For $$0 < a < x,$$ the minimum value of the function $$lo{g_a}x + {\log _x}a$$ is $$2$$.

Evaluate the following $$\int\limits_0^{{1 \over 2}} {{{x{{\sin }^{ - 1}}x} \over {\sqrt {1 - {x^2}} }}dx} $$

Four person K, L, M, N are initially at the four corners of a square of side d. Each person now moves with a uniform speed v in such a way that K always moves directly towards L, L directly towards M, M directly towards N, and N directly towards K. The four person will meet at a time _____________.

A projectile fired from the ground follows a parabolic path. The speed of the projectile is minimum at the top of its path.