Write down the M.O. electron distribution of O₂. Specify its bond order and magnetic property.

Calculate the energy required to excite one litre of hydrogen gas at 1 atm and 298 K to the first excited state of atomic hydrogen. The energy for the dissociation of H-H bond 436 kJ mol^-1.

If $$\alpha ,\,\beta $$ are the roots of $$a{x^2} + bx + c = 0$$, $$\,\left( {a \ne 0} \right)$$ and $$\alpha + \delta ,\,\,\beta + \delta $$ are the roots of $$A{x^2} + Bx + c = 0,$$ $$\left( {A \ne 0\,} \right)\,$$ for some contant $$\delta $$, then prove that $${{{b^2} - 4ac} \over {{a^2}}} = {{{B^2} - 4Ac} \over {{A^2}}}$$.

A coin has probability $$p$$ of showing head when tossed. It is tossed $$n$$ times. Let $${p_n}$$ denote the probability that no two (or more) consecutive heads occur. Prove that $${p_1} = 1,{p_2} = 1 - {p^2}$$ and $${p_n} = \left( {1 - p} \right).\,\,{p_{n - 1}} + p\left( {1 - p} \right){p_{n - 2}}$$ for all $$n \ge 3.$$