The balanced reaction between KMnO\(_4\) and ferrous ammonium sulfate in the presence of H\(_2\)SO\(_4\) is:
\[
2 \, \text{KMnO}_4 + 10 \, \text{Fe}^{2+} + 8 \, \text{H}_2\text{SO}_4 \rightarrow 2 \, \text{Mn}^{2+} + 10 \, \text{Fe}^{3+} + 8 \, \text{H}_2\text{O} + \text{K}_2\text{SO}_4 + 2\text{H}_2\text{SO}_4.
\]
This equation shows that for every 2 molecules of KMnO\(_4\), 8 molecules of water are produced.
Thus, for 2 molecules of KMnO\(_4\), the total number of water molecules produced is:
\[
8 \, \text{water molecules}.
\]
At STP \(x\) g of a metal hydrogen carbonate (MHCO$_3$) (molar mass \(84 \, {g/mol}\)) on heating gives CO$_2$, which can completely react with \(0.02 \, {moles}\) of MOH (molar mass \(40 \, {g/mol}\)) to give MHCO$_3$. The value of \(x\) is:
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:
The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is: