Solution: Decacarbonyldimanganese(0), represented as Mn(CO)10, is a complex where manganese is surrounded by ten carbon monoxide ligands.
Coordination Number: In this complex, manganese exhibits a coordination number of 10, which typically leads to an octahedral geometry.
The carbon monoxide ligands are arranged around the manganese atom in a way that minimizes repulsion between the ligands, resulting in an octahedral shape.
Geometry: The octahedral arrangement is characteristic of complexes with a higher coordination number, where the ligands occupy the vertices of an octahedron around the central metal atom.
Thus, the coordination geometry around manganese in Mn(CO)10 is: Octahedral
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: