- Electronic configuration of each complex: - \( [FeF_6]^{3-} \): \( [Ar] 3d^5 4s^0 \).
There are 5 unpaired electrons in the \( 3d \)-orbitals.
- \( [CoF_6]^{3-} \): \( [Ar] 3d^6 4s^0 \).
There are 4 unpaired electrons in the \( 3d \)-orbitals.
- \( [Ni(CO)_4] \): \( [Ar] 3d^8 4s^2 \).
The \( CO \) ligand is a strong field ligand, so the pairing of electrons leads to 0 unpaired electrons.
- \( [Ni(CN)_4]^{2-} \): \( [Ar] 3d^8 4s^0 \).
The \( CN \) ligand is a strong field ligand, leading to the pairing of all electrons, so there are 0 unpaired electrons.
Thus, the order of the unpaired electrons is: \[ [FeF_6]^{3-}>[CoF_6]^{3-}>[Ni(CN)_4]^{2-} = [Ni(CO)_4] \]
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: