In what ratio does the $$x$$-axis divide the area of the region
bounded by the parabolas $$y = 4x - {x^2}$$ and $$y = {x^2} - x?$$

A normal is drawn at a point $$P(x,y)$$ of a curve. It meets the $$x$$-axis at $$Q.$$ If $$PQ$$ is of constant length $$k,$$ then show that the differential equation describing such curves is $$y = {{dy} \over {dx}} = \pm \sqrt {{k^2} - {y^2}} $$

Find the equation of such a curve passing through $$(0,k).$$

If two events $$A$$ and $$B$$ are such that $$P\,\,\left( {{A^c}} \right)\,\, = \,\,0.3,\,\,P\left( B \right) = 0.4$$ and $$P\left( {A \cap {B^c}} \right) = 0.5,$$ then $$P\left( {B/\left( {A \cup {B^c}} \right)} \right.$$$$\left. \, \right] = $$ ............

Let $$A, B, C$$ be three mutually independent events. Consider the two statements $${S_1}$$ and $${S_2}$$
$${S_1}\,:\,A$$ and $$B \cup C$$ are independent
$${S_2}\,:\,A$$ and $$B \cap C$$ are independent
Then,