$\mathrm{KMnO}_{4}$ acts as an oxidising agent in acidic medium. ' X ' is the difference between the oxidation states of Mn in reactant and product. ' Y ' is the number of ' d ' electrons present in the brown red precipitate formed at the end of the acetate ion test with neutral ferric chloride. The value of $\mathrm{X}+\mathrm{Y}$ is _______ .
1. Oxidation states of Mn:
- Reactant: $\mathrm{Mn}^{7+}$
- Product: $\mathrm{Mn}^{2+}$
- Difference in oxidation states: $X = 7 - 2 = 5$
2. Brown red precipitate: - The brown red precipitate is $\mathrm{Fe}(\mathrm{OH})_2(\mathrm{CH}_3\mathrm{COO})_n$.
- $\mathrm{Fe}^{3+}$ has 5 d-electrons. - Therefore, $Y = 5$.
3. Calculate $\mathrm{X}+\mathrm{Y}$: \[ \mathrm{X} + \mathrm{Y} = 5 + 5 = 10 \]
Therefore, the correct answer is (10).
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is: