The trigonometric equation $${\sin ^{ - 1}}x = 2{\sin ^{ - 1}}a$$ has a solution for :

If the function $$f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,$$ where $$a>0,$$ attains its maximum and minimum at $$p$$ and $$q$$ respectively such that $${p^2} = q$$ , then $$a$$ equals

If $$A = \left[ {\matrix{ a & b \cr b & a \cr } } \right]$$ and $${A^2} = \left[ {\matrix{ \alpha & \beta \cr \beta & \alpha \cr } } \right]$$, then

If $$1,$$ $$\omega ,{\omega ^2}$$ are the cube roots of unity, then

$$\Delta = \left| {\matrix{ 1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr {{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr {{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr } } \right|$$ is equal to