
In a dilute alkaline solution, the permanganate ion \( \text{MnO}_4^- \) undergoes a chemical reduction reaction to form the manganate ion \( \text{MnO}_4^{2-} \). This process can be explained by analyzing the changes in oxidation states and balancing the resultant half-reaction in a basic environment.
When potassium permanganate \( \text{KMnO}_4 \) is dissolved in a dilute alkaline solution, the \(\text{MnO}_4^-\) ion is reduced. The reaction can be represented as follows:
Reduction half-reaction:
\( \text{MnO}_4^- + e^- \rightarrow \text{MnO}_4^{2-} \)
In this reaction, the oxidation state of manganese changes from +7 in \( \text{MnO}_4^- \) to +6 in \( \text{MnO}_4^{2-} \). This change indicates a gain of one electron, demonstrating a reduction process. In an alkaline medium, the addition of hydroxide ions \( \text{OH}^- \) assists in maintaining charge balance:
\( \text{MnO}_4^- + \text{e}^- + \text{OH}^- \rightarrow \text{MnO}_4^{2-} + \text{OH}^- \)
Thus, in dilute alkaline conditions, \( \text{MnO}_4^- \) will most commonly convert into \( \text{MnO}_4^{2-} \), which is known as the manganate ion.

$\mathrm{KMnO}_{4}$ 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 _______ .
Given below are two statements:
Statement (I): The first ionization energy of Pb is greater than that of Sn.
Statement (II): The first ionization energy of Ge is greater than that of Si.
In light of the above statements, choose the correct answer from the options given below:
A solid cylinder of mass 2 kg and radius 0.2 m is rotating about its own axis without friction with angular velocity 5 rad/s. A particle of mass 1 kg moving with a velocity of 5 m/s strikes the cylinder and sticks to it as shown in figure. 
The angular velocity of the system after the particle sticks to it will be: