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

The linear system of equations

$$\left. \matrix{ 8x - 3y - 5z = 0 \hfill \cr 5x - 8y + 3z = 0 \hfill \cr 3x + 5y - 8z = 0 \hfill \cr} \right\}$$ has