If $${f_r}\left( x \right),{g_r}\left( x \right),{h_r}\left( x \right),r = 1,2,3$$ are polynomials in $$x$$ such that $${f_r}\left( a \right) = {g_r}\left( a \right) = {h_r}\left( a \right),r = 1,2,3$$
and $$F\left( x \right) = \left| {\matrix{ {{f_1}\left( x \right)} & {{f_2}\left( x \right)} & {{f_3}\left( x \right)} \cr {{g_1}\left( x \right)} & {{g_2}\left( x \right)} & {{g_3}\left( x \right)} \cr {{h_1}\left( x \right)} & {{h_2}\left( x \right)} & {{h_3}\left( x \right)} \cr } } \right|$$ then $$F'\left( x \right)$$ at $$x = a$$ is ...........

If $$f\left( x \right) = {\log _x}\left( {In\,x} \right),$$ then $$f'\left( x \right)$$ at $$x=e$$ is ................

The set of all real numbers $$a$$ such that $${a^2} + 2a,2a + 3$$ and $${a^2} + 3a + 8$$ are the sides of a triangle is ...........

In a triangle $$ABC$$, if cot $$A$$, cot $$B$$, cot $$C$$ are in A.P., then $${a^2},{b^2},{c^2}$$, are in ............... progression.