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.
<h3>Decide if the "personal a" is necessary in the following sentence. No recuerdo _____ tu nombre.</h3>
Answer: 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".