Given below are two statements :

Statement I : A hypothetical diatomic molecule with bond order zero is quite stable.
Statement II : As bond order increases, the bond length increases.
In the light of the above statements, choose the most appropriate answer from the options given below :

The correct option is Both Statement I and Statement II are false.

Let us evaluate each statement step-by-step using molecular orbital theory and bond concepts:

Statement I: A hypothetical diatomic molecule with bond order zero is quite stable.
According to molecular orbital theory, the bond order of a molecule is given by:
$\mathrm{Bond\; Order}=\frac{1}{2}(N_b-N_a)$
where $N_b$ is the number of electrons in bonding molecular orbitals and $N_a$ is the number of electrons in antibonding molecular orbitals.
A bond order of zero means that the stabilizing effect of bonding electrons is completely counteracted by the destabilizing effect of antibonding electrons. Consequently, no chemical bond is formed between the two atoms, making the molecule highly unstable and unable to exist under normal conditions. Thus, Statement I is false.

Statement II: As bond order increases, the bond length increases.
Bond order indicates the number of shared electron pairs (bonds) between two atoms. As the bond order increases (for example, from a single bond to a double bond, and then to a triple bond), the attractive force pulling the nuclei together becomes stronger.
A stronger force of attraction pulls the atoms closer to each other, which decreases the distance between the nuclei. Therefore, as bond order increases, the bond length decreases (it does not increase). Thus, Statement II is false.

Conclusion: Since both Statement I and Statement II are false, the most appropriate option is "Both Statement I and Statement II are false".