is given on the circumference of a circle of radius $$r$$. Chord $$QR$$ is parallel to the tangent at $$P$$. Determine the maximum possible area of the triangle $$PQR$$.

If $$\int {{{4{e^x} + 6{e^{ - x}}} \over {9{e^x} - 4{e^{ - x}}}}\,dx = Ax + B\,\,\log \left( {9{e^{2x}} - 4} \right) + C,} $$ then
$$A = .....,B = .....$$ and $$C = .....$$