Statement I: True. Fusion of MnO$_2$ with KOH in the presence of oxygen gives K$_2$MnO$_4$ (potassium manganate), which is dark green in color: \[ \text{MnO}_2 + 4\text{KOH} + \text{O}_2 \xrightarrow{\text{fusion}} 2\text{K}_2\text{MnO}_4 + 2\text{H}_2\text{O}. \]
Statement II: True. In alkaline medium, the manganate ion (MnO$_4^{2-}$) undergoes electrolytic oxidation to form the permanganate ion (MnO$_4^-$): \[ \text{MnO}_4^{2-} \rightarrow \text{MnO}_4^- + e^-. \]
Both statements are correct as they describe valid chemical processes.
Let one focus of the hyperbola $ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $ be at $ (\sqrt{10}, 0) $, and the corresponding directrix be $ x = \frac{\sqrt{10}}{2} $. If $ e $ and $ l $ are the eccentricity and the latus rectum respectively, then $ 9(e^2 + l) $ is equal to: