- The melting point of Mn is lower than Fe because Mn has a lower atomic number and a less stable crystal structure compared to Fe.
- The melting point of Tc is lower than Ru due to the differences in atomic structure and bonding.
- The melting point of Re is lower than Os because Os has a higher atomic number and stronger metallic bonding, leading to a higher melting point. Thus, the correct answer is \( {Mn}<{Fe}, \ {Tc}<{Ru} \ {and} \ {Re}<{Os} \).
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: