The stability of complexes is often related to the value of \( \Delta \), which is the energy difference between the d-orbitals in the ligand field. Higher \( \Delta \) values typically correspond to more stable complexes.
Based on the \( \Delta \) values:
- \( [{Fe(CN)}_6]^{3-} \) has the highest \( \Delta \) value due to the strong field ligand \( {CN}^- \), making it the most stable complex.
- \( [{Co(CN)}_6]^{3-} \) is slightly less stable compared to \( [{Fe(CN)}_6]^{3-} \).
- \( [{Mn(CN)}_6]^{3-} \) has the lowest \( \Delta \) value and is the least stable among these complexes. Thus, the correct increasing order of stability is \( {III}<{II}<{IV}<{I} \).
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:
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: