Explanation of Assertion (A):
In aqueous solutions, $\text{Cr}^{2+}$ acts as a reducing agent and is oxidised to $\text{Cr}^{3+}$. This is because $\text{Cr}^{3+}$ has a stable $d^3$ electronic configuration. Conversely, $\text{Mn}^{3+}$ acts as an oxidising agent and is reduced to $\text{Mn}^{2+}$, which has a stable half-filled $d^5$ electronic configuration.
Explanation of Reason (R):
The half-filled electronic configuration provides extra stability due to symmetrical distribution of electrons and exchange energy. This explains why $\text{Cr}^{3+}$ and $\text{Mn}^{2+}$ are more stable compared to their respective other oxidation states.
Conclusion:
Both Assertion (A) and Reason (R) are true. The reason given (R) correctly explains why $\text{Cr}^{2+}$ is reducing and $\text{Mn}^{3+}$ is oxidising, as it is related to the stability of the resulting electronic configurations.
Let a line passing through the point $ (4,1,0) $ intersect the line $ L_1: \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} $ at the point $ A(\alpha, \beta, \gamma) $ and the line $ L_2: x - 6 = y = -z + 4 $ at the point $ B(a, b, c) $. Then $ \begin{vmatrix} 1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c \end{vmatrix} \text{ is equal to} $