Let f be a polynomial function such that

f (3x) = f ' (x) . f '' (x), for all x $$ \in $$ R. Then :

A line drawn through the point P(4, 7) cuts the circle x² + y² = 9 at the points A and B. Then PA⋅PB is equal to :

If 2x = y$${^{{1 \over 5}}}$$ + y$${^{ - {1 \over 5}}}$$ and

(x² $$-$$ 1) $${{{d^2}y} \over {d{x^2}}}$$ + $$\lambda $$x $${{dy} \over {dx}}$$ + ky = 0,

then $$\lambda $$ + k is equal to :

A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1) = 1, then the curve also passes through the point :