With respect to oxygen maximum valency is shown by

Halogen family

The valency of an element with respect to oxygen depends on the number of electrons it can gain, lose, or share to form compounds. The maximum valency of an element is determined by the highest oxidation state it can exhibit. This oxidation state is directly linked to the total number of valence electrons of the element.

To answer this question, let us examine the oxidation states of elements from the options provided:

• Halogen Family (Option a): Halogens like chlorine can form compounds such as Cl₂O₇, where chlorine exhibits a maximum oxidation state of +7. This is because halogens have seven valence electrons and can share all of them in covalent bonding.
• Oxygen Family (Option b): Elements like oxygen have a maximum oxidation state of 0 in O₃ (ozone) or -2 in most compounds. This is because oxygen can gain or share a limited number of electrons due to its electron configuration.
• Nitrogen Family (Option c): Nitrogen can exhibit a maximum oxidation state of +5, as seen in N₂O₅. This is because nitrogen has five valence electrons and can share them in covalent bonding.
• Boron Family (Option d): Boron can exhibit a maximum oxidation state of +3, as seen in B₂O₃. This is because boron has three valence electrons and can only form three covalent bonds.

Analysis: Among the given families, the halogen family shows the highest oxidation state (+7) in compounds like Cl₂O₇. This is higher than the maximum oxidation states exhibited by elements in the oxygen, nitrogen, or boron families.

The group that shows the maximum valency with respect to oxygen is: 
a) Halogen family