The maximum value of

$$f(x) = \left| {\matrix{ {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\cos 2x} \cr {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\cos 2x} \cr {{{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr } } \right|,x \in R$$ is :

S_n(x) = log_a^1/2x + log_a^1/3x + log_a^1/6x + log_a^1/11x + log_a^1/18x + log_a^1/27x + ...... up to n-terms, where a > 1. If S₂₄(x) = 1093 and S₁₂(2x) = 265, then value of a is equal to ____________.

If the distance of the point (1, $$-$$2, 3) from the plane x + 2y $$-$$ 3z + 10 = 0 measured parallel to the line, $${{x - 1} \over 3} = {{2 - y} \over m} = {{z + 3} \over 1}$$ is $$\sqrt {{7 \over 2}} $$, then the value of |m| is equal to _________.

Let $${1 \over {16}}$$, a and b be in G.P. and $${1 \over a}$$, $${1 \over b}$$, 6 be in A.P., where a, b > 0. Then 72(a + b) is equal to ___________.