In \([\text{Ni}(\text{CO})_4]\), nickel is in the zero oxidation state and has a \(3d^{10}\) electron configuration, leading to a fully paired electron configuration, making it diamagnetic.
In contrast, in \([\text{NiCl}_4]^{2-}\), nickel is in the \(+2\) oxidation state with a \(3d^8\) electron configuration. The chloride ligands are weak-field ligands and do not pair up the electrons, resulting in unpaired electrons and a paramagnetic nature.
Thus, the correct answer is:
\[ [\text{Ni}(\text{CO})_4] \text{ is diamagnetic and } [\text{NiCl}_4]^{2-} \text{ is paramagnetic.} \]Let \( S = \left\{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \right\} \), where
\[ A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \]Then \( n(S) \) is equal to ______.
Let \( T_r \) be the \( r^{\text{th}} \) term of an A.P. If for some \( m \), \( T_m = \dfrac{1}{25} \), \( T_{25} = \dfrac{1}{20} \), and \( \displaystyle\sum_{r=1}^{25} T_r = 13 \), then \( 5m \displaystyle\sum_{r=m}^{2m} T_r \) is equal to: