Explanation: CuSO$_4$ $\cdot$ 5H$_2$O (copper(II) sulfate pentahydrate) contains the Cu$^{2+}$ ion, which has a partially filled $d$-orbital. The electronic configuration of Cu$^{2+}$ is [Ar] $3d^9$. In an aqueous environment, the Cu$^{2+}$ ion forms a complex with water molecules, creating a distorted octahedral geometry. The $d$-orbitals of the Cu$^{2+}$ ion split into two energy levels due to the ligand field created by the surrounding water molecules. The absorption of visible light causes electrons to transition between these $d$-orbital energy levels ($d$-$d$ transitions), resulting in the characteristic blue colour of the compound.
Other Options: K$_2$Cr$_2$O$_7$ (potassium dichromate), K$_2$CrO$_4$ (potassium chromate), and KMnO$_4$ (potassium permanganate) show colour due to charge transfer transitions, not $d$-$d$ transitions. In these compounds, the chromate and permanganate ions are highly coloured due to the transfer of electrons between the metal and oxygen atoms.
Conclusion: The compound CuSO$_4$ $\cdot$ 5H$_2$O shows colour due to $d$-$d$ transitions.
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: