Let $$\alpha$$ and $$\beta$$ be the roots of x² $$-$$ 6x $$-$$ 2 = 0. If a_n = $$\alpha$$ⁿ $$-$$ $$\beta$$ⁿ for n $$ \ge $$ 1, then the value of $${{{a_{10}} - 2{a_8}} \over {3{a_9}}}$$ is :

If 0 < x, y < $$\pi$$ and cosx + cosy $$-$$ cos(x + y) = $${3 \over 2}$$, then sinx + cosy is equal to :

cosec$$\left[ {2{{\cot }^{ - 1}}(5) + {{\cos }^{ - 1}}\left( {{4 \over 5}} \right)} \right]$$ is equal to :

Let A be a 3 $$\times$$ 3 matrix with det(A) = 4. Let R_i denote the i^th row of A. If a matrix B is obtained by performing the operation R₂ $$ \to $$ 2R₂ + 5R₃ on 2A, then det(B) is equal to :