Let f(x) = log_e(sin x), (0 < x < $$\pi $$) and g(x) = sin^–1 (e^–x ), (x $$ \ge $$ 0). If $$\alpha $$ is a positive real number such that a = (fog)'($$\alpha $$) and b = (fog)($$\alpha $$), then :

The distance of the point having position vector $$ - \widehat i + 2\widehat j + 6\widehat k$$ from the straight line passing through the point (2, 3, – 4) and parallel to the vector, $$6\widehat i + 3\widehat j - 4\widehat k$$ is :

The sum of the real roots of the equation
$$\left| {\matrix{ x & { - 6} & { - 1} \cr 2 & { - 3x} & {x - 3} \cr { - 3} & {2x} & {x + 2} \cr } } \right| = 0$$, is equal to :

The smallest natural number n, such that the coefficient of x in the expansion of $${\left( {{x^2} + {1 \over {{x^3}}}} \right)^n}$$ is ⁿC₂₃, is :