Suppose

det$$\left| {\matrix{ {\sum\limits_{k = 0}^n k } & {\sum\limits_{k = 0}^n {{}^n{C_k}{k^2}} } \cr {\sum\limits_{k = 0}^n {{}^n{C_k}.k} } & {\sum\limits_{k = 0}^n {{}^n{C_k}{3^k}} } \cr } } \right| = 0$$

holds for some positive integer n. Then $$\sum\limits_{k = 0}^n {{{{}^n{C_k}} \over {k + 1}}} $$ equals ..............

Five persons A, B, C, D and E are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is ............

The value of the integral $$ \int\limits_0^{\pi /2} {{{3\sqrt {\cos \theta } } \over {{{(\sqrt {\cos \theta } + \sqrt {\sin \theta } )}^5}}}} d\theta $$ equals ..............

The value of

$${\sec ^{ - 1}}\left( \matrix{ {1 \over 4}\sum\limits_{k = 0}^{10} {\sec \left( {{{7\pi } \over {12}} + {{k\pi } \over 2}} \right)} \sec \left( {{{7\pi } \over {12}} + {{(k + 1)\pi } \over 2}} \right) \hfill \cr} \right)$$

in the interval $$\left[ { - {\pi \over 4},\,{{3\pi } \over 4}} \right]$$ equals ..........