A tangent is drawn to the parabola $$y^2 = 6x$$ which is perpendicular to the line $$2x + y = 1$$. Which of the following points does NOT lie on it?

The parabola is $$y^2 = 6x$$, so $$4a = 6$$ and $$a = \frac{3}{2}$$.

The given line $$2x + y = 1$$ has slope $$-2$$. A line perpendicular to it has slope $$m = \frac{1}{2}$$.

The equation of the tangent to $$y^2 = 6x$$ with slope $$m$$ is $$y = mx + \frac{a}{m} = \frac{x}{2} + \frac{3/2}{1/2} = \frac{x}{2} + 3$$.

Multiplying through by 2: $$2y = x + 6$$, or equivalently $$x - 2y + 6 = 0$$.

Now we check which point does NOT lie on this line. For $$(0, 3)$$: $$0 - 6 + 6 = 0$$ (lies on it). For $$(4, 5)$$: $$4 - 10 + 6 = 0$$ (lies on it). For $$(5, 4)$$: $$5 - 8 + 6 = 3 \neq 0$$ (does NOT lie on it). For $$(-6, 0)$$: $$-6 - 0 + 6 = 0$$ (lies on it).

Therefore, the point that does NOT lie on the tangent is $$(5, 4)$$.