In the titration of $$\mathrm{KMnO}_{4}$$ and oxalic acid in acidic medium, the change in oxidation number of carbon at the end point is ___________.

Optical activity of an enantiomeric mixture is $$+12.6^{\circ}$$ and the specific rotation of $$(+)$$ isomer is $$+30^{\circ}$$. The optical purity is __________$$\%$$.

In the following reaction

The $$\%$$ yield for reaction I is $$60 \%$$ and that of reaction II is $$50 \%$$. The overall yield of the complete reaction is __________ $$\%$$. [nearest integer]

Let $$R_{1}$$ and $$R_{2}$$ be two relations defined on $$\mathbb{R}$$ by

$$a \,R_{1} \,b \Leftrightarrow a b \geq 0$$ and $$a \,R_{2} \,b \Leftrightarrow a \geq b$$

Then,