Let $$A = \left[ {\matrix{ 1 & { - 1} & 1 \cr 2 & 1 & { - 3} \cr 1 & 1 & 1 \cr } } \right]$$ and $$10B = \left[ {\matrix{ 4 & 2 & 2 \cr { - 5} & 0 & \alpha \cr 1 & { - 2} & 3 \cr } } \right]$$

If B is the inverse of A, then the value of $$\alpha$$ is

If $$x \in \left( {0,{\pi \over 2}} \right)$$, then the value of $${\cos ^{ - 1}}\left( {{7 \over 2}(1 + \cos 2x) + \sqrt {({{\sin }^2}x - 48{{\cos }^2}x)\sin x} } \right)$$ is equal to

A running track of 440 ft is to be laid out enclosing a football field, the shape of which is a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum, then the lengths of its side are

$$\left( {{{dy} \over {dx}}} \right)\tan x = y{\sec ^2}x + \sin x$$, find general solution