If $$f(x) = \left| {\matrix{ {\cos 2x} & {\cos 2x} & {\sin 2x} \cr { - \cos x} & {\cos x} & { - \sin x} \cr {\sin x} & {\sin x} & {\cos x} \cr } } \right|$$,

then

Let O be the origin and $$\overrightarrow{OX}$$, $$\overrightarrow{OY}$$, $$\overrightarrow{OZ}$$ be three unit vectors in the directions of the sides $$\overrightarrow{QR}$$, $$\overrightarrow{RP}$$, $$\overrightarrow{PQ}$$ respectively, of a triangle PQR.

If the triangle PQR varies, then the minimum value of cos(P + Q) + cos(Q + R) + cos(R + P) is

Let O be the origin and $$\overrightarrow{OX}$$, $$\overrightarrow{OY}$$, $$\overrightarrow{OZ}$$ be three unit vectors in the directions of the sides $$\overrightarrow{QR}$$, $$\overrightarrow{RP}$$, $$\overrightarrow{PQ}$$ respectively, of a triangle PQR.

|$$\overrightarrow{OX}$$ $$ \times $$ $$\overrightarrow{OY}$$| = ?

Let p, q be integers and let $$\alpha $$, $$\beta $$ be the roots of the equation, x² $$-$$ x $$-$$ 1 = 0 where $$\alpha $$ $$ \ne $$ $$\beta $$. For n = 0, 1, 2, ........, let a_n = p$$\alpha $$ⁿ + q$$\beta $$ⁿ.

FACT : If a and b are rational numbers and a + b$$\sqrt 5 $$ = 0, then a = 0 = b.

a₁₂ = ?