Review concepts of reduction potentials, stability of oxidation states in d-block elements, and the relationship between unpaired electrons and magnetic moment
The reduction potential for the M3+/M2+ couple for manganese is greater than that for iron:
\[ E^\circ_{\text{Mn}^{3+}/\text{Mn}^{2+}} = +1.57 \, \text{V}, \, E^\circ_{\text{Fe}^{3+}/\text{Fe}^{2+}} = +0.77 \, \text{V} \]
Therefore, this statement is incorrect.
Higher oxidation states of first-row d-block elements are stabilized by oxide ions (O2−) due to the formation of strong metal-oxygen bonds. This statement is correct.
Chromium in the Cr2+ oxidation state can reduce H+ to H2 in aqueous solution:
\[ \text{Cr}^{2+} + \text{H}^+ \rightarrow \text{Cr}^{3+} + \frac{1}{2}\text{H}_2 \]
The reduction potential \( E^\circ_{\text{Cr}^{3+}/\text{Cr}^{2+}} = -0.26 \, \text{V} \) confirms this. This statement is correct.
V2+ has three unpaired electrons, resulting in a magnetic moment of approximately 3.87 BM, which is not within the range of 4.4-5.2 BM. This statement is incorrect.


Let $ A \in \mathbb{R} $ be a matrix of order 3x3 such that $$ \det(A) = -4 \quad \text{and} \quad A + I = \left[ \begin{array}{ccc} 1 & 1 & 1 \\2 & 0 & 1 \\4 & 1 & 2 \end{array} \right] $$ where $ I $ is the identity matrix of order 3. If $ \det( (A + I) \cdot \text{adj}(A + I)) $ is $ 2^m $, then $ m $ is equal to:
A square loop of sides \( a = 1 \, {m} \) is held normally in front of a point charge \( q = 1 \, {C} \). The flux of the electric field through the shaded region is \( \frac{5}{p} \times \frac{1}{\varepsilon_0} \, {Nm}^2/{C} \), where the value of \( p \) is: