If $${S_r} = \left| {\matrix{ {2r} & x & {n(n + 1)} \cr {6{r^2} - 1} & y & {{n^2}(2n + 3)} \cr {4{r^3} - 2nr} & z & {{n^3}(n + 1)} \cr } } \right|$$, then the value of

$$\sum\limits_{r = 1}^n {{S_r}} $$ is independent of

If the following three linear equations have a non-trivial solution, then

x + 4ay + az = 0

x + 3by + bz = 0

x + 2cy + cz = 0

The least positive integer n such that $${\left( {\matrix{ {\cos \pi /4} & {\sin \pi /4} \cr { - \sin {\pi \over 4}} & {\cos {\pi \over 4}} \cr } } \right)^n}$$ is an identity matrix of order 2 is

If the polynomial $$f(x) = \left| {\matrix{ {{{(1 + x)}^a}} & {{{(2 + x)}^b}} & 1 \cr 1 & {{{(1 + x)}^a}} & {{{(2 + x)}^b}} \cr {{{(2 + x)}^b}} & 1 & {{{(1 + x)}^a}} \cr } } \right|$$, then the constant term of f(x) is