Evaluate :$$\,\,\int {\left[ {{{{{\left( {\cos 2x} \right)}^{1/2}}} \over {\sin x}}} \right]dx} $$

$$f\left( x \right) = \left| {\matrix{ {\sec x} & {\cos x} & {{{\sec }^2}x + \cot x\cos ec\,x} \cr {{{\cos }^2}x} & {{{\cos }^2}x} & {\cos e{c^2}x} \cr 1 & {{{\cos }^2}x} & {{{\cos }^2}x} \cr } } \right|.$$
Then $$\int\limits_0^{\pi /2} {f\left( x \right)dx = .......} $$

A man takes a step forward with probability $$0.4$$ and backwards with probability $$0.6$$ Find the probability that at the end of eleven steps he is one step away from the starting point.

A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane. It follows that :