Decide if the "personal a" is necessary in the following sentence.
no recuerdo tu nombre.
the "personal a" is necessary.
the "personal a" is not necessary.

The "personal a" is not necessary in that sentence. If you say: "No recuerdo tu nombre" it means "I don't remember your name". If you say "No recuerdo a tu nombre" it is not grammatically correct.

Given that including the "personal a" is grammatically incorrect, then the personal "a" is not necessary.

The "personal a" is not necessary.

No recuerdo tu nombre.

I do not remember your name.

Explanation: "tu nombre" is the direct object of the verb "recordar". In Spanish the preposition "a" is used when the direct object is a person and is omitted when the direct object is a thing. There may be exceptions: when a direct object is a thing but is personified, it may carry the preposition "a", and on the other hand if a direct object is a person, but is considered as an object of use, then is not used the preposition "a".